The physics of structural phase transitions

Basic Concepts

Introduction

Phase transitions can be understood through thermodynamics, though a detailed understanding of the transition mechanisms is crucial in critical regions While many thermodynamics textbooks focus on phase equilibria in isotropic media, structural phase transitions in crystals are often only briefly addressed Various types of phase transitions exist in nature, including conductor-to-insulator transitions and normal-to-superconducting transitions in metals Ehrenfest classified phase transitions based on discontinuities in thermodynamic potential derivatives at critical temperatures Notably, second-order phase transitions, marked by continuous changes in Gibbs potential at critical temperatures, are significant due to their relation to lattice stability in crystals This chapter explores continuous phase transitions through thermodynamic principles, acknowledging that critical anomalies and domain structures in ordered phases extend beyond classical thermodynamics.

Landau developed a thermodynamic theory for continuous phase transitions in binary systems, highlighting the significance of an order parameter at the critical temperature (T c) that indicates the ordered phase through its nonzero values, which are linked by inversion symmetry He suggested that the Gibbs potential's variation below T c can be represented as a power series of the order parameter, indicating that the ordering process is fundamentally nonlinear While this theory is widely accepted for uniform phases, it requires redefinition for anisotropic systems, and it fails to account for critical anomalies, partly due to its neglect of inhomogeneities during critical transitions.

The 6 1 Thermodynamical Principles and the Landau Theory address the effects of distributed spontaneous strains in uniform crystals Landau identified limitations in his initial abstract theory and proposed incorporating spatial derivatives of the order parameter to enhance the description of phase transitions This revised Landau expansion includes a Lifshitz term, which accounts for lattice modulation However, it remains uncertain whether anomalies stem from the dynamic behavior of the order parameter Ultimately, phase transitions are significant phenomena observed on a macroscopic scale.

In noncritical phases, thermodynamic properties can be effectively described by the ergodic average of microscopic variables representing the order parameter However, in the critical region, short-range correlations among these variables lead to slow motion, making ensemble averages inadequate The Landau-Ginzburg theory modifies this understanding by incorporating derivatives of the order parameter to account for spatial inhomogeneity, yet critical anomalies remain partly unexplained due to time-dependent fluctuations Currently, a reliable model for the transition mechanism is still lacking, but a thermodynamic approach offers a preliminary framework for addressing the issue Experimentally, it is essential to identify the order variable within a system based on its constituent ions and molecules, while also visualizing their behavior in anisotropic lattices through observed results.

This chapter focuses on key thermodynamic principles to address phase transitions, recognizing that Landau theory has limitations in anisotropic systems Given the abundance of literature on liquid and magnetic systems, including Stanley's notable monograph, our exploration of isotropic systems will be succinct and to the point.

Phase Equilibria in Isotropic Systems

The thermodynamic properties of an isotropic and chemically pure substance are represented by the Gibbs potential G(p, T), where pressure (p) and temperature (T) are external variables in equilibrium with the substance In equilibrium, the substance is uniform, as G(p, T) depends solely on these variables However, in scenarios like liquid-vapor equilibrium, the substance may not be homogeneous under specific p-T conditions Therefore, a condensing system requires two potentials, G1 and G2, to individually represent the distinct phases that can coexist in equilibrium within a certain range of pressure and temperature.

T, being maintained by exchanging heat and mass Accordingly, these Gibbs potentials of coexisting phases should be involved in different internal mechanisms specified by the numbers of constituent particlesN 1 andN 2 whilep andT remain as common external variables On the other hand, for a crystal, the structural detail should specify the Gibbs potential, although insignificant for thermal properties, as discussed later Hence, the knowledge of isotropic equilibria provides a useful guideline for structural phase transitions.

The thermodynamical equilibrium under a given p-T condition is determined by minimizing the Gibbs potential For a two-phase system, we minimize the total Gibbs functionG= G 1 +G 2 , where G 1 =G 1 (N 1 , p, T) and

Therefore, the phase equilibrium can be speciﬁed by

The derivative à = (∂G/∂N) p,T is known as the chemical potential, equivalent to the Gibbs potential per particle The equilibrium condition for two phases can be expressed as à1(p, T) = à2(p, T), indicating that pressure (p) and temperature (T) for phase equilibria are interdependent As shown in Fig 1.1, the two phases are graphically represented by areas divided by the curve defined by this equilibrium condition, where all points (p, T) along the curve signify that the phases are in equilibrium.

When comparing phase equilibria at two close temperatures, T and T+δT, along the equilibrium line, a pressure difference δp = (dp/dT)δT is anticipated due to the small temperature variation The slope dp/dT can be derived from the changes in pressure and temperature at a specific point (p, T), assuming the continuity of chemical potential across the line, represented by δà1(p, T) = δà2(p, T).

Here,s i=−(∂ài /∂T) p andv i = (∂ài /∂p) T are speciﬁc entropies and volumes of the phases i = 1 and 2, respectively From (1.2) we can derive the Clausius- Clapayron relation dp/dT =(s 1 −s 2 )

8 1 Thermodynamical Principles and the Landau Theory

The phase diagram of H2O illustrates the chemical potentials of ice, liquid water, and vapor in the pressure-temperature (p-T) plane Notably, the phase boundary between ice and water displays a significant negative slope rather than being perfectly vertical Additionally, the equilibrium line separating water and vapor concludes at the critical point (p c, T c) The terms ∆s and ∆v represent the structural differences between the two phases, indicating that the finite entropy difference corresponds to the latent heat per particle.

L = T∆s, and the finite volume difference indicates a packing difference. Equation (1.3) determines the rate at which the equilibrium pressure varies with the equilibrium temperature in thep-T diagram.

The reciprocal rate dT/dp indicates how the transition temperature changes with pressure, showing a positive correlation for liquid-vapor transitions due to the relationship v vapor > v liquid, where the vapor absorbs latent heat Additionally, the boiling point of a liquid increases with rising pressure, while the freezing temperature during a liquid-solid transition may either increase or decrease, depending on the sign of ∆v during solidification.

To calculate the vapor pressure of a liquid, we can integrate the differential equation dv vapor/dT = L, while disregarding the negligible volume of the liquid compared to the larger volume of the vapor This approach allows us to derive a practical expression for the vapor pressure.

The equation T v vapor = Lp vapor k B T² describes the relationship between vapor pressure and temperature, assuming the vapor behaves according to the ideal gas law, expressed as v vapor = kB T / p vapor, where kB is the Boltzmann constant By further assuming that L remains constant regardless of temperature, this equation can be integrated to yield the vapor pressure as p vapor = p o exp.

In the context of isothermal condensation, the vapor pressure of an ideal gas remains constant, represented by the integration constant \( p_o \), which equals \( p_{\text{vapor}} \) when \( L = 0 \) This observation enhances the understanding of the van der Waals isotherm and supports the mathematical conjecture in the equation of state discussed in Section 1.4.

Phase Diagrams and Metastable States

A phase diagram effectively illustrates the behavior of chemical potentials for two coexisting phases near equilibrium While the Gibbs potential (G) is suitable for a uniform substance, chemical potentials are more practical for systems with multiple phases in equilibrium Typically, the chemical potential (μ) varies continuously with pressure (p) and temperature (T), forming a smooth mathematical surface in the three-dimensional μ-p-T space In the context of liquid-vapor equilibrium, the surfaces representing the two phases intersect along a curve where their chemical potentials are equal This intersection indicates that the two phases can coexist, whereas at points off the equilibrium line, only one phase with a lower chemical potential remains stable.

In the case of a simple isotropic substance like water, which has three phases—solid, liquid, and vapor—these phases can intersect to form three equilibrium curves in the pressure-temperature (p-T) space A unique point, known as the triple point, exists where all three phases coexist For practical purposes, phase diagrams are typically represented in two dimensions, using pressure and temperature as variables while keeping a third variable constant This allows for the projection of the three-dimensional p-T surface onto the p-T plane Additionally, similar projections can be created for the pressure-enthalpy (p-η) and enthalpy-temperature (η-T) planes, yielding valuable phase diagrams under constant temperature and pressure conditions.

In phase diagrams, intersecting curves indicate accessible equilibrium states, but practical systems often exhibit metastable states, like point x on the extension of a constant à-line These metastable states, deviating from the vapor-liquid equilibrium curve, can appear stable despite being thermodynamically unstable For example, vapor can be compressed beyond its vapor pressure in the absence of significant condensation nuclei, which represent unavoidable impurities that play a crucial role in the condensation process.

The chemical potential surfaces for two phases in equilibrium are illustrated in Fig 1.2, with the intersection at point A indicating the critical point (c.p.) The thick broken line represents this equilibrium state, while points B and D depict potential metastable states at a constant pressure.

The triple point (Tt) and critical point (Tc) in a three-phase system are crucial for understanding phase transitions A point on the solid-vapor equilibrium line indicates a supersaturated state, where the metastable vapor is prone to instability This supersaturated vapor can react to external disturbances, such as shock waves, leading to sudden condensation.

A metastable state can be illustrated using a phase diagram, as depicted in Figure 1.4b, which presents an à-T diagram In this diagram, the à-curves 1 and 2 intersect at a specific temperature T x while maintaining a constant pressure p This unique crossing point is defined by the chemical potential à, highlighting that the transition between two phases typically occurs discontinuously in relation to the Gibbs potential.

In Fig 1.4b for a à-pdiagram, drawn asà2 Tc) indicate uniform states characterized by pressure (p) and volume (V), while those below the critical temperature (T < Tc) represent a condensing state with distinct vapor and liquid phases However, there are mathematical conjectures in equation (1.4a) that conflict with physical realities It is important to note that the isotherm, as described by equation (1.4), transitions continuously between these two categories as the temperature changes.

T c The three roots below T c become equal to V c , when the critical point is approached from below Therefore, in the limit of T → T c , (1.4b) should be

Fig 1.5 Van der Waals’ isotherms for vapor liquid equilibria in a p-T diagram.

At the critical temperature (T c), the liquid and vapor phases achieve a critical equilibrium defined by critical pressure (p c) and critical temperature (T c) Below T c, these two phases coexist along a horizontal line, represented by their respective chemical potentials, denoted as (A) for the liquid phase and (E) for the vapor phase.

AE at a constant vapor pressurep o The ﬁgure shows thatp o can be determined as area(ABC) = area(CDE). written as

Therefore, critical values p c , V c and T c are all determined by the molecular constantsaandb; that is p c = a

Using these results, we can conﬁrm that

= 0, which are the requirements for the inﬂection point with horizontal tangent at

It is realized that such a continuity of the van der Waals isotherm at

The concept of T c arises from the mean-field assumption regarding molecular interactions, treating the entire system as homogeneous This perspective, however, contradicts the existence of two distinct phases, specifically liquid droplets coexisting with vapor at the condensation threshold.

Isothermal p-V curves must adhere to the principle that pressure decreases as volume increases However, van der Waals isotherms below the critical temperature (T c) present a contradiction, as the segment marked BD in Fig 1.5 exhibits a positive slope This observation highlights a mathematical inconsistency within the van der Waals theory.

In Section 1.2, we demonstrated using the Clausius-Clapeyron equation that vapor pressure remains constant during isothermal condensation Consequently, with the assumption of a constant vapor pressure \( p_o \), all actual states between points B and D should align along the straight horizontal line \( p = p_o \) rather than following a curved trajectory.

In the context of an isotherm at temperature T, the straight line EA indicates equilibrium states of the vapor-liquid mixture at pressure p0 Points E and A mark the initial and final stages of condensation, while point P represents the coexistence of phases within the mixture.

When the vapor is compressed from a state Y along an isotherm below

During the condensation process, liquid begins to form at point E, and as compression continues to point P, additional liquid is created while coexisting with vapor at a constant pressure At this stage, the volume of the liquid remains constant, while the volume of the vapor decreases as the total volume is reduced This relationship between the two phases can be represented using Gibbs potentials for both the vapor and liquid.

G vapor(E) =G vapor(P) +G liquid(P)−p o(V 3 −V P), where lim

G liquid (P) =G liquid (A) for isothermal compression atT, so that we have the relation

Assuming, on the other hand, that the van der Waals equation is acceptable over the entire region of isotherms forT < T c , we may write a therma- dynamical relation

Therefore, in the limit of P→A, we can write

The thermodynamical principles, alongside the Landau Theory, indicate that the areas ABC and CDE, situated above and below the horizontal line p=p0, are equal in magnitude but opposite in sign This relationship holds true specifically at the temperature T = T_c.

At the critical temperature (T c), the Gibbs free energy of the liquid phase equals that of the vapor phase, indicating a continuous transition In contrast, below T c, the phase transitions are characterized by a discontinuity represented by the work done, p o (V 3 − V 1) By utilizing chemical potentials and the number of molecules in each phase, the Gibbs functions can be effectively expressed.

It is noted thatN liquid =N at state A, andN vapor =N at state E, so that

In the analysis of phase equilibrium, the Gibbs potentials for the liquid and vapor phases, represented as G liquid (A) and G vapor (E), demonstrate a relationship where àvapor(E) aligns with àliquid(A) along the equilibrium line depicted in Fig 1.5 However, there is a discontinuity in the Gibbs potentials at points E and A, quantified by p o (V 3 − V 1) on this isotherm This phenomenon is illustrated by the Maxwell equal-area construction, which indicates that the horizontal line p = p o is drawn in accordance with the thermodynamic equilibrium condition (1.1).

The van der Waals equation incorporates molecular interactions in the gas phase using a mean-field approximation, assuming gas molecules are in random motion while condensed phase molecules exhibit near order The liquid phase is treated as uniform, defined by a parameter v liquid, and the mixed phase is characterized by the difference in specific volumes (∆v) at temperatures near the critical temperature (T c), serving as the order parameter for phase transitions However, these assumptions are overly simplistic for the critical region, necessitating a more detailed approach Despite this, the specific density difference (∆ρ), where ρ = 1/v, is more practical than ∆v, allowing the definition of the order parameter η = ∆ρ = ρ liquid - ρ vapor ≈ ρ liquid This definition holds for a uniform state and is valid under mean-field accuracy for the mixed state below T c The order parameter is crucial for understanding phase transitions, typically defined as η = ρ order - ρ disorder ≈ ρ order, where the liquid phase is considered ordered and the vapor phase disordered.

Second-Order Phase Transitions and the Landau Theory

Figure 1.4b illustrates the equilibrium between different phases under constant pressure conditions in the T-Δ plane In this representation, two chemical potential curves, Δ1(T) and Δ2(T), intersect at a temperature Tx, where their slopes are typically unequal, indicating that (∂Δ1/∂T)p does not equal (∂Δ2/∂T)p This discontinuity in the slope signifies a critical point in the phase transition.

At temperature T x, a finite change in entropy, represented as ∆s = s 1 − s 2, indicates a latent heat of T x ∆s during the phase transition In the à-p diagram, where two à-curves intersect at point p x under constant temperature, the discontinuity in the slope (∂à/∂p) T at T x signifies the transition This process requires external work, calculated as -p x ∆v, for mass transfer between the two phases, as outlined in van der Waals theory Figure 1.6 illustrates the behavior of isotherms in the à-p diagram as the critical region is approached from below T x, with transitions at pressure p o characterized by discontinuous first-order derivatives (∂à/∂p) T.

ﬁrst order, whereas at the critical pointp c , the transition is continuous, where the two phases cannot be distinguished by the ﬁrst derivatives However, the

The two-phase equilibria in the à-p plane at varying temperatures illustrate the behavior of the intersection A (equivalent to E in Fig 1.5) between the curves à1(p) and à2(p) This analysis reveals that transitions are generally discontinuous, with the exception occurring at the critical point (c.p.).

The Landau Theory and thermodynamical principles allow for the distinction of two phases when higher-than-first-order derivatives of Gibbs potentials differ at the critical point Ehrenfest categorized these transitions as second- and higher-order phase transitions, based on the order of the lowest nonvanishing derivative.

Figure 1.7a depicts a transition in a T-α diagram marked by a common tangent at T_c If the curvatures at T_c are unequal, one phase remains more stable than the other on both sides of T_c, indicating no phase transitions occur Conversely, analyzing a system as a "single" phase across all temperatures allows us to mathematically fulfill Ehrenfest’s criterion for second-order phase transitions In this scenario, the transition is spontaneous and is characterized by a discontinuous change in curvature.

T c , i.e., ∆(∂ 2 à/∂T 2 ) p,T = T c = 0, and ∆(∂à/∂T) p,T = T c = 0 However, for such a system, the number of constituents is constant and insigniﬁcant, and so the

A second-order phase transition is depicted in the à-T diagram, illustrating the equilibrium of two phases, and in the G-T diagram, showcasing a continuous change in entropy (S) with a notable discontinuity in curvature at the critical temperature (T c), represented by two circles of varying radii This transition can be effectively described using the Gibbs function.

Denoting the Gibbs potential under a constant pas G o (T) forT > T c , the second-order transition toG(T) on lowering temperature can be expressed as

Here, the ﬁrst-order term of T c − T is absent in the expansion because

∆(∂G/∂T) p = 0 atT =T c , where the leading term proportional to (T c −T) 2 represents a change of the curvature at T c that corresponds physically to discontinuity in the heat capacity ∆C p

While the argument presented is valid for understanding second-order phase transitions, the anomalies observed in the critical region are more complex than this basic theory suggests Furthermore, despite the possibility that higher-order terms could play a significant role in the expansion, there is currently no empirical evidence from practical systems documented in the literature to support this notion.

In the realm of second-order phase transitions, it is essential to recognize that metastable states are absent, as evidenced by the lack of hysteresis in experimental observations.

Second-order phase transitions are commonly observed in systems experiencing order-disorder transitions, where the order parameter η can exhibit an inversion mechanism While η is typically a scalar in isotropic systems, it may behave as a vector quantity in crystalline systems, reflecting the directional movement of active species Thermodynamic properties do not inherently reveal the domain structure resulting from the intrinsic inversion mechanism, although they can be influenced by external fields or stresses In binary systems, ordered states may manifest in two forms without external influence due to inversion or reflection symmetry, leading to thermodynamic properties, represented by the Gibbs potential, that remain invariant under the inversion or reflection of η in the mirror plane.

Landau further postulated for a binary system that the Gibbs potential can be expanded into an inﬁnite power series ofη, which is expressed as

The thermodynamical principles outlined in the Landau Theory indicate that the Gibbs free energy, G, is defined at the critical temperature, T_c, with coefficients A, B, C, etc., being smooth functions of temperature Notably, odd-power terms are excluded from the expansion to satisfy invariance requirements As the temperature approaches T_c, the magnitude of the order parameter, η, becomes infinitesimal, allowing the series expansion to be truncated at the quartic term.

4 Bη 4 for the critical region without losing accuracy Therefore, at near T c , the order parameter can be determined by minimizing the truncated Gibbs potential

In this case, the order parameter in thermal equilibrium can be determined from ∂G

∂η =Aη+Bη 3 =η(A+Bη 2 ) = 0, yielding simple solutions; i.e., either η= 0, (1.14a) or η=±

(1.14b) near the critical temperature The solution η = 0 of (1.14a) is the value of the order parameter in the disordered state aboveT c , where the minimum of

G(η) occurs at η = 0, as characterized by A > 0 and B = 0 at T c On the other hand, the other solution (1.14b) can be real ifA < 0 and B > 0 and assigned to the ordered phase belowT c.

In the disordered phase, the Gibbs potential is defined as G(η) = 1/2 Aη², where equilibrium occurs at η = 0, indicating stability against fluctuations due to a positive A However, as the temperature drops below T_c, A becomes negative, causing the equilibrium to shift to η₀ = ±(−A/B)¹/², influenced by a positive quartic potential 1/4 Bη⁴ that arises at T_c This quartic potential is significant as it relates to correlations among microscopic order variables, reflecting the change in the sign of the coefficient A at T_c, as described by Landau.

The relationship A = A(T - T_c) indicates that A is positive when T is greater than T_c and negative when T is less than T_c Additionally, the coefficient B remains zero and positive in the respective temperature ranges above and below T_c Consequently, the equilibrium value of the order parameter η_o is defined as η_o = 0 for temperatures above T_c, and η_o = ±{(A/B)(T_c - T)}^{1/2} for temperatures below T_c.

Fig 1.8.Changes of the Gibbs potentialG(η) in the vicinity ofT c: from a parabolic aboveT c to a double-well potential belowT c, where the equilibrium is speciﬁed by η= 0 andη=±ηo, respectively.

Figure 1.8 illustrates the Gibbs potential curves as a function of η near the critical temperature (T c) Above T c, equilibrium is established at η = 0, while below T c, the equilibrium gradually shifts to ±η as the temperature decreases.

In a macroscopically homogeneous system, internal interactions can be assessed using the mean-field approximation However, due to binary symmetry, the ordered phase typically divides into two equal-volume domains below the critical temperature (T c), resembling phases with opposite polarizations While the Gibbs potentials G(±ηo) indicate that the properties of these domains are identical, thermodynamic explanations for the coexistence of opposite sublattices remain elusive Notably, these oppositely polarized domains can be transformed into one another through the application of an external field or stress.

In a magnetic system, transformations resemble vapor-liquid transitions that occur through vapor compression or decompression This first-order, irreversible conversion experiences energy loss during hysteresis cycles due to interactions with extrinsic agents like lattice defects When the order parameter responds to external actions, domain conversion can be effectively used to test theories in single-domain samples.

Susceptibilities and the Weiss Field

The interaction of the order parameter with an external field or stress (F) can influence the behavior of a crystal, particularly above the critical temperature (T c) When an external force is applied, the Gibbs potential becomes non-invariant under inversion, leading to unequal domain volumes and a non-continuous transition between domains Conversely, when F is zero, the Gibbs function remains invariant, resulting in a continuous transition where the domains for ±η occupy equal volumes This section examines the effects of a weak external force on the order parameter in the vicinity of T c.

In the context of the Landau theory, domains under the influence of an external field (F) can be characterized by susceptibility, reflecting their linear response Weiss emphasized that the internal field (F int) plays a crucial role in the singular behavior of susceptibility, which is instrumental in determining the transition temperature (T c) when the magnitude of the parameter (η) is finite.

1.6.1 Susceptibility of an Order Parameter

In the presence ofF, we can write the Gibbs potentials for two domains as

G(±η) =G o + 1 2 Aη 2 + 1 4 Bη 4 ∓αηF, (1.17) which is truncated at η 4 for small η, considering F as a suﬃciently weak perturbation Obviously, the system of dipolar η is forced to be ordered by

In a system subjected to a force F, clear transition temperatures above Tc are absent, leading to a diffuse transition Despite this, the equilibrium state at a specific force F can be established by minimizing G(±η) as described in equation (1.17) with respect to η.

∂η =A(±η) +B(±η) 3 ∓αF = 0 should be solved for the equilibrium value of η Ignoring further the term

Bη 3 for a smallη, we can immediately obtain expressions for the response at temperatures very close toT c; that is, χ=αη/F = (α 2 /A )/(T−T c) and χ(α 2 /A )/(T c −T) (1.18) for T > T c and T < T c , respectively, which are known as the Curie-Weiss law The susceptibilityχ goes to inﬁnity as T c is approached, which should be observed with a very small applied ﬁeld F ≈ 0, thereby identifying the transition as second order On the other hand, the responses below and above

The critical temperature \( T_c \) should be fundamentally distinct due to the overlooked correlation term \( B \) below \( T_c \) in the previous derivation Furthermore, in the critical region, the observed susceptibility \( \chi \) deviates from the Curie-Weiss law, as illustrated in Figure 1.9c, highlighting the limitations of the mean-field approximation Empirical observations indicate that \( 1/\chi \) can be represented using critical exponents \( \gamma \) and \( \gamma' \).

1/χ∝(T−T c ) γ and 1/χ∝(T c −T) γ , in the regions above and below T c , respectively The exponentsγ and γ are both equal to 1 in the mean-ﬁeld approximation.

1.6.2 The Weiss Field in a Ferromagnetic Domain

A real gas condenses at temperatures below the critical temperature (T c), which is determined by the molecular constants a and b, as described by van der Waals theory The transition between vapor and liquid at a constant temperature below T c can occur through external work In the context of binary phase transitions, the Landau theory provides only an approximate explanation, leaving the origin of the transition temperature T c unclear For magnetic systems, Weiss introduced the concept of a molecular field to represent average magnetic interactions within a crystal, attributing singular behavior in susceptibility to an internal field.

In a uniformly magnetized magnet, the internal magnetic field \( B_{int} \) is defined by the equation \( B_{int} = \lambda M \), where \( M \) represents the magnetization and \( \lambda \) is the Weiss constant Heisenberg later theorized that microscopic magnetic interactions arise from quantum-mechanical exchange mechanisms between adjacent magnetic ions These interactions can be quantified by the correlation energy term \( -\sum_{i,j} J_{ij} s_i \cdot s_j \), where \( J_{ij} \) is the exchange integral reflecting the correlation between spins \( s_i \) and \( s_j \) This expression can be reformulated to show that the sum \( \sum_j J_{ij} s_j \) corresponds to the local field \( B_{int}(i) \) at spin \( s_i \) Consequently, the average internal magnetic field \( B_{int} \) in the mean-field approximation can be expressed as \( B_{int} = \frac{1}{N} \sum_i \left( \sum_j J_{ij} s_j \right) = \lambda M \), encompassing long-range contributions, despite the original Heisenberg formula focusing on short-range spin correlations Additionally, the internal field \( B_{int} \) can be represented more broadly as a power series of \( M \), in line with Landau's theory.

In the presence of an applied field B o, the spin s iis considered to be in the effective field B o+ B int, so that the Weiss relation (1.19a) can be modified as

The thermodynamic principles, particularly the Landau theory, describe the paramagnetic susceptibility (χo) that follows the Curie law (χo = C/T), indicating the fundamental response from uncorrelated spins By integrating these concepts, the magnetic susceptibility can be reformulated as χo = M/Bo = C/(T - Cλ) = C/(T - To), which aligns with the Curie-Weiss law for temperatures above To, where To = Cλ signifies the transition temperature in the mean-field approach Consequently, the linear Weiss field is pivotal in explaining the unique behavior of magnetization (M) at the transition temperature (T = To).

The Weiss field B, as defined by equations 1.19a and 1.19b, remains a theoretical concept unless validated through experimental evidence However, in ferroelectric crystals, an analogous internal electric field has been successfully detected using polar molecular probes, lending credence to the notion of a real Weiss field in magnetized crystals Furthermore, the nonzero constant λ suggests that the transition temperature Tc is related to the formation of a minimum spin cluster with short-range correlations, analogous to the initial condensation of a gas into a liquid droplet This concept will be explored further in Chapter 3, where we discuss how second-order phase transitions are initiated by the formation of such ordered clusters with minimum correlations.

The Weiss fields, +B_int and -B_int, correspond to +M and -M in their respective domains, while the external field B_o influences the entire crystal Consequently, the Gibbs potentials for these domains under the external field B_o can be expressed accordingly.

G + =G o −(+ M ).(+ B int + B o ) and G − =G o −(−M ).(−B int+ B o ), whereG o is the potential for B o = 0 and, hence,

Similar to the condensation process described in equation (1.8), equation (1.21) illustrates a "phase transition" between two magnetization domains (M), influenced by external work represented as -M.Bo Consequently, a magnetized object can transition to a single domain when subjected to a sufficiently strong external magnetic field (Bo).

The total energy for domain transformation in soft magnets is represented by the equation (1.21), which accounts for the energy contributions of the domains under the influence of an external magnetic field Notably, this transformation occurs without the need for energy to move domain walls, highlighting the ability to switch domains effectively in soft magnetic materials.

The behavior of permanent magnets is complex due to the locking of magnetic domains within the crystal structure by lattice defects At the critical temperature (T c), the transition is classified as a thermodynamically second-order transition, where the Gibbs free energy (G) is equal for both states when the magnetic field (B o) is zero However, for temperatures below T c, the conversion of magnetic domains occurs as a first-order transition, as illustrated in Figure 1.11 for an idealized ferromagnet.

The comparison of magnetization curves for a soft ferromagnet reveals distinct behaviors under varying external magnetic fields When no external field (B₀ = 0) is applied, the transition in magnetization is sharp, whereas it becomes more diffuse with a weak applied field (B₀ > 0) As B₀ varies, the magnetic domains in the material exhibit characteristics akin to two phases in a first-order equilibrium, transitioning between states A and B.

Critical Anomalies, Beyond Classical Thermodynamics

In the critical region of a second-order phase transition, anomalies observed cannot be fully explained by thermodynamic principles, as they show significant deviations from mean-field theory in experimental results These anomalies can be attributed to correlated order variables, despite the unknown dynamical origins of the fluctuations Before delving into arguments that extend beyond classical thermodynamics, it is worthwhile to speculate on the potential causes of these critical anomalies It is logical to consider that internal fluctuations may arise from the system itself while maintaining equilibrium with its surroundings at a given pressure and temperature.

T It is noted that the order parameterηat parabolic minima of the Gibbs potential G(η) can be subjected to a harmonic motion rather than random ﬂuctuations In addition, such a change inηshould be described in terms of a space-time variation in the lattice structure, where ∆Gcan be expressed as a function ofδη=η−(0,±ηo) For a small deviationδη, we can write

Corresponding to these excitations ∆G, the kinetic energy 1 2 m(dδη/dt) 2 can be considered, where m is the eﬀective mass Therefore, we may deﬁne the

The thermodynamical principles and Landau theory indicate that the characteristic frequency of an oscillator is given by ϖ = (A/m)^(1/2) According to Landau's expression for A, the frequency of harmonic fluctuations is represented as ϖ ∝ (T−T₀)^(1/2) for T > T₀ and ϖ ∝ (T₀−T)^(1/2) for T < T₀, highlighting the significance of these softening frequencies in indicating phase transitions These fluctuations are associated with interactions within the lattice that undergoes symmetry changes, a concept introduced by Cochran through the notion of soft modes Furthermore, it is possible to detect the average change in Gibbs free energy (∆G) over a brief observation period (t₀).

The nonzero and detectable change in Gibbs free energy, ∆Gdt, occurs when the softening frequency, ϖ, is less than 1, while it diminishes over time if ϖ is greater than or equal to 1 As the temperature approaches T o, the softening frequency can decrease significantly, causing the characteristic time of fluctuations, τ = 2π/ϖ, to compete with the time scale t o This results in a measurable average ∆G t, which may indicate critical anomalies where the spatial profile of fluctuations becomes evident Consequently, the crystal exhibits inhomogeneity as the temporal fluctuations slow down.

Landau [7] recognized such nature of critical ﬂuctuations and described the spatial inhomogeneity in terms of distributed Gibbs potentials,

The average of local Gibbs free energy, denoted as ∆G t i ∆g i t, is determined at each position i by considering a sufficiently large volume to achieve a meaningful macroscopic average This approach acknowledges the inhomogeneity of the substance, characterized by local pressure p i and temperature T i For a nonzero value of ∆g i, it is essential that these local conditions differ from the surrounding pressure and temperature (p and T), allowing us to establish a significant relationship between them.

The equation ∆g i = g i (p i , T i )−g i (p, T)≥ −(T i −T)∆s i + (p i −p)∆v i illustrates the relationship between changes in local entropy (∆s i) and volume (∆v i) during an irreversible process This inequality indicates that local entropy production is inherently irreversible Additionally, the observed symmetry change at the transition suggests that lattice excitations play a crucial role in the intrinsic mechanism driving entropy production.

Critical fluctuations during second-order phase transitions differ significantly from random thermodynamic fluctuations caused by varying external parameters like pressure and temperature In typical crystalline states, thermodynamic fluctuations are minimal, allowing for thermal properties to be effectively described by ergodic averages However, critical fluctuations are linked to spatial deformations of the lattice, resulting in mechanical inhomogeneity, and are not ergodic Recent studies indicate that these critical fluctuations exhibit a sinusoidal nature rather than being random.

Remarks on Critical Exponents

In Landau's thermodynamic interpretation, critical anomalies are characterized by the spatial average of Gibbs potentials that deviate from equilibrium Near the critical temperature (T c), critical fluctuations occur over a long timescale (τ), resulting in a distinct spatial profile observable within a shorter timescale (t o ≤ τ) As a result, these anomalies cannot be adequately described using traditional thermodynamic concepts, as they are associated with the deformed lattice in the critical region, which will be explored further in Chapter.

3 Nevertheless, observed anomalies ofη,χandC p are empirically analyzed in thermodynamic terms with critical exponents on ∆T =T c −T, as expressed by η∝(∆T) β , χ > ∝(−∆T) −γ , χ < ∝(∆T) −γ , and

Exponential expressions are used to address anomalies in thermodynamics, although they lack rigorous justification and remain hypothetical This means that a primary mechanism is sought to explain phase transitions While the physical implications are not entirely clear, these formulas effectively illustrate deviations from mean-field predictions using critical exponents Importantly, there are universal relationships among these exponents across different systems, forming the foundation of scaling theory for phase transitions.

The scaling theory addresses spatial inhomogeneity by renormalizing microscopic variables at lattice sites, allowing for the consideration of ordering dimensionality, akin to short-range interactions in anisotropic crystals This approach enables the system to be treated as quasi-uniform, facilitating thermodynamic descriptions of phase transitions However, while collective modes from ordered clusters are significant, the scaling method may be overly simplistic for anisotropic correlations in real crystals Therefore, we advocate for a crystallographic model of short-range correlations as a more suitable framework for discussing structural phase transitions Despite its relevance in modern statistical physics, the scaling theory is not the primary focus of this monograph.

Order Variables, Their Correlations and

Statistics: the Mean-Field Theory

Order Variables

In Chapter 1, we defined the order parameter as a macroscopic variable indicative of phase transitions, derived from microscopic variables σm associated with ions or molecules at lattice sites m The ensemble average of these microscopic variables represents the macroscopic order parameter η, which is meaningful only in sufficiently uniform systems Furthermore, if σm varies with space-time coordinates at long wavelengths, statistical averaging may not be applicable in inhomogeneous crystal states.

Above the critical temperature (T c), the variables σm are in rapid random motion, resulting in a time-averaged value that is negligible at each lattice site Conversely, below T c, these variables exhibit slow correlated motion, leading to a range of averaged values across the crystal Near T c, the crystal displays partial order and topological inhomogeneity, which can manifest as domains or a sublattice structure as temperature decreases Consequently, ensemble averages should be computed for specific subsystems rather than the entire crystal, and observed results must be interpreted in the context of the conditions under which they were measured.

Identifying the active group within a system can be challenging, as the variable σm is not always apparent from the chemical formula or unit-cell structure, except in simple cases To understand σm, it is often necessary to examine the dynamical behavior in the critical region In the context of structural phase transitions, the collective motion of these variables σm is crucial and serves as a primary focus of our research To differentiate, we will refer to the microscopic σm as the order variable, distinguishing it from the macroscopic order parameter η.

The schematic representation of a binary system in two dimensions illustrates two ordered phases: (a) an ordered phase characterized by intermingling sublattices, and (b) an ordered phase featuring two contrasting domains The boundaries between these domains are indicated by broken lines.

In a statistical framework, the relationship between η and σm can be expressed through the spatial average of σm over a subsystem of N lattice sites, assuming σm values are uncorrelated or exhibit weak correlation Conversely, when σm values are locally correlated, the time average must first be computed for a cluster of correlated σm, which is subsequently averaged across the lattice space within the subsystem Thus, the equation can be represented as η = Σ(ηi * σm) / N, where ηi denotes the cluster of σm values.

The state of a subsystem can be regarded as thermodynamically uniform if correlated clusters are predominant, as postulated in the renormalization group theory However, during the ordering process, the crystal is generally inhomogeneous, requiring precise knowledge of correlated collective motion to understand its behavior.

Ordering processes in crystals are significantly more intricate than in isotropic materials due to the influence of the strained lattice Experimental observations reveal that the collective mode of σm changes gradually, as indicated by critical anomalies across various timescales While solid-state processes typically occur at a slow pace, the timescales for observation are often overlooked.

2.2 Probabilities, Correlations, and the Mean-Field Approximation 33 in the critical region In this chapter, we review existing statistical theories on binary systems, which are discussed in light of a slow variation In solid-states, values of σm tare usually calculated by using probabilities at sites m, which are in fact a valid concept in fast processes, where the timescale is assumed asinﬁnitywith the ergodic hypothesis that is the basis for statistical theories of random processes.

Probabilities, Short- and Long-Range Correlations, and the Mean-Field Approximation

In a binary alloy like Cu-Zn (β-brass), atomic ordering occurs spontaneously as the temperature decreases past a critical point (T c), driven by atomic rearrangements within the lattice When the rate of these rearrangements is sufficiently rapid relative to the observation timescale, the mean occupancy (σm) can be expressed through the probabilities of atomic presence at specific sites, denoted as p m (A) and p m (B) However, there is uncertainty regarding whether these rearrangements happen quickly enough under realistic conditions While individual thermal rearrangements can occur rapidly, collective atomic motion may slow down the process Traditional statistical theories assume that these rearrangements are fast enough to validate the probabilistic model, despite the complexities involved.

In a disordered phase where atoms are uncorrelated, the occupancy probabilities for sites can only be either 1 or 0, indicating whether a site is occupied by atom A or not For example, if site m is occupied by atom A, then p m (A) = 1, and p m (B) = 0 In this state, the probabilities are independent across all lattice sites, which are filled exclusively with either atom A or B However, below the critical temperature (T c), atomic correlations lead to dependencies between different sites, allowing the probabilities p m (A) and p m (B) to take on any continuous value between 0 and 1 due to varying atomic arrangements in neighboring sites Consequently, these probabilities and the variable σm can be treated as continuous functions of space-time coordinates, referred to as classical variables, in contrast to quantum-mechanical variables like spins, which exhibit discrete values when uncorrelated but behave like classical variables when correlated.

In systems characterized by slow-moving order variables σm, these variables exhibit a quasi-static distribution across lattice sites, allowing for time averaging due to their sufficiently rapid variation The order parameter η can be derived as a mean-field average, which is meaningful only when the spatial variance of the distribution is minimal The validity of this order parameter is assessed through the correlation function Γ(r mn), defined as Γ(r mn) = (σm − η)(σn − η) = σmσn − η² δmn, where r mn represents the distance between σm and σn In this expression, δmn is the Kronecker delta, equating to 1 when m = n and 0 otherwise In cases of complete disorder, the correlation function yields Γ(r mn) = 0 for all pairs where m ≠ n, indicating that σmσn = 0 and signifying a lack of correlations.

In an ordering process, the correlation function Γ(r mn) remains nonzero for all pairs, indicating that the product σmσn is significant Consequently, the correlation energy can be expressed as being proportional to σmσn.

The equation E mn = −J mn σmσn illustrates the correlation between variables σm and σn, with the coefficient J mn dependent on the distance r mn, indicating the strength of this correlation The negative sign in the equation signifies that σm and σn are in a stable arrangement, existing at a lower energy state While this relationship is typically considered for correlated pairs, it can also be directly derived for simple systems characterized by short-range energies.

The short-range interaction energy, denoted as Em for σm, can be articulated using the interaction energies between two atoms at specific sites m and n, represented as εAB(m,n) and εAA(m,n) This relationship is defined through the probabilities outlined in equations (2.2) and (2.3).

E m n E mn n[p m(A)εAA(m,n)p n(A) +p m(B)εBB(m,n)p n(B) +p m (A)εAB(m,n)p n (B) +p m (B)εBA(m,n)p n (A)], (2.6)

In a cubic lattice where only nearest neighbor interactions are significant, the equation can be simplified by omitting the site specifications for the energy parameters εAB, εAA, and εBB By utilizing the order variables σm and σn, the probabilities can be expressed as p m (A) = 1/2 (1 + σm), p m (B) = 1/2 (1 - σm), p n(B) = 1/2 (1 + σn), and p n(A) = 1/2 (1 - σn).

2.2 Probabilities, Correlations, and the Mean-Field Approximation 35 Substituting these relations inE mn, we have

E mn = 1 2 (2εAB+εA+εB) + 1 4 (εAA −εBB)(σm+σn) + 1 4 (2εAB −εAA −εBB)σmσn

K= 1 4 (εAA −εBB) and J = 1 4 (εAA+εBB −2εAB).

In the context of binary correlations with nearest neighbors, the parameter J signifies their magnitude, aligning with J mn in equation (2.5) The factor K can be zero when εAA equals εBB, which is common in most binary systems, while it approaches zero for alloys with similar atoms where εAA is approximately equal to εBB, rendering the K term generally insignificant Furthermore, the first constant term in equation (2.7) is independent of order variables, contributing to its insignificance Consequently, equation (2.7) has been validated as essentially identical to equation (2.5).

2.2.2 The Concept of a Mean Field

Order variables σ_min crystals are defined as statistical variables that utilize occupation probabilities at lattice sites These correlations among lattice site variables represent short-range molecular interactions, which can be interpreted in terms of probabilities for similar or dissimilar arrangements of atoms, despite the unspecified ranges in the analysis.

The internal field acting on σm can be represented by the quantity n J mn σn=F m, which is summed over effective ranges of r mn This leads to the expression E m =−σm F m By considering distances r mn for both nearest and next-nearest neighbors, we can analyze the correlation energy.

In the context of statistical mechanics, the short-range interaction energy at site m, denoted as E_m, allows us to compute the effective local field F_m(t) below the critical temperature T_c By averaging these local fields over the entire subsystem, we can derive the effective macroscopic field Within the mean-field approximation, this long-range average becomes a significant quantity that adheres to thermodynamic principles Consequently, an ordered system is characterized by the presence of a macroscopic internal field, represented as F = (ΣF_m s m F_m t) / N, in the mean-field approximation.

In a binary system, probabilities can be represented by long-range averages, where p(A) and p(B) satisfy the equation p(A) + p(B) = 1 across all sites The order parameter for the two subsystems is defined as η1 = η = p(A) - p(B) and η2 = -η = p(B) - p(A).

It is noted in general that 1≥ η1 ≥ 0 and 0 ≥η2 ≥ −1, where 1 ≥p(A), p(B)≥0 For complete disorder, η1=η2= 0, and so p(A) =p(B) = 1 2 On the other hand, for complete order, η1 = 1 andη2 =−1, which correspond top(A) = 1,p(B) = 0 andp(B) = 1,p(A) = 0, respectively.

If long-range correlations are signiﬁcant, the binary ordering in an alloy

In mean-field accuracy, the interaction between average probabilities p(A) and p(B) can be analyzed through the parameter J When J > 0, attracting unlike atoms at minimal distances decrease interaction energy by -J, while repelling like pairs become unstable, increasing energy by +J Conversely, intermingling sublattices can achieve stability in an anti-ordered crystal structure Focusing solely on nearest neighbors reveals the average number of interacting A atoms.

B and B-A pairs can be represented by the formula 2N zp(A)p(B), where z denotes the number of lattice sites at the shortest distance, and N represents the total number of sites in each domain, calculated as N = 1/2 N Consequently, this leads to the determination of the number of unlike pairs and their associated interaction energy.

N AB = 2N p(A)p(B) = 1 2 N z(1−η 2 ) and the total ordering energy is

The equation E = E₁ + E₂ = const + 2J{1/2 Nz(1−η²)} illustrates the relationship between energy states, indicating that in a disordered state with η = 0, the energy simplifies to E = const + 1/2 NzJ Notably, in a completely ordered state where η = ±1, the condition N_AB = 0 leads to a constant energy, while the energy difference between ordered and disordered states, -1/2 NzJ, quantifies the macroscopic energy reduction from disorder Therefore, during the ordering process, the energy associated with partial order is reflected in the η-dependent term in the equation.

Statistical Mechanics of an Order-Disorder Transition

The factor α is utilized for unit adjustments, and similar to the magnetic Weiss field B in λ M, the internal field F int is typically not directly measurable in standard conditions Nevertheless, it is important to note that this internal field F int can be effectively combined with an external field.

F, when dealing with the response of order variables toF In this context, the Weiss field is not a mere theoretical concept, but representing a real internal field in an ordered phase In fact, as will be discussed in Chapter 9, the internal electric fieldF int in some ferroelectric crystals was detected by dipolar paramagnetic probes in magnetic resonance experiments.

2.3 Statistical Mechanics of an Order-Disorder

The concept of long-range order, introduced by Bragg and Williams in their statistical theory of binary alloys, posits that the thermal properties of a partially ordered alloy can be defined by the order parameter η and the macroscopic correlation energy −E(η) They suggest that the ordering system functions as a canonical ensemble governed by statistical principles In this context, the ordered state arises from statistically correlated A-B pairs at nearest-neighbor sites, resulting in a vast number of combinations g(η) This significant "degeneracy" of the energy −E(η) is directly related to the system's entropy.

S(η) = k B lng(η) under a constant-volume condition, and we minimize the Helmholtz free energyF(η) =E(η)−T S(η) to obtain the equilibrium value ofηat a given temperatureT.

In a single-domain crystal, in order for N lattice sites to be occupied by either A or B atoms with no vacancies, the combination number is given by g(η) N N p(A)

The free energy can be expressed with the partition function

From the condition (∂F/∂η)V= 0, we obtain

Using the Stirling formula for a large N, the term lng(η) can be evaluated approximately using

Hence, the equilibrium order parameter at T can be determined from the equation zJ kB Tη= ln1 +η

Equation (2.11) can be solved graphically by ﬁnding the intersection between the straight line y= zJ

In the η-y plane, the relationship between 2kBTη and the curve η = tanh(η) is depicted in Fig 2.2a It is observed that when 2kBT/zJ ≥ 1, the only intersection occurs at the origin η = 0 Conversely, when 2kBT/zJ < 1, an additional intersection appears within the range 0 < η ≤ 1, indicating a state of partial order, with η approaching 1 signifying complete order This diagram illustrates the transition between these states.

The graphical analysis of the spontaneous order parameter reveals that for temperatures greater than the critical temperature (T > T c), the intersection point A between the line y = (zJ/2kB T)η and the curve η = tanhy results in a real solution of η = 0 Conversely, when the temperature is below the critical threshold (T < T c), the only intersection occurs at η = 0 Additionally, for ferromagnetic order, the intersection point B consistently exists below T c, where the line y = (T /T c)η − ηo intersects the curve η = tanhy, while no solutions are found above T c.

The Ising Model for Spin-Spin Correlations

between disordered to ordered states can be speciﬁed as 2kB T c /zJ = 1, where the unit slope at the origin gives the transition temperatureT c, namely

T c η= tanh − 1 η≈η+η 3 /3 for a smallη, from which an approximate relation η 2 ≈3(T c −T)

T c forT < T c (2.12b) can be derived Hence for a small η, the order parameter shows a parabolic temperature-dependence, which is a consequence of the mean-ﬁeld approximation.

The heat capacity for ordering can be calculated easily with the above results; that is

= 3Nk B , when T c is approached closely from below In the disordered phase,C V = 0 asη= 0, and, hence, the discontinuity at T c is ∆C V = 3Nk B

2.4 The Ising Model for Spin-Spin Correlations

In ferromagnetic crystals, internal magnetic interactions are quantum mechanical, and expressed by the Heisenberg exchange energy between spins s m and s n , i.e.

The Hamiltonian for a uniaxial magnetic crystal is given by H mn = −2J mn s m s n, where J mn represents the exchange integral between unpaired electrons of magnetic ions at lattice sites m and n In this system, the spin vectors precess around the unique z-axis at a constant frequency ωm, maintaining constant components s mz This behavior occurs as long as the spin-spin interactions, represented by the terms s mx s nx + s my s ny, are considered perturbative.

In this case, known as therandom phase approximation, the spin-spin correlations are described by the time average H mn tcalculated over the timescale t o of observation Namely,

H mn t=−2J mn[s mz s nz+s mx s nx+s my s ny t], where the second term vanishes if 2π/ωmand 2π/ωnare both shorter thant o.

It is noted that if theseprecessions can be assumed at random in phase, we can write in the zero order

The Ising model, represented by the equation H mn t=−2J mn s mz s nz, focuses on the significance of the z components of spin vectors, highlighting the impact of random phases in spin precessions In this context, the spin component s mz can be associated with the probabilities of two quantum states, ± 1/2, at site m, which parallels the classical binary variable for ordering This interpretation allows for a classification of magnetic ordering types, including antiferromagnetic, ferrimagnetic, spiral, and others Although not explicitly stated, the choice of a unique z-axis is generally linked to the considerable magnetic anisotropy present in a specific crystal.

The Ising model, initially developed for simplified ferromagnetic systems, can also be applied to various binary systems, such as binary alloys, where atomic rearrangements lead to ordering In these contexts, Ising spins serve as a statistical tool to describe the system's behavior, with classical spin variables defined at specific sites.

|m=a m |+ 1 2 +b m | − 1 2 , (2.15a) where | ± 1 2 are the wavefunctions for an uncorrelated spin s m, and the coeﬃcientsa m andb m are normalized as a m 2 +b m 2 = 1 (2.15b)

In this case, a m 2 andb m 2 are interpreted as the probabilities for the site m to be occupied by + 1 2 and − 1 2 spins, i.e p(+ 1 2 ) and p(− 1 2 ), respectively We can therefore deﬁne the order variable by σm=a m 2 −b m 2 (2.15c)

Assuming nearest-neighbor interactions, the short-range interaction energy is expressed as

+a m 2 b n 2 +− |s mz s nz |+−+b m 2 a n 2 −+|s mz s nz | −+ ], where we have consideredz= 8 andJ mn =J for a cubic lattice For spins 1 2 , these matrix elements are

The Role of the Weiss Field in an Ordering Process

+− |s mz s nz |+−=−+|s mz s nz | −+ =− 1 4 , hence

By substituting the factor 1 2 J with 2J mn, we express E mn as E mn = −J mn σmσn, which aligns with equation (2.5) Unlike the spin s mz that has two states | ± 1 2 , the binary order variable σm is defined by two probability values of ±1, and is referred to as pseudospin.

In the mean-field approximation, the internal field at all sites within the crystal is represented by the spatial average \( F_{\text{int}} = F_m \), which considers the sum \( \sum F_{m n} J \sigma_n \) for the local field at a specific site \( m \).

The internal ordering energy in each of two domains can then be expressed by the identical formula, that is

Accordingly, the internal energy of the whole crystal is

In conventional notation, the magnetic moment and internal magnetic field are represented as \( m_z = \frac{1}{2} \sigma_m \), where \( a_m = g \beta s m_z \) and \( B_{m_z} = \frac{2}{g \beta} F_m \) This leads to the expression \( E = -M B_{int} \), with \( B_{int} \) being the Weiss field and \( M \) the macroscopic magnetization Although derived from magnetic spins using random phase approximation, the Ising spin \( \sigma_m \) is effectively utilized to describe binary correlations in scenarios where probabilities are significant.

2.5 The Role of the Weiss Field in an Ordering Process

In the mean-field approximation, spontaneous ordering, represented by η, is influenced by the Weiss internal field F int, both of which emerge at the critical temperature T o and increase in magnitude as the temperature decreases Although initially derived for binary alloys, the equation can be adapted using binary probabilities p(±1) for Ising's spin states, which are defined by Boltzmann statistics By considering the internal field F int from the previous equation, and simplifying with α set to 1, the relationship can be expressed as η = tanh(zJ).

−F int k B T is the partition function for energies±F int of the Ising spinσ=±1 in the ﬁeld

In mean-field theory, the order parameter η is derived from the difference in Boltzmann probabilities for the states, expressed as p(+1) = Z^(-1) exp(+F_int / k_B T) and p(−1) = Z^(-1) exp(−F_int / k_B T) This leads to the relationship η = p(+1) − p(−1) = σ_m s Importantly, within the mean-field approximation, the spatial average is governed by the thermal average of the pseudospin energy in the internal field.

F int , although F int may remain as a conjecture unless supported by experimental evidence.

In a uniaxial ferromagnet, the order parameter is defined as the average of pseudospins, represented by η = σm s, while the internal energy is expressed as −1/2 N zJη² When a magnetic field B₀ is applied, the internal energy within each domain can be formulated accordingly.

E 2 (−η) =− 1 2 N 2 zJη 2 +N 2 (gβη)B o , whereN 1 andN 2 are not equal to 1 2 N Therefore, we can write the internal energies per order variable as ε+ =E 1 (+η)/ N 1 andε − =E 2 (−η)/ N 2 ; that is, ε+=− 1 2 zJ−gβB o and ε − =− 1 2 zJ+gβB o

We may consider the probabilities for these states as given by the Boltzmann statistics, i.e. p(+1) =Z − 1 exp

1−η, where the last expression was derived from the deﬁnition ofp(±1) in Section 2.2 From this relation, we obtain η= tanh

2.5 The Role of the Weiss Field in an Ordering Process 43 which is the equation to be solved forηin the ﬁeld B o Equation (2.17) can be solved forη in exactly the same manner as (2.11), by ﬁnding graphically the intersection between the straight line y zJ

2k B T η+gβB o k B T and the curvey= tanh − 1 η WritingzJ/2kB =T c as deﬁned in (2.12a), these are reexpressed as η T

Figure 2.2b depicts intersecting lines in the η-y plane, where the line at T = T c intersects the η axis at ηo = −gβB o /k B T c, a value that is typically very small For instance, with β set to 1 Bohr’s magneton and B o assumed to be 3 weber/m² at T c around 10³ K in a standard ferromagnet, ηo is approximately 10⁻² It is important to note that the presence of B o ensures a real solution for equation (2.18) at all temperatures, indicating the absence of a critical temperature, although η becomes singular at T c when B o equals zero.

Above T c , when the temperature is close to T c , we can set y ≈ η and

T c η−gβB o k B T c , which is the Curie-Weiss formula as written in the form χ=N gβ

The Weiss field, although primarily observed in ferroelectric crystals, is believed to represent a genuine internal field in ordered magnetic crystals as well, based on the preceding analysis.

This article reviews the statistical theories of binary ordering using the mean-field approximation, focusing on the Ising spin σm at lattice site m It highlights that statistical variables at lattice points may not exhibit periodic functions if their correlations are weak In mean-field theory, all σm are represented by the spatial average σm s, suggesting that the ordering is independent of the lattice structure The binary values of σm s are determined by Boltzmann probabilities, indicating that the ordering is a thermal process influenced by temperature Although the lattice's role is implicit, the thermal accessibility of these states is linked to random collisions with phonons, which contribute to the free energy as expressed by TdS under constant volume.

The inadequacy of mean-field theory in critical regions indicates that the ordering process occurs slowly, making probability an unreliable concept In displacive systems, collective displacements dominate, leading to violations of translational symmetry at lattice sites, which can create strains when correlated incommensurately This phenomenon is essential for understanding the symmetry change at the critical temperature (T_c) Below T_c, the free energy transitions through internal mechanical work (dW), resulting in an ordered crystal structure with a deformed lattice.

In lattice dynamical theory, normal crystals exhibit three independent acoustic modes at long wavelengths, alongside numerous high-frequency modes that represent thermal vibrations of the lattice These lattice modes can influence the Gibbs free energy based on external variables such as pressure and temperature, with mechanical and thermal contributions denoted by dW and TdS, respectively Chapter 3 delves into these order variables within displacive systems.

Collective Modes of Pseudospins in Displacive Crystals and the Born-Huang Theory

Displacive Crystals

In a stable crystal, ions or molecules are systematically organized at lattice sites, with thermodynamic properties only indirectly reflecting lattice symmetry An idealized crystal, depicted by a unit cell, is considered macroscopically uniform when surface imperfections and lattice defects are minimal.

In "perfect" crystals, the Gibbs potential is primarily a function of external variables such as pressure and temperature, similar to isotropic systems Structural transitions between crystalline phases involve symmetry changes, which are not the sole mechanism for the transition Many crystals experience reconstructive structural changes during phase transitions, while others see active ions or groups continuously displace from their lattice sites Thermodynamically, reconstructive transitions are classified as first order, whereas displacive transitions are second order In reconstructive transitions, the symmetries of the phases above and below the transition temperature are generally unrelated, while continuous transitions maintain a relationship between the two phases, highlighting theoretical interest in structural instability.

In a displacive system, structural changes occur through spontaneous linear or angular displacements of active ions or groups at the transition temperature (T c) As the temperature decreases, these displacements become more pronounced, which has been supported by findings from diffuse X-ray diffraction and magnetic resonance studies.

[20], [21] In a continuous phase transition, such displacements are believed to occur collectively, resulting in a change of macroscopic symmetry.

Perovskite crystals exemplify displacive phase transitions, characterized by their chemical formula ABO3 In the normal phase, the unit cell features an octahedral BO6 2− complex surrounded by eight A2+ ions located at the corners of a cubic cell The perovskite family showcases diverse structural changes, demonstrating various displacement schemes A notable example is the ferroelectric phase transition observed in BaTiO3.

The perovskite structure features distinct unit cells, illustrated in three forms: (a) the normal phase, (b) showcasing linear ionic displacements along the C4 axis in BaTiO3, and (c) depicting a rigid-body rotation of TiO2 6− octahedra around the C4 axis in SrTiO3 crystals.

At 405K, the structural change in SrTiO3 is characterized by an off-center displacement of the central Ti4+ ion along the C4 axis, which aligns with one of the cubic axes Conversely, the phase transition occurring at 105K involves a rotational displacement of the TiO6^2- octahedra around the C4 axis.

The low-temperature phase, referred to as a binary transition, is defined by two opposing directions of the order parameter, which are connected through inversion or reflection across the crystal's mirror plane This results in a microscopic order variable denoted by σmz = ±1 of a classical vector sm, indicating that the order at different lattice sites remains uncorrelated.

In the context of BaTiO3 and SrTiO3, σmz denotes the linear displacements of Ti4+ ions along the ±z directions and small-angle rotations ±δϕ around the z-axis, respectively While Landau theory treats these displacements as scalars, they function as vector order parameters within the lattice At the critical temperature (Tc), the local potential V(sm) for these vector order parameters experiences a change, which is reflected in the Gibbs potential change (∆G) through the spatial average of V(sm) s.

In the normal phase T > T c , ∆G should be zero, whereas the Gibbs potentialGis contributed by distributed local potential where the minimum is ats m= 0; that is,

The potential energy function V > (s m) is defined by the equation V > (s m) = 1/2 a x σmx² + 1/2 a y σmy² + 1/2 a z σmz², where the coefficients are positive, ensuring the stability of the lattice at s m = (σmx, σmy, σmz) = 0 Consequently, at this point, V > (s m) is equal to zero, indicating that the normal phase is characterized by s m t = 0 and s m s n t = 0 It is important to note that the local symmetry axes x, y, and z may differ from the lattice symmetry axes.

Below the critical temperature (T c), the local potential's minimum shifts from the origin to new positions, specifically at σ mz = ±σ o along the z-axis or at angles ±δϕ o, indicating a twist around the z-axis in the local environment.

Fig 3.2 Local crystalline potentials in a quasi-two-dimentional lattice (a) a paraboloidalV > aboveT c; (b) a double-well paraboloidalV < belowT c. potential Therefore, such a potential forT < T c can be written as

The potential function V < is expressed as V < (s m) = 1/2 a x σmx² + 1/2 a y σmy² + 1/2 a z σmz² + 1/4 b z σmz⁴, where a z < 0 and b z > 0, while a x and a y remain unchanged Figures 3.2a and 3.2b depict the potentials V > and V < for a y = 0, revealing two minima in V < at σmz = ±σo = ±(−a z /b z)¹/², indicating static equilibrium positions akin to those shown in Fig 1.8 Dynamically, fluctuations around ±σo and between +σo and −σo can occur, potentially due to quantum-mechanical tunneling or classical barrier jumping When variables at different sites m and n are correlated, the relationship m·s n = 0 leads to collective displacements, emphasizing the importance of nearest neighbor interactions Over an extended timescale, probabilities p m (+σo) and p m (−σo) can be defined for the variables m to occupy either minimum +σo or −σo, expressed as σmz σo.

For uncorrelated pseudospins, we have σmz t= 0 and p(+σo) s=p(−σo) s= 1 2

In the context of classical motion within crystal space, the transversal components σmx and σmy play a significant role when correlated For instance, when considering continuous motion in the xz plane, the direction of the spin magnetization (s m) can be represented by an angle θm from the z-axis This relationship leads to the equations σmz = σo cos θm and σmx = σo sin θm, where cos θm is defined as p m (+σo) - p m (−σo) and sin θm is expressed as {2p m (+σo)p m (−σo)}^(1/2).

The angle θm varies across different sites, indicating that the correlated pseudospins sm exhibit collective motion within the lattice It is important to note that the local potentials remain invariant under the inversion of classical pseudospins (sm → −sm) The onset of nonzero correlations sm · sn for collective pseudospins at the critical temperature Tc signifies a displacive phase transition At this temperature, the local potentials V(s m) and V(s n) begin to exhibit quartic anharmonicity, as described by the final term in equation (3.1b), which is responsible for the correlation between displacements at sites m and n.

Depending on the observation timescale, a slow classical displacement vector may not be influenced by thermal probabilities, rendering the Ising spin model insufficient By drawing an analogy to the Heisenberg exchange interaction, we propose a Hamiltonian to account for correlations between these displacements.

The correlation energy, represented by the equation H mn = -J mn s m s n, highlights the significance of all components of a displacement vector in classical pseudospins s m and s n at sites m and n This contrasts with Ising spins, which are only characterized by their z-component.

The Landau Criterion for Classical Fluctuations

In the following discussions, we postulate the equations (3.3a) and (3.3b) for pseudospin correlations responsible for structural phase transitions, leaving values ofJ mn to empirical evaluation.

3.2 The Landau Criterion for Classical Fluctuations

In a crystal, displacive variables m are located at regular lattice points and can function as a periodic element of the lattice period However, these finite displacements m can disrupt local lattice symmetry, leading to significant strains within the structure If such distortions persist over a long timescale, they may destabilize the lattice Although Born and Huang theoretically examined strained crystals, they did not focus on structural instability during phase transitions, which is crucial for understanding spontaneous structural changes Therefore, we must explore this issue further, beginning with the Landau criterion for a classical order variable, as it provides valuable insights into the interactions between order variables and the hosting lattice regarding structural stability.

In distorted crystals, low-frequency dynamical displacements manifest as acoustic excitations with long wavelengths, leading to stationary waves that create an appearance of inhomogeneity due to uneven density distributions Landau identified this spatial inhomogeneity in stressed crystals, conceptualizing them as comprising numerous small volumes dV(r, t) at position r, where thermodynamic properties deviate from equilibrium He noted that variations in classical order variables stem from density deviations defined by ∆ρ(r, t) = dσ(r, t)/dV, which are influenced by distributed thermodynamic probabilities ∆p(r, t) = p₀ − p(r, t), with p₀ representing the uniform distribution in an undistorted crystal This framework allows for the consideration of negative distributed entropy per volume for the distributed probabilities −∆p, as articulated by the Boltzmann relation.

In the context of thermodynamics, assuming that the pressure change (∆p/p) is close to 1, the maximum decrease in entropy (∆s(r, t)) can be approximated as being on the order of kB This results in a heat energy transfer of T_c ∆s = kB T_c to the lattice at temperature T_c, which subsequently leads to an excitation (∆ε(r, t)) within the crystal structure It is important to note that this energy transfer process is expected to be thermodynamically irreversible, allowing us to express this relationship as an inequality.

In the critical phase, the lattice experiences excitation due to displacive ordering, as indicated by the condition ∆ε( r, t)≤T c ∆s( r, t) = k B T c This phenomenon leads to negative entropy production, which is linked to changes in local volume dV i( r, t) The work done, represented by −p idV i, arises from spontaneous local stresses within the lattice structure.

On the other hand, classically such an excitation energy ∆ε can be described as related to a decay of the collective order variables( r, t); that is,

The equation ∂t = −s/τ suggests that the ordering energy is relaxed to the lattice excitation This energy transfer, denoted as ∆ε, is fundamentally quantum mechanical, even though the classical decay represented by the equation should be considered in the limit as ¯h approaches zero The Hamiltonian density, H, characterizes the interaction between s and the lattice, and the quantum-mechanical order variable operators must adhere to the Heisenberg equation i¯h.

= [H , s op] t , (3.6) where the time derivative in the classical approach should be evaluated by the average over the relaxation time interval ∆τ, allowing one to replace

∂ s op /∂t t by ∂ s /∂t in (3.5) From (3.6) we arrive at the uncertainty relation [17] for small variations ∆s and ∆ε in classical variables s( r, t) and ε( r, t), respectively; that is,

The relation τ∆ε∼¯h mirrors the traditional uncertainty principle when ∆s / s is approximately 1 Landau proposed the equation τT c ¯h/k B ∼10 − 11 (secK), indicating that this timescale τ supports the classical nature of s at the critical temperature T c, where the fluctuation ∆s can be negligible, thus affirming that s behaves as a classical variable Although thermally inaccessible, the minor classical excitation ∆ε, κ > for temperatures above T c and A (∆q,∆ω)² = A > (T - T₀) + κ > ∆q² and ϖ < (∆q,∆ω)² = A < (T - T₀) + κ < ∆q², respectively.

In neutron inelastic scattering experiments, soft modes can be identified through anomalies in scattering intensities at a fixed value of ∆q However, without scanning scattering angles, we anticipate that no insights into breaking spatial symmetry can be obtained from these intensity anomalies While we will address the issue of spatial fluctuations later, this discussion focuses on the detection of soft modes in standard scattering experiments conducted at a constant q.

The critical region is characterized by long-wave fluctuations at ∆q, where in a simple dielectric crystal, we approximate sm as pm e(u+ - u-)m At the phase transition when G = 0, small fluctuations can be expressed as ∆q = q, with the Fourier transform p±q influenced by the internal field E±q resulting from correlations with distant pm When an external field E is applied, the effective field becomes E±q = E + E±q, allowing us to investigate the singular behavior of p±q through susceptibility as E approaches zero The equation of motion for dipolar oscillations in the crystal is represented as d²p±q/dt² + γ dp±q/dt + ω²p±q = (e²/m)E±q exp(-iωt).

The susceptibility can then be deﬁned as χ ±q (ω) =χ ±q (ω)−iχ ±q (ω) = lim

The dielectric analysis of the soft mode is governed by the equations \( p ±q /E ±q = (e^2/m)/(ϖ^2 − ω^2 + iγω) \) and its real and imaginary components, expressed as \( χ ±q (ω) = (e^2/m)(ϖ^2 − ω^2)/{(ϖ^2 − ω^2)^2 + γ^2 ω^2} \) and \( χ ±q (ω) = (e^2/m)γω/{(ϖ^2 − ω^2)^2 − γ^2 ω^2} \) The characteristic frequency \( ϖ \) can be determined from the peak or the inflection point of \( χ ±q (ω) \), which occurs at \( ω = ϖ \) under undamped conditions (i.e., \( γ < ϖ - 1 \)) In contrast, when damping is significant, the behavior transitions to a relaxational decay.

In neutron inelastic scattering, soft modes can typically be observed at nonlattice points where G i = 0, as the wavevector of thermal neutrons is comparable to lattice constants Neutrons interact with heavy nuclei or magnetic spins located at lattice points, making them effective probes for phonon spectra The finite G i, which is similar in magnitude to the neutron wavevector, plays a crucial role in defining the scattering geometry in accordance with the conservation law of wavevectors For example, when scattering occurs at a zone-boundary point G i = 1/2 G, the exact scattering geometry can be expressed as K 2 - K 1 = 1/2 G, which relates to the energy conservation in the process.

4.5 Observation of Soft-Mode Spectra 85 byε2 −ε1=εo ∓∆ε Here K 1 ,ε1and K 2 ,ε2are wavevectors and energies of incident and scattered neutrons, respectively, andεo is the lattice excitation energy associated with 1 2 G At the ﬁxed geometry, phase ﬂuctuations are expected as related to loss and gain of the neutron energy∓∆ωduring inelastic scattering process The scattering intensity is generally expressed by the time average of correlated amplitudes of scattered neutrons;

The quantityA 1 / 2 G, m is called thescattering amplitudefrom the nucleus at a site m and the total scattering amplitude is given by

A 1 / 2 G ∝ m u m expi[−(ε2 −ε1 −εo ±∆ε)t m /¯h] m u mexpi[(ω2 −ω1 −ωo)t m ] expi(±∆ω.t m ), where these energies are expressed in frequenciesω1 , 2=ε1 , 2 /¯handωo=εo /¯h.

Therefore, the scattering anomaly can be expressed as

The equations of motion for the fields \( u_o \) and \( u_o^* \) are influenced by effective fields \( F_o \exp(\mp i \Delta \omega t) \) due to their coupling with pseudospins These can be expressed as \( \frac{d^2 u_o}{dt^2} + \gamma \frac{du_o}{dt} + \omega^2 u_o = F_o \exp(-i \Delta \omega t) \) and \( \frac{d^2 u_o^*}{dt^2} + \gamma \frac{du_o^*}{dt} + \omega^2 u_o^* = F_o^* \exp(i \Delta \omega t) \) The steady-state solutions arise from these equations.

Fig 4.5 Phonon energy in K2SeO4 measured by neutron inelastic scattering at

G i = 0.7 a ∗ Curves 1, 2, 3 and 4 were obtained at 250, 175, 145 and 130K, respectively (From M Iizumi, J D Axe, G Shirane and K Shimaoka, Phys Rev B15,

In the study of neutron scattering, the time correlation function Γ(t) is defined as Γ(t) = sin(∆ω·t₀)/(∆ω·t₀), evaluated at the observation timescale t₀ Notably, in the critical region where ∆ω·t₀ < 1, the value of Γ(t) approaches 1 Furthermore, the product u₀* · u₀t is directly proportional to the imaginary part of the susceptibility χ(∆ω), as defined in equation (4.23b) This relationship highlights the scattering anomaly observed at the reciprocal lattice vector 1/2 G.

∆I( 1 2 G,∆ω) = 2|F o | 2 χ 1 / 2 G (∆ω), (4.24) allowing one to identify the soft frequency from the peak of scattering anomalies that occur when ∆ω=ϖ.

Neutron inelastic scattering at arbitrary points within the Brillouin zone, such as G i = 0.7 a ∗, reveals soft modes in orthorhombic K 2 SeO 4 crystals, as illustrated in Figures 4.5 and 4.6a Additionally, scattering results at the zone boundaries in materials like SrTiO 3 and KMnF 3 further exemplify these phenomena Moreover, ferroelectric anomalies identified in the dielectric spectra ε(ω) of TSCC, as reported by Sawada and Horioka, are interpreted as mixed with zero-frequency anomalies, which will be elaborated on in the following section.

Susceptibility indicates how the order variable responds linearly to the mean field, increasing with stronger correlations As the critical temperature for a continuous structural change is approached, soft modes become evident, arising from the quartic potential linked to the fluctuating lattice potential.

The Central Peak

Fig 4.6 (a) Soft-mode spectra from SrTiO3 and KMnO3 (From S M Shapiro,

J D Axe, G Shirane and T Riste, Phys Rev B6, 4332 (1972).); (b) Oscillator- relaxator behavior in the dielectric response from TSCC nearT c= 130K (From A. Sawada and M Horioka, Jpn J Appl Phys 24-2, 390 (1985).)

Recent studies have revealed an anomalous absorption peak near zero frequency in phonon susceptibility curves of practical crystals, in addition to a temperature-dependent soft mode This sharp zero-frequency peak, known as the central peak, has garnered significant attention despite its unclear origin Riste and colleagues identified this central peak in the phonon spectrum of SrTiO3 crystals during a cell-doubling phase transition near the critical temperature Shapiro et al expanded the investigation of the central peak phenomenon across various systems, illustrating the spectra in their research The zero-frequency absorption line indicates a relaxation to the lattice, likely caused by imperfections, although its precise origin remains unidentified due to the featureless decay The measured relaxation time is typically around 10^-9 seconds, often constrained by instrumental resolution In dielectric studies of the ferroelectric phase transition in TSCC, Sawada and Horioka analyzed the dielectric spectra, linking the soft lattice mode to the central peak and interpreting the anomalies as a decay of the lattice mode as frequency approaches zero.

Damping in lattice potentials typically arises from anharmonic potentials of odd power, linked to strains within the crystal structure However, decay at zero frequency indicates an additional damping mechanism likely caused by lattice imperfections This leads to the consideration of the damping mechanism for the lattice mode \( u_q \), which involves both damping from lattice strains and an additional relaxational mode \( v_q \) The equation of motion is expressed as \( \frac{d^2 u_q}{dt^2} + \gamma \frac{du_q}{dt} + \gamma \frac{dv_q}{dt} + \omega^2 u_q = F_q e^{-i\omega t} \), while the relaxational mode \( v_q \) follows the equation \( \frac{dv_q}{dt} + \frac{v_q}{\tau} = F_q e^{-i\omega t} \), where \( \tau \) represents the relaxation time Assuming a simple coupling \( v_q = c u_q \), the steady-state solutions can be derived as \( u_{q0} (-\omega^2 - i\omega\gamma + \omega^2) - i\omega\gamma v_{q0} = F_q \).

Therefore, the susceptibility foru q is given by χ q (ω) = 1/{ϖ 2 −ω 2 −iωγ−icγ F q ωτ/(1−iωτ)}, or by lettingcγ F q =δ 2 for convenience we have the formula for a so-called coupled oscillator-relaxator: χ q (ω) = 1/{ϖ 2 −ω 2 −iωγ−δ 2 ωτ/(1−iωτ)} (4.25)

Particularly, if the conditions γ δ 2 τ and ϖ τ − 1 are fulﬁlled [39], the imaginary part of (4.25) can be shown to be χ q (ω) = ω ϖ 2 −ω 2 δ 2 ϖ 2 τ

Symmetry-Breaking Fluctuations in Binary Phase Transitions 89

The second term in equation (4.26) indicates absorption due to a soft mode at frequency ω=ϖ, while the first term reflects a Debye-type relaxation that becomes significant at ω=δ A key aspect of this equation is that the soft mode transitions to a nonzero frequency ϖ=δ, subsequently followed by the relaxation mode The dielectric dispersion spectra of TSCC, depicted in Fig 4.6b, are primarily influenced by a relaxational mode In SrTiO3, the linear extrapolation of ϖ² toward T_c does not clearly reveal a small non-zero δ², but the estimated value aligns with the magnitude derived from related anomalies in EPR spectra For TSCC, Sawada and Horioka reported estimates of δ=0.6 cm⁻¹ and τ=0.9 s based on dielectric measurements.

Fujimoto and his team assessed the soft-mode frequency to be around 20GHz, suggesting a non-zero terminal frequency in experimental findings This indicates a finite coupling (δ) between soft and relaxational modes, highlighting the significant impact of lattice imperfections on structural transformations However, the specific mechanisms behind these transformations remain unclear based on the observed central peaks.

4.7 Symmetry-Breaking Fluctuations in Binary Phase Transitions

Critical fluctuations in thermodynamics can be described by the Gibbs potential variation, δG, which occurs near equilibrium due to momentum-energy exchanges between pseudospins and soft phonons These fluctuations are characterized as sinusoidal patterns in space-time.

∆ q and ∆ω or described by the ﬂuctuating phase Therefore, as signiﬁed in part by ±∆ q , the spatial variation in the critical region can be revealed

The plot of the squared soft-mode frequency (ϖ²) against the temperature difference (T - Tc) reveals insights from neutron inelastic scattering experiments conducted on SrTiO3 at the reciprocal lattice point G i = (1/2, 1/2, 3/2) This analysis is achieved through the examination of scattering experiments or by utilizing magnetic resonance probes to sample the condensate.

Caused by the couplingwwith a lattice mode in near phase, binary pseudospins ﬂuctuate around (±q,∓ω), for which the corresponding kinetic energies can be written as ε( q±∆ q ) = (¯h 2 /2m)( q±∆ q ) 2 , ε(−q±∆ q ) = (¯h 2 /2m)(−q±∆ q ) 2

The interaction with the lattice mode results in variations of ±∆q and corresponding energy changes ε(q±∆q) - ε(q) = ∓ℏ∆ω These pseudospin modes can be described by the equations s(q±∆q) = sq exp{i[(q±∆q).r - (ω±∆ω)t]} and s(-q±∆q) = s-q exp{i[(-q±∆q).r + (ω±∆ω)t]}, where sq and s-q represent the amplitudes of the pseudospin modes.

A binary crystal system exhibits reflection symmetry on a crystallographic mirror plane, where pseudospin modes propagate in opposite directions and are related by the condition s(r) → −s(−r) Due to their sinusoidal nature, the inversion in crystal space (r → −r) corresponds to wavevector inversion in reciprocal space (q → −q) Below the critical temperature (T_c), the inversion symmetry is disrupted, resulting in the formation of two opposite domains that reflect broken reflection symmetry for the pseudospin modes Consequently, the inversion relations q → −s −q must be applied to these amplitudes.

The kinetic energies of fluctuations are identical when ∆q = 0, indicating that ε(q) equals ε(−q) To break reflection symmetry, it is essential to identify asymmetrical fluctuations between +∆q and −∆q Furthermore, the inversion of pseudospins is largely independent of the harmonic lattice, suggesting that these fluctuations are linked to anharmonic interactions For convenience, we express K as ±q ∓ ∆q, and plot the kinetic energies of fluctuations ε(±K) = ¯h²K²/2m against ±K As shown in Fig 4.8, two parabolic curves for ε(K) and ε(−K) intersect at K = 0, where ε(0) equals ε(q) and ε(−q), specifically ¯h²q²/2m when ∆q = 0.

When a perturbing potential is present at K = 0, the previously independent states become coupled, leading to fluctuations that can be described by these combined states This scenario resembles a well-known level-crossing problem, where the degeneracy at K = 0 can be resolved through the introduction of a perturbing anharmonic potential.

In the vicinity of K = 0, the perturbed pseudospin modes (x, t) can be represented as a linear combination of two propagating modes at K = ±q ∓ ∆q and ∆ε = ℏ∆ω When considering the propagation direction x as perpendicular to the mirror plane, the fluctuating mode is expressed as s(x, t) = c+ s₀ exp(i(Kx - ∆ωt)) + c− s₀ exp(i(-Kx + ∆ωt)).

4.7 Symmetry-Breaking Fluctuations in Binary Phase Transitions 91

Fig 4.8.(a) Critical spatial ﬂuctuations near the minimum condensate energies at

K=±k; (b) a magniﬁed view of the circled part in (a) showing an energy gap at

In a perturbing quartic potential, the phase of fluctuations near K = 0 is represented as φ = Kx - ∆ω.t, with the expression s(φ) = c + s₀ exp(iφ) + c₋ s₀ exp(-iφ) The mixing constants c₊ and c₋ are normalized such that c₊² + c₋² = 1 For small values of K, φ can be treated as a continuous function that spans angles from 0 to 2π The fluctuating lattice energy associated with the pseudospin mode can be expressed as δU = 1/2 A u K u - K + 1/2 κ.

The equation K ,−K u K u −K represents the potential and kinetic energies associated with harmonic distortion, with the third term indicating anharmonicity in the fourth order In the context of this quartic potential, it is essential to focus on phonon scattering.

K+ (−K)→K + (−K ), to obtain a secular perturbation, as in the Cowley theory If the phonon scattering (K ,−K ) is regarded as independent from the scattering (K,−K), the factor

In the context of quartic energy, the terms K, −K, u, and K can be simplified using the mean-field average u²K, represented as −A/B in Landau’s expansion This simplification, known as the Wick approximation, streamlines calculations by reducing perturbations to a quadratic form Consequently, the quartic potential energy can be expressed as δUₚ = −1/4 A.

The approximation K,−K u K u −K (4.28a) is employed, disregarding all other terms near K= 0 This method mirrors the approach taken in Subsection 4.4.2, which addresses the symmetry change during the ferroelectric phase transition in TSCC.

Writing (4.28a) with indexes K=q±∆qexplicitly, we have δU p =− 1 4 A[u q± ∆ q ∗ u q± ∆ q +u −q± ∆ q ∗ u −q± ∆ q

(ϖ−∆ω)t}, and so forth Considering only those terms for±q∓∆q modes, the ﬂuctuation potential energy can be expressed as δU p =− 1 4 A u 2 o [2 + 4 cos(2qx) + 2 cos{2(q+ ∆q)x−2∆ω.t}

In the context of phase matching for perturbed pseudospins near K = 0, only the last two terms are relevant For pseudospin modes at K = 0, we can align the partial potential V p (2φ) with the quartic lattice potential δU p, represented by the phase 2φ = 2(Kx−).

∆ω.t) Thus, we arrive at the perturbing potential energy fors(φ) atK= 0:

V(φ) =C∆ocos 2φ, (4.29) for which ∆ o cos 2φcan be regarded as an eﬀective displacement due to the quartic strains andC is a constant proportionality factor.

For the unperturbed modess(φ) ands(−φ) with degenerated energies at

K= 0, we calculate the matrix element ofV(φ):

4.7 Symmetry-Breaking Fluctuations in Binary Phase Transitions 93

Writing C =C/8π for brevity, for the energy ε(φ) = ε(−φ) = ¯h 2 q 2 /2m at

K= 0, the degeneracy will be lifted as calculated with the secular equation ε(φ)−ε C ∆ o

Solving this equation, we obtain ε=ε ± = 1 2 {ε(φ) +ε(−φ)} ± 1

= ¯h 2 q 2 /2m±C ∆o , (4.30) which gives an energy gapε+ −ε − = 2C ∆ o atK= 0 Corresponding to these energiesε ± separated by 2C ∆ o , the pseudospin modes of (4.27) are given by symmetric and antisymmetric combinations ofs(±φ), i.e 1 2 { s(φ)± s(−φ)}.

The energies of the combined modes remain unchanged upon reflection, with mixing constants \( c_+ \) and \( c_- \) determined to be ±1, based on the normalization condition \( c_+^2 + c_-^2 = 1 \) The normalized functions can be represented as \( s_{\pm}(\phi) = \frac{1}{2} \{ s(\phi) \pm s(\pi - \phi) \} = \frac{1}{2} \{ s(\phi) \mp s(-\phi) \} \), corresponding to the perturbed levels \( \epsilon_{\pm} \) The antisymmetric function \( s_{-}(\phi) \) is associated with the lower level \( \epsilon_{-} \), while the symmetric function \( s_{+}(\phi) \) corresponds to the upper level \( \epsilon_{+} \), which are expressed in terms of the phase and amplitude modes, represented by \( \cos(\phi) \) and \( \sin(\phi) \) respectively.

Macroscopic Observation of a Binary Phase Transition;

Fig 4.9 (a) The phonon dispersion curve in ThBr4 crystals at 81K, showing resolved amplitude and phase modes (From L Bernard, R Currat, P Delanoye, C.

M E Zeyen, S Hubert and Kouchkovsky, J Phys C16, 433 (1983).) (b) Phonon energy curve observed by Raman scattering from K2SeO3 The phase between T i andT cis incommensurate, and the phase belowT cis ferroelectric (From M Wada,

H Uwe, A Sawada, Y Ishibashi, Y Takagi and T Sakudo, J Phys Soc Japan,

4.8 Macroscopic Observation of a Binary Phase

Transition; l -anomaly of the Speciﬁc Heat

Pseudospin condensates exhibit notable thermal stability, primarily attributed to low-damping soft modes These collective pseudospin modes can be observed within a brief timescale of t ≤ 2π/ϖ Close to a phase transition around 100K, low-energy excitations indicate mechanical strains within the lattice structure, which defy statistical analysis based on the ergodic hypothesis Despite not sharing thermal energies with the crystal, these nonergodic excitations remain thermally inaccessible, while underdamped soft modes suggest a gradual energy transfer between the condensates and their environment.

The phonon dispersion curves and pseudospin variations illustrate the amplitude (a) and phase (p) modes in a crystal system The Gibbs free energy serves as a valid thermodynamic potential, considering that a crystal's volume can fluctuate under internal stresses Phase transition thresholds are influenced primarily by short-range correlations, where condensates achieve mechanical equilibrium with lattice strains Additionally, the specific heat of this system is observed over a significantly longer timescale compared to the characteristic time of the condensates, represented as 2π/ϖ.

In Section 4.7, the ﬂuctuating modes(±φ) was considered as perturbed by spontaneous strains expressed by an eﬀective displacement ∆(φ) = ∆ o cosφ.

The internal strain energy, denoted as δU strain, acts as the "sink" for the ordering energy, as indicated by the lattice strain energy Despite being thermally isolated from the rest of the crystal, this energy can be expressed in terms of the strains ∆ K (φ) as δU strain = − 1/2 α.

∆ K = ∆ocos 2φ, ∆ K = ∆ocos 2φ , φ=Kx−Ωt and φ =Kx −Ωt

Here, K = q±∆q, Ω = ϖ∓∆ω, and α is the proportionality constant, therefore, δU strain=− 1 2 α∆ 2 o cos 2φcos 2φ

4.8 Macroscopic Observation of a Binary Phase Transition 97

As indicated by (4.31) and (4.32), critical space-time ﬂuctuations are observed in two independent modes as cosφand sinφat constant Ω and constant

K In the trigonometric formula 2 cos 2φcos 2φ= cos 2(φ+φ ) + cos 2(φ−φ ), we notice that the ﬁrst term depends on the average phase φo = 1 2 (φ+φ ), that is however constant and hence insigniﬁcant On the other hand, the second cosine term determined by cos 2Ω(t −t ) at x = x leads to the unvanishing time average 1 2 , if integrated over a long timescale, and for thermal observation the space-time average of δU strain can be expressed as δU strain t=−(1/4)α∆ 2 o +const.Hence, we can write the corresponding strain energy of the lattice as

W u = 1 4 A∆ 2 o + const., (4.34) whereAis a constant after adjusting the magnitude to the macroscopicW u

We consider that the ﬂuctuating pseudospin mode exchanges the energies ±¯h∆ωwithW u that can be interpreted as the internal “heat sink”.

In (4.30), correlation energies of collective pseudospins are expressed in two separate modes of kinetic energies ε ± (φ) For the ﬂuctuating wavevector

The equation K = q o ∓ ∆q can be expressed as ε ± (q o ∓ K, ∆o) = ε ± (±∆q, ∆o) = 1/2 (X o + X) ± (X o X + C²∆²o)¹/², utilizing the convenient abbreviations X o = ¯h²q o²/m and X = ¯h²K²/m It is important to note that this formulation is based on fixed spacetime coordinates (x, t), while the wavevector q or K is subject to fluctuations.

The Gibbs free energy of a system of pseudospins and the lattice strains can be written as

2π +W u , which can be minimized with respect to ∆ o independently for the two states of a condensate in thermal equilibrium with the surroundings For the lower state, we can write dG − d∆ o = 2 q o o

WritingX o X=ξ 2 , where ξ= (¯h 2 /m)Kq o , we have dK= (q o /X o )dξ, and so q o

C ∆ o , and hence the equilibrium condition can be expressed as

From this relation, the strain amplitude ∆oin equilibrium can be determined from ¯ h 2 q 2 o mC ∆ o = sinh

In a binary system, when the argument of the sinh-function exceeds 1, we observe a discontinuity in the Gibbs free energy, quantified as 2C ∆o at the critical temperature T c This conclusion can also be reached by minimizing the function G.

The specific heat curve exhibits a λ-anomaly, characterized by a distinct shape resembling the Greek letter "lambda," as shown in Fig 1.10 Utilizing the condensate model, the sharp increase in the curve at T_c, observed from above, indicates a sudden discontinuity (∆G= 2C ∆o) linked to the onset of correlated motion, resulting in (∆C_p) T_c = ∞ In contrast, the gradual tail observed below T_c reflects the slow ordering process occurring in and beneath the critical region.

In the non-critical region below the critical temperature (T c), the ordering process aligns with the temperature dependence of the order parameter for long-range order, represented by η ∝ (T o − T) 1 / 2 This suggests that the temperature-dependent Weiss field (E int) plays a key role in the gradual changes observed in the specific heat curve As discussed in Section 5.9, it can be inferred that the pseudospin energy is transferred incrementally to the lattice.

∆σE int (T) ∝ ∆T in the soliton potential, representing deformed levels of the structure Therefore, the change in the Gibbs potential below T c for a temperature step ∆T can be expressed as

E int dT where α is a constant related to previously deﬁned α in (1.17) Assuming

E int ∝(T o −T) 1 / 2 , and writingkfor the front factor of the integral to simplify the expression, we obtain

= 2C ∆o+ 1 2 k/(T o −T) 1 / 2 , which agrees at least qualitatively with the gradual tail in observed curves, except that it becomes inﬁnity asT →T o instead of the transition temperature

T c Presumably, in the critical region the distributedE intis responsible for the anomaly betweenT o andT c Nevertheless, using a empirical critical exponent

4.8 Macroscopic Observation of a Binary Phase Transition 99 α , we may write ∆C p ∝(T c −T) −α for observed anomalous curves, where

E int ∝η(T c −T) β can be considered to cover the region betweenT o andT c with the exponentsα andβ Hence, we have a relationshipβ−1 =−α , which gives speciﬁcallyα =β= 1 2 andT c =T o in the mean-ﬁeld approximation.

Dynamics of Pseudospins Condensates and the Long-Range Order

Imperfections in Practical Crystals

In the critical region, pseudospin condensates exhibit sinusoidal fluctuation modes characterized by σocosφ and σosinφ, where the amplitude σo approaches infinitesimal values at the threshold The phase φ, defined as ∆q.x + ∆ω.t + φo, indicates propagation along a specific direction x in an anisotropic crystal, with a speed v equal to ∆ω/∆q In idealized crystals, the phase constant φo remains undetermined unless boundary conditions are applied at specific space-time coordinates (xo, to) of a lattice site It is important to note that while light- and neutron-scattering experiments can observe small values of ∆q and ∆ω, they do not conclusively validate the existence of pseudospin condensates without proper identification of the scatterers Conversely, magnetic resonance sampling allows for the visualization of collective pseudospins in slow motion, providing a reliable representation of condensates within the laboratory frame of reference.

Real crystals are inherently imperfect due to surface irregularities and unavoidable defects that disrupt lattice periodicity While practical crystals can be considered perfect under high-energy observation, low-energy condensates encounter significant challenges from these lattice imperfections, which can immobilize them by pinning them in place In high-quality crystals with low defect density, these pinned condensates can be effectively studied, particularly in ferroelectric crystals, where the quality can be assessed through the coercive force values in the hysteresis curve of dielectric polarization.

In practical crystals, imperfections significantly disrupt lattice periodicity, primarily at their sites, as modeled by point imperfections While perfect crystals remain a theoretical concept, these point defects play a crucial role in pinning condensates, enabling the exploration of their intrinsic properties.

102 5 Dynamics of Pseudospins Condensates and the Long-Range Order condensates in relation to a given lattice structure Pinned condensates con- stitute primarily a subject for experimental investigation of structural phase transitions.

The Pinning Potential

In this analysis, we focus on the critical region where the collective pseudospins exhibit sinusoidal propagation A stationary point defect located at lattice site \( r_i \) is characterized by a local field \( F(r - r_i) \), which reflects the symmetry at the defect site and constitutes a subgroup of the crystal's point group This local field interacts with a pseudospin \( \sigma_j \) at site \( r_j \) (where \( j = i \)), representing the local lattice distortion By examining the correlations between pseudospins along a specific direction \( x \) in an anisotropic crystal, we can express the attractive potential \( V(x, t; x_i) \) for a collective pseudospin mode at the defect site \( x_i \) as \( dV(x, t; x_i) = -\sigma(\phi) F(x - x_i) dx \).

Further, we consider that the ﬁeldF is symmetrical with respect to the defect centerx i; that is

The relationship F(x−x i) = F(x i −x) indicates a symmetry in the defect, which may be reduced if the coordinate x i shifts away from its original lattice point near a vacant site Furthermore, it is reasonable to assume that the field is highly localized around x i, allowing us to express it in a simplified manner.

F(x−x i) =Fδ(x−x i), (5.2) where the delta function signiﬁes a localized ﬁeld at x=x i with a strength

F We can then deﬁne the pinning potential at x i and t by the integral

{∂V(x, t;x i )/∂x}dx For binary sin- and cos-modes, we deﬁne pinning potentials

V P(x i , t) =− σo Fcos(∆q.x−∆ω.t+φo)δ(x−x i)dx, respectively Using (5.1), these pinning potentials can be expressed as

The phases φi are randomly distributed within the crystal, leading to virtually continuous spatial phases ∆q.xi due to small ∆q and random xi This allows φi to be treated as a continuous variable across the entire angular range Therefore, instead of using the discrete phases in a system of pinned condensates, we can define a continuous phase variable φ = ∆q.x - ∆ω.t + φo, where 0 ≤ φ ≤ 2π This continuous phase representation enables the expression of pinning potentials effectively.

In the context of symmetrical defects described by equation (5.1), the potential values are defined as V A (φ) = −V o sinφ and V P (φ) = −V o cosφ, where V o is set to σo F for simplicity At φ = 0, the potential V P (0) is negative, indicating that a condensate in the cos mode can be stabilized, while the sin mode remains unstable Additionally, V A (0) equals zero, highlighting the distinct stability characteristics of these modes at this specific angle.

Asymmetrical defects create pinning potentials that stabilize the sin mode at φ = 1/2 π, with the normal defect represented by a symmetrical potential An external electric field (E) introduces an asymmetric potential (−σo E), leading to equilibrium at φ = 1/2 π This external electric field significantly influences ferroelectric phase transitions by inducing polar ordering alongside the spontaneous mechanism Additionally, the internal field generated by long-range order may also create an asymmetrical potential, exhibiting similar behavior.

Dynamically, a pinned phase mode σP(φ) at symmetrical defects should ﬂuctuate in an oscillatory motion around the equilibriumφ= 0, for which the restoring force f R =−∂V P (φ)

∂φ =−∆qV o sinφ is responsible Therefore, for a small phase variationδφ=φ−0 in the vicinity ofφ= 0, the pinning potential can be written as

V P (δφ)≈V o + 1 2 V o (δφ) 2 , which can be responsible for dynamic ﬂuctuations of a pinned condensate. Obviously, such a ﬂuctuationδφ should be originated from the phonon interaction in the condensates.

Rice explored the oscillatory behavior of a charge-density-wave condensate under an applied electric field, represented as E = E₀ exp(−iωt), and derived a corresponding susceptibility formula Assuming E₀ indicates the amplitude of the applied field at the wavevector q and neglecting damping for simplicity, the equation of motion can be expressed in the potential V₀ = 1/2 mω₀²(δφ)².

The dynamics of pseudospin condensates and their long-range order are characterized by the effective mass and charge of the charge density wave (CDW) condensate By expressing the fluctuation as δφ = (δφ)o exp(−iωt), the steady-state solution reveals the susceptibility χ(ω) = (δφ)o /Eo = (e/m)/(ω²o − ω²), which illustrates the dielectric response of the pinned condensate and indicates a singularity at ω = ωo Notably, Pawlacyk et al identified a low-frequency fluctuation mode at ωo = 0.1 GHz in the dielectric spectra of TSCC crystals near the critical temperature (Tc) of approximately 130 K, distinguishing it from the soft-mode frequency of around 20 GHz.

Supported by experimental evidence, such a pinning potential as a function of the internal variation δφ permits a thermodynamical description of ﬂuctuations in terms of a dynamic Gibbs functionG(δφ), i.e.

(dx/L), (5.7a) whereκis related to the kinetic constant proportional to the inverse mass of a condensate andLis the length of integration.

In the critical region of a ferroelectric phase transition, the application of a static electric field E alters the pinning scheme of condensates, allowing the sin mode σA(δφ) to also experience pinning Near the point where φ = 0, the pinning potentials can be mathematically expressed.

For a uniform applied ﬁeldE =−dV /dx=−∆qdV /dφ, where the potential functionV is antisymmetric with respect toφandx, in contrast to symmetric defect potentials Hence the ﬁrst integral vanishes, because

Ecos(δφ)dx=− dV dφ cos(δφ)dφ=− dVcos(δφ)

=−∆V cos(δφ)−(−∆V) cos(−δφ) = 0, whereas the second integral is not zero;

Esin(δφ)dx=−∆Vsin(δφ)−(−∆V) sin(−δφ) =−2∆V sin(δφ).

Here, 2∆V = −2(E/∆q)δφ represents the potential diﬀerence between the phase limits±δφin these integrals Therefore, althoughV P (δφ, E) is virtually unchanged by a weak ﬁeldE on, these pinning potentials can be written as

The pinning equilibrium for the sine mode is established at φ = 1/2 π, indicating that this sine mode functions similarly to a cosine mode when the phase is shifted by 1/2 π As the electric field E increases, the behavior of V A (δφ, E) becomes indistinguishable.

The equilibrium phases of V P (δφ) are confined within the range of 0 to 2π, specifically at 1, 2, and π In this context, σA can be associated with a weak applied field E, despite its lineshape lacking distinct features Detailed experimental insights on field pinning will be provided in Chapter 9 Additionally, the Gibbs function for fluctuating σA and P in the presence of an applied field E can be expressed mathematically.

For amplitude and phase modes, such Gibbs functions can be minimized independently, and where the dynamic ﬂuctuations are described as a harmonic phase variationδφ.

The Lifshitz Condition for Incommensurate Fluctuations

In Chapters 3 and 4, we explored the modulated structures of collective pseudospins arising from competing short-range correlations, raising questions about their classification as macroscopic phases If a state defined by continuous variables (φ) is stable under specific temperature and pressure conditions, the crystal can be regarded as a thermodynamic phase, with the Gibbs potential G(s(φ)) represented as a function of entropy, pressure, and temperature Real crystal systems provide examples of modulated phases where pinned pseudospins maintain a stationary structure that is incommensurate with the lattice period Although this modulated structure is fundamentally time-dependent due to lattice interactions, it appears to be steadily modulated within the crystal space over short timescales.

Incommensurate fluctuations were initially detected at microwave frequencies during the critical phase transition of TSCC While stable incommensurate crystal phases have been identified in various systems, Lifshitz established a thermodynamic criterion for these phases This section explores the Lifshitz condition for incommensurability, which can be derived from correlated pseudospins, although the source of fluctuations remains unspecified in Lifshitz's argument.

In crystals, the variables (φ) experience phase fluctuations when subjected to a pinning potential These fluctuations can be attributed to correlations among the pseudospins By examining the sinusoidal pseudospin modes (φ) = s o exp(iφ), it becomes evident that the correlations primarily occur between different phases, φ1 and φ2, while the amplitude, s o, remains constant under stable pressure and temperature conditions Consequently, the fluctuating phase difference plays a significant role in the overall behavior of the system.

The dynamics of pseudospin condensates exhibit significant long-range order in the direction x, while the influence of the critical region is minimal We analyze the density correlations arising from notable interferences between nonlattice points x1 and x2, particularly within a timescale of to ≤ t1 − t2 Consequently, the free energy of pseudospins is enhanced by an additional term, GL, which is influenced by these density correlations This term is proposed to be proportional to s*(φ1)s(φ2)t, calculated as a time average over the timescale to.

For a one-dimensional correlations alongx, such correlations can be written as s ∗ (φ1)s(φ2) t=s 2 o ± exp{(±i∆q)(x 2 −x 1 )}exp{(∓i∆ω)(t 2 −t 1 )} t

, in which the time correlation factor can be expressed as Γt= exp{∓i∆ω(t 2 −t 1)} t=t − o 1 t o

0 cos(∆ω.τ)dτ=sin(t o ∆ω) t o ∆ω , where the variableτ=t 2 −t 1 is in the range 0≤τ≤t o

Due to time-reversal symmetry, the function Γ t is real and approaches a value close to 1 when the condition t o ∆ω 1 is satisfied This scenario, referred to as "slow," leads to a quasi-static condition dominated by the symmetrical spatial correlation factor cos{∆q(x 2 −x 1 )} It is important to note that this spatial correlation function for regular lattice points becomes zero when x 2 − x 1 equals an integer multiplied by the lattice constant, while it remains nonzero for nonlattice points, provided that t o ∆ω 1 holds true.

In this case, the ﬂuctuations are revealed as incommensurate.

Denoting small deviations from a regular lattice point xas x 1 =x−δx andx 2 =x+δx, where|δx|< a, the lattice constant, the correlation function can be written as Γ(δx) ={ s(x 1 , t 1 )− s(x, t)} ∗ { s(x 2 , t 2 )− s(x, t)} t s ∗ (x, t)∂ s(x, t)

In order for Γ(d x) to indicate nonvanishing correlations when d x = 0, the quantity within the brackets must be nonzero It is evident that Γ(0) remains nonzero regardless of the bracketed quantity Lifshitz suggests that incommensurability is guaranteed if the Gibbs potential includes an additional term, G L, that is proportional to Γ(δx).

The Lifshitz condition for incommensurability is expressed as L, (5.9) and G L = 0, where the coefficient iD/2 incorporates δx, and the factor 1 2 i is used for convenience This formulation assumes a continuum crystal, which is applicable within the long-wave approximation.

The dynamic Gibbs function for a fluctuating state encompasses both the kinetic energy for propagation and the correlation term from Landau's expansion, allowing for a comprehensive understanding of equilibrium.

L +G L , (5.10) where κ =mc 2 o and c o the speed of propagation Assuming that Γ t = 1 for brevity, (5.10) can be written fors=s oexpiφas

The pseudospin system achieves thermal equilibrium with lattice excitation, represented by G S This equilibrium condition is defined by the equation d{G(s) + G S } = 0, indicating that G S is largely independent of s To find the equilibrium values of s o and φ, one must solve the simultaneous equations ∂G/∂ s o = 0 and ∂G/∂φ = 0, which leads to the expression a s o + b s 3 o + κ d 2 s o dx 2.

The equation (ii) reveals that the wavevector is generally irrational, as indicated by the relationship dφ/dx = -Dκ = q, where D and κ are parameters primarily dependent on temperature and unrelated to lattice periodicity The term κ(d²so/dx²) in equation (i) is negligible, leading to the conclusion that the amplitude so is a function of temperature and pressure, expressed as so = -(a - D²/κ)/b Although the origin of this relationship is not specified, the coupling with the soft mode accounts for incommensurate fluctuations in practical systems.

108 5 Dynamics of Pseudospins Condensates and the Long-Range Order

A Pseudopotential for Condensate Locking and

Pseudospins are active groups located at regular lattice points, making their collective motion sensitive to even minor deviations from periodic structures While point defects can disrupt translational symmetry, another important aspect involves pseudo structures that may not be easily identifiable through crystallographic methods, as they are often too small for X-ray diffraction to resolve However, when the wavelength aligns with the repeat unit of a pseudo structure, pseudospin fluctuations can be stabilized, leading to commensurate lattice modulation Transitions from incommensurate to commensurately modulated phases, observed at specific temperatures, are marked by a shift from continuous phase variables to discrete angles in practical systems.

An example of pseudo potentials is screw symmetry, characterized by the sequential rotation of active groups along a specific direction This phenomenon is commonly observed in twofold or threefold screw axes, where constituent molecules rotate by π or 2π/3 over two or three unit cells, respectively.

In certain cases, the repeat unit of pseudo symmetry can extend two to three times longer than the standard lattice constant This leads to a phase transition from incommensurate to commensurate, which is essential for achieving phase matching between the pseudospin mode and a screw potential aligned in the same direction This modulated structure involves an excitation energy, attributed to a pseudolattice potential U m, characterized by an m-fold screw axis along a specific crystallographic axis, such as the b axis.

In the context of pseudopotential U m for active groups at each site p, transversal vectors u ⊥p are utilized for successive rotation and translation along the axis, defined by u ⊥ p = u ⊥o expiθp, where θp = ±2π mp and p = 1,2,…,m Consequently, the pseudopotential for screw symmetry can be formulated accordingly.

U m ∝ ± pu ⊥ p ± pu o {exp(iθp) + exp(−iθp)}, or

The potential U m exhibits distinct maxima determined by successive rotations θ p at x p, as described by the equation (5.11) By applying the phase matching rule, the phase φ of a pseudospin mode can synchronize with the phase of U m when mφ equals the phase G b x p, leading to the relationship φ = 2πp/m for p values ranging from 0 to m−1.

In the study of collective pseudospin modes, we relate the potential \( V_m \) to the lattice distorting potential \( U_m \), allowing for phase matching between \( s \) and \( V_m \) at a specific temperature \( T_i \) For precise phase matching, we express the potential \( V_m \) as proportional to \( \cos(m\phi) \).

V m (φ) =ρ{ s m + (s m ) ∗ }/m 2ρ m σ m o cos(mφ), (5.12) whereρis constant The incommensurate pseudospin mode s(φ) is therefore perturbed by the pseudopotentialV m (φ) as the temperature is lowered through

T i , where the dynamic Gibbs function fors(φ) can be expressed as

The collective pseudospin mode is identified as a nonergodic variable, indicating that its temperature dependence arises from long-range correlations, which will be elaborated in Section 5.6 Consequently, the thermodynamic properties represented by the variables σo and φ in the Gibbs function are intrinsically linked to temperature, albeit in an implicit manner as shown in equation (5.13) Furthermore, the Gibbs potential outlined in (5.13) can be reformulated accordingly.

The variation principle allows for the minimization of the function G(σo, φ) to achieve thermal equilibrium by addressing arbitrary variations δσo and δφ This involves solving the simultaneous equations ∂G/∂σo = 0 and ∂G/∂φ = 0, which leads to the expression aσo + bσo^3 + 2ρσm−o cos(mφ) + κσo dφ/dx.

= 0 and κσ 2 o d 2 φ dx 2 + 2ρσ m o sin(mφ) = 0 (5.14a) Using the abbreviationsψ= mφandζ= (2mρ/κ)σ m o − 2 , the second equation can be expressed as d 2 ψ dx 2 −ζsinψ= 0, (5.14b) which is known as the sine-Gordon equation Integrating (5.14a) once, we have

110 5 Dynamics of Pseudospins Condensates and the Long-Range Order representing the energy relation betweens(φ) and the pseudo-potential U m.

It is noted that the kinetic energy of phase ﬂuctuation given by the ﬁrst term in (5.14c) originates from an energy exchange between V m and U m at

T i , although remaining implicit in the above, and onlyφis considered for the phase transition.

The sine-Gordon equation characterizes nonlinear motion with a finite amplitude, σ₀, influenced by the magnitude, ρ, of the pseudopotential Frank and van der Merwe explored this dynamic issue, providing insights applicable to the current study of nonlinear fluctuations Based on Böttiger’s textbook, we can express this relationship in a specific mathematical form.

The equation \(2\left(\frac{d\psi}{dx}\right)^2 - \zeta \cos \psi = E\) describes a system where \(E\) is an integration constant determined by specific values of \(\psi\) and \(\frac{d\psi}{dx}\) at a point \(x = x_0\), similar to initial conditions in pendulum motion In this context, \(E\) represents energy and \(\zeta\) indicates potential height on a reduced scale The motion is oscillatory with finite amplitude when \(E < \zeta\), while it becomes nonoscillatory when \(E \geq \zeta\) In the oscillatory case, the mode \(\psi\) is stabilized by the potential \(-\zeta \cos \psi\) in phase with the pseudoperiod, whereas in the nonoscillatory case, the potential does not hinder free propagation.

The solution of (5.15) can be expressed in terms of an elliptic integral x−x o ψ 0

[2(E+ζcosψ)] − 1 / 2 dψ, which can be rewritten in the standard form as x−x o ζ 1 / 2 κ φ o

The integral defined by (1−κ² sin² Θ)dΘ, where κ² = 2ζ/(E+ζ) represents the squared modulus of the elliptic integral and Θ = 1/2 ψ, has a lower limit of Θ = 0 and an upper limit, ϕ, corresponding to a specific value of Θ at a given x Additionally, this integral can be rewritten in a reversed form as sinϕ = snκ(x−x₀) ζ^(1/2).

The analysis of equations (5.16) and (5.17) reveals that the variable x−x o exhibits periodic behavior as the angle ϕ changes according to sinϕ, provided that the modulus is within the range of 0 < κ < 1 In contrast, when κ is greater than or equal to 1, x−x o does not demonstrate periodicity This leads to the identification of a lock-in phase transition at temperature T i, which occurs at E = ζ or κ = 1, distinguishing between the phases defined by E < ζ and E > ζ The specific angle ϕ at this temperature T i is denoted accordingly.

The kink solutions of the sine-Gordon equation are illustrated graphically in Figure 5.1, showcasing their behavior as a function of x - x₀ for various values of m (5) and p (0, 1, 2, 3, 4) The "wavelength" (Λ) is defined as the distance between adjacent kinks that exhibit the same derivatives The phase transition can be characterized by the kink solution represented by the equation sin(θ) = tanh{(x - x₀)/ζ^(1/2)} This leads to the expression θ = sin^(-1)[tanh{(x - x₀)/ζ^(1/2)} + θₚ], where θₚ = pπ and p takes on integer values from 0 to m - 1.

Transitions occur when the function ifs(φ) is trapped within the potential V m (φ), as evidenced by m kinks at specific angles θp= 0, π, 2π, , (m−1)π, or through the relationship mφp=ψp= 2θpatφp= 2πp/m These discrete kinks, noted as σ(φ p), have been observed in K2ZnCl4 and Rb2ZnCl4 crystals as discommensuration lines, which can be qualitatively explained by a simplified one-dimensional model.

For such a modulated structure characterized by these kinks, it is useful to deﬁne the distance between kinks as Λ(κ), which is expressed from (5.16) as κΛ(κ)/ζ 1 / 2 = 2 π 2

Jacobi’s complete elliptic integral, denoted as K(κ), is represented by the equation κΛ(κ)/ζ 1 / 2 = 2K(κ) In this context, the parameter Λ(κ) resembles the wavelength in a sinusoidal wave, indicating the repeat length in an elliptical wave Figure 5.1 illustrates numerical plots of the periodic elliptic function for values of p equal to 1, 2, 3, and 4, serving as a mathematical illustration.

Propagation of a Collective Pseudospin Mode

The critical region of binary phase transitions is dominated by slow ﬂuctu- ations in phase reversal φ ↔ −φ, whereas the collective pseudospins s(φ)

Fig 5.2 (a) A dark-ﬁeld image from satellite reﬂections from (100) plane of a

At 205K, K2ZnCl4 crystals exhibit discommensuration stripes aligned with the b-direction A dark-field micrograph taken at 208K reveals distinct discommensuration lines within a circled area of the pair structure Additionally, images displaying a "vortex" formation of three pairs of discommensuration lines show a noticeable splitting of the outer pairs, captured sequentially as the duration of electron irradiation increases.

The dynamics of pseudospin condensates and long-range order reveal a nonlinear propagating mode in each domain, particularly as temperature decreases This study focuses on the propagation of collective pseudospin modes within high-quality crystals that have low defect densities, minimizing obstacles to propagation At the transition threshold, a collective mode emerges at infinitesimal amplitude, which increases with enhanced correlations as temperature drops Below the critical temperature (T c), these collective pseudospins exhibit propagating motion in low-dimensional anisotropic crystals The mode can be described by the expression s = s₀ f(φ), where the amplitude s₀ and phase φ remain constant at a specific temperature Empirical observations indicate that thermodynamic quantities display temperature dependencies related to (T c - T), although the underlying mechanisms require further investigation This initial analysis employs the long-wave approximation to examine the dynamics of correlated pseudospins in one dimension, focusing on phase behavior at finite amplitudes while deferring temperature dependence for future exploration.

In a displacive system, each lattice site m has a pseudospin influenced by a potential V_m = 1/2 a s_m² + 1/4 b s_m⁴, which is further perturbed by the correlation potential -n J_mn s_m s_n in one dimension along the x direction The binary correlation energies at site m can be reformulated as n = m J_mn s_m s_n = 1/2 n = m J_mn (s_m - s_n)² - n J_mn s²_n, where the first term dominates while the second remains constant at a specific temperature under mean-field conditions For nearest neighbor correlations between m and n, we can approximate displacements as m - s_n = (∂s/∂x)_m (x_m - x_n) in the long-wave limit, suggesting that the interaction behaves elastically Thus, by ignoring the constant term, we can simplify our analysis.

With such interactions, the Hamiltonian can be written for a chain crystal as

The effective mass of a pseudospin in a chain crystal has been explored by researchers such as Krumshansl and Schrieffer, as well as Aubry Their work focuses on the dynamics of an infinite number of particles, which can be described using a long-wave approximation through an integral of the Hamiltonian density.

V{σ(x, t)}= 1 2 aσ(x, t) 2 + 1 4 bσ(x, t) 4 andc o = (2LC/m) 1 / 2 is the speed of propagation Here, for the constants a andb, we considera >0 andb= 0 forT > T c , whereasa 0 for

T < T c , to be consistent with the Landau theory Here, the momentump(x, t) is canonically conjugate to σ(x, t) and related to the Hamiltonian density

H = H(σ, ∂σ/∂x) by the canonical transformations dp dt =−∂H

∂p, thereby obtaining the equation for propagation forT < T c m

∂σ =−aσ−bσ 3 (5.19a) This equation can be reduced to the ordinary diﬀerential equation d 2 Y dφ 2 +Y −Y 3 = 0, (5.19b) by using rescaled variables,

At the transition temperature \( T_c \), the velocity \( v \) equals the critical velocity \( c_o \) and the parameter \( k \) is zero (\( k_o = 0 \)) In contrast, below \( T_c \), the velocity \( v \) is less than \( c_o \) while \( k \) is greater than or equal to \( k_o \) By defining the frequency as \( \omega = vk \) and utilizing the parameter \( k \) from the second expression in (5.19d), we derive the dispersion relation \( \omega^2 = c^2_o (k^2 - k^2_o) \) as shown in (5.19e).

The dispersive property is a key feature of nonlinear propagation, significantly influencing the behavior of collective pseudospins below the critical temperature (T c), particularly in relation to soliton potential According to Landau's theory, the parameter 'a' changes sign at the transition point, where a = 0 indicates T = T c In alignment with soft-mode theory, k=k o is associated with ω= 0, highlighting that a nonzero k o initiates modulation at T c, while k = 0 results in the absence of any modulated structure.

Although soluble analytically, (5.19b) can be simply solved for a small amplitude, ignoringY 3 In this case, the linear equation d 2 Y dφ 2 +Y = 0 has a sinusoidal solution

Y =Y osin(φ+φo), whereφo is a phase constant, andY o is the inﬁnitesimal amplitude.

On the other hand, (5.19b) can be analytically solved for a ﬁniteY, using Jacobi’selliptic function Integrating (5.19b) once, we obtain

The integration constant α, defined as (dY/dφ) at φ = 0, indicates the slope of Y(φ) at that point and can vary based on the amplitude σo, as illustrated in Fig 5.3 By integrating the equation once more, the phase φ can be represented through an integral known as the elliptic integral of the first kind.

The equation (1−ξ²)(1−κ²ξ²)⁻¹/² dξ defines the relationship where ξ equals Y/λ, and κ represents the modulus λ/à The phase φ is determined by the integral's upper limit at ξ=ξ₁ The parameters λ, à, κ, and α are influenced by the derivative (dY/dφ) evaluated at φ=0, which can be established through σ₀, indicating that these parameters are temperature-dependent For convenience, λ and à can be expressed in terms of the modulus κ, specifically λ=2¹/²κ.

The reverse form of (5.21a) is written as ξ= sn(àφ/2 1 / 2 ), (5.21b)

Fig 5.3.Numerical plots ofY =λsn(2 −1/2 àφ) for various values of the modulusκ. which is the elliptic sn-function Deﬁning the angular variable Θ byξ= sin Θ, (5.21a) can be written as àφ1 /2 1 / 2 Θ 1 0

(1−κ 2 sin 2 Θ)dΘ, where the upper limit Θ 1 is speciﬁed by the relationξ1= sin Θ 1 = sn(àφ1 /2 1 / 2 ). Therefore, we can write the relation σ1=λσosin Θ 1 =λσosnàφ1

The longitudinal component σ1 of classical vectors, represented with an amplitude λσo, forms an angle θ with the chain direction, as depicted in Figs 5.4a and 5.4b As the temperature rises, σ1 increases with λ ranging from 0 to 1, while the direction of s rotates by θ along the x-axis.

Jacobi's sn-function exhibits periodicity for moduli in the range of 0 < κ < 1, as illustrated in Fig 5.3; however, it lacks periodicity when κ equals 1 The period of the function can be defined as 4K(κ).

(1−κ 2 sin 2 Θ) − 1 / 2 dΘ (5.23) is the complete elliptic integral In the process forκto approach 0,κ→0 (or λ→0 andà= 2 1 / 2 ) corresponds to a periodic solution, whereas for the other extreme case ofκ= 1 (λ=à= 1), (5.21b) takes the speciﬁc form

Fig 5.4 (a) A pseudospin mode in a quasi-one-dimensional lattice; (b) pseudospin vectors in a collective mode; (c) longitudinal components σ1 as given by

The relationship Y = σ1 / σo = tanh(2 - 1/2 φ) and the transversal components σ ⊥ / σo = sech²(2 - 1/2 φ) exhibit values ranging from -1 to +1 at φ = 0, as illustrated in Fig 5.4c This behavior signifies a kink in the pseudospin variable, aligning with mirror reflection across the plane of φ = 0, which is perpendicular to the x-axis Consequently, this plane at the kink can be interpreted as a domain boundary.

The existing theory inadequately addresses certain aspects of domain formation in real crystals For a planar domain boundary to be accurately represented by φ= 0, it is essential to account for significant interchain correlations, which are overlooked in the one-dimensional theory Furthermore, the pinning mechanism caused by random defects, previously discussed in Section 5.1, requires reevaluation in relation to the current argument regarding domain boundaries.

In classical displacements, the transversal component plays a significant role in interchain correlations Alongside the longitudinal component, defined as σ1 = λσo cosθ, the transversal component is expressed as σ⊥ = λσo sinθ, which can further be defined as σ⊥(φ) = λσo cn(φ).

In the context of κ= 1, the transversal σ ⊥ represents a solitary pulse, particularly notable near φ= 0, where φ transitions from -1 to +1 During this process, the classical vector s undergoes a directional reversal, which necessitates a certain amount of energy If we assume that σ ⊥ can assume all perpendicular directions to the x-axis with equal probability, the energy required to reverse s can be expressed as proportional to π k² σ ⊥², with k representing the wavevector of propagation.

F s(φ) or the corresponding potentialV s(φ) =−dF s /dxproportional tok 2 σ ⊥ 2 should be involved in reversing the propagating pulse ofσ ⊥ Hence, writing

The potential V s(φ) is proportional to -k² sech²(φ/2¹/²), leading to the relationship F s ∝ tanh(φ/2¹/²) ∝ σ₁ This can be interpreted as an internal field, as defined in Chapter 3 (equation 3.31) Known as the soliton in nonlinear dynamics, the potential V s(φ) represents the internal field primarily arising from short-range correlations within the chain, while also accounting for contributions from distant pseudospins due to dipolar interactions, similar to those found in ferroelectric crystals.

Generally, for 00 can be replaced by Eckart potentials

−n(n+1) sech 2 zapproximately In this approximation, eigenvalues p = n−m, where m = 0,1, ,n−1, are shown here by horizontal levels in these wells, but should be broadened into bands due to overlaps between Eckart potentials.

The stepwise changes between energy levels, indicated by the soliton potential (∆V o) when ∆p=1, can be attributed to "de-excitation" facilitated by the dipole field (E dip) This process transfers energy, represented as (∆V o) ∆p=1 = ∆s.E dip, to the lattice.

The statistical average of the dipole moment change, ∆s dip, is approximated by the relationship ∆s ≈ ∆s dip = E dip / λ By assuming random orientation of ∆s dip, we can express this average as proportional to ∆s² dip, which relates to the difference in temperature between two states, represented as kB(Tp − Tp+1), according to the equipartition theorem for the ergodic quantity ∆s dip.

Assuming that cp remains constant with respect to p, we derive the relationship V(n) ∝ To − T, which aligns with the observation that ∆s dip ∝ (To − T)^(1/2) as described in Landau theory This proposed mechanism also supports Cowley’s theory of phonon scattering influenced by quartic potential, particularly near the temperature To While we simplify the analysis by isolating the dipolar component ∆s dip from a polar crystal, this argument also applies to nonpolar cases where a stable modulated structure can occur Consequently, the actual transition temperature Tc, defined for minimum correlations, is notably lower than expected.

T o, indicating that E dip is meaningful only at temperatures lower thanT o.

In the soliton potential V(z,κ) for κ ≤ 1, the levels n of stabilized σ(z,κ) broaden into a band structure due to the overlap of adjacent Eckart wells, facilitating thermal fluctuations of pseudospin modes Additionally, when considering planar domain boundaries in practical crystals, the interchain correlations for κ ≤ 1 must be accounted for to enable random fluctuations of E dip Assuming that ∆s dip E dip represents the primary temperature-dependent interaction energy in the non-critical region, the broad tail of the specific heat below T c can be explained as discussed in Section 4.8 This indicates that the ergodic E dip ordering can progress thermally even beyond the critical region.

Soliton theory reveals that the unit defined by (∆Vo) ∆p=1 acts like an independent particle, allowing us to conceptualize the system as a gas of soliton particles This characteristic can be validated through the “two-soliton solution” of the Korteweg-de Vries equation, although we will not delve into the intricate mathematical details Importantly, this soliton gas can be viewed as a quasi-ergodic system capable of achieving thermal equilibrium with the lattice at a specific temperature (T) Properly expressing the relationship between entropy (s(φ)) and temperature (T) provides the “equation of state” for the ordering system.

Structural phase transitions in crystals arise from the complex interplay between order variables and their lattice In the harmonic approximation, the basic excitations in these subsystems are independent, leading to experimental results that often seem incompatible For instance, some ferroelectric phase transitions are described as displacive based on soft-mode results, while dipolar ordering indicates a polar phase below the transition temperature T_c However, neither perspective adequately addresses the critical anomalies resulting from interactions between order variables and soft phonons Therefore, traditional classifications like displacive or order-disorder are not entirely logical without a proper understanding of these critical anomalies.

In Chapter 4, we examined the order-variable condensate that exists in the critical region, characterized by its long-lived mobility due to low damping of soft modes around the critical temperature (T c) At and just below T c, the collective pseudospin variables exhibit phase fluctuations between binary states in a partially ordered system These fluctuations occur on a timescale that is very slow compared to microwave measurements, resulting in the crystal appearing quasi-statically modulated, while the phonon spectra provide insights into the temporal profile of motion.

The dynamics of collective motion in pseudospin condensates can be investigated through various experimental techniques, including light and neutron inelastic scattering, magnetic resonance sampling, and inelastic neutron impacts These methods allow for the examination of the pseudospin phase and amplitude, providing insights into the nature of the condensate Furthermore, nuclear spin relaxation analysis can reveal the coupling between pseudospins and soft phonons, while dielectric and Brillouin light-scattering experiments offer valuable information on condensate dynamics in crystals By combining these complementary data, researchers can gain a deeper understanding of the condensate's behavior in the critical region.

Modulated Crystals

An ideal crystal composed of infinite arrays of ions and molecules appears macroscopically uniform, with its thermodynamic properties at a specific pressure and temperature described by the Gibbs potential G(p, T) While surfaces and lattice defects exist, the bulk properties dominate in large crystals, characterized by internal translational symmetry and periodic boundary conditions During structural changes, crystal phases above and below the transition temperature T_c remain idealized in the initial approximation However, in the transition region, the crystal exhibits spontaneous inhomogeneity due to locally violated lattice symmetry, which lacks a comprehensive thermodynamic description Notably, when subjected to diffraction experiments, this spontaneously modified crystal reveals a modulated structure.

An idealized crystal exhibits a periodic structure defined by three fundamental translational vectors, a1, a2, and a3, aligned along its symmetrical axes This arrangement allows for a continuous periodic function f(r) at any position r, which remains unchanged under basic translations.

R =n 1 a 1 +n 2 a 2 +n 3 a 3 , (6.1a) wheren 1 ,n 2 andn 3 are integers specifying a lattice point For such a function, we have a relation f( r ) =f( r + R ) (6.1b)

A perfect crystal can also be characterized by the Fourier transformg( k ) that is deﬁned by g( k ) = f( r ) exp(−ik.r )d 3 r or f( r ) = g( k ) exp(ik.r )d 3 k (6.2)

142 6 Diﬀuse X-ray Diﬀraction and Neutron Inelastic Scattering

By combining equations (6.1b) and (6.2), we establish that exp(ik.R) equals 1 at every lattice point R, indicating that the vector k must take on discrete values, specifically k = G This condition ensures that G.R equals 2π multiplied by an integer, with the vector G signifying a translation within the reciprocal lattice.

In a crystal, the relationship between the reciprocal lattice and the translational vectors is expressed as G = ha1* + ka2* + la3*, where a1*, a2*, and a3* are defined by the equations a1* = (2π/Ω)(a2 × a3), a2* = (2π/Ω)(a3 × a1), and a3* = (2π/Ω)(a1 × a2) Here, Ω represents the unit-cell volume, and h, k, and l are integers that index the lattice points in reciprocal space In a uniform crystal, all unit cells are identical, leading to the invariance of functions f(r) and g(k) in the normal and reciprocal lattices, respectively This results in the relationship g(k) = g(k + G) in the reciprocal lattice, analogous to the equation for the normal lattice.

In a modulated crystal, the order variable does not exhibit periodicity in relation to lattice translation; however, its Fourier transform can show periodicity concerning a non-lattice point G i in the reciprocal lattice This point G i cannot be defined using integral indices, as at least one index, such as h, must be irrational in the unit of a 1 ∗, signifying an incommensurate modulation along the symmetry axis a 1 Consequently, the lattice modulation can alternatively be represented by a vector Q derived from the nearest reciprocal lattice point.

The equation Q = G i − G = ma 4 ∗ illustrates a one-dimensional modulation represented by four indices (h, k, l, m) within the reciprocal lattice De Wolff and colleagues have introduced a group-theoretical approach to multidimensional spaces characterized by such wavevectors, referred to as superspace, which is applicable to aperiodic crystals discussed in contemporary literature However, this monograph focuses on examining the impact of disrupted translational symmetry in real crystals, thus adhering to the conventional framework that includes a reciprocal lattice (a 1 ∗, a 2 ∗, a 3 ∗) along with an additional modulation vector a 4 ∗.

Lattice modulation at an irrational vector Q indicates the existence of excitation energy ε(Q) within a crystal, which arises during the crystal's ordering process This excitation energy must be balanced by lattice distortions under equilibrium conditions Therefore, determining the values of Q and the corresponding ε(Q) is crucial for modulated crystals In inelastic neutron scattering experiments, the scattering geometry is typically arranged so that K₀ - K = Q, allowing for the determination of Q at the peak intensity of scattered neutrons Meanwhile, ε(Q) can be derived from the phonon spectra at Q in the scattering setup Conversely, unmodulated crystals with Q = 0 produce a collimated X-ray beam that generates a diffraction pattern, revealing well-defined crystal planes characterized by a rational G, which serves as a method for structural analysis.

The Bragg Law of X-ray Diﬀraction

This section outlines the principle of Bragg diffraction from an ideal crystal before addressing modulated crystals When a collimated X-ray beam strikes a crystal, it produces a diffraction pattern that reflects the three-dimensional structure of the crystal This pattern can be analyzed through the concept of reflection by numerous parallel crystal planes The interaction between X-ray photons and orbiting electrons can be understood classically as elastic collisions, indicating that there is no loss of X-ray energy during the interaction, which allows the lattice structure to remain virtually intact.

A regular crystal structure can be visualized as consisting of numerous sets of parallel planes filled with identical atoms Each set of these parallel planes is defined by a common normal vector, denoted as n, which has a unit length This normal vector is also parallel to the reciprocal lattice vector, highlighting the geometric relationship within the crystal's structure.

G Denoting an arbitrary lattice point on a crystal plane by R , and the distance between adjacent planes by d =dn , the equation of the plane can be expressed as n.( R−d ) = 0; hence, n.R = n.d =d.

In this case, it is obvious that n G and henceG= (2π/d)×integer, because of the relation G.R = 2π×integer, and the gap between adjacent planes can be calculated as

The Bragg law of X-ray diffraction is derived from classical radiation theory, utilizing the conservation of energy and momentum during photon interactions with crystal planes This approach demonstrates that X-rays behave similarly to optical beams, adhering to the reflection law, which highlights the importance of geometrical crystal planes in understanding diffraction patterns.

G parallel to n represents a “crystal momentum”, so that the Bragg law

In two-dimensional Bragg diffraction, two diffracted rays with wavevectors K and K exhibit a path difference of dcosθ, leading to constructive interference when dcosθ equals 1/2λ multiplied by an integer This phenomenon can be understood in the context of momentum conservation during elastic impacts The typical lattice constant is approximately 5 Å, suggesting that the X-ray energy required for diffraction is in the range of 10 to 50 keV When electrons are excited to higher atomic levels due to impact, they eventually return to their ground state by emitting photons that carry the same energy as the initial excitation.

In contrast, heavy nuclei forming the lattice remain unchanged during impact Figure 6.1 illustrates X-ray diﬀraction as interpreted with the concept of crystal planes.

The rigorous radiation theory, while complex, can be simplified for distant observations An incident X-ray beam is represented as a plane wave interacting with a target ion, inducing an oscillating electric dipole moment proportional to the charge density This induced dipole radiates a spherical wave, with its amplitude at a distant point being proportional to the wavevector and the distance from the target Consequently, the scattered amplitude at the observation point can be expressed in a specific mathematical form.

≈E o [ d 3 r o ρ( r o )][expi( K.r−ωt)/r] expi{( K o −K ).r o }, where we considered the approximation|r−r o | ∼rforrr o The scattering amplitude relative to the incidentE ocan then be deﬁned as

The equation A o /E o = ρ( r o ) expi( K o −K ).r o d 3 r o illustrates that, according to the conservation law, the maximum amplitude approaches 1 for small values of |r o| This implies that exp(−iG.r o) is approximately equal to 1, allowing the reflection law to be effectively applied to a crystal plane represented by the vector G, behaving similarly to a rigid reflector.

A collimated beam interacts with a specific area of the crystal plane, where multiple scatterers are positioned near the lattice points These scatterers reflect X-ray photons in phase, leading to constructive interference of the reflected waves, which is described by the total amplitude.

A o m A om , and the practical scattering amplitude is given by

A o /E o m f m ( G ) exp(−iG.R m ), (6.7a) where f m ( G ) = ρ( r om) exp(−iG.r om)d 3 r om (6.7b)

The atomic scattering factor, denoted as m(G), is defined in equation (6.7b) However, the overlapping charge densities, ρ(r om), lead to a significant overestimation of A o when calculated using equation (6.7a) Thus, it is more accurate to calculate reflections from all atoms distributed throughout the target area By employing a delocalized coordinate vector, s, A om can be effectively expressed.

= ρ( s−r om ) exp(−iG.( s−r om )d 3 ( s−r om )×exp(−iG.r om )

In this case, the quantity

The structural form factor, denoted as S(G) = m f m(G) exp(−iG·r om), is defined using the indices h, k, and l of the vector G, which corresponds to a specific set of parallel crystal planes In this context, the position vector r om can be expressed as r om = x m a1 + y m a2 + z m a3.

(6.8a) The electric ﬁeld of scattered X-ray at a distancerr o is expressed as

146 6 Diﬀuse X-ray Diﬀraction and Neutron Inelastic Scattering where the relative intensityI( G ) is given by

I( G )/I o ( G ) = E G ∗ ( r ) E G ( G )/E o 2 =r − 2 S ∗ ( G )S( G ), (6.9) indicating that the diﬀraction intensity from crystal planes G is determined by the structure factorS( G ).

Crystal planes are typically viewed as rigid reflectors for X-ray diffraction; however, it is essential to recognize that crystals undergo thermal motion, which impacts the observed diffraction patterns By considering simple harmonic vibrations of each constituent, the scattered intensity is adjusted through the Debye-Waller factor, as detailed in the following discussion.

In the Einstein model of solids, the position of a constituent scatterer is defined as \( r_o(t) = r_o + u(t) \), where \( u(t) \) represents the instantaneous displacement from the equilibrium position \( r_o \) Under this model, it is assumed that \( u(t) = 0 \) and \( r(t) = r_o \) For small displacements, the exponential factor in \( E_G(r) \) can be expanded as \( \exp(iG \cdot r_o(t)) = \exp(iG \cdot r_o)[1 - \frac{1}{2} |G \cdot u(t)|^2 + \ldots] \), allowing the second term on the right side to be substituted accordingly.

|G.u (t)| 2 = 1 3 G 2 u(t) 2 , if the ﬂuctuations can be assumed as isotropic in three-dimensional crystals. Hence the scattering intensity ratio between thermal and rigid crystals given by (6.9) is expressed by

The Debye-Waller factor, crucial in understanding atomic vibrations, can be derived from the Einstein model By applying the equipartition theorem to a harmonic oscillator of mass M, the average displacement squared, u(t)², is determined as (3/2)k_B T / (1/2 Mω²) This relationship highlights the connection between thermal energy and atomic motion, emphasizing the significance of the Debye-Waller factor in materials science.

This implies that the scattering intensity decreases with increasing temperature and that the diﬀraction from crystal planes at lowGshows less thermal broadening than from those of higherG.

Diﬀuse Diﬀraction from Weakly Modulated Crystals

In a modulated crystal at an irrational vector Q , the original lattice periodicity is modiﬁed; hence, unit cells are not all identical Accordingly the vector

G i = G + Q does not represent crystal planes If, however, |Q| |G|, X- ray diﬀraction exhibit anisotropically broadened patterns orsatellite spots in some cases, where the concept of a crystal plane is acceptable approximately.

For phase transitions occurring at the Brillouin-zone boundary or nonlattice points, where the magnitude of |Q| is significant, X-ray diffraction is not an effective technique Instead, neutron inelastic scattering offers a direct approach for investigating modulated crystals.

Binary structural changes at the center Q = 0 or at the zone boundary

In the Brillouin zone, Q = 1 2 G displays anomalies near transition temperatures, indicative of small fluctuations between q and −q These fluctuations, evidenced by soft modes in the critical region, arise from momentum and energy exchanges between order variables and soft phonons in condensates Figures 6.2a and 6.2b depict the scattering of X-rays and neutrons at the zone center and boundary, respectively, with an unspecified direction of q Notably, small q fluctuations are crucial in the critical region, leading to a discussion on diffuse X-ray diffraction experiments from weakly modulated crystals at Q = 0, while strongly modulated cases are reserved for neutron-scattering studies in Section 6.5 The conservation laws for X-ray scattering at Q = 0 from a binary crystal can be articulated as follows.

The equations K o − K = G±q and ε( K o )−ε( K ) =∓∆ε( q ) describe the relationship between X-ray energies before and after impact, with ε( G ) = 0 indicating the crystal's ground state To enhance the Bragg theory presented in Section 6.2, we redefine the variables by expressing ω as ε( K o)/¯h and ω as ε( K )/¯h, leading to the relation ω−ω =∓∆ω, which replaces energy terms with frequency terms This modification allows for the expression of the scattered amplitude in the new notational framework.

Using the conservation laws (6.11ab), the ﬁeldE G ( r ) of a scattered beam at a distant point r is expressed as

E G ( r )/E o ∝ expi( K.r−ω t) r × m[f( G + q ) expi( q.R m −∆ω.t o ) +f( G−q ) expi(−q.R m+ ∆ω.t o )], (6.12) where f( G±q ) = d 3 r o ρ( r o ) expi( G±q ).r o (6.13)

Fig 6.2.(a) The Bragg diﬀraction K o = K + G , where G is a lattice translation vector (b) Scattering by a nonlattice vector G i= G + Q , where Q in a modulation vector The small vector q represents ﬂuctuations in a modulated lattice.

G = 0, and the scattering intensity can be expressed by

The observed intensity is influenced by the average of the equation I(G)/I₀ + 2r − 2|f(G)|² × m = n cos{q.(Rₘ − Rₙ) − ∆ω(tₒₘ − tₒₙ)}, where the phase of fluctuations on the crystal plane G is represented by φ = q.(Rₘ − Rₙ) − ∆ω(tₒₘ − tₒₙ) Intensity anomalies occur due to the second term in the equation, with zero anomalies observed when ∆ωτ > 1, while unvanished anomalies arise for ∆ω < 1/τ, where τ is a distributed variable The limit of τ at the long end corresponds to the measurement timescale tₒ, indicating that observable fluctuations are determined by ∆ω < 1/tₒ Additionally, Rₘ − Rₙ specifies a continuous spatial range in a particular direction x, and the phase φ varies continuously between 0 and 2π.

(6.15) where S is the eﬀective area for X-ray impact on the plane of G , hence the integral represents the spatial average of cosφ(x,τ) t.

Assuming that S is a rectangular areaL x L y , the average in (6.15) can be evaluated as

L − x 1 cosφdx= (qL x ) − 1 cosφdφ= (qL x ) − 1 (sinφ2 −sinφ1)

The average phase φ, defined as φ = 1/2 (φ1 + φ2), is influenced by the X-ray beam, while the difference in phase ∆φ = φ2 - φ1 = qLx is random and continuous within the range of 0 ≤ φ ≤ 2π By substituting φ with qx - ∆ωτ, where 0 ≤ τ ≤ to, the time average can be determined using the integral t - o 1 cosφ dτ sin(1/2 ∆ω to).

Therefore, the observable intensity anomaly in a diﬀraction pattern can be expressed as

The phase angle ϕ is redefined as ϕ = qx - 1/2 ∆ω.t₀, where it ranges from 0 to 2π For small values of ∆φ and ∆ω.t₀, the front factors in the square brackets approach 1, based on the limit lim θ→ 0 (sinθ/θ) → 1 The condition ∆ω.t₀ < 1 is crucial for visualizing the spatial phase distribution of ∆φ within the timescale t₀ Additionally, similar diffraction anomalies may occur during phase transitions at zone boundaries; however, at arbitrary points in the Brillouin zone, such anomalies can only be detected through neutron inelastic scattering experiments, as elaborated in Section 6.5.

The Laue Formula and Diﬀuse Diﬀraction from Perovskites

Phase transitions at irrational points in the Brillouin zone are not suitable for X-ray studies; however, the diffraction method can be applied to certain crystals transitioning to low-dimensional order In perovskite crystals, the structural phase transition from cubic to tetragonal can be interpreted as two-dimensional order in planes perpendicular to the tetragonal axis, influenced by one-dimensional correlations along two symmetry directions This perspective suggests that below the transition temperature, such a plane behaves like a modulated crystal plane, leading to intensity anomalies in Bragg diffraction.

In Chapter 3, we explored pseudospin correlations in perovskites, which lead to two incommensurate directions that are perpendicular to the tetragonal axis in the low-temperature phase Notably, diffuse diffraction was observed in X-ray studies conducted by Comes et al on NaNbO3 at 700 °C, as depicted in the corresponding diffraction photograph Additionally, M¨uller and his team identified anomalies in magnetic resonance lines from SrTiO3 at 105 K, indicating modulation in two independent directions This section aims to explain the observed diffuse diffraction pattern.

The image in Figure 6.3 displays a two-dimensional diffuse diffraction pattern from perovskite NiNbO3 at a temperature of 700 °C, as sourced from the work of R Comes et al in Ferroelectrics This pattern is analyzed using a model that features two fluctuating one-dimensional condensates that are oriented perpendicularly to each other.

In this study, we analyze the behavior of periodically arranged correlated pseudospins along the repeat unit axis \( a_1 \), which become active during structural changes For pseudospins positioned at \( r_i = n a_1 \), where \( n \) is an integer, we define the reciprocal vector \( G = h a_1^* \), although it does not represent crystal planes due to the linear correlation of these pseudospins The scattered radiation arises from dipole moments along \( a_1 \) induced by incident radiation When the incident radiation is perpendicular to \( a_1 \) (i.e., \( K_0 \perp a_1 \)), scattered beams exhibit constructive or destructive interference at various angles \( K \) on a conical surface defined by an apex angle \( \theta_1 \) The conditions for scattering are given by \( K \cdot a_1 = (2\pi a_1 / \lambda) \cos \theta_1 \), where \( \theta_1 \) must satisfy either \( a_1 \cos \theta_1 = \lambda \) (integer) or \( a_1 \cos \theta_1 = \frac{1}{2} \lambda \times \text{integer} \) This scattered X-ray flux can be captured on a photographic plate positioned perpendicular to \( K_0 \), resulting in an image of symmetric hyperbolas, as shown in Fig 6.4 In three dimensions, these hyperbolas from two perpendicular dipole lines intersect on the plate, creating a distinctive pattern of diffuse diffraction, known in X-ray crystallography as the Laue construction of diffraction patterns.

The Laue method is employed to analyze perovskite crystals by aligning the incident X-ray beam parallel to one of the cubic axes, such as K o a 3, while examining the diffraction cones around the a 1 and a 2 axes In these materials, diffraction spots are typically represented by the indices (h+ 1 2 , k+ 1 2 , l), which signify a phase transition at the M-point of the Brillouin zone Notably, the diffuse diffraction spots and hyperbolic lines observed in NaNbO 3 crystals at 700 ◦ C provide compelling evidence of two one-dimensional correlations that remain largely independent along the a 1 and a 2 axes, yet are interconnected through inversion symmetry, as suggested by the magnetic resonance anomalies.

The Laue condition for constructive interference is expressed as 1cosθ1=hλ for correlated pseudospins along the a1 axis, which can be reformulated to (2π/λ) cosθ1 = 2πh/a1 This leads to the relationship |K|cosθ1=ha1*=G1, indicating that the wavevector K comprises components K=G1 and K⊥⊥a1 Scattering from a linear chain exhibiting binary fluctuations is represented by the structural form factor f(G1±q) = d3r₀ ρ(r₀) expi(G1±q)r₀.

Here, assuming q G 1 for weak modulation, we have f(G 1 ±q)≈ f(G 1).

In Fig 6.4b, the fluctuations of scattered X-rays are depicted by shaded regions along the vectors K and ∆K By also taking into account temporal fluctuations, we can derive the intensity anomalies of a beam that is scattered from a group of particles.

Fig 6.4 (a) The Laue diffraction from a one-dimensional lattice along the a 1 direction X-ray beam ⊥a 1 (b) Diffreaction from a fluctuating one-dimensional lattice. identical pseudospins as

The observed anomaly can be expressed by averaging the phase difference, represented by the equation I(G 1 ±q)−I(G 1 )∝I o m =ncos{q(x m −x n )−∆ω(t om −t on )}, where the phase q(x m −x n )−∆ω(t om −t on ) is considered continuous This averaging is conducted with respect to the distributed position x = x om −x on at random times t om −t on=τ of impact, within the ranges 0< x < L and −t o < τ < t o.

∆ω.t o cosϕ, where 0 ≤ ϕ ≤ 2π and L represents the size of the X-ray beam As in the previous argument, the condition ∆ω.t o ≤1 is essential for the anomaly to be observed in the timescalet o.

Neutron Inelastic Scattering

A structural phase transition occurs at a specific temperature \( T_c \), where correlation energies reach their minimum due to singular variable correlations This behavior can manifest not only at the center of the Brillouin zone (\( Q = 0 \)) but also at a specific irrational wavevector (\( Q_c \)), where the corresponding energy \( \epsilon(Q_c) \) is nonzero This phenomenon is likely linked to the emergence of pseudosymmetry at a certain temperature, leading to fluctuations in the condensate and causing anomalies at wavevectors close to \( Q_c \) A notable example of this phase transition is observed in K\(_2\)SeO\(_4\) crystals at \( T_c = 130K \).

Q c = 0, where so-called phonon-dispersion curvesε=ε( Q ) were observed at temperatures near 130K as shown in Fig 4.5, where there is a notable dip at

Q c ∼2 a ∗ /3 when the temperatureT c is approached.

The transition vector \( Q_c \) is comparable to the wavevector of thermal neutrons, allowing for the study of energy fluctuations around \( \epsilon(Q_c) \) through neutron inelastic scattering experiments In this context, neutron scattering reveals the wavevector and associated energy as excitations within the "phonon spectra," where the minimum \( \epsilon = \epsilon(Q_c) \) indicates a phase transition Consequently, near \( \epsilon(Q_c) \), critical anomalies can be understood as fluctuations arising from interactions with pseudospins in the lattice, expressed as \( \epsilon(Q) = \epsilon(Q_c) \mp \Delta\epsilon \).

Q = Q c ±q On the other hand, incident neutrons can either lose or gain their kinetic energies during impact, so that scattered neutrons in the direction of

The modulation of Q at ε(Q) occurs at ±q and ±∆ε, where these modulated quantities, indicated by "primes," reflect phonon fluctuations in scattered neutrons For simplicity in the calculations, we can omit these primes while preserving the signs related to momentum-energy exchange in the critical region.

The conservation laws for neutron inelastic scattering can be expressed as

E( K o )−E( K ) =ε( G i ±q ) =∓∆ε( q ) (6.17b) in the reciprocal space, where G i = G + Q c and ∆ε( q ) can be expressed as

2κ q 2 , representing the kinetic energy of ﬂuctuations if|q| N_{-\frac{1}{2}} \), this configuration facilitates the absorption of radiation quanta \( \hbar \omega \) when \( \omega = \omega_L \) The average pumping rate per cycle is defined as \( \frac{dW}{dt} \).

(πω 2 L B 1 2 /k B T)|à + − | 2 f(ωL), which is equivalent to the macroscopic expression dW dt t

2πωL |à+ − | 2 f(ωL)/k B T (8.11) The result indicates that the magnetic resonance absorption can be apprecia- ble at lower temperatures due to a higher population diﬀerence.

The observation of magnetic resonance in systems of magnetic moments requires a macroscopic interpretation, focusing on magnetization (M) as the key variable characterized by two relaxation times, T1 and T2 At resonance, the system achieves equilibrium with radiation quanta from the high-frequency field B1 at a specific temperature T This phenomenon can be mathematically represented by the rate equation for the population difference, n = N + 1/2 − N − 1/2, expressed as d(n−n₀)/dt = −(n−n₀).

T 1 , where n o is the value of n with no radiation at T, and is identical to (8.5).

The relaxation process outlined by T2 in equation (8.4) signifies the duration required for microscopic moments undergoing Larmor precessions in random phases at temperature T to synchronize with the applied magnetic field B1 once it is activated, or to revert to a state of randomness when the field is deactivated.

Magnetic Resonance Spectrometers

Magneticmagnetic resonance spectrometers resonance is signiﬁed by a maximum absorption of radiation energy of the oscillating ﬁeldB 1 atω=γB o in

184 8 The Spin-Hamiltonian and Magnetic Resonance Spectroscopy

Fig 8.2.(a) A radio-frequency resonator consisting of a capacitorCand an inductor

Lwith a sample crystal.B 1represents magnetic rf lines (b) A microwave resonator.

In resonance experiments, a sample can be strategically positioned at the center of the resonator box to maximize the microwave field amplitude (B1) The adjustable iris allows for the selection of the desired quality factor (Q) By utilizing a uniform magnetic field (Bo), resonance experiments can be conducted by either varying the frequency (ω) while maintaining a constant Bo or adjusting Bo while keeping ω fixed Although both methods are theoretically simple, the latter is more practical and is commonly employed in most spectrometers that utilize conventional laboratory magnets capable of generating a uniform field up to approximately 15 kG With these magnets, the frequency range for nuclear resonance typically falls between 1 and 100 MHz, while paramagnetic resonance experiments conventionally operate within the 5 to 40 GHz microwave frequency range.

For small sample experiments, observing magnetic resonance at a fixed position in a resonator where B1 is maximized is practical At radio frequencies, the sample is positioned in a resonator with inductance Lo and capacitor C, with the tuning frequency defined by ω = (LoC) − 1/2 Conversely, at microwave frequencies, the sample is placed within a cavity resonator at the peak of the standing wave, affecting the resonant frequency based on the sample's finite partial volume Vs within the inductor.

When a sample is placed in such a resonator, the inductance can be expressed as

L= (1 +χ)L o = (1 +χ )L o −iχ L o , where the value of the susceptibility χ = χ −iχ depends on the sample volume The impedance of a “loaded” resonator can therefore be expressed by

The impedance of a circuit can be expressed as Z = R + iωL + 1/iωC, where R represents the effective resistance and ωχLo accounts for energy loss due to magnetic resonance The resonant frequency can be approximated by setting the imaginary part to zero, leading to the equation ω²o = 1/(1 + χ)LoC If the magnetic field B1 is assumed to be uniform across the sample volume, the susceptibility is defined as (Vs/Vm)χ, with Vm being the effective volume for maximum response.

B 1 The volume ratioα=V s /V m is called theﬁlling factor, which is less than

In practical applications, the energy loss in a resonator can be described in terms of thequality factorQthat is deﬁned by

The energy loss per cycle, denoted as Q, is calculated by the ratio of total energy loss to energy stored, and is compared to the unloaded resonator's quality factor, Q₀ The electromagnetic energy stored in each cycle is represented by the formula ω(1/2 a₀ B 1/2 Vₘ) / Q₀, while the energy loss from magnetic resonance is given by 1/2 ωL a₀ χ(ωL) B 1/2 Vₛ Consequently, the energy loss at resonance, where ω equals ωL, is articulated as a fractional change in the quality factor.

Q o =αχ (ωL), being proportional to the imaginary part ofχ(ωL) In practical observation,

The parameter ∆Q/Q₀ is determined from χ(B₀) through impedance measurements at a constant frequency ωL, while B₀ is varied around ωL/γ A typical magnetic resonance spectrometer, illustrated in Figure 8.3, features an impedance bridge that compares the sample resonator's impedance Z(B) with a reference impedance Z₀ Near the magnetic resonance, the sample impedance can be accurately expressed.

In this type of bridge, the reflected waves from Z(B) and Z o can be balanced either in phase or out of phase, which affects the detector signal's relationship to χ or χ Specifically, for B o = ωL / γ ± ∆B o, the detector signal is proportional to these parameters.

Although independently measurable, these χ and χ are related by theKramers-Kr¨onig formula, and so only one of these is suﬃcient for magnetic resonance to be detected.

Fig 8.3.A block diagram of a microwave bridge spectrometer.

The Crystalline Potential

Magnetic probes with a spin greater than 1/2 are ideal for investigating structural phase transitions, though they necessitate complex spectral analysis influenced by the applied magnetic field B0 As discussed in Chapter 9, lattice modulation effects can be effectively evaluated with first-order accuracy, providing ample information for various applications While a comprehensive spectral analysis could be performed, it may not be essential for structural studies The subsequent sections will outline the method for spin-Hamiltonians in normal crystals before addressing modulated crystals through magnetic resonance parameters.

In typical crystals, the electronic state of a paramagnetic probe with unpaired spins is mainly characterized by its total orbital and spin angular momenta, L and S These momenta are influenced by the surrounding crystalline potential at the probe site This approach, known as the Russell-Saunders scheme in atomic spectroscopy, effectively describes the behavior of ions within crystals.

“transition elements” commonly used as magnetic resonance probes.

In an orthorhombic crystal, electrons orbiting around a nucleus at a lattice site experience perturbation from a crystalline potential This potential can be represented as a power series of coordinates (x, y, z), reflecting the local lattice symmetry At the lowest order, the crystalline potential is expressed as a quadratic form.

The static potential V(x, y, z) is defined by the equation V(x, y, z) = Ax² + By² + Cz², where the coefficients A, B, and C must satisfy the condition A + B + C = 0 to comply with the Laplace equation (∆V = 0) In the case of uniaxial symmetry, specifically for tetragonal or trigonal systems, the coefficients can be set as A = B, leading to C = -2A Consequently, the potential can be expressed in a simplified form.

V(x, y, z) =A(x 2 +y 2 −2z 2 ) =A(r 2 −3z 2 ), wherer 2 =x 2 +y 2 +z 2 , being uniaxial along thez direction.

Another example is the quartic potential

The cubic symmetry is represented by the equation V(x, y, z) = D(x^4 + y^4 + z^4), indicating that the lowest order potential is quartic While a quadratic potential can be expressed as A(x^2 + y^2 + z^2) or Ar^2, it fails to capture the essence of crystalline potential Therefore, it is crucial to recognize that only a deviation from spherical symmetry is necessary for crystalline potential, reinforcing that a cubic potential must be quartic at the lowest order.

To ensure consistency with crystal symmetry, the axes x, y, and z should align with the symmetry axes a, b, and c of the crystal when only one probe fits within a unit cell Conversely, if multiple crystallographically equivalent sites are available for a probe, the coordinates x, y, and z must be aligned with the symmetry of the probe site The concept of crystalline potential, established by Bethe in 1929, has been refined for various solid-state applications Notably, for magnetic issues, Abragam and Pryce have developed the foundational method of spin Hamiltonians, which serves as the basis for magnetic resonance spectroscopy.

The Zeeman Energy and the g Tensor

The orthorhombic symmetry of crystalline structures often allows for a simplified analysis through a unique axis, leading to quantized orbital angular momentum (L) along this axis This results in the splitting of perturbed ionic energy into multiple levels In cases where the quadratic potential constant A exceeds kBT, the ground state remains distinct from the upper levels, linking the magnetic moment solely to the spin angular momentum (S) and forming a doubly-degenerate ground state known as the Kramers doublet Additionally, the influence of spin-orbit coupling (λ L·S), with λ ranging from 100 to 800 cm−1 for iron-group elements, must be considered, as it creates a significant interaction between S and the crystal field, despite being typically smaller than the crystalline field splitting.

The Spin-Hamiltonian in magnetic resonance spectroscopy describes how an excited state ψ ε is influenced by a perturbation λ L.S in the second order This interaction is represented by the off-diagonal elements ψ ε ∗ L ⊥ ψodv, leading to a modified wavefunction expressed as ψ = ψo + ε λ.

The energy gap (∆ε) between the excited and ground states is crucial for understanding the behavior of ions in a magnetic field In this context, the Zeeman energy of the ion can be effectively described when subjected to a uniform magnetic field (B₀).

The magnetic moment is described by the equation H Z = −m e B o, where m e is defined as m e = −β ψ ∗ (g e S)ψ dv, with β representing the Bohr magneton (β = e¯h/2m e c = −0.927 × 10^−20 emu) and g e being the Landé factor for a free electron (g e = 2.0023) When a magnetic field B o is applied parallel to the z axis of a crystalline potential, the perturbation arises from the off-diagonal element of L ⊥, leading to the expression ψ ε ∗ L ⊥ ψ o dv = 2 exp(i∆εt/¯h), where only the component along the z-axis remains stationary Consequently, the effective g z factor in the Zeeman energy can be articulated.

For B o applied parallel to x or y axis, we consider that L ⊥ =L y +iL z or

L z+iL x, respectively, for the perpturbing spin-orbit coupling, and we obtain similar expressions of Zeeman energies, namely

If B o is applied in a general direction n , the eﬀectiveg factor behaves as a tensor quantityg, and the Zeeman energy can be written as

In equation (8.13a), the vectors S and B o are represented as row and column matrices, respectively, enabling the formation of a 3×3 matrix in product form This representation simplifies matrix calculations, allowing for a more straightforward manipulation of the equation.

The equation H Z=β S| g |B o (8.13b) represents a scalar product of row and column vectors, highlighting the interaction between the effective magnetic moment and the effective magnetic field This relationship can be interpreted as either the effective magnetic moment precessing around the static magnetic field B o or the spin S precessing around the effective magnetic field g |B o Experimentally, the first interpretation is often more practical, as B o serves as a stable reference direction within the laboratory Additionally, the crystal field axes provide another useful reference, allowing the static field direction B o to be expressed as B o =B o n, while the effective field for quantizing S can be defined as B =g B o.

B =g n B o where g n 2 = n| gg |n ˆ , (8.14) where ˆgexpresses a “transposed matrix” of g Physicallygis a symmetrical tensor, so that ˆggis identical tog 2 The Zeeman energy in a ﬁeldB o |n can therefore be expressed as

The magnetic moment is effectively expressed by the formula H Z = βg n S n B o, indicating the role of the spin S quantized along the direction n This equation serves as a valuable tool in magnetic resonance, where the g n-factor is adjusted according to the squared tensor g 2 Importantly, the symmetry axes of the crystalline potential can be identified through the principal axes of g 2, which can be derived from a coordinate transformation based on the experimentally obtained quadratic form n| g 2 |n.

In a typical magnetic resonance practice, spectra are collected by rotating a sample crystal around a crystallographic axis, allowing for the measurement of various directions of B0 in a fixed laboratory frame This process yields three elliptical angular variations of g²n that exhibit sinusoidal curves in symmetry planes By fitting these variations to the equation of an ellipse, we can extract three on-axis elements (gii²) and three off-axis elements (gij) for the 3×3 symmetrical tensor g² This tensor can then be numerically diagonalized to identify the principal axes (X, Y, and Z), resulting in zero off-diagonal elements The variation of g²n can be expressed in principal form as g²n = gX²n²X + gY²n²Y + gZ²n²Z, with the constraint n²X + n²Y + n²Z = 1, where (nX, nY, nZ) are the direction cosines relative to the principal axes It is essential that the principal values gX, gY, and gZ are all positive, directly derived from gX², gY², and gZ².

A signiﬁcant feature of the gtensor is traceg=g X+g Y+g Z= 3g e , (8.16a)

The Spin-Hamiltonian and Magnetic Resonance Spectroscopy reveal that deviations from the free-electron value of g (g_e = 2.0023) serve as significant indicators of symmetry within the crystalline potential To assess this, we experimentally determine the tensor ∆g, defined as g - g_e, where e represents the unit tensor (e_ij = δ_ij) Consequently, it is evident that the trace of ∆g equals zero.

As will be discussed later, these equations (8.16a) and (8.16b) are important formula for a three-dimensional analysis of observed Zeeman terms.

The Fine Structure

In Section 8.5, we explored how spin-orbit coupling affects the ionic ground state in Zeeman levels, leading to modifications in the g-factor due to crystalline potential Although the crystal potential significantly quenches the orbital momentum in the first order, an anisotropic g shift emerges in the second order from the λ L.S interaction, highlighting the symmetry of the crystalline environment.

The spin-orbit coupling causes a significant second-order deformation of the charge cloud, especially in ions not in S states This deformation can be interpreted classically as an electric quadrupole moment induced by the crystalline potential Quantum mechanically, this quadrupole energy is derived from the second-order perturbation of the spin-orbit coupling In the context of magnetic resonance spectroscopy, the quadrupole energy of an ion influenced by the crystalline potential is commonly known as the fine structure.

The spin-orbit coupling in a crystalline potential can generally be written as

The Hamiltonian \( H_{LS} \) is expressed as \( H_{LS} = \lambda (L_X S_X + L_Y S_Y + L_Z S_Z) \), where the coupling constant \( \lambda \) is considered isotropic, similar to that of an unperturbed ion In the context of the crystal field, the orbital angular momentum \( L \) is quenched in the first order, allowing for the calculation of the second-order energy of \( H_{LS} \).

The energy E LS (2) is influenced by the non-vanishing off-diagonal elements of L, which connect the ground state and the excited state separated by an energy difference ∆ε This energy can be represented as a quadratic form in relation to the spin components S X, S Y, and S Z.

(8.18b) are elements of the ﬁne-structure tensorD.

In equation (8.18b), it is established that D ij equals D ji, indicating that D is a symmetrical tensor characterized by being traceless The trace of D is expressed as traceD i D ii = (λ 2 /∆ε) i 0|L i ∗ |ε ε|L i |0, which simplifies to (λ 2 /∆ε) i 0|L i ∗ L i |0 This expression can be verified to equal zero for any value of L, confirming that traceD equals zero.

Transforming aDtensor to the principal form, we can express (8.19) as

D X +D Y +D Z = 0, (8.19a) which is held for the diagonal elements Therefore, in a uniaxial crystal field, we have specific relationsD X =D Y andD Z =−2D X , and the fine-structure energy can be expressed by

= 1 2 D Z {3S Z 2 −S(S+ 1)}, (8.20) where S 2 = S X 2 +S Y 2 +S Z 2 The ﬁne structure is therefore determined effectively by the spin S and the component S Z , which however vanishes if

The principal directions indicate the symmetry of an ionic charge cloud that is altered by crystalline potential According to classical theory, this charge cloud can be described as non-spherically deformed and is represented through an electric quadrupole moment in the second-order approximation.

The g2 tensor's quadratic form, n|D|n, is generally proportional to the direction of the applied field Bo, indicating a deformation in the charge cloud When Bo is applied on the XZ plane, the magnitude Dn changes elliptically with the direction of |n as the crystal rotates around the b axis, revealing an elliptically deformed charge This deviation is expressed by an electric quadrupole moment, with the charge cloud deviating positively or negatively from a circular shape along the X and Z axes Three-dimensional measurements can capture these elliptic loci on three symmetry planes, representing projections of the ellipsoidal charge, as illustrated in the elliptical variation on the ab plane.

In magnetic resonance of a paramagnetic ion, a strong applied field Bo has been considered, along which the electronic spin S is quantized, reflecting the local symmetry at the site of a magnetic probe However, the tensors Δg and D may exhibit varying symmetries depending on the strength of the applied field Bo, necessitating a more general approach to account for these differences.

The ellipsoidal deformation of an electronic charge, influenced by crystalline potential, leads to the fine structure observed in the spin-Hamiltonian This effect is illustrated by an elliptic cross-section formed between the ellipsoid of the bilinear n|D|n and an observation plane, where the applied magnetic field Bo is rotated.

When the Zeeman term \( H_Z \) exceeds the fine-structure energy \( H_F \), the spin \( S \) undergoes precession around the effective field \( |B = g |B_0 = g_n B_0 |n \) In contrast, the spin vector \( |S \) in \( H_F \) is effectively quantized along the direction of \( B_0 |n \), leading to the expression \( |S = |S/g_n = (S_n/g_n)g |n \) Consequently, equation (8.21) can be reformulated to reflect this relationship.

Both the ∆g and D tensors arise from the spin-orbit coupling mechanism within a crystalline potential, characterized as coaxial when H F > H Z Consequently, we can express D as D = gDg /g n 2 g 2 D /g n 2 in equation (8.23), where the principal axes X, Y, and Z are typically considered common to both g and D tensors Using these common axes, equation (8.23) can be rewritten as g n 2 D n = g X 2 D X + g Y 2 D Y + g Z 2 D Z, as shown in (8.24).

Eigenvalues of the spin-Hamiltonian (8.22) are determined by the magnetic quantum numberM Sfor the spin stateS n:

E(M S) =g nβB o M S+D n M S 2 , (8.25) where D n is such an effective fine-structure parameter as defined above for the direction|n, and the magnetic resonance occurs when the selection rule

The resonance conditions between energy levels are defined by the equation ¯hω = E(M S + 1) - E(M S) = g n βB o + D n (2M S + 1), where ∆M S = ±1 This equation reveals the fundamental resonance line ¯hω = g n βB o for spin levels M S = ±1/2, along with a series of additional lines that are evenly spaced by D n.

In cases where the Hamiltonian H F H Z allows for the spin S to be quantized along the direction of the largest principal value of D, and H Z acts as a perturbation, first-order calculations may not yield accurate results Higher-order perturbation calculations are often necessary, but extracting structural information from g and D derived from high-order calculations can be complex Therefore, it is advisable to select experimental probes that facilitate easier interpretation Consequently, this discussion on spectroscopy will not be extended further, as general references on magnetic resonance spectroscopy can provide additional insights However, low-field analysis remains essential in specific applications, such as Zeeman studies of nuclear quadrupole resonance and triplet states in molecular probes.

Tiêu đề	The Physics of Structural Phase Transitions
Tác giả	Minoru Fujimoto
Trường học	University of Guelph
Chuyên ngành	Physics
Thể loại	thesis
Năm xuất bản	2005
Thành phố	Guelph

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Số trang	283
Dung lượng	2,42 MB