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Paramagnetic fluctuations in cubic Van-Vleck pnictides
1158
PARAMAGNETIC FLUCTUATIONS IN CUBIC VAN-VLECK PNICTIDES*
V. ZEVIN, D. DAVIDOV, R. L E V I N and D. S H A L T I E L Racah Institute o[ Physics, The Hebrew University of Jerusalem, Israel The lineshape of the paramagnetic fluctuation spectra in cubic Van-Vleck compounds is discussed in conjunction with the relaxation of an impurity or that of the host nucleus in these systems.
The present work discusses the fluctuations in single ground state systems with emphasis on relaxation of impurity [1, 2] and the host nuclei [3]. The fluctuation spectra of the host paramagnetic ions in singlet ground state systems with cubic structure can be described by the spectral functions Kpp(to) (p = x , y,z). These spectral functions are expressed as the sum Kpp(to) = Zk,jKpp(to)~J where Kpp(to)kJ is the Fourier transform of the time correlation function (Jtpk)(t)J~p)) and ~k) is the p component of the angular momentum of the kth host magnetic ion. For the case of ESR (NMR), the indices k and j run over several RE ions surrounding the impurity (or host nucleus). It has been demonstrated that while the N M R relaxation is determined by the to = 0 component of Kpp(to), the ESR relaxation is dominated by fluctuations at both to = 0 and to = too (too is the impurity resonance frequency). The method of moments was chosen to study the spectral functions at low frequencies. This method is based on the Hamiltonian of the system without any approximation. It yields limited information concerning the lineshape of the fluctuation spectra but is valid for any temperature. The fluctuation spectra were calculated assuming a standard form of cubic crystalline field and bilinear isotropic host exchange interaction. We assumed also a weak host exchange coupling, ~, with respect to the crystalline field splitting, A. In this case the fluctuation spectra can be separated into low and high frequency components where the high frequency components are located at frequency, to, to = A. Only the low frequency part of the fluctuation spectra is of interest to us. To calculate the second moment, M2, of the low frequency part we have defined a * Supported by the US-Israel Binational Science Foundation.
Physica 86-88B (1977) 1158-1159 © North-Holland
spectral function F(to) = K~L . F . (to)
L F K~z" '(to)dto,
(1)
where K~L.F.(to) is the low frequency part of K~z(to). Using the projection operator technique and the truncation procedure [4, 1] we found for M2, the autocorrelation part of the second moment of F(to), the following expression 1 ME-
z E po I t r j . E')lrj.>l 2 a
+Z'E
e f~f~
(l(rjo,
1,i)
I r j . ,r rJ t)l 2
x [(rJ'lYz"lrj;) - (rj l,i)
I
t
+ (1 - 6,~)l(Fof,, Fcf~l~x Irate, r j o ) l
×
(rjo
2
(2)
where ~tCe~x i) is the exchange interaction between the ion labelled " 1 " and the ith ion, ~¢~Li)= --(gj--1) 2 . f l . J ( ~ ) . J " ) . The wave function (roL, r~f~l represents "two-ions" wave function where the first two indices characterize the ion labelled " r ' and the last two indices characterize the wave function appropriate to the ith ion. Z is the partition function, E~ are crystalline field splitting levels and p~ is the population of the F~ crystalline field splitting level. Explicit expressions for the case of Tm or Pr Van-Vleck compounds can be derived from (2) [1, 3]. A fourth moment calculation has shown that MJ3M~ -~ 0.7 at T = 0 for P r cubic compounds. This indicates that the low frequency fluctuation spectra exhibit a Gaussian lineshape and thus F(0) = (2¢rM2)-in.
!159 The second moment of the low frequency part of the fluctuation spectra enables us to give a formula for the NMR relaxation rate in VanVleck compounds or even the ESR relaxation rate of impurities in the limit Ato ~> too [where Ao~ is the width of F(to)]. We found the following general expression 1 _ 27r(gj -
PrSb Gd
/
900 -
Z
/
800 700
1)z Zk (~)2
Xoopo I
Iv jo ;,I ,
T 600
I-a
(3) where ~ is the host rare-earth-host nucleus interaction in the case of NMR or the host rare-earth-impurity exchange interaction for the case of ESR. Expression (3) depends explicitly on the host exchange, ~, the value of ~ ' and the host crystalline field splitting level, E~. Thus knowledge of E, (from neutron scattering experiments), ~ ' from hyperfine constant [3] or ESR g shift [2, 6] enables one to extract the host exchange. This procedure is carried out for the case of ESR in PrSb:Gd. The energy levels were taken from Birgeneau et al. [7]. The fit of the theory to the experimental linewidth [1, 2] is shown in fig. 1. This enables us to extract the host exchange, ~, to be ~ ~ 0.9 meV. For comparison fig. 1 exhibits also a fit with the theory of Moriya and Obata [2]. This theory is phenomenological and does not yield the host exchange interaction. In conclusion, we have demonstrated that ESR or NMR relaxation provided information on the low frequency part of the fluctuation spectra Kpp(to) in Van-Vleck compounds. In inelastic neutron scattering [8], on the other hand, information was provided on the host susceptibility x ( q , to) and, especially, on its high frequency part [9]. Thus inelastic neutron scattering and ESR (NMR) are complementary techniques. References [1] D. Davidov, V. Zevin, R. Levin, D. Shaltiel and K. Baberschke, Phys. Rev., to be published.
500 w Z
-~ 400
300
/
7
2OO 1O0 ~
0
J
!
I
I
I
I
I
L
20 40 60 80 100 120 140 160 180 200 TEMPERATURE (K)
Fig. 1. The ESR linewidth of Gd in PrSb (after subtracting the residual width). The open squares represent data taken from ref. 2. The closed squares and circles were taken from re~. 1. The solid line is the best fit of (3) to the experimental results; the dashed line is the fit of the theory of Moriya and Obata (ref. 2). [2] S. Sugawara, C.Y. Huang and B.R. Cooper, Phys. Rev. BII (1975) 4450, T. Moriya and Y. Obata, J. Phys. Soc. Japan 13 (1958) 1333. [3] S. Myers and N. Narath, Phys. Rev. B9 (1974) 207. [4] M. McMillan and W. Opechowski, Can. J. Phys. 389 (1963) 2915. V. Zevin and B. Shanina, Ukr. Fiz. J. 11 (1966) 1089 [Sov. Phys.]. [5] V. Zevin, D. Davidov, R. Levin, D. Shaltiel and K. Baberschke, to be published. [6] C. Rettori, D. Davidov, A. Grayevskey and W.M. Walsh, Phys. Rev. B l l (1975) 4455. [7] K.C. Tuberfield, L. Passel, R.J. Birgeneau and E. Bucher, Phys. Rev. Lett. 25 (1970) 752; J. Appl. Phys. 42
(1971) 1746. [8] R.J. Birgeneau, J. Als-Nielsen and E. Bucher, Phys. Rev. Lett. 27 (1971) 1630. R.J. Birgeneau, J. Als-Nielsen and E. Bucher, Phys. Rev. B6 (1972) 2724. [9l W.J.L. Buyers, T.M. Holden and A. Perreault, Phys. Rev. BII (1975) 266. W.J.L. Buyers, AIP conf. Proc. 24 (1974) 27 and references therein.
PARAMAGNETIC FLUCTUATIONS IN CUBIC VAN-VLECK PNICTIDES*
V. ZEVIN, D. DAVIDOV, R. L E V I N and D. S H A L T I E L Racah Institute o[ Physics, The Hebrew University of Jerusalem, Israel The lineshape of the paramagnetic fluctuation spectra in cubic Van-Vleck compounds is discussed in conjunction with the relaxation of an impurity or that of the host nucleus in these systems.
The present work discusses the fluctuations in single ground state systems with emphasis on relaxation of impurity [1, 2] and the host nuclei [3]. The fluctuation spectra of the host paramagnetic ions in singlet ground state systems with cubic structure can be described by the spectral functions Kpp(to) (p = x , y,z). These spectral functions are expressed as the sum Kpp(to) = Zk,jKpp(to)~J where Kpp(to)kJ is the Fourier transform of the time correlation function (Jtpk)(t)J~p)) and ~k) is the p component of the angular momentum of the kth host magnetic ion. For the case of ESR (NMR), the indices k and j run over several RE ions surrounding the impurity (or host nucleus). It has been demonstrated that while the N M R relaxation is determined by the to = 0 component of Kpp(to), the ESR relaxation is dominated by fluctuations at both to = 0 and to = too (too is the impurity resonance frequency). The method of moments was chosen to study the spectral functions at low frequencies. This method is based on the Hamiltonian of the system without any approximation. It yields limited information concerning the lineshape of the fluctuation spectra but is valid for any temperature. The fluctuation spectra were calculated assuming a standard form of cubic crystalline field and bilinear isotropic host exchange interaction. We assumed also a weak host exchange coupling, ~, with respect to the crystalline field splitting, A. In this case the fluctuation spectra can be separated into low and high frequency components where the high frequency components are located at frequency, to, to = A. Only the low frequency part of the fluctuation spectra is of interest to us. To calculate the second moment, M2, of the low frequency part we have defined a * Supported by the US-Israel Binational Science Foundation.
Physica 86-88B (1977) 1158-1159 © North-Holland
spectral function F(to) = K~L . F . (to)
L F K~z" '(to)dto,
(1)
where K~L.F.(to) is the low frequency part of K~z(to). Using the projection operator technique and the truncation procedure [4, 1] we found for M2, the autocorrelation part of the second moment of F(to), the following expression 1 ME-
z E po I t r j . E')lrj.>l 2 a
+Z'E
e f~f~
(l(rjo,
1,i)
I r j . ,r rJ t)l 2
x [(rJ'lYz"lrj;) - (rj l,i)
I
t
+ (1 - 6,~)l(Fof,, Fcf~l~x Irate, r j o ) l
×
(rjo
2
(2)
where ~tCe~x i) is the exchange interaction between the ion labelled " 1 " and the ith ion, ~¢~Li)= --(gj--1) 2 . f l . J ( ~ ) . J " ) . The wave function (roL, r~f~l represents "two-ions" wave function where the first two indices characterize the ion labelled " r ' and the last two indices characterize the wave function appropriate to the ith ion. Z is the partition function, E~ are crystalline field splitting levels and p~ is the population of the F~ crystalline field splitting level. Explicit expressions for the case of Tm or Pr Van-Vleck compounds can be derived from (2) [1, 3]. A fourth moment calculation has shown that MJ3M~ -~ 0.7 at T = 0 for P r cubic compounds. This indicates that the low frequency fluctuation spectra exhibit a Gaussian lineshape and thus F(0) = (2¢rM2)-in.
!159 The second moment of the low frequency part of the fluctuation spectra enables us to give a formula for the NMR relaxation rate in VanVleck compounds or even the ESR relaxation rate of impurities in the limit Ato ~> too [where Ao~ is the width of F(to)]. We found the following general expression 1 _ 27r(gj -
PrSb Gd
/
900 -
Z
/
800 700
1)z Zk (~)2
Xoopo I
Iv jo ;,I ,
T 600
I-a
(3) where ~ is the host rare-earth-host nucleus interaction in the case of NMR or the host rare-earth-impurity exchange interaction for the case of ESR. Expression (3) depends explicitly on the host exchange, ~, the value of ~ ' and the host crystalline field splitting level, E~. Thus knowledge of E, (from neutron scattering experiments), ~ ' from hyperfine constant [3] or ESR g shift [2, 6] enables one to extract the host exchange. This procedure is carried out for the case of ESR in PrSb:Gd. The energy levels were taken from Birgeneau et al. [7]. The fit of the theory to the experimental linewidth [1, 2] is shown in fig. 1. This enables us to extract the host exchange, ~, to be ~ ~ 0.9 meV. For comparison fig. 1 exhibits also a fit with the theory of Moriya and Obata [2]. This theory is phenomenological and does not yield the host exchange interaction. In conclusion, we have demonstrated that ESR or NMR relaxation provided information on the low frequency part of the fluctuation spectra Kpp(to) in Van-Vleck compounds. In inelastic neutron scattering [8], on the other hand, information was provided on the host susceptibility x ( q , to) and, especially, on its high frequency part [9]. Thus inelastic neutron scattering and ESR (NMR) are complementary techniques. References [1] D. Davidov, V. Zevin, R. Levin, D. Shaltiel and K. Baberschke, Phys. Rev., to be published.
500 w Z
-~ 400
300
/
7
2OO 1O0 ~
0
J
!
I
I
I
I
I
L
20 40 60 80 100 120 140 160 180 200 TEMPERATURE (K)
Fig. 1. The ESR linewidth of Gd in PrSb (after subtracting the residual width). The open squares represent data taken from ref. 2. The closed squares and circles were taken from re~. 1. The solid line is the best fit of (3) to the experimental results; the dashed line is the fit of the theory of Moriya and Obata (ref. 2). [2] S. Sugawara, C.Y. Huang and B.R. Cooper, Phys. Rev. BII (1975) 4450, T. Moriya and Y. Obata, J. Phys. Soc. Japan 13 (1958) 1333. [3] S. Myers and N. Narath, Phys. Rev. B9 (1974) 207. [4] M. McMillan and W. Opechowski, Can. J. Phys. 389 (1963) 2915. V. Zevin and B. Shanina, Ukr. Fiz. J. 11 (1966) 1089 [Sov. Phys.]. [5] V. Zevin, D. Davidov, R. Levin, D. Shaltiel and K. Baberschke, to be published. [6] C. Rettori, D. Davidov, A. Grayevskey and W.M. Walsh, Phys. Rev. B l l (1975) 4455. [7] K.C. Tuberfield, L. Passel, R.J. Birgeneau and E. Bucher, Phys. Rev. Lett. 25 (1970) 752; J. Appl. Phys. 42
(1971) 1746. [8] R.J. Birgeneau, J. Als-Nielsen and E. Bucher, Phys. Rev. Lett. 27 (1971) 1630. R.J. Birgeneau, J. Als-Nielsen and E. Bucher, Phys. Rev. B6 (1972) 2724. [9l W.J.L. Buyers, T.M. Holden and A. Perreault, Phys. Rev. BII (1975) 266. W.J.L. Buyers, AIP conf. Proc. 24 (1974) 27 and references therein.