The Normal or Gaussian Distribution
Statistical physics focuses on systems with numerous identical components, where certain system parameters are derived from the aggregation of individual element parameters For instance, Brownian motion exemplifies how a particle's movement is influenced by collisions with gas atoms, while a system of free particles demonstrates that the total energy results from the sum of individual particle energies, with each particle's momentum fluctuating randomly due to collisions Our objective in both scenarios is to determine the particle's position displacement in the first case and the momentum variation in the second case after numerous collisions Ultimately, we aim to calculate the probability of a variable z achieving a specific value after n1 steps, given that each step's distribution is random and the particle parameter changes are defined.
The function f(z, n) represents the probability of a variable achieving a specific value after n steps, while ϕ(z_k)dz_k indicates the probability that the variable's value falls within the range of z_k to z_k + dz_k after the k-th step Both functions, f(z) and ϕ(z), are probabilities that adhere to normalization conditions.
−∞ ϕ(z)dz= 1 From the definition of the above functions we have: f(z, n) ∞
−∞ ϕ(z) exp(−ipz)dz (2.2) The inverse operation yields: f(z) = 1
Principles of Statistical Physics: Distributions, Structures, Phenomena,
Kinetics of Atomic Systems Boris M Smirnov
Copyright © 2006 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim
8 2 Basic Distributions in Systems of Particles
−∞ zϕ(z)dz=iz k ; g (0) =−z k 2 (2.3) wherez k andz 2 k are the mean shift and the mean square shift of the variable after one step. From the formulae (2.1) and (2.3) there follows:
Sincen 1, the integral converges at smallp Expandinglngin a series over smallp, we have lng= ln
In the context of statistical analysis, the mean shift of a variable after a specified number of steps is represented by \( z = n z_k \), while the mean square deviation is defined as \( n \Delta^2 = n z_k^2 - (z_k)^2 \) For systems comprising numerous identical elements, this deviation is referred to as the fluctuation of the quantity The formula associated with this concept is known as the normal distribution.
The Gaussian distribution is applicable when small values significantly contribute to the integral, as indicated by the condition z k p1, z k 2 p 2 1 This relationship suggests that the integral is influenced by nz 2 k p 2 being approximately equal to 1, reinforcing the validity of the Gaussian distribution for a large number of steps or elements, denoted as n1.
Specifics of Statistical Physics
Statistical physics analyzes systems with numerous elements, allowing for the use of average values rather than full distributions for certain parameters For instance, we can illustrate this concept through the distribution of identical particles within a defined volume Ω, which contains a constant number N of free particles.
find the distribution of a number of particles located in a small partΩ o Ωof this volume.
Assuming a large mean number of particles, the probability \( W_n \) of finding \( n \) particles in a specific volume can be expressed as the product of the probability of locating those \( n \) particles within the volume \( (\Omega_o / \Omega)^n \), the probability of locating the remaining \( N - n \) particles outside this volume \( (1 - \Omega_o / \Omega)^{N - n} \), and the number of combinations \( C_{N}^{n} \) for this arrangement.
This probability satisfies the normalization condition n
W n = 1. Let us consider the limitn1,n=N Ω Ω o 1, nN, n 2 N Then we have
This formula is called thePoisson formula.
In the case considered,n1, n1, the functionW n has a narrow maximum atn=n.
, n1 (2.6) we find that the expansion ofW n nearnhas the form lnW n = lnW o −(n−n) 2
2n (2.7) whereW o = (2πn) −1 / 2 , and the fluctuation of the number of particles in a given volume equals
This article illustrates a fundamental principle of statistical physics by dividing a total volume into cells, with the average number of particles in the ith cell represented as ni = N/Ωi, where N is the total particle count in the volume Ω By focusing on the mean particle numbers ni and disregarding fluctuations, we find that the distribution of particles in each cell tends to cluster around the average It is evident that these fluctuations remain relatively small, confirming the validity of this principle when the number of particles in the cells is sufficiently large (ni > 1).
The distribution of particles within cells can be determined using two methods, both of which approximate the average values while neglecting fluctuations The first method involves measuring the number of particles in each cell, denoted as n_i, which aligns with the average value up to the fluctuation size The second method tracks a test particle in a specific cell over a time interval, resulting in the number of particles in that cell being represented as N t_i / t, which also corresponds to n_i within the same accuracy limits Therefore, in statistical physics, average values are often used in the first approximation, disregarding fluctuations.
10 2 Basic Distributions in Systems of Particles
Temperature
In a system of free atoms, collisions lead to a specific distribution of atomic energies This distribution allows us to define the temperature of the atoms, T, based on the relationship ε z = 1.
The average kinetic energy of a single atom in motion along the z-direction is represented as εz (2.9), and due to the symmetry in all three spatial directions, the overall average kinetic energy of the atom can be expressed as ε = 3.
Temperature is typically measured in kelvins (K), with the value k_B T often utilized in equations instead of T, where k_B = 1.38 × 10^(-16) erg/K is the Boltzmann constant, serving as a conversion factor between erg and K This constant's role in physical equations stems from historical perspectives on temperature and energy as distinct dimensional values For our purposes, we will consider kelvins as an energetic unit and will not apply the conversion factor Table 2.1 illustrates the relationship between this energetic unit and other measurement units.
Table 2.1.Conversion factors between kelvins (K) and other energetic units.
Energy unit erg eV cal/mol cm −1 Ry
In an ensemble of n free atoms at temperature T, the distribution of the system over the total kinetic energy of the atoms can be described using formula (2.4) Instead of the variable z, we focus on the total kinetic energy, denoted as E, with its average value calculated accordingly.
2nT (2.10b) and the mean squared deviation of the total kinetic energy is
The equation ∆ 2 = n ε 2 − ε 2 illustrates the relationship between the average energy (ε) and the average energy squared (ε 2) for individual atoms It is evident that ε 2 is proportional to the square of the temperature (T 2), indicating a direct correlation Additionally, the relative width of the distribution function for the total kinetic energy of the atoms is represented by δ, which is approximately equal to ∆.
E ∼√1 n i.e this value is small if there are a large number of atoms in the system.
The Gibbs Principle
Statistical physics plays a crucial role in analyzing distribution functions for particles within an ensemble, focusing on the energy distribution of weakly interacting particles in a closed system By considering a large ensemble of particles, we can categorize them into states defined by a set of quantum numbers Each particle's state is characterized by its internal quantum numbers, such as the electron shell state in atoms and the vibrational and rotational states in molecules.
In this article, we explore the coordinates and momenta of particle states, denoted as r and p We consider a range of particle positions (∆r) and momenta (∆p), leading to multiple states represented by the index i, with a significant number of states g_i Our objective is to determine the average number of particles within each group of states We focus on a system of free particles that predominantly remain free during most observation periods, only interacting briefly with surrounding particles or vessel walls This interaction is crucial as it establishes equilibrium within the particle system, although we analyze individual particles under the assumption of their free state at any given moment.
In a gaseous system composed of numerous free particles, weak interactions and infrequent collisions result in a specific distribution of particles across various states, influenced by the system's parameters Assuming the total number of particles, denoted as n, remains constant over time, we can express the number of particles in the ith state as n i Consequently, the conservation of the total number of particles is represented by the equation n = Σ n i.
In a closed system of particles, where there is no exchange of energy with external systems, the total energy of the particles must be conserved.
In a closed system, the energy of a particle in the ith state is represented by ε i (2.12) While individual particles may transition between states during the system's evolution, the average number of particles in each state remains relatively constant This phenomenon is known as thermodynamic equilibrium.
The transitions of individual particles between states occur due to collisions with other particles The likelihood of a particle existing in a specific state, along with the average number of particles in that state, is directly proportional to the number of possible configurations that can occur This concept, known as the Gibbs principle or the principle of homogeneous distribution, underpins statistical physics According to this principle, the probability of a particle system being in a particular state correlates with the number of configurations that result in that distribution.
Denote byP(n 1 , n 2 , ã ã ã n i ã ã ã)the number of ways thatn 1 particles are found in thefirst group of states,n 2 particles are found in the second group of states,n i particles are found
In systems of particles, the calculation of basic distributions within the ith group of states is essential To determine the number of possible distributions, we assume that the position of one particle does not affect the positions of others Consequently, the total number of ways to achieve a specific distribution of particles across different groups of states is derived from the product of distributions within each individual group.
P(n 1 , n 2 , n i , ) =p(n i )p(n 2 )ã ã ãp(n i )ã ã ã (2.13) wherep(n i )is the number of ways to distributen i particles inside a given group of states Let us perform this operation successively.
First, taken 1 particles for the first state from the total number ofnparticles There are
The total number of ways to distribute particles into different states can be calculated using the formula C(n, n1) = (n - n1)! / n1! To find the distribution for the second state, we select n2 particles from the remaining n - n1 particles, which can be achieved in C(n - n1, n2) ways By continuing this process, we can derive the probability associated with the specific distribution of particles across the states.
The formula (n i !) (2.14) includes a normalization constant and is based on the premise that particles are independent, meaning the state of one particle does not affect the distribution of the others.
The Boltzmann Distribution
To find the most probable number of particles \( n_i \) in a state \( i \) for a system of weakly interacting particles, we analyze the distribution \( P \) of these particles The maximum of \( P \) occurs at \( n_i = n_i \), and by introducing a small variation \( dn_i = n_i - n_i \) while assuming \( n_i \) is significantly larger than \( dn_i \), we can expand the logarithm of \( P \) near this maximum Utilizing the approximation \( \ln n! \approx n \ln n - n \), we derive insights into the behavior of the system at equilibrium.
0 lnxdx, we have dln n!/dn= lnn.
On the basis of this relation, we obtain from formulae (2.4) and (2.14): lnP(n 1 , n 2 ,ã ã ãn i ,ã ã ã) = lnP(n 1 , n 2 ,ã ã ãn i ,ã ã ã)− i lnn i dn i − i dn 2 i
The condition for the maximum of this value gives: i lnn i dn i = 0 (2.15)
Alongside this equation, we take into account the relations which follow from equations (2.11) and (2.12): i dn i = 0 (2.16)
Equations (2.15) to (2.17) enable the calculation of the average number of particles in a specific state By multiplying equation (2.16) by -lnC and equation (2.17) by 1/T, where C and T represent key parameters of the system, we can sum the resulting equations to derive the expression: i lnn i − lnC + ε i.
To satisfy the equation for any value of \( n_i \), the expression within the parentheses must equal zero Consequently, this results in the derived formula for the most probable number of particles in a specific group of states, expressed as \( n_i = C \exp \).
The Boltzmann distribution formula describes the probability of finding a particle in a specific state, assuming that this probability is independent of the states of other particles This distribution is applicable under certain statistical conditions, particularly when the average population of any given state is small This criterion is essential for the validity of the Boltzmann distribution.
Let us determine the physical nature of the parametersCandT in equation (2.18), which follows from the additional equations (2.11) and (2.12) From equation (2.11) we have
C i exp(−ε i /T) = N, so that the valueC is the normalization constant The energetic parameterTis the temperature of the system One can see that this definition of the tempera- ture coincides with (2.9).
Let us prove that at large¯n i the probability of observing a significant deviation fromn¯ i is small According to the above equations this value equals (compare with (2.7) and (2.15)):
In fact, this formula coincides with the Gaussian distribution (2.4) From this it follows that a shift of n i from the average value n i , at which the probability is not so small, is
|n i −n i | ∼ 1/√ n i Sincen i 1, the relative shift of a number of particles in one state is small: |n i −n i |/n i ∼ 1/√ n i Thus the observed number of particles in a given state differs little from its average value.
In a system of weakly interacting particles with a large number of particles, one can establish key characteristics, including the introduction of a distribution function that represents the number of particles across various states at a given time, assuming no interactions occur This distribution remains conserved over time with an accuracy of approximately 1/√n_i, where n_i indicates the average number of particles in a specific group of states Additionally, by observing a single particle over an extended period, the distribution function can also be defined based on the total time the particle occupies different states Within the aforementioned accuracy, both definitions of the distribution function align, demonstrating the principle known as the ergodic theorem, which highlights the equivalence between phase space averaging and long-term observation of an individual particle.
14 2 Basic Distributions in Systems of Particles
Statistical Weight, Entropy and the Partition Function
In equations (2.15) and (2.19), the subscript denotes a group of particle states, and we now examine a scenario where this includes a set of degenerate states We define the statistical weight \( g_i \) of a state as the number of degenerate states associated with it For instance, a diatomic molecule in a rotational state characterized by the rotational quantum number \( J \) has a statistical weight given by \( g_i = 2J + 1 \), representing the number of momentum projections along the molecular axis Incorporating the statistical weight, equation (2.12) can be expressed as \( n_i = C g_i \exp \).
T whereCis the normalization factor In particular, this formula gives the relation between the number densities of particles in the groundN o and excitedN i states:
T (2.20) whereε i is the excitation energy, andg o andg i are the statistical weights of the ground and excited states.
Let us introduce theentropyS i of a particle which is found in a given group of states:
The entropy of a particle in a specific state is defined as S_i = ln g_i, assuming equal probability for its location across these states When another particle occupies a state from group j, the total statistical weight for their locations becomes g_ij = g_i g_j, leading to the overall entropy of the system being S_ij = ln(g_i g_j) = S_i + S_j This demonstrates that entropy is an additive function By generalizing this definition, we can derive a new formula for cases where a particle may exist in multiple states.
The probability of a particle being in a specific state is represented by S ln 1 w i, where w i denotes the likelihood of the particle's location This expression can be reformulated to reflect averages taken over various states.
Transferring to a system composed of a specific number of particles or subsystems allows for the definition of entropy with a constant accuracy for that particle system Consequently, this expression can be reformulated for a system of n particles.
S=− i n i lnn i (2.22) wheren i is the number of particles located in a given state (or the distribution function of particles over states).
2.6 Statistical Weight, Entropy and the Partition Function 15
To analyze a system of weakly interacting particles, we can define the partition function \( z_i \) for each individual particle, representing its position in the ith state This is expressed as \( z_i = g_i \exp \), where \( g_i \) denotes the degeneracy of the state.
The average number of particles in the ith state is expressed as n_i = n z_i / z, where z_i represents the partition function for a specific particle For an ensemble of n identical atomic particles, the total partition function is crucial for understanding the system's statistical properties.
Z i (2.23b) whereZ i =nz i is the partition function of a given state (or group of states) for an ensemble ofnparticles.
The partition function serves as a fundamental tool for expressing various average parameters in a many-particle system Specifically, it allows for the calculation of the total energy of particles, especially within the context of the Boltzmann distribution.
∂lnT (2.24) where we use the relation from formula (2.23a)
In thermodynamic equilibrium, an ensemble of weakly interacting particles at temperature T shows that the most probable energies cluster around a specific value, E By exploring various combinations of particle distributions across states with total energies close to E, we can derive the total partition function for this ensemble.
−E T Γ whereΓ i g i is the total number of ensemble states with this internal energy By analogy with formula (2.21), one can introduce theentropyof this particle ensemble as
Then, using formula (2.24), we obtain from this relation the following connection between the entropy and partition function of this particle ensemble:
16 2 Basic Distributions in Systems of Particles
To determine the statistical weight of states in the continuous spectrum, we start by examining the wave function of a free particle moving along the x-axis, represented as exp(ipx/ħ) for positive momentum and exp(−ipx/ħ) for negative momentum, where ħ is the reduced Planck constant When placed in a potential well with infinitely high walls, the particle can move freely within the region 0 < x < L, with the wave function being zero at the walls By constructing a wave function that meets these conditions, we derive ψ = Csin(px/ħ) Applying the boundary conditions ψ(0) = 0 and ψ(L) = 0 leads to the quantization condition pL/ħ = πk, where k is an integer, allowing us to calculate the energies of quantum states for a particle confined in a rectangular potential well.
The number of states available for a particle with momentum in the range \( p_x \) to \( p_x + dp_x \) is expressed as \( dg = \frac{L \, dp_x}{2\pi} \), considering both directions of momentum Additionally, when the spatial interval is \( dx \), the number of particle states is given by \( dg = dp_x \, dx \).
Generalizing this to the three-dimensional case, we obtain for the number of states of a test particle dg= dp x dx
In this article, we denote the phase space element as \( d p = d p_x d p_y d p_z \) and \( d r = d x d y d z \) The product \( d p d r \) represents an element of the phase space, and the statistical weight of the continuous spectrum, as indicated in formula (2.24), corresponds to the number of states associated with this phase space element.
To derive the Boltzmann formula (2.18), we categorize states into groups, each containing a significant number of states denoted as g_i Each group corresponds to a specific element of the phase space, represented as p d r /(2π)³ for a particle, and encompasses certain internal states Therefore, g_i represents the statistical weight of each group of states.
To determine the heat capacity of an ensemble of Boltzmann particles, we start with the average energy formula, C = dE/dT We assume that temperature changes do not affect the energy levels (ε i) of the system, indicating a constant volume during these variations By applying the relevant formulas and considering the relationship ∂Z i /∂T = ε i Z i /T² for Boltzmann particles, we can derive the heat capacity of the system effectively.
The Maxwell Distribution
The energetic parameters of a system can be expressed as ε_i = ε_i Z_i / Z and ε²_i = ε²_i Z_i / Z, where the bar indicates an average over the ensemble Additionally, the heat capacity of the system is directly proportional to the number of particles, n, due to the weak interactions among them.
The velocity distribution of free particles resulting from collisions affects the energy of individual particles, as described by the Boltzmann formula In a one-dimensional scenario, the energy of a particle is expressed as mv²x/2, with the statistical weight of this state being proportional to dvx Consequently, the number of particles with velocities within the range from vx to vx + dvx is represented by the equation f(vx)dvx = C exp.
2T dv x whereCis the normalization factor Correspondingly, in the three-dimensional case we have: f(v)d v=Cexp
The kinetic energy of a particle, expressed as mv²/2, is the total of its kinetic energies across all motion directions, represented by the vector v with components vx, vy, and vz By normalizing the distribution function to the particle number density N, we derive the relationship f(v) = N/m, adhering to the normalization condition.
Introduce the functionϕ(v x ) ∼ f(v x ), which is normalized to unity:
These distribution functions of particles on velocities are called theMaxwell distributions Let us determine the average kinetic energy of particles on the basis of formula (2.29a) We have: mv 2
18 2 Basic Distributions in Systems of Particles
In this context, the constant does not vary with temperature, leading to the conclusion that the kinetic energy of particles per degree of freedom is T/2 Consequently, the average kinetic energy of particles in three-dimensional space is expressed as mv²/2 = 3T/2 These relationships serve as the foundation for defining temperature in equations (2.9) and (2.10), aligning with the temperature definition utilized in deriving the Boltzmann formula (2.18).
Introducing the distribution functionf(ε)on kinetic energiesε=mv 2 /2of free particles, which is normalized by the condition
0 f(ε)ε 1 / 2 dε= 1 (2.31a) we obtain for this distribution function in the case of the Maxwell distribution of free particles: f(ε) = √ 2 πT 3 / 2 exp
Mean Parameters of an Ensemble of Free Particles
An ensemble of free particles represents the simplest particle system, characterized by particles occupying a defined volume with minimal interaction We can derive average parameters from the distribution function of free particles, leading to the conclusion that the average energy of a Maxwell particle ensemble is given by ε = 3T.
4 wherenis the number of ensemble particles Then formula (2.28) gives for the heat capacity of an ensemble of Maxwell particles
On the other hand, we get the same result by introducing the heat capacity of the particle ensemble as
2n whereE=nε= 3T n/2is the average energy of an ensemble ofnfree particles.
Fermi–Dirac and Bose–Einstein Statistics
We calculate the partition function for a particle ensemble that follows the Maxwell distribution of kinetic energies, which corresponds to an ideal monatomic gas This analysis is based on the definition of the partition function, as outlined in equation (2.23).
(2π) 3 wherenis a number of particles, the total energy of particles equalsE i ε i according to formula (2.12), and the subscripticorresponds to parameters of theith particle The factor
1/n!accounts for the identical nature of the particles Usingε i =p 2 i /(2m), we have
Z =z n /n! wherezis the partition function of an individual particle which equals z exp
HereΩis the system volume Using the Stirling formulan! ≈ (2πn) −1 / 2 (n/e) n for large n, whereeis the natural logarithm base, we finally obtain for the partition function of the ensemble of structureless particles
(2.32) whereN is the number density of particles Note that the valuelnZis the additive function of individual particles, in this case of a weak interaction between particles.
2.9 Fermi–Dirac and Bose–Einstein Statistics
In a system of identical particles, their interactions depend on the statistics they follow Bose-Einstein statistics apply to particles with integer spin, resulting in a symmetric total wave function that allows multiple particles to occupy the same state Conversely, Fermi-Dirac statistics govern particles with half-integer spin, leading to an antisymmetric wave function that enforces the Pauli exclusion principle, which prohibits two particles from occupying the same state This principle is particularly significant for electrons, as it dictates their behavior at low temperatures and underpins the structure of atomic systems.
As a matter of fact, a certain symmetry of the total wave function of many identical parti- cles means the existence of an interaction between particles called theexchange interaction.
20 2 Basic Distributions in Systems of Particles
The exchange interaction causes repulsion that prevents electrons from getting too close to one another To analyze this effect, we can calculate the average number of particles occupying a specific state based on the symmetry of their wave function, as outlined in formula (2.18) for noninteracting particles According to the Boltzmann formula, the probability of finding a particle in a particular state of a group can be expressed as \( w_i = \exp\left(-\frac{\epsilon_i}{T}\right) \).
In Bose–Einstein statistics, the probability of a state being occupied by a particle is represented by \( w_i \), where \( \alpha \) denotes the chemical potential and \( C = \exp(\alpha/T) \) To calculate the average number of particles in a given state, we consider \( w_i \) as the probability of one particle occupying that state Since Bose–Einstein statistics allows for multiple particles to occupy the same state, the probability for two particles is \( w^2_i \), and for \( m \) particles, it is \( w^m_i \) Consequently, the average number of particles in this state can be expressed as \( n_i = g_i \sum_{m=1}^{\infty} w^m_i = w_i \).
Note that we refer the indexito a group of states which number isg i
To derive the Bose–Einstein distribution, we can employ a method similar to that used for the Boltzmann formula We begin by categorizing the states of the particle system into groups and applying the Gibbs principle alongside a specific formula that enables independent particle distribution across these groups We focus on determining the number of ways, denoted as p(n_i), to distribute n_i particles among g_i states within a particular group By treating n_i particles and g_i states as elements of a set, we can construct sequences where a state occupies the first position, with the remaining elements arranged randomly The count of particles located between successive states indicates the distribution of particles in each state The total number of distinct distributions is given by (g_i + n_i - 1)!, accounting for identical distributions resulting from the permutation of states or particles Consequently, the Bose–Einstein statistics yield the distribution formula: p(n_i) = (n_i + g_i - 1)! / (n_i! (g_i - 1)!).
Thus, instead of equation (2.15) we get in this case,g i 1,n i 1 i dlnp(n i ) dn i (n i =n i )dn i i
Repeating the operations which were used to deduce formula (2.18) and denoting C exp(−à/T), we obtain on the basis of this relation the Bose–Einstein formula (2.34) instead of (2.18).
In the case of Fermi–Dirac statistics, we repeat the derivation of the Boltzmann formula(2.18), taking into account the exchange interaction of particles which prohibits the location
Fermi-Dirac and Bose-Einstein statistics describe the distribution of particles among various energy states For a given energy state ε_i with g_i available states, n_i particles can be arranged in p(n_i) = C(g_i, n_i) = g_i! / (n_i!(g_i - n_i)!) ways, provided that n_i is less than or equal to g_i The total arrangement of particles across states is expressed through a specific formula, and in scenarios where g_i is significantly larger than n_i, the Fermi-Dirac statistics can be represented by the equation i dlnp(n_i) / dn_i = ln(n_i) / (g_i - n_i).
Repeating the operations for deducting formula (2.18), using (2.16) and (2.17), we obtain finally theFermi–Dirac distribution n i = g i exp ε i −à
+ 1 (2.35) where we introduce the chemical potentialàinstead of the constantCby analogy with Bose– Einstein statistics This gives for the average number of particles in one state: n i = 1 exp ε−à
In the limitn i 1formulae (2.34) and (2.36) for the population numbers of the Bose– Einstein and Fermi–Dirac statistics transform into the Boltzmann formula (2.18) This limit corresponds to the criterion: exp ε i −à
The chemical potential, as defined in formulas (2.33) to (2.36), along with the constant C in the Boltzmann formula (2.18), is established through the normalization condition Specifically, for a system of free particles that follow Bose–Einstein or Fermi–Dirac statistics, this normalization condition takes a particular form.
The number density of particles, denoted as HereN, is influenced by their statistical weight, represented by g, which is determined by their spin S (where g = 2S + 1) The energy of the particles, ε, is related to their momentum, p, through the equation ε = p²/(2m), with the sign ± varying based on the type of statistics applied By introducing the parameter z = à/T, this relationship can be reformulated accordingly.
When the probability of a particle's location in a particular state approaches one, the distribution of particles across states is influenced by their statistical properties Conversely, in scenarios where this probability is minimal, the distribution of particles aligns with the Boltzmann distribution.
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Distribution of Particle Density in External Fields
The Boltzmann formula enables the analysis of particle distribution in external fields, exemplified by particles in a gravitational field In this context, the formula expresses the distribution function as N(x) ∼ exp(−U/T), where U represents the potential energy of a particle Specifically, for a gravitational field, the potential energy is defined as U = mgh, with m being the particle mass, g the acceleration due to gravity, and h the altitude above the Earth's surface Thus, the Boltzmann formula takes on this particular form when applied to gravitational scenarios.
The barometric distribution describes the molecule number density, N(z), at various altitudes, z At room temperature, the atmospheric pressure decreases significantly with altitude, with a gradient of mg = 0.11 km⁻¹, indicating that pressure drops noticeably within several kilometers above the surface.
In a quasineutral plasma, the Coulomb field of a charged particle influences the distribution of surrounding charged particles, leading to a shielding effect at a certain distance For illustration, consider a positively charged plasma particle, with surrounding particles having a charge of ±e (the charge of an electron) In a vacuum, the electric potential (ϕ) of this test charged particle at a distance (r) is expressed as ϕ = e/r.
In the presence of other charged particles in the plasma, the electric potential of a test particle is determined by the Poisson equation
HereN − ,N + are the number densities of negatively and positively charged plasma particles, which according to formula (2.18) are equal to
−eϕ T whereN o is the average number density of charged particles of the plasma andT is the plasma temperature, i.e the temperature of charged particles Thus the Poisson equation takes the form:
At significant distances from a test particle, the equation holds true when compared to the average spacing between charged particles, represented as N o −1 / 3 However, at closer ranges where no other charged particles are present, the right side of the Poisson equation equals zero, leading to the Coulomb electric potential of the test particle, as defined by formula (2.39).
Fluctuations in a Plasma
Taking into account that the electric potential of the particle does not depend on angle, we have for distances whereeϕT:
The solution of this equation, which is transformed into (2.39) at small distancesr, has the form: ϕ= e rexp
The Debye–Hückel radius, denoted as D, is a fundamental plasma parameter that defines the distance over which electric fields are shielded in plasma An ionized gas is classified as plasma when its Debye–Hückel radius is significantly smaller than the overall dimensions of the system.
The shielding of the particle field occurs when the shielding distance D is significantly larger than the average distance between charged particles in the plasma, represented as N o −1 / 3 This relationship can be simplified to the criterion e 2 N o 1 / 3, highlighting the essential conditions for effective shielding in plasma environments.
In a plasma, the interaction energy between charged particles at an average distance is relatively small, approximately 2 N o 1/3, compared to the thermal energy of the particles, which is around ∼T When this criterion holds true, the charged particles in the plasma are predominantly free, indicating a state similar to a gas This relation serves as the defining criterion for an ideal plasma, where the behavior of charged particles is largely uninhibited.
An ideal quasineutral plasma consists of neutral particles, electrons, and ions with identical charge density, where the mean potential energy of charged particles is significantly lower than their kinetic energy, resembling a gas of charged particles In this plasma, the number density of neutral particles can surpass that of charged particles; however, the properties are primarily influenced by the charged particles due to their long-range interactions A key characteristic of ideal plasma is the screening of electric fields, which occurs as charged particles displace in response to these fields This screening is quantified by the Debye–Hückel radius, which defines the typical distance over which charged particles can influence one another, establishing an ideal plasma as one where a significant number of charged particles are contained within this radius.
24 2 Basic Distributions in Systems of Particles
In an ideal plasma, the movement of charged particles generates a plasma potential, which alters the energy of a charged particle entering the plasma through its boundary This article aims to calculate the average plasma potential and the distribution function related to these potentials Based on formula (2.41), the interaction energy of a test ion with other ions is defined by this potential.
− r r D so that the mean potential energy of a test ion in an ideal plasma is
In an ideal plasma, when the condition outlined in equation (2.42) is satisfied, the mean potential energy of a charged plasma particle, as described in equation (2.43), is significantly lower than its thermal energy, assuming that the temperatures of electrons and ions are equal.
Note that this potential energy is identical for positively and negatively charged particles (for ions and electrons).
Using the same method one can find the mean square for the ion or electron potential energy
T 2 ∼ e 2 r D T 1 for an ideal plasma Next,
U 2 = 16πN o r D 3 1 according to the definition of an ideal plasma, because many charged particles are located inside a sphere of radius r D Therefore we ignore the valueU in the expression for the
2.11 Fluctuations in a Plasma 25 distribution function over the potential energiesf(U)of charged particles, and the distribution function in accordance with formula (2.4) has the form f(U)dU= √ 1
The fluctuation of potential energy for a charged particle in an ideal plasma leads to the displacement of surrounding particles, resulting in shielding of the particle field by plasma particles Additionally, these surrounding charged particles generate random fields near a test charged particle, causing significant fluctuations relative to the mean particle energy.
In an ideal plasma, the mean free path of a charged particle, such as an electron or ion, is determined by its scattering in a random field influenced by a plasma potential that varies by approximately ∆U over a distance of about rD The energy of the particle changes by roughly ∆U across this distance, leading to a typical energy of the charged particle, denoted as ∼T, which results from approximately (T/∆U)² scattering events Consequently, we estimate the mean free path (λ) of charged particles to be approximately equal to rD, indicating the distance over which the particle's energy varies by ∼T.
The mean free path of a charged particle in an ideal plasma is influenced by plasma non-uniformities, demonstrating an inverse relationship with the mean number density of charged particles and a direct relationship with the square of the temperature.
Laws of Black Body Radiation
This article explores systems of particles and quasiparticles governed by Bose-Einstein statistics, including photons and phonons, which are the vibrational excitations of solid molecules and atoms By considering the general interactions within these systems, we find that particles remain largely free during most observation periods, enabling an effective distribution across various states The weak interactions present in these systems lead to a distribution that follows Bose-Einstein statistics.
We first consider a system of photons as elementary particles of an electromagnetic field.
Equilibrium radiation refers to the number of photons of a specific frequency, which is influenced by their interaction with a gas or the walls of a vessel When considering a vessel with walls at temperature T, the radiation field within is affected by the absorption and emission of radiation by these walls This interaction dictates the number of photons, making it independent of external conditions The radiation present in the vessel is known as black body radiation According to the Boltzmann formula, the probability of finding n photons of energy ω in a particular state is given by exp(−ωn/T) Consequently, the average number of photons nω in that state can be expressed as nω = (1 / (exp(ω/T) - 1)).
The Planck distribution is associated with the Bose-Einstein distribution, where the chemical potential is zero, indicating that the number of particles in the system is not conserved In this scenario, the distribution of particles across states is determined using equations (2.12) and (2.15), while equation (2.11) is not considered Consequently, the absence of the chemical potential in the expression for state population is confirmed, leading to the conclusion that the chemical potential equals zero.
The spectral radiation density \( U_\omega \) represents the energy of radiation per unit time, volume, and frequency range It can be expressed as the radiation energy within a frequency range from \( \omega \) to \( \omega + d\omega \), calculated as \( \Omega U_\omega d\omega \), where \( \Omega \) is the volume of the vessel Alternatively, this value is represented by \( \frac{2\omega n_\omega \Omega d k}{(2\pi)^3} \), with the factor of 2 accounting for the two polarizations of electromagnetic waves Here, \( k \) denotes the photon wave number, \( \frac{\Omega d k}{(2\pi)^3} \) signifies the number of states in a specific phase space element, and \( n_\omega \) indicates the number of photons per state It is important to note that the electromagnetic wave is transverse, meaning that the electric field strength \( E \) is oriented perpendicular to the direction of propagation defined by the vector \( k \).
Principles of Statistical Physics: Distributions, Structures, Phenomena,
Kinetics of Atomic Systems Boris M Smirnov
Copyright © 2006 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim
28 3 Bose–Einstein Distribution ω=ckbetween the frequencyωand wave vectorkof the photon (cis the velocity of light), we have from the above relations
By replacing the Planck distribution (3.1) in formula (3.2), we obtain thePlanck radiation formulawhich has the form
In the classical limiting caseω T this formula is transformed into theRayleigh–Jeans formula
Since this formula corresponds to the classical limit, it does not contain the Planck constant. The other limiting case yields theWien formula
The radiation flux emitted by a black body surface can be evaluated using formula (3.3), which defines it as the radiation flux originating from a hole in an opaque-walled cavity containing black body radiation This black body surface is characterized by its ability to absorb all incident radiation Additionally, it emits an isotropic energy flux represented by cU ω dω per frequency interval dω, resulting in the energy flux dΘ.
The elementary solid angle, denoted as U ω dω, is emitted at an angle Θ = dϕ dcosθ By projecting the elementary radiation fluxes onto the resultant flux that is perpendicular to the emitting surface, we can derive the resultant radiation flux exiting the emitting surface.
U ω dω=σ T 4 (3.6) whereθ is the angle between the normal to the surface and the direction of motion of an emitting photon The constantσ is called theStefan–Boltzmann constant We use
Spontaneous and Stimulated Emission
Equation (3.6) is called theStefan–Boltzmann law.
The dependence of radiation flux can be derived through dimensional analysis, highlighting its relationship with key parameters: radiation temperature (T), the Planck constant, and the speed of light (c) By analyzing these parameters, we find that there is only one viable combination that yields the dimensions of flux, resulting in the expression J ∼ T^4 c^(-2), which aligns with the established formula (3.6).
We assess the partition function for equilibrium radiation within a closed vessel of volume V, maintained at a wall temperature T The partition function for a single oscillation mode with frequency ω is given by specific thermodynamic principles.
The total partition function of the radiation field is determined by the number of excited vibrations, denoted as n Since the individual oscillations of different modes operate independently, the overall partition function can be expressed as the product of the partition functions for each mode Consequently, the logarithm of the total partition function, lnZ, is the sum of the logarithms of the individual partition functions.
Herek=ω/cis the photon wave vector, the factor 2 accounts for two polarizations of pho- tons, andσis the Stefan–Boltzmann constant.
Equilibrium radiation arises from interactions with vessel walls and the absorption and emission of photons by atomic particles The most efficient processes occur during transitions between discrete atomic states, as illustrated by the reaction scheme ω + A ↔ A*.
The Bose-Einstein Distribution is satisfied when the number density of atoms (N), the typical system size (L), and the atomic absorption cross section for photons at frequency ω (σ abs(ω)) are considered In this context, equilibrium radiation occurs at specific frequencies where this condition holds, indicating that the equilibrium is maintained by certain processes Notably, the temperature of this equilibrium radiation aligns with the temperature of atomic excitations.
In analyzing the equilibrium of an atom within a radiation field, we denote the number of photons in a specific state of a group as nω This number increases by one when a transition occurs from the ground state and decreases by one upon the absorption of a photon The absorption rate is directly proportional to the number of photons present in the gaseous volume, allowing us to express the probability of photon absorption by a single atom per unit time accordingly.
The equation W(i, n ω → f, n ω−1) = An ω illustrates the transition probabilities between atomic states, where the subscript i represents the lower state and f denotes the upper state Formula (3.9) highlights that transitions are nonexistent without photons (n ω = 0) and that only single-photon transitions are considered Notably, the constant A is independent of the strength of the electromagnetic field and is solely determined by the characteristics of the atomic particle.
The probability per unit time of an atomic transition with emission of a photon can be represented in the form:
The reciprocal lifetime (1/τ) represents the spontaneous emission rate of an excited atom without an external field, while the quantity B pertains to stimulated radiation induced by an external electromagnetic field Both of these values are intrinsic properties of the atomic particle, and A and B are collectively referred to as the Einstein coefficients.
The relationship among the parameters 1/τ, A, and B can be derived from thermodynamic equilibrium analysis, which includes both atomic particles and photons According to Boltzmann's law, the number densities of atomic particles in the excited state (Nf) and the ground state (Ni) are related, highlighting the fundamental principles of statistical mechanics.
In a system where the statistical weights of the ground and excited states are represented by T whereg i andg f, the photon energy ω corresponds to the energy difference between these states The average number of photons in a specific state follows the Planck distribution At thermodynamic equilibrium, the rate of emission transitions matches the rate of absorption transitions per unit time This relationship can be applied to a unit volume for further analysis.
On the basis of formulae (3.10) this relation can be transformed to the following form:
Vibrations of Diatomic Nuclei
By applying the established formulas that connect the equilibrium number densities of atomic particles with the average number of photons in a specific state, we derive the Einstein coefficients, where A equals g f divided by (g i τ) and B equals 1/τ Consequently, this results in the formulation of rates for one-photon processes.
Note that the condition of thermodynamic equilibrium requires the presence of stimulated radiation, which is described by the last term and is of fundamental importance.
The Planck formula is applicable to all systems of harmonic oscillators, and we will illustrate its validity using diatomic molecules as an example We introduce the interaction potential U(R) between two atoms separated by a distance R, which typically resembles a potential well that reaches its minimum at the equilibrium distance R = R_e Near this minimum, the interaction potential can be expressed in a specific form.
The interaction potential in diatomic molecules is defined by the equation ∂R 2 | R=R e (3.13), where D represents the depth of the potential well corresponding to pair interactions, equating to the dissociation energy in classical terms This potential leads to the Schrödinger equation, which governs the wave function Ψ, describing the relative motion of atoms within diatomic molecules.
The equation 2x 2 Ψ = EΨ, where x = R - Re and à = m/2 represents the reduced mass of nuclei with m as the mass of a single atom, leads to the determination of the molecular vibration spectrum through its solutions.
2κ m (3.14) wherevis the so-calledvibrational quantum number(an integer), and the energy of the vibra- tional states starts from the bottom of the potential well.
A test molecule surrounded by atomic particles experiences collisions that establish a specific temperature, T According to the Boltzmann formula, the probability of the molecule occupying the vth vibrational level is determined by this temperature.
From this we find the average number of molecule excitations v v vP v v
The Bose-Einstein distribution aligns with the Planck formula, highlighting a key distinction from radiation fields In this context, we focus on oscillators of a single frequency, unlike electromagnetic fields that encompass a range of frequencies.
Structures of Solids
The interaction of atoms leads to the formation of condensed systems, characterized by two types of bonding In metallic systems, valence electrons are shared among all atomic nuclei, significantly influencing the system's properties and atomic binding Conversely, in systems where individual atomic identities are preserved, valence electrons partially transfer to neighboring atoms, resulting in interactions primarily between atoms or ions This is evident in rare gas solids, which exhibit short-range interactions between neighboring atoms We will utilize this system for demonstration purposes.
This article explores the fundamental concepts of atomic crystal structures, focusing on the processes that occur within these systems In solids or crystals, which consist of a vast number of atoms, the atoms arrange themselves into a crystal lattice characterized by translational symmetry A lattice exhibiting this periodicity is referred to as a Bravais lattice, with the positions of atoms defined by specific coordinates within the Bravais lattice.
The equation R = n₁a₁ + n₂a₂ + n₃a₃ represents a lattice, where n₁, n₂, and n₃ are integers, and the vectors a₁, a₂, and a₃ serve as the basis for this lattice By using the unit vectors along the x, y, and z axes, denoted as i, j, and k, one can express the basis vectors of a specific lattice For reference, Table 3.1 provides the expressions for the basis vectors of the simplest crystal lattices.
Close-packed structures consist of atoms bonded through short-range interactions, typically represented by identical hard spheres confined within rigid walls In these arrangements, each internal sphere is surrounded by 12 nearest neighbors, illustrating the efficient packing and spatial organization of the atoms within the structure.
Table 3.1 illustrates the basis vectors for the simplest lattice structures, where 'a' denotes the lattice constant The terms 'fcc' and 'bcc' refer to the face-centered cubic and body-centered cubic lattices, respectively, while 'hex' pertains to the hexagonal lattice.
The close-packed structures of crystal lattices include the face-centered cubic (fcc) and hexagonal structures, as illustrated in Figure 3.1 These structures are formed by arranging spheres in a compact manner on a plane, ensuring each sphere touches six neighboring spheres The subsequent layers are constructed by placing spheres in the gaps of the previous layer, with the third layer allowing for two configurations: hexagonal or fcc Notably, the projections of the spheres in the first and third layers overlap in the hexagonal structure, while they differ in the fcc structure, as depicted in Figure 3.2 Additionally, Table 3.1 presents the fcc structure using two coordinate systems, highlighting its high symmetry and periodicity, which are maintained through specific transformations.
Because of the high symmetry of the fcc structure, it occurs more often than the hexagonal one.
Figure 3.2.Hexagonal (a) and face-centered cubic (b) structures of hard balls.
Let us introduce theinverse lattice Kwith respect to the Bravais lattice such that the plane waveexp(i KR)takes identical values at atom positions, i.e.exp(i KR) = 1or
34 3 Bose–Einstein Distribution wheremis an integer Representing the inverse lattice vector in the form
K=k 1 b 1 +k 2 b 2 +k 3 b 3 (3.18) wherek 1 ,k 2 andk 3 are whole numbers, we have for the basis vectorsb 1 ,b 2 andb 3 of the inverse lattice b 1 = 2π [a 2 ∗ a 3 ] a 1 [a 2 ∗ a 3 ], b 2 = 2π [a 3 ∗ a 1 ] a 2 [a 3 ∗ a 1 ], b 3 = 2π [a 1 ∗ a 2 ] a 3 [a 1 ∗ a 2 ] (3.19) and we have from the above formulae
KR= 2π(n 1 k 1 +n 2 k 2 +n 3 k 3 ) (3.20) in accordance with formula (3.17).
The Wigner–Seitz cell is constructed around the origin of the inverse crystal lattice, ensuring that all points within the cell are closer to a test atom than to any other atoms This cell represents the first Brillouin zone of a specific lattice For the face-centered cubic (fcc) lattice, the Wigner–Seitz cell takes the shape of a regular truncated octahedron, characterized by its surface, which comprises six squares and eight regular hexagons, resulting in a total of thirty-six identical edges formed by the intersection of these geometric figures.
Figure 3.3.The basic Brillouin zone for the face-centered cubic crystal lattice.
Crystal planes are characterized by parameters known as Miller indices, which describe the direction of a vector perpendicular to the plane and originating from the origin These indices, represented as (m1, m2, m3), denote the minimal whole number components of the perpendicular vector b = m1i + m2j + m3k Importantly, the value of m1 is expressed as a positive integer, replacing any negative sign.
Elementary crystal cells and small crystalline particles exhibit specific symmetry in their surface planes For instance, a truncated octahedron, which features a face-centered cubic structure, exemplifies this symmetry characterized by (3.16) The surface of this geometric figure comprises six square faces and eight hexagonal faces, with the squares identified by the Miller indices (100), (010), (001), and (100).
(010),(001)(where1means−1) The sum of these directions is denoted as{100}, i.e it is accepted that squares have the directions{100} In the same manner, the regular hexagons
Structures of Clusters
The surface planes of the truncated octahedron align with the {111} direction It is important to note that all planes within this direction are interchanged through the transformations outlined in equation (3.16) for the face-centered cubic structure.
Solid clusters are defined as crystal systems composed of a finite number of bound atoms, where surface effects play a significant role, unlike in bulk crystal systems The binding energy of n bound atoms in an optimal configuration is denoted as E_n, while the binding energy of the nth atom is represented as ε_n, calculated using the formula ε(n) = E_n - E_(n-1).
The maximum binding energy is achieved when atoms form optimal configurations based on their interactions This energy is greater for completed geometric arrangements of atoms compared to configurations with fewer or more neighboring cluster atoms, as indicated by the relationships ε(n m ) > ε(n m −1) and ε(n m ) > ε(n m + 1).
Such numbers of cluster atoms are calledmagic numbers.
A cluster of 13 atoms serves as an example where nearest neighbor interactions are predominant Figure 3.4 illustrates the arrangement of these atoms, highlighting the significance of short-range interactions This cluster can adopt two distinct structures derived from face-centered cubic (fcc) or hexagonal crystal lattices.
36bonds between nearest neighbors of such clusters From Figure 3.4 and formula (3.21) it follows for this and neighboring clusters thatε(13) = 5Dandε(12) =ε(14) = 4D, where
Dis the binding energy per bond We have from this in accordance with formula (3.22) that n= 13is a magic number for cluster atoms.
Figure 3.4.Close-packed structures of a cluster consisting of
The structure consists of 13 atoms arranged in a basic plane oriented along the {111} direction, as indicated by Miller indices When the projections of atoms from the upper and lower layers onto this plane align, a hexahedron is created by connecting the centers of the nearest surface atoms, characterizing a hexagonal structure Conversely, if these projections differ, a cuboctahedron forms, representing the elementary cell of the face-centered cubic lattice.
We will assess the total binding energy \(E_f^{cc}\) for a cuboctahedral cluster consisting of 13 atoms, utilizing a pair interaction potential \(U(R)\) between the atoms By defining the distance \(a\) between the nearest neighbors in this cluster, we can analyze the results as illustrated in Figure 3.4.
The (3.24) model serves as an interaction potential for atoms, where R e represents the equilibrium distance in diatomic molecules and D signifies the dissociation energy of classical molecules This study calculates the total energy of atoms within a cuboctahedral cluster consisting of 13 atoms using this pair interaction potential.
From optimization of this expression we have the optimal distance between nearest neighbors a= 0.990R e and the cluster binding energyE f cc = 40.88D.
The binding energies of the fcc (cuboctahedral) and hexagonal cluster structures can be analyzed by examining a cluster of 13 atoms in a hexagonal arrangement The variation in binding energies arises from the interactions between atoms in the upper and lower layers Assuming that the distance between nearest neighbors is the same for both structures, the difference in atomic binding energies can be expressed in terms of the pair interaction potential U(R) of the atoms.
The binding energy of a hexagonal cluster of 13 atoms, represented by the formula 3a − 6U(2a), is influenced by the Lennard–Jones interaction potential This results in a binding energy difference of ∆E = 0.15D, indicating that the face-centered cubic (fcc) structure is favored However, the difference in binding energies is minimal, approximately 0.4%, suggesting that this variation can often be disregarded in practical applications.
Icosahedral structures, while exhibiting pair interactions among atoms, are not advantageous for bulk systems An example is shown in Figure 3.5, which depicts an icosahedral cluster made up of 13 atoms Figure 3.6 illustrates the geometric form of the icosahedron along with the locations of its vertices Unlike close-packed structures such as face-centered cubic (fcc) and hexagonal arrangements, the icosahedral structure lacks periodic symmetry and is distinguished by two distinct nearest-neighbor distances Specifically, the distance between nearest atoms within the same layer is greater than that between nearest atoms across neighboring layers.
The icosahedral cluster with the smallest magic number contains 13 atoms, featuring 12 bonds of length R1 connecting the central atom to the surface atoms, and 30 bonds of length R2 linking the nearest surface atoms.
In the case of pair interaction between cluster atoms, the total binding energy of atoms in the cluster is
Figure 3.5.The icosahedral cluster consisting of 13 atoms.
In particular, in the case of the Lennard–Jones interaction potential (3.24) between atoms, from optimization of this formula we geta= 0.964R e ,E ico = 44.34D.
In a study of a cluster comprising 13 atoms using the Lennard–Jones interaction potential, it was found that the icosahedral cluster structure is more favorable than close-packed configurations This preference arises from the higher number of nearest neighbors in the icosahedral structure (42) compared to 36 in face-centered cubic (fcc) or hexagonal clusters While each internal atom in all structures has 12 nearest neighbors, the differing distances between nearest neighbors in icosahedral clusters make them less advantageous for larger clusters or bulk systems However, for moderate-sized clusters, the icosahedral structure can compete effectively with close-packed arrangements, highlighting the greater variety of structures available in smaller clusters compared to bulk systems.
Figure 3.6.The icosahedron as a geometric figure – positions of its center and vertices: (a) side view;(b) top view; (c) developed view of a cylinder in which surface pentagons of the icosahedron are inscribed.
Vibrations of Nuclei in Crystals
In solids, atomic positions exhibit a specific symmetry, indicating a long-range order among the furthest atoms At low temperatures, the nuclei of condensed atomic systems vibrate around their equilibrium positions, with the nature of these vibrations influenced by internal interactions These interactions also determine the atomic structure within the crystal This article will focus on the vibrational behavior of nuclei in a crystal, specifically considering pair interactions between nearest atoms.
At absolute zero, the optimal distance between nearest atoms is determined by the minimum of the pair interaction potential The oscillation frequency (ω₀) of a diatomic molecule, influenced by this atomic interaction, serves as a key parameter for the vibrations of a system with multiple bound atoms In the classical framework, the relative motion of diatomic atoms is described by Newton's equation, leading to a solution of the form x = Csin(ω₀t + α), where ω₀ = √(k/m), which characterizes the classical vibrations of diatomic nuclei.
Now let us consider classical vibrations of nuclei for a system of many bound atoms. Newton’s equation for the vibration of theith nucleus has the form m r i +κ j
The equation \( R_{ij} - a_{ij} = 0 \) represents the distance between the \( i \)th and \( j \)th nuclei, where \( a_{ij} \) denotes their equilibrium positions, and \( r_i \) indicates the deviation of the \( i \)th atom from this equilibrium By considering these deviations to be small and defining \( x \) as the direction of vibrations, we can derive Newton's equations in the form \( m \ddot{x}_i + \kappa_j \).
In the context of nuclear physics, the deviations from equilibrium positions for nuclei i and j are represented as Herex i andx j The equilibrium distance between these nuclei can be expressed as a = i X ij + j Y ij + k Z ij, where i, j, and k are unit vectors aligned with the x, y, and z axes, respectively The components X ij, Y ij, and Z ij denote the projections of the vector a ij, which connects nuclei i and j, onto these axes.
In crystal physics, the vibrations of nuclei are viewed as waves that propagate within the crystal lattice, known as phonons By representing the wave parameters as \( x_i = C \exp(-i\omega t + i kR) \), we aim to establish the relationship between wave frequency \( \omega \) and wave vector \( k \), referred to as the dispersion relation By substituting this wave parameter into Newton’s equation, we derive the dispersion relation in the form \( m\omega^2 = \kappa j \).
3.6 Vibrations of Nuclei in Crystals 39
Below we consider long-wave vibrations, so that ka1 (3.26)
This criterion facilitates the expansion over a small parameter, revealing two types of waves The longitudinal wave propagates in the direction of the nuclei's vibrations, while the dispersion relation is expressed as ω² = 1, in accordance with the established criterion.
Assuming the crystal exhibits symmetry, the terms ∼ ika in equation (3.25) are mutually canceled out, a condition met in all examples discussed Similarly, for long transverse waves with the wave vector aligned along the y and z axes, the relationship is described by ω² = 1.
The dispersion relations for longitudinal and transverse vibrations can be expressed as ω = ck, similar to the relationship for photons In this context, the speeds of sound for longitudinal (cl) and transverse (ct) acoustic waves are defined, with the equation c²l = 1 illustrating the fundamental properties of these wave types.
In a one-dimensional crystal structure, wave propagation is analyzed with each atom having two nearest neighbors, where the interaction between them is represented by \( X_{ij} = a \) Consequently, the relationship derived from the formula \( c_l = \frac{\kappa m a}{\omega_0} \) illustrates the connection between wave speed, spring constant, mass, and atomic spacing.
In a cubic crystal lattice, when the cube's z-axis forms an angle θ with the vibration direction, and vibrations occur in the plane defined by the z and x axes, the speeds of longitudinal and transverse waves can be expressed using specific formulas The longitudinal wave speed is given by \( c_l^2 = \omega_o^2 a^2 (\cos^4 \theta + \sin^4 \theta) \), while the transverse wave speeds are represented as \( c_{1t}^2 = 2\omega_o^2 a^2 \sin^2 \theta \cos^2 \theta \) and \( c_{2t}^2 = \omega_o^2 a^2 \cos^2 \theta \) This indicates that the longitudinal speed of sound varies with the angle θ, demonstrating the relationship between wave propagation and crystal structure.
2up toω o /2, and the transverse speed of sound varies from zero up toω o /√
In a face-centered crystal structure with short-range atomic interactions limited to nearest neighbors, the centers of the 12 nearest neighbors of a test atom, positioned at the origin, are found on a sphere with a radius of 'a' By assuming a random distribution of these nearest neighbors on the sphere and averaging their positions, we derive the longitudinal and transverse speeds of sound, represented by the formula c²ₗ = 3ω₀²a².
In this analysis, we adopt a reference frame with axes aligned to the {100} direction in Miller indices notation The coordinates of the 12 nearest neighbors of the test atom are identified as (0, ±1, ±1), (±1, 0, ±1), and (±1, ±1, 0), with all coordinates measured in units of a/√.
2 Taking thex,y andzaxes as the directions of the vibrations and the propagation of waves, we obtain for the sound velocity c 2 l =1
In the Miller indices notation, the {111} plane serves as the reference frame, where the 12 nearest neighbors of a test atom are positioned as illustrated in Figure 3.4 The test atom, located at the center of the basis layer, is surrounded by six nearest neighbors that form a regular hexagon Additionally, three atoms from both the lower and upper layers occupy the hollows of the basis layer atoms There are two configurations for the positioning of these atoms: if the projections of the lower and upper layer atoms align on the basis plane, they collectively create the elementary cell of a hexagonal lattice, resulting in a hexahedron shape Conversely, if the projections do not align, the arrangement corresponds to the elementary cell of a face-centered cubic lattice, leading to the formation of a cuboctahedron when the centers of the nearest surface atoms are connected.
In a specific reference frame, a test atom is positioned at the origin, with two axes aligned in the basic {111} plane, one of which intersects an atom The vibrations and wave propagation are oriented along these axes According to the established formulae, the sound speeds are determined based on the vibration directions, yielding the relationship: \( c^2_l = \omega^2_0 a^2 = 5 \).
Sound velocities vary based on the directions of wave vibration and propagation, yet they tend to cluster within a narrow range By considering the average values of these speeds, we can gain a clearer understanding of their behavior.