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Magnetic excitations 267 Fig. 9. Double logarithm plot of the linewidth (FWHM) of the spin wave excitations in EuS versus wave vector at T =14.1 K. The solid line is a guide to the eye. The straight parts of the curve indicate a slope proportional to q 3 or q 4 . The change in slope occurs near q ≈0.7Å −1 ,where ¯ hω ≈ k B T (from Bohn et al. [19]). Fig. 10. Linewidth (FWHM) at T = T C of EuS as a function of q in a double logarithmic scale. The straight line represents the best fit of the data giving a dynamical scaling exponent of z = 2.09 ±0.06 (from Bohn et al. [19]). 268 T. Chatterji Fig. 11. Double logarithm plot of the linewidth versus q of EuO at T C . The prediction of the dynamical scaling theory (straight line) is obeyed in d decades in energy (from Böni et al. [23]). 3.4. Spin waves in Heisenberg antiferromagnets We already noted that there are only a few insulating Heisenberg ferromagnets. But there exist several insulating ionic antiferromagnets that are well described by a Heisenberg Hamiltonian with the addition of some single-ion anisotropy terms. It is convenient to start with a more general Hamiltonian that can describe ferrimagnets and as well as antiferro- Magnetic excitations 269 magnets: H = m,r J(r)S m ·S m+r + n,r J(r)S n ·S n+r + m,R J 1 (R)S m ·S m+R + n,R J 2 (R)S n ·S n+R −g 1 µ B (H +H A,1 ) m S z m −g 2 µ B (H −H A,2 ) n S z n , (48) where m and n are lattice vectors connecting the sites of the two interpenetrating sublat- tices, the up sublattice (lattice vector m) and the down sublattice (lattice vector n). R con- nects sites on the same sublattice and r connects sites on the opposite sublattices. The spins on the m sublattice have magnitude S 1 and those on n magnitude S 2 . J 1 and J 2 are exchange parameters within each sublattice. An uniaxial anisotropy is incorporated by the effective magnetic fields, H A,1 and H A,2 and g 1 and g 2 are the gyromagnetic ratios of the two types of ions. Following a similar procedure as indicated in the case of the spin waves in Heisenberg ferromagnets, one arrives at the spin wave dispersion equations 2 ¯ hω q,0 =(a 1 +a 2 +b 1 −b 2 ) +2Ω(q), (49) 2 ¯ hω q,1 =−(a 1 +a 2 +b 1 −b 2 ) +2Ω(q), (50) where 2Ω(q) = (a 1 −a 2 +b 1 +b 2 ) 2 −S 1 S 2 4J (q) 2 1/2 , (51) a 1 =g 1 µ B H, (52) a 2 =g 2 µ B H, (53) b 1 =2S 1 σ J (0) −2S 1 J 1 (0) −J 1 (q) +g 1 µ B H A,1 , (54) b 2 =2S 1 J (0) −2S 1 σ J 2 (0) −J 2 (q) +g 2 µ B H A,2 . (55) The above equations mean that the Hamiltonian (48) has two linear spin wave modes given by (49) and (50) which are in general not degenerate. So far the equations are quite general in that we have assumed that the magnetic ions in the two sublattices are not the same. For an antiferromagnet, we have g 1 =g 2 =g, H A,1 =H A,2 =H A , S 1 =S 2 =S and J 1 = J 2 =J . Therefore, a 1 =a 2 =gµ B H, (56) b 1 =b 2 =2SJ (0) −2S J (0) −J (q) +gµ B H A =b. (57) So we have the dispersion equation ¯ hω q,a =(−1) a gµ B H +Ω(q), a = 0, 1, (58) Ω(q) ≡ b 2 − 2SJ (q) 2 1/2 . (59) 270 T. Chatterji For cubic crystals and for small values of q J (q) = r J(r) exp(−iq ·r) J (0) − 1 6 q 2 r r 2 J(r) (60) and defining J (2) = r r 2 J(r) (61) we have J (0) −J (q) 1 6 q 2 J (2) , (62) and similarly J (0) −J (q) 1 6 q 2 J (2) . (63) Although the above two equations (62) and (63) hold strictly only for cubic crystals at small q, it is often a good approximation for other cases. So using (62) and (63) in (58) we get ¯ hω q,a = gµ B H A 4SJ (0) +gµ B H A + 1 3 q 2 S 4SJ (0) J (2) −J (2) −2gµ B H A J (2) 1/2 +(−1) a gµ B H, (64) which simplifies to ¯ hω q,a =q 4s 2 3 J (0) J (2) −J (2) 1/2 +(−1) a gµ B H, (65) when H A =0. For antiferromagnets for which only the nearest-neighbor exchange coupling J is dom- inant, one gets from (58) with H = 0 the degenerate spin wave dispersion ¯ hω q,a =2rJS (1 +h A ) 2 −γ q 1/2 =2rJSε q , (66) where a reduced anisotropy field h A = gµ B H A /2rJS is defined. One notes that the anisotropy field H A produces a gap in the spin wave spectrum at q =0. The spin wave dispersion of several insulating Heisenberg antiferromagnets have been investigated by inelastic neutron scattering. The classic example is MnF 2 and therefore Magnetic excitations 271 Fig. 12. Schematic illustration of the crystal and magnetic structures of MnF 2 . we will discuss this in some details. MnF 2 crystallizes with tetragonal rutile type structure (space group D 4h 14 ,P4 2 /mnm) with lattice parameters a = 4.873,c= 3.130 Å. There are 2 formula units in the unit cell and the atomic coordinates are give by: Mn: (2a) 0, 0, 0; 1 2 , 1 2 , 1 2 ,F:(4f ) x,x,0; ¯x, ¯x,0; 1 2 + x, 1 2 − x, 1 2 ; 1 2 − x, 1 2 + x, 1 2 . The only reflection condition is given by h0l: h + l =2n. So nuclear reflections are absent for h0l: h +l = 2n +1. The shortest (nearest-neighbor) Mn–Mn distance is along 001, with next nearest neighbors along 111 and third nearest neighbors along 100 and 010.MnF 2 orders at T N ≈67.5 K to an antiferromagnetic phase. The magnetic moment of all corner Mn atoms ((2a) 0, 0, 0) in the unit cell is parallel to 001 and that of Mn atom at the body centered position ((2a) 1 2 , 1 2 , 1 2 ) is oppositely oriented. The propagation vector of this magnetic structure is k = 0. The reflection condition of the magnetic reflections is h+k +l = 2n+1. So in general magnetic reflections in neutron diffraction are superimposed on the structural reflections. However, h0l reflections with h +l = 2n +1 have no structural contributions and are purely magnetic. Figure 12 shows schematically the crystal and magnetic structures of MnF 2 . To measure spin wave dispersions along the two principal symmetry directions [100] and [001] in MnF 2 it is convenient to orient the MnF 2 single crystal such that the scatter- ing plane contains the (h0l) zone of the reciprocal space shown in Figure 13. This zone contains pure magnetic reflections (h + l = 2n + 1) and also some pure nuclear reflec- tions (h + l = 2n). The large open circles show the positions of the nuclear reflections whereas the large filled circles show the positions of the magnetic reflections. They are also indicated by the suffixes N and M. The dashed lines show the antiferromagnetic zone boundaries. The small circles and squares indicate positions for well-focused constant-Q scans of spin waves propagating along [100] and [001] directions. Figure 14 shows such constant-Q scans from MnF 2 and Figure 15 shows dispersions of spin waves in MnF 2 along [100]and [001] directions. Let us now discuss the exchange couplings in MnF 2 .The strongest exchange interaction in MnF 2 is between the next nearest-neighbor Mn atoms sit- 272 T. Chatterji Fig. 13. (h0l) zone of the reciprocal lattice of MnF 2 . The large solid circles represent nuclear reflections, whereas the large open circles represent magnetic reflections. Small circles and squares represent positions for well-focused constant-Q scans of spin waves propagating along the [100]and [001] directions, respectively (from Shirane et al. [2]). Fig. 14. Constant-Q scans of spin waves of MnF 2 at 10 K (from Shirane et al. [2]). uated at (000) and ( 1 2 , 1 2 , 1 2 ) positions. They belong to two different magnetic sublattices. This strong interaction, denoted by J 2 , is the antiferromagnetic superexchange interaction via the fluorine ligands. The nearest-neighbor Mn atoms along [001] are coupled by the ferromagnetic exchange interaction J 1 , which is by about a factor five less than the next nearest-neighbor superexchange interaction J 2 shown from the inelastic neutron scattering investigation of the spin wave dispersion of MnF 2 by Okazaki et al. [24]. The same study Magnetic excitations 273 Fig. 15. (Top) Spin wave dispersion of MnF 2 along the [100] and [001]directions obtained from scans like those shown in Figure 14. The solid lines are fits to (67). (Bottom) Corresponding integrated intensities corrected for the magnetic form factor and also the Q-dependent factor involved in the scan process (from Shirane et al. [2]). also showed that the interaction between the third nearest-neighbor Mn atoms along [100] has insignificant contribution to the spin wave dispersion. The zone center spin wave gap of the dispersion clearly shows the existence of substantial single-ion anisotropy energy D, which can be explained by dipole–dipole interactions. The spin wave dispersion [24] re- lation for MnF 2 corresponding to the Heisenberg Hamiltonian including a dipolar term is given by ¯ hω q = ¯ hω 2 (1 +ζ q ) 2 −γ 2 q , (67) where ζ q = D +2 ¯ hω 1 sin 2 q z c 2 ¯ hω 2 , (68) γ q =cos q x a 2 cos q y a 2 cos q z a 2 (69) and ¯ hω i =2Sz i J i . (70) 274 T. Chatterji Here a and c are the tetragonal lattice parameters of MnF 2 , z 1 =2 is the number of nearest neighbors and z 2 =8 is the number of second nearest-neighbors. If the dipolar term and the nearest-neighbor ferromagnetic exchange interaction are neglected, then by setting D =0 and J 1 =0 we get ζ q =0. The dispersion equation then reduces to ¯ hω q = ¯ hω 2 1 −γ 2 q . (71) For small q we get ¯ hω q ≈ qa 2 √ 2 (72) ignoring the tetragonal distortion of the lattice, i.e., assuming a = c. So for small q a linear dispersion is obtained. This is typical for the spin wave dispersion of antiferro- magnets as opposed to the quadratic dispersion in ferromagnets in the same limit. The anisotropy energy D introduces an energy gap in the spin wave spectrum, while a finite value of the nearest-neighbor interaction J 1 causes the dispersion to be different along the [100] and [001] directions. 3.5. Two-magnon interaction in Heisenberg antiferromagnets The effects of temperature on the spin wave dispersion of Heisenberg antiferromagnets can be evaluated by extending the method used in Heisenberg ferromagnets. For simplicity one considers the case of the nearest-neighbor exchange coupling. We recall that for non- interacting spin waves in a nearest-neighbor Heisenberg model the dispersion equation can be written as ¯ hω q =2rJS (1 +h A ) 2 −γ 2 q ≡2rJSε q . (73) Considering two-magnon interactions it can be shown [25] that the spin wave dispersion relation is modified to ¯ hω q (T ) =2rJS 1 −C(T ) 1 +h A (T ) 2 −γ 2 q , (74) where C(T) is given by the integral equation C(T ) = 1 2NS q {1 −C(T )}(1 −γ 2 q ) +h A ε q (T ) coth ¯ hω q (T ) 2k B T −1. (75) The two-magnon interaction also renormalizes the spin wave dispersion of Heisenberg antiferromagnets like in the case of Heisenberg ferromagnets. But in the case of Heisenberg antiferromagnets there is the additional feature of renormalizing the anisotropy field. To calculate the temperature dependence of the dispersion given by (74) one has to solve a Magnetic excitations 275 complicated integral equation (75) by numerical methods. However one can simplify the situation by assuming h A =0 in (75). C(T ) is then given by C(T ) = 1 2NS q 1 −γ 2 q −1 + 1 NS q 1 −γ 2 q 1 exp{ ¯ hω q (T )/(2k B T)}−1 . (76) The dispersion relation is also simplified to ¯ hω q (T ) =2rJS 1 −C(T ) 1 −γ 2 q . (77) For small values of q we can write ¯ hω q (T ) =D(T )q, (78) where D(T ) =D 1 −C(T ) (79) and D is given by ε q =q ρ 2 3 1/2 = Dq 2rJS . (80) Low [26] has made a detailed test for the validity of the renormalization of the spin wave dispersion in MnF 2 . Figure 16 shows the measured dispersions of MnF 2 [24] at T =4.2, 49.5 and 62.0 K along with the calculations [26] based on the renormalized spin wave theory outlined above. The two sets of curves for T =49.5 and 62.0 K correspond to calculations with the renormalization factor in ¯ hω q multiplied by that indicated in the figure. 3.6. Spin waves in Heisenberg ferrimagnets The ferrimagnetic materials consist of two magnetic sublattices occupied by two different types of ions having unequal moments oriented in the opposite directions. They have like ferromagnets net magnetization for a particular domain. Equations (48)–(50) are quite gen- eral and are valid for two sublattices which are not in general equal. So they are valid for ferrimagnets. These equations mean that the Hamiltonian given by (48) leads to two linear spin wave modes with energy ¯ hω q,0 and ¯ hω q,1 . These two modes are not equal in the case of ferrimagnets with two unequal sublattices. A ferrimagnet can possess under certain con- ditions, thermodynamic properties similar to those of a ferromagnet. To illustrate this we 276 T. Chatterji Fig. 16. The measured dispersions of MnF 2 [24] at T = 4.2, 49.5 and 62.0 K along with the calculations [26] based on the renormalized spin wave theory outlined in above. The two sets of curves for T = 49.5 and 62.0 K correspond to calculations with the renormalization factor in ¯ hω q multiplied by that indicated in the figure (from Low [26]). set H A,1 = H A,2 = 0,H = 0 and also J 1 = J 2 = 0. The dispersion equation for the two spin wave modes is then reduced to ¯ hω q,a =(−1) a J (0)(S 2 −S 1 ) + J (0)(S 1 +S 2 ) 2 −4S 1 S 2 J 2 (q). (81) Writing J (0) −J (q) 1 6 q 2 J (2) , (82) we have then J 2 (q) J 2 (0) − 1 3 q 2 J (0)J (2) . (83) By using (81)–(83) the dispersion equation can be written as ¯ hω q,a −J (0)(S 2 −S 1 )(−1) a + J (0)(S 1 −S 2 ) 2 − 16 3 S 1 S 2 q 2 J (0)J (2) J (0)(S 1 −S 2 ) 1 +(−1) a+1 +q 2 (2/3)S 1 S 2 J (2) (S 1 −S 2 ) . (84) Considering the acoustic mode for which a = 0 we get ¯ hω q,0 2 3 S 1 S 2 J (2) S 1 −S 2 q 2 . (85) [...]... 290 T Chatterji Fig 24 Calculated spin wave dispersion of Fe for q along [100 ] (Blackman et al [42]) Fig 25 Contours of the scattered neutron intensities from magnetic excitations in Fe for q along [100 ] (from Blackman et al [42]) Magnetic excitations 291 Fig 26 Contours of the calculated scattered neutron intensities from magnetic excitations in Fe for q along [111] (Blackman et al [42]) tut Laue–Langevin... expects to observe in a neutron scattering experiment from at least a partly itinerant magnetic electron system As shown schematically in Figure 18, a spin wave mode starts at hω = 0 and q = 0 and increases in energy with the increase ¯ of momentum transfer to meet finally the single particle Stoner excitations shown by the + shaded area The single-particle excitations result from the term Im χk,0 discussed... ferromagnetic transition metal like Fe The ferromagnetic transition temperature TC = 631 K of Ni is however much lower than that (TC = 104 3 K) of iron The crystal 292 T Chatterji Fig 27 Experimental scattered intensity contours along [100 ] from Fe (from Paul et al [44]) Fig 28 Experimental scattered intensity contours along [111] from Fe (from Paul et al [44]) Magnetic excitations 293 structure of Ni at... The magnetic moment of b.c.c iron is about 2.2 µB and is reasonably large for measuring spin dynamics of Fe However the magnetic form factor decreases rapidly with increasing Q This makes only the first two zone centers, ( 110) and (002), and nearby regions suitable for the measurements of magnons in Fe The ferromagnetic transition temperature of b.c.c iron is TC = 104 3 K A detailed inelastic neutron scattering. .. term + − Im{χQ (ω) + χ−Q (ω)} gives the transverse part To determine the neutron scattering crosssection we need to determine the imaginary part of the susceptibility given by + Im χk (ω) = + Im χk,0 (ω) + + {1 − [I /(gµB )2 ] Re χk,0 (ω)}2 + {[I /(gµB )2 ] Im χk,0 (ω)}2 (102 ) For the small q case setting the excitation energy to hω and using (101 ) and (102 ) one ¯ gets e2 d 2σ = γ dΩ dE mc2 × 2 F (Q)... susceptibilities To compare the results of theoretical calculations with those of neutron scattering experiments it is therefore necessary to express the neutron scattering crosssection in terms of generalized susceptibilities We note that the spin correlation function Magnetic excitations 283 used to calculate the neutron scattering cross-section is related to the generalized susceptibilities through... arises strictly due to the band effect The results of neutron scattering investigations by Mook and Paul [51] shown in Figure 32 agree qualitatively with these calculations The experiment was done on a highquality 400 g 60 Ni single crystal which gave scattering intensities free from incoherent scattering and highly reduced inelastic scattering from phonons The experiment was performed on the triple-axis... the spin wave scattering in the energy range 40–160 meV decreases monotonically with increasing energy Such a decrease in inten- Magnetic excitations 289 Fig 23 Spin wave dispersion of pure iron at T = 10 K The continuous curve shows the result of fitting the dispersion relation E = D|q|2 (1 − β|q|2 ) with D = 307 ± 15 meV Å2 and β = 0.32 ± 0 .10 Å2 (from Loong et al [41]) sity is expected from itinerant... susceptibility They were chosen so that the calculated ferromagnetic band structure gives the correct magnetic moment and the correct t2g and eg character of the moment as observed in the neutron form factor measurements No other adjustable parameters were used in the calculations Cooke et al [50] have calculated the neutron scattering cross-section from Ni for energy transfers up to 1 eV Figure 31(a) shows... relation for Ni at several temperatures The solid curves are the fit to (105 ) Above the transition temperature there is no shift in the position of scattering as seen by constant-E scans (from Lynn and Mook [48]) Fig 30 Integrated intensity of spin wave scattering as a function of energy at different temperatures (from Lynn and Mook [48]) Magnetic excitations 295 Fig 31 (a) Calculated spin wave dispersion . moment per atom. Equation (103 ) gives us an idea of the spin excitations one expects to observe in a neutron scattering experiment from at least a partly itinerant magnetic electron system. As. +χ − −Q (ω) . (101 ) The term connected with Imχ zz (ω) gives the longitudinal part, whereas the term Im{χ + Q (ω) +χ − −Q (ω)} gives the transverse part. To determine the neutron scattering cross- section. ground state properties of transition elements. Magnetic excitations 281 4.1. Generalized susceptibility and neutron scattering cross-section Neutron scattering cross-sections given in the previous
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