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Line shape of magnetic excitations in singlet-ground-state systems
1166 L I N E SHAPE OF M A G N E T I C E X C I T A T I O N S IN S I N G L E T - G R O U N D - S T A T E SYSTEMS* PER BAKt Brookhaven National Laboratory, Upton, N Y 11973, USA The excitation spectrum in a paramagnetic singlet doublet system is calculated using a diagrammatic expansion technique, and the theoretical predictions are compared with experiments on praseodymium. The theory gives an accurate description of the dramatic temperature dependence of the energies and lineshapes for the exciton modes.
For the last several years there has been a great interest in the properties of localized magnetic s y s t e m s which possess a nonmagnetic singlet ground state. The ordering in such s y s t e m s occurs as an exchange polarization of the ground state, provided the exchange interaction between the magnetic ions exceeds a certain threshold value. H e r e we shall present a calculation of the paramagnetic excitation s p e c t r u m in a singlet doublet system using a diagrammatic G r e e n ' s function expansion method. A similar technique has been applied to a f e r r o m a g n e t by Vaks, Larkin and Pikin [1], and to the Ising model in a t r a n s v e r s e field by S t i n c h c o m b e [2]. In contrast to other techniques, based on various decouplings of equations of motion for spin operators, this kind of theory gives a well defined high-density expansion p a r a m e t e r ( I / Z ) where Z is an effective n u m b e r of interacting neighbors. The approximations are based on physical, not technical reasons. P r a s e o d y m i u m is an e x a m p l e of a singlet ground state magnet in which the exchange is only slightly undercritical with respect to magnetic ordering. The ground state on the hexagonal sites is the pure IJz= 0) singlet, and the first excited state is the doublet Ix), lY), where Ix) = I / V ~ (11>÷1-1)) and lY)---i/x~2 (11)-I-1)). The paramagnetic excitation spectrum has recently been m e a s u r e d by H o u m a n n et al. [3] using inelastic neutron scattering technique. The lowest lying m o d e shows a clear tendency towards softening as the t e m p e r a t u r e is lowered towards 0 K, and a dramatic increase of the intrinsic linewidth takes place as the t e m p e r a t u r e increases f r o m 6 K to 30 K, where well-defined m o d e s cease to exist, dhcp * Work supported by Energy Research and Development Administration. t Present address: NORDITA, Blegdamsvej 17, 2100 Copenhagen ¢, Denmark.
Physica 86-88B (1977) 1166--1167 © North-Holland
p r a s e o d y m i u m s e e m s to be the simplest real singlet-ground-state magnet, and significantly m o r e information is now available on the excitations in this material than in any other paramagnetic system. This m a k e s the element P r almost ideal for a confrontation between e x p e r i m e n t and theoretical model calculations. The effective Hamiltonian describing the magnetic ions on the hexagonal sites in P r may be written :
A(S, i
2-
_,,
: x, y,
ij,aB
A is the crystal field splitting (= 3.2 meV). We introduce the G r e e n ' s functions
Ga~(rl, ~'l; r2, r2) = (T,S~(rl, zOS~(r2, T2)), where ( T , . . . ) denotes the thermal average of the T ordered product of operators in the interaction representation. The G r e e n ' s function can be represented by a sum of connected diagrams consisting of singlecell blocks joined by interaction lines, J(q). F o r a detailed description of the technique, see ref. 4. The zeroth order diagrams (RPA) simply consist of noninteracting G r e e n ' s function connected with interaction lines. The poles of the R P A - p r o p a g a tors, which give the energy s p e c t r u m are (to~) 2 = a 2 _ 4AR~ N (q), where R = {1 - exp ( -/3A)} {1 + 2 exp ( -/3A)} -~ and oCN(q) are eigenvalues of a 4 x 4 matrix describing the exchange interactions between the ions. The m e a s u r e d dispersion relations were fitted to this expression and interatomic exchange p a r a m e t e r s were derived. Fig. 1 shows the polarization of two of the four m o d e s as a function of q in the F - K - M direction. The polarizations change rapidly as a function of q:
1167 q
0
0.5
0.6
0.8
0.9
1.0
I.I
1.2
1.5
c,l @OooooO@
1.5
tion are c o m p a r e d with e x p e r i m e n t in fig. 2. Almost complete a g r e e m e n t b e t w e e n positions of peaks, intensities and lineshapes is o b s e r v e d at any temperature. The a g r e e m e n t is least perfect at the highest temperatures, where the linewidth, as calculated to first order in the ( l / Z ) expansion, is c o m p a r a b l e to the energy as cal-
1.7
@1
Fig. 1. Polarization of exciton modes in praseodymium as a
function of wavevector, q, in the FKM direction. I
I
I
I
I
I
I
I
I
I
I
I
I
I
i
I
-410 T=GK
z
T=IIK
T=ISK
T=20K
T=25K
T=50K
~ . ~o,o,o)
zoo
g o
L_ I 2
• II 51
I 2
I 3
I I
I 2
. [ 5
[ I ENERGY
I 2 (meV)
I 3
.. [ [
I 2
I 5
. I I
2
.: 5
4
Fig. 2. Spectral functions for magnetic excitations in Pr. Points: Neutron measurements (ref. 5); lines: theoretical
curves folded with experimental resolution. at the zone center the polarization is linear in the x or y direction, but in general the m o d e s are elliptically polarized in the xy plane. The eigenvectors of the m o d e s can be m e a s u r e d by neutron scattering technique. The first order diagrams give rise to small shifts of the excitation energies and, m o r e interesting, b y extending the theory one order b e y o n d the R P A we obtain the leading contributions to the lifetimes of the excitations. It is interesting that the damping occurs to first order in the expansion. F o r boson or f e r m i o n systems, the lowest order imaginary part of the selfenergy occurs in second order in the high density expansion due to interaction b e t w e e n excitations. The spectral functions, which are proportional to the neutron scattering crosssection, can easily be e x p r e s s e d in terms of the G r e e n ' s functions using the fluctuation-dissipation theorem. The theoretical lineshapes convoluted with the experimental resolution func-
culated to zeroth order. In addition to the diagrams considered here, there are also diagrams which, to first order in 1/Z, describes quasielastic contributions to S(q, to). Recently, a w e a k central p e a k has been o b s e r v e d in P r at low t e m p e r a t u r e s [6]. It would be interesting to calculate this effect within the present formalism and c o m p a r e with experiment. Re|erences [1] V.G. Vaks, A.I. Larkin and S.A. Pikin, Sov. Phys. JETP
26 (1968) 188, 647. [2] R.B. Stinchcombe, J. Phys. C6 (1973) 2484. [3] J.G. Houmann, M. Chappelier, A.R. Mackintosh, P. Bak, O.D. McMasters and K.A. Gschneidner, Jr., Phys. Rev. Lett. 34 (1975) 587. 14] Per Bak, Phys. Rev. Lett. 34 (1975) 1230; Phys. Rev. BI2 (1975) 5203. [5] J.G. Houmann and A.R. Mackintosh, private communication. [6] J.G. Houmann, B. Lebech, A.R. Mackintosh, W.J.L. Buyers, O.D. McMasters and K.A. Gschneidner, Jr., this conference.
For the last several years there has been a great interest in the properties of localized magnetic s y s t e m s which possess a nonmagnetic singlet ground state. The ordering in such s y s t e m s occurs as an exchange polarization of the ground state, provided the exchange interaction between the magnetic ions exceeds a certain threshold value. H e r e we shall present a calculation of the paramagnetic excitation s p e c t r u m in a singlet doublet system using a diagrammatic G r e e n ' s function expansion method. A similar technique has been applied to a f e r r o m a g n e t by Vaks, Larkin and Pikin [1], and to the Ising model in a t r a n s v e r s e field by S t i n c h c o m b e [2]. In contrast to other techniques, based on various decouplings of equations of motion for spin operators, this kind of theory gives a well defined high-density expansion p a r a m e t e r ( I / Z ) where Z is an effective n u m b e r of interacting neighbors. The approximations are based on physical, not technical reasons. P r a s e o d y m i u m is an e x a m p l e of a singlet ground state magnet in which the exchange is only slightly undercritical with respect to magnetic ordering. The ground state on the hexagonal sites is the pure IJz= 0) singlet, and the first excited state is the doublet Ix), lY), where Ix) = I / V ~ (11>÷1-1)) and lY)---i/x~2 (11)-I-1)). The paramagnetic excitation spectrum has recently been m e a s u r e d by H o u m a n n et al. [3] using inelastic neutron scattering technique. The lowest lying m o d e shows a clear tendency towards softening as the t e m p e r a t u r e is lowered towards 0 K, and a dramatic increase of the intrinsic linewidth takes place as the t e m p e r a t u r e increases f r o m 6 K to 30 K, where well-defined m o d e s cease to exist, dhcp * Work supported by Energy Research and Development Administration. t Present address: NORDITA, Blegdamsvej 17, 2100 Copenhagen ¢, Denmark.
Physica 86-88B (1977) 1166--1167 © North-Holland
p r a s e o d y m i u m s e e m s to be the simplest real singlet-ground-state magnet, and significantly m o r e information is now available on the excitations in this material than in any other paramagnetic system. This m a k e s the element P r almost ideal for a confrontation between e x p e r i m e n t and theoretical model calculations. The effective Hamiltonian describing the magnetic ions on the hexagonal sites in P r may be written :
A(S, i
2-
_,,
: x, y,
ij,aB
A is the crystal field splitting (= 3.2 meV). We introduce the G r e e n ' s functions
Ga~(rl, ~'l; r2, r2) = (T,S~(rl, zOS~(r2, T2)), where ( T , . . . ) denotes the thermal average of the T ordered product of operators in the interaction representation. The G r e e n ' s function can be represented by a sum of connected diagrams consisting of singlecell blocks joined by interaction lines, J(q). F o r a detailed description of the technique, see ref. 4. The zeroth order diagrams (RPA) simply consist of noninteracting G r e e n ' s function connected with interaction lines. The poles of the R P A - p r o p a g a tors, which give the energy s p e c t r u m are (to~) 2 = a 2 _ 4AR~ N (q), where R = {1 - exp ( -/3A)} {1 + 2 exp ( -/3A)} -~ and oCN(q) are eigenvalues of a 4 x 4 matrix describing the exchange interactions between the ions. The m e a s u r e d dispersion relations were fitted to this expression and interatomic exchange p a r a m e t e r s were derived. Fig. 1 shows the polarization of two of the four m o d e s as a function of q in the F - K - M direction. The polarizations change rapidly as a function of q:
1167 q
0
0.5
0.6
0.8
0.9
1.0
I.I
1.2
1.5
c,l @OooooO@
1.5
tion are c o m p a r e d with e x p e r i m e n t in fig. 2. Almost complete a g r e e m e n t b e t w e e n positions of peaks, intensities and lineshapes is o b s e r v e d at any temperature. The a g r e e m e n t is least perfect at the highest temperatures, where the linewidth, as calculated to first order in the ( l / Z ) expansion, is c o m p a r a b l e to the energy as cal-
1.7
@1
Fig. 1. Polarization of exciton modes in praseodymium as a
function of wavevector, q, in the FKM direction. I
I
I
I
I
I
I
I
I
I
I
I
I
I
i
I
-410 T=GK
z
T=IIK
T=ISK
T=20K
T=25K
T=50K
~ . ~o,o,o)
zoo
g o
L_ I 2
• II 51
I 2
I 3
I I
I 2
. [ 5
[ I ENERGY
I 2 (meV)
I 3
.. [ [
I 2
I 5
. I I
2
.: 5
4
Fig. 2. Spectral functions for magnetic excitations in Pr. Points: Neutron measurements (ref. 5); lines: theoretical
curves folded with experimental resolution. at the zone center the polarization is linear in the x or y direction, but in general the m o d e s are elliptically polarized in the xy plane. The eigenvectors of the m o d e s can be m e a s u r e d by neutron scattering technique. The first order diagrams give rise to small shifts of the excitation energies and, m o r e interesting, b y extending the theory one order b e y o n d the R P A we obtain the leading contributions to the lifetimes of the excitations. It is interesting that the damping occurs to first order in the expansion. F o r boson or f e r m i o n systems, the lowest order imaginary part of the selfenergy occurs in second order in the high density expansion due to interaction b e t w e e n excitations. The spectral functions, which are proportional to the neutron scattering crosssection, can easily be e x p r e s s e d in terms of the G r e e n ' s functions using the fluctuation-dissipation theorem. The theoretical lineshapes convoluted with the experimental resolution func-
culated to zeroth order. In addition to the diagrams considered here, there are also diagrams which, to first order in 1/Z, describes quasielastic contributions to S(q, to). Recently, a w e a k central p e a k has been o b s e r v e d in P r at low t e m p e r a t u r e s [6]. It would be interesting to calculate this effect within the present formalism and c o m p a r e with experiment. Re|erences [1] V.G. Vaks, A.I. Larkin and S.A. Pikin, Sov. Phys. JETP
26 (1968) 188, 647. [2] R.B. Stinchcombe, J. Phys. C6 (1973) 2484. [3] J.G. Houmann, M. Chappelier, A.R. Mackintosh, P. Bak, O.D. McMasters and K.A. Gschneidner, Jr., Phys. Rev. Lett. 34 (1975) 587. 14] Per Bak, Phys. Rev. Lett. 34 (1975) 1230; Phys. Rev. BI2 (1975) 5203. [5] J.G. Houmann and A.R. Mackintosh, private communication. [6] J.G. Houmann, B. Lebech, A.R. Mackintosh, W.J.L. Buyers, O.D. McMasters and K.A. Gschneidner, Jr., this conference.