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Anisotropic exchange contributions to the magnetic susceptibility of transition metal ions in hexagonal close packed metals
Solid State Communications,Vol. 16, pp. 1301—1304, 1975.
Pergamon Press.
Printed in Great Britain
ANISOTROPIC EXCHANGE CONTRIBUTIONS TO THE MAGNETIC SUSCEPTIBILITY OF TRANSITION METAL IONS IN HEXAGONAL CLOSE PACKED METALS J.M. Dixon and R. Dupree Department of Physics, University of Warwick, Coventry, England (Received 20 December 1974 by B. Muhlschlegel)
When the effect of anisotropic exchange, together with crystal-field potentials is included in the Hamiltonian for magnetic impurity ions in a non-cubic metal, an additional contribution to the magnetic susceptibility is obtained. This produces a change in the temperature dependence of the susceptibility The anisotropy at temperatures gives an anisotropy varying lIT2. in low exchange, deducedand from experiment, is 0.0086 perascent for Zn: Mn and between 2.4 and 7.2 per cent for Zn : Cr with J 11 > ~ in both cases.
1. INTRODUCTION RECENT measurements of the magnetic susceptibility and anisotropy of single crystals of zinc containing up to 600 p.p.m. mar~anesehave been interpreted in terms of a crystal field which splits the spin ground state of the manganese ion.1 These experiments have recently been extended by Hedgcock et al.2 to include the low field magnetic anisotropy of Zn—Cr and the high field anisotropy Zn—Mn.susceptibility At low temperatures anisotropy in theofmagnetic is foundthe to contam terms which have a T2 temperature dependence, for both alloy systems, in addition to the Kondo-like terms in log (TITh). This su~estsa term in S~in the Hamiltonian for the system and indeed Hedgcock et al. did interpret their results in terms of the Hamiltonian H = ASZ2 + g 1$H~S~g11(3H~S~
One would expect to fmd very small zero field splittings for Zn: Mn because, as 6S, theall ioncontributions has a zero to such momentum a splitting will be due to non-vanishing perturangular ground state bative matrix elements between 6S and excited states of the ion. The ground term cannot be split by spin— orbit coupling to any order4 and the only term which can be mixed into the ground term, by the spin—orbit coupling, is 4P 512 in first order. However, are6S, no 4P northere within crystal matrix so any field splitting will elements be due towithin a high order interaction. The mechanisms which produce such splittings are not completely understood even in insulators and several have been suggested59 such as spin—orbit, spm—spm and relativistic orbit lattice interactions. In the case of Zn : Cr it is difficult to see why one
—
should obtain magnetic susceptibility behaviour simi-
—
where A was a “crystal field” parameter and ~ and g1 are the g-factors parallel and perpendicular to the 3 z axis (which wasmeasured taken to be c-axis).susceptibility Camwell has also recently thethe magnetic of single crystals of Zn: Cr up to concentrations of 400 p.p.m. over a wider temperature range than Hedgcock et al. The results agree at low temperatures but at higher temperatures Camwell fmds a change in the sign of the anisotropy presumably due to higher levels becoming populated.
lar to that for Zn: Mn since in this case one expects that the momentum chromium ions will ground retain atstates. least Even some iforbital angular in their the ground state was an orbital singlet as in Mn one would expect second order spin—orbit coupling matrix elements to be non-negligible as the energy separations to excited states, in the presence of orbital angular momentum, will be much smaller than in the case of Mn.
1301
1302
ANISOTROPIC EXCHANGE IN HEXAGONAL CLOSE PACKED METALS
In the past there have been two main approaches to the impurity problem. In the Friedel—Anderson approach10—13 one argues that the interaction between conduction electrons the ion is so strong cannot be treated as aand perturbation. One thenthat triesitto solve the coupled system of ions and conduction electrons together. This leads to the conclusion that the magnetic electrons of the impurity occupy broadened virtual bound states which could in principle be of the order of 1 eV. wide. It is extremely difficult to account for the observed anisotropy in the magnetic susceptibiity of either Zn : Cr or Zn : Mn using this model. In the ion-based approach’4”5 it is argued that a useful zero order Hamiltonian is provided by the ion and the conduction electrons separately, without any interaction. The unperturbed ionic states are those of the free ion including appropriate configuration, term, spin—orbit and crystal-field splittings. In this communication we shall use this latter approach, as it would seem more physically appropriate than the former, to discuss the magnetic susceptibilities of Zn: Mn and Zn: Cr. We shall not attempt a full discussion of this general approach and fully realise that the ionic model for the case of ions in metals has not been justified. Nevertheless is is of interest to see if such an approach can provide a useful physical interpretation of the experimental data.
c-axis throughout and Hmt has been written with J
11 * J1 so that the anisotropic nature of the host is reflected in the symmetry of ~ 1618 We assume that the spin—orbit coupling is isotropic. To calculate the magnetic susceptibility we have used the Kubo method used by Kondo19 and Borchi and Gennaro2°using the exchange Hmt and the spin— orbit coupling as perturbations on the hexagonal crystal field states. The subsequent analysis applies to Zn: Cr with an orbital singlet lowest (this seems likely since the anisotropy in the susceptibility is small) and also to Zn : Mn with L = 0. The parallel and perpendicular susceptibilities may be written
f
2
x~ (~—~—) =
dA (e~’(L~ + 2S~)e~”(L~ + 2Sf))
and 2
Xi
k-i—) f dX (e~’(L~+ 2S~)e~H(Lx + 2S~))
=
where H is the total Hamiltonian which we write as H = A [3L~ —L(L + 1)1 + B[35L~ 30L(L + 1)L~ —
+ 25L~
—
6L(L + 1) + 3L2 (L + 1)2 1 + HF
+ XL~S~ =
An interpretation in which the interaction with conduction electrons is neglected, retaining spin—orbit coupling and using spin and orbital angular momenta appropriate for the free ion, has been found to be completely inadequate to describe the results for Zn : Cr. In order to explain the low temperature experimental results, it was found to be sufficient only to have a ground ionic state which was almost degenerate (~1 cm’) with two other states. For spin—orbit
Ho+XL-S+H~~.
The terms in A and B represent the hexagonal crystal field in a manifold with orbital angular momentum L, HF describes the Fermi sea of conduction electrons
and the third term is the spin—orbit coupling. Using Fermion annihilation and creation operators for the Fermi sea we may re-write Hmt as ~
=
Nk,k’ [Ji 1SZ(~tC~’t
coupling parameters of the same order magnitude 2~,Aof—60 cm1) it as those found into insulators(for Cr situation unless was not possible produce the above the crystal field was an order of magnitude greater than one would expect using a point-charge-like-model. Therefore to make our model more physically realistic we have introduced the interaction Hmt with the conduction electrons, which we have taken for simplicity to be of Heisenberg type, and write Hmt
=
Vol. 16, No. 12
—
C4C~’~)
+ JiS+Ck~CJ~t + .JjS_C~tC~’j where N is a constant. To second order in ~ + XL- S
one fmds the separable form XII = X + x~ + x~°~ + x~ (1) and similarly for x~, where ~ is the zeroth order contribution to the susceptibility from the orbital angular momentum, ~ is the free spin susceptibility,
J 11S2c,~+ ~J1(S÷a_+ S_o+)
where S and a are the ion and conduction—electron spins respectively. The z-axis will be taken to be the
x~°~ is due to the spin—orbit coupling alone and x~ the susceptibility from ~ alone. There are no terms in first order from exchange although spin—orbit coupling contributes a small term for Zn : Cr if the
Vol. 16, No.12
ANISOTROPIC EXCHANGE IN HEXAGONAL CLOSE PACKED METALS
orbital state is a singlet. (If the lowest orbital state is degenerate such a term could be large and the perturbation series may be non-converging). The reason for
this is that
2 S~S+1) 3kT
N~ F_4p2
—
P0H~F,,H~Po +... — E0
~
~
(4)
0
where H0 = H0 — HF, P0 is a projection operator for the FermiseaandP~thatforthe presence of elementary excitations. Fermiseain The third term the
1~—J~)S~ + ~(J~ —/D)
_~p2~(log~ 4) 7S(S + 1)J~.
(5)
(ZMB)
=
x (0.432_lose [~~})_~~kT then
(~i~
=
+ N2 2 ~ZIzB)2 k2T2
[s(s+ 1
15
1)
p2D~1oge4)~J~f ~Th ______
.
(3S~+ 3S
—
1)—
(s(s+
=
E
1))2]
(2) and
The second (5) produces only a constant energy shiftterm in theineffective Hamiltonian and as this this term. Our effective Hamiltonian can therefore will not affect the magnetic susceptibility, we drop be written 11 = 110 + XLS+ES~ with
2
+ ~(J~ —
XFC
perturbation and use the degenerate perturbation formalism of Bates et al.22 We find 11 = H+?L•S+P 0H~F0
in (4)~.p2D(1og~ vanishes and the fourth becomes 4)(J
~-~—j)
and
1303
22 (Si.zB) k2T2 2D (loge 4)(j p 1t — J~2) Xf + N ~S(S+i) X ~ 30 (2S2 + 2S + l) (s(s+ l))2] _____
~p2D(log~
=
4XJ~—J.?).
(6)
IfJ 11 >Jj, E < 0 and its effect will be to compensate terms from second order spin—orbit coupling in S~ the ML = ±1 crystal field states are ~ cm’ higher in and powers of S~from higher orders (for Zn: Cr, if energy than the singlet ground state such a term is 3X2S~/L~~). To agree with experimentE can be adjusted until the three states lowest in energy are degenerate in which case, at low temperatures, if we use an ef—
(3) where MB is the Bohr magneton, k is Boltzmann’s constant, p is the assumed constant density of states over the band width D.
fective spin Hamiltonian E ‘S2 with S’ = I (for Zn: Cr), E’ will be small, a complicated function of A and E and a high temperature approximation for the susceptibility can be used. To first order in AL• S + ESL the term in E contributes x~ and x’ to the susceptibility, given by
Unfortunately one cannot add ~ + + x~°. to (2) to obtain the total susceptibility because A i~ not small enough (if A 60 cm’), compared with the unperturbed energy differences ofH 0, (forusing esti- a Xi mation purposes these can be those calculated point charge model) so the perturbation expansion for x should go to a higher order in A before the necessary convergence is obtained. Expressions for xff, for /ll = Jj have been given by Yosida and Okiji21 using infinite order ~rturbation theory. To find an effective Hamiltonian, H, for the ion we treat HMt only as a
2132E {~s(s+ 1X3S2 + 3S — 1) =
—
— and
xl =
(gl.LB)
IS(S + l)\21 ~ )
,~
— .g~B2~2E fS(s+ IS(S+ l)\2) -~
3
(7) 1)(~2 +
2S + 1)
1304
ANISOTROPIC EXCHANGE IN HEXAGONAL CLOSE PACKED METALS
Vol. 16, No. 12
Comparing equations (7) with (2) and (3) we see that contributions to x~,apart from the terms are the same as those using H, to first order, so for Zn : Cr we
For Zn : Mn, assuming the Kondo temperature TK = 0.25 K, and the free electron band width, we obtain an anisotropy of 0.0038 per cent, with J11 > J~.In the
can approximate and include higher order terms in A by diagonalising H in (6) by computer and then add x~’~ afterwards,
case of Zn: Cr the situation is more complicated because of the presence of spin—orbit coupling and the possibility of low-lying crystal field states. The ground orbital state of Zn—Cr we believe to be ML = 0) since if either the IML = ±1) or ML = ±2) state were lowest a much higher degree of anisotropy would be observed. The value deduced from experiment for E depends on the value taken for the spin—orbit coupling and also on the disposition of higher levels. However, we find that E is between 15 and 45 cm~and hence the anisotropy in J is between 2.4 and 7.2 per cent. These values for the anisotropy in the exchange seem reasonable in view of the anisotropic nature of the host. Further details of the crystal fields in Zn—Cr will be presented elsewhere.
We believe that the origin of the Si-like term in both the case of Zn—Mn and Zn—Cr is the anisotropic exchange interaction rather than a crystal-field mcclianism suggested by Hedgcock et a!. Our coeffIcient E is not to be equated with the A of the latter authors since they have used a phenomenological Hamiltoman with an effective spin S” = 2 for Mn and S” = 4 for Cr. However, we expect this to be proportional in the case of Mn but not in the case of Cr because of additional contributions from non-vanishing spin—orbit coupling,
REFERENCES 1.
HEDGCOCK FT., LI P.L., MUIR W.B. and STROM-OLSEN J.O., Phys. Rev. Lett.. 31, 29(1973).
2. 3.
HEDGCOCK F.T., LENIS S., LI P.L., STROM-OLSEN J.O. and WASSERMANN E.F., Can. J. Phys., 52, 1759 (1974). CAMWELL P.L., Ph.D. Thesis University of Warwick (1974).
4.
GRIFFITH J.S., The Theory of Transition Metal Ion. Cambridge University Press (1964).
5.
CHATTERJEE R., NEWMAN Di. and TAYLOR C.D., J. Phys. C6, 706 (1973).
6.
LULEK T.,Acta. Phys. Polon. A36, 551 (1969).
7. 8.
LACROIX R. and EMCH G.,Helv. Phys. Acta 35, 592 (1962). BUCKMASTER H.A., CHATTERJEE R., SHING Y.H., Can. J. Phys. 50, 78(1972).
9.
HAN P.S., DAS T.P. and RETTIG M.F.,J. Chem. Phys. 49, 4665 (1968).
10. 11. 12. 13.
ANDERSON P.W.,Phys. Rev. 124, 41 (1961). FRIEDELJ.,Nuovo CinentoSuppl. 7,287(1958); Can. J. Phys. 34,1190(1956); BLANDIN A. and FRIEDEL J.,J. Phys. Radium 19, 573 (1958). YOSIDA K., OKIJI A. and CHICKAZUMI S., Frog. Theor. Phys. 33, 559 (1965). COQBLIN B. and BLANDIN A.,Adv. Phys. 17, 281 (1968).
14.
SCHRIEFFER J.R.,J. App!. Phys. 38, 1143(1967).
15.
FIIRST L.L.,Phys. Cond. Mar. 11,255(1970).
16.
RIMMER D.E. and JOHNSTON D.F., Proc. Phys. Soc. 89, 943 (1966).
17. 18.
MCFARLANE R.M.,J. Chem. Phys. 42,442(1965). KAMIMURA H.,Phys. Rev. 128, 1077 (1962).
19.
KONDO J.,Solid State Physics (Edited by SEITZ F., TURNBULLD. and EHRENREICH H.) Vol. 23, p. 183. Academic Press, New York (1969). BORCHI E. and DE GENNARO S.,Phys. Rev. B. 9, 209 (1974). YOSIDA K. and OKIJI A.,Progr. Theoret. Phys. (Kyoto)34, 505 (1965). BATES C.A., DIXON J.M., FLETCHER J.R. and STEVENS K.W.H.,J. Phys. C. Ser. 2, Vol. 1,859(1968).
20. 21. 22.
Pergamon Press.
Printed in Great Britain
ANISOTROPIC EXCHANGE CONTRIBUTIONS TO THE MAGNETIC SUSCEPTIBILITY OF TRANSITION METAL IONS IN HEXAGONAL CLOSE PACKED METALS J.M. Dixon and R. Dupree Department of Physics, University of Warwick, Coventry, England (Received 20 December 1974 by B. Muhlschlegel)
When the effect of anisotropic exchange, together with crystal-field potentials is included in the Hamiltonian for magnetic impurity ions in a non-cubic metal, an additional contribution to the magnetic susceptibility is obtained. This produces a change in the temperature dependence of the susceptibility The anisotropy at temperatures gives an anisotropy varying lIT2. in low exchange, deducedand from experiment, is 0.0086 perascent for Zn: Mn and between 2.4 and 7.2 per cent for Zn : Cr with J 11 > ~ in both cases.
1. INTRODUCTION RECENT measurements of the magnetic susceptibility and anisotropy of single crystals of zinc containing up to 600 p.p.m. mar~anesehave been interpreted in terms of a crystal field which splits the spin ground state of the manganese ion.1 These experiments have recently been extended by Hedgcock et al.2 to include the low field magnetic anisotropy of Zn—Cr and the high field anisotropy Zn—Mn.susceptibility At low temperatures anisotropy in theofmagnetic is foundthe to contam terms which have a T2 temperature dependence, for both alloy systems, in addition to the Kondo-like terms in log (TITh). This su~estsa term in S~in the Hamiltonian for the system and indeed Hedgcock et al. did interpret their results in terms of the Hamiltonian H = ASZ2 + g 1$H~S~g11(3H~S~
One would expect to fmd very small zero field splittings for Zn: Mn because, as 6S, theall ioncontributions has a zero to such momentum a splitting will be due to non-vanishing perturangular ground state bative matrix elements between 6S and excited states of the ion. The ground term cannot be split by spin— orbit coupling to any order4 and the only term which can be mixed into the ground term, by the spin—orbit coupling, is 4P 512 in first order. However, are6S, no 4P northere within crystal matrix so any field splitting will elements be due towithin a high order interaction. The mechanisms which produce such splittings are not completely understood even in insulators and several have been suggested59 such as spin—orbit, spm—spm and relativistic orbit lattice interactions. In the case of Zn : Cr it is difficult to see why one
—
should obtain magnetic susceptibility behaviour simi-
—
where A was a “crystal field” parameter and ~ and g1 are the g-factors parallel and perpendicular to the 3 z axis (which wasmeasured taken to be c-axis).susceptibility Camwell has also recently thethe magnetic of single crystals of Zn: Cr up to concentrations of 400 p.p.m. over a wider temperature range than Hedgcock et al. The results agree at low temperatures but at higher temperatures Camwell fmds a change in the sign of the anisotropy presumably due to higher levels becoming populated.
lar to that for Zn: Mn since in this case one expects that the momentum chromium ions will ground retain atstates. least Even some iforbital angular in their the ground state was an orbital singlet as in Mn one would expect second order spin—orbit coupling matrix elements to be non-negligible as the energy separations to excited states, in the presence of orbital angular momentum, will be much smaller than in the case of Mn.
1301
1302
ANISOTROPIC EXCHANGE IN HEXAGONAL CLOSE PACKED METALS
In the past there have been two main approaches to the impurity problem. In the Friedel—Anderson approach10—13 one argues that the interaction between conduction electrons the ion is so strong cannot be treated as aand perturbation. One thenthat triesitto solve the coupled system of ions and conduction electrons together. This leads to the conclusion that the magnetic electrons of the impurity occupy broadened virtual bound states which could in principle be of the order of 1 eV. wide. It is extremely difficult to account for the observed anisotropy in the magnetic susceptibiity of either Zn : Cr or Zn : Mn using this model. In the ion-based approach’4”5 it is argued that a useful zero order Hamiltonian is provided by the ion and the conduction electrons separately, without any interaction. The unperturbed ionic states are those of the free ion including appropriate configuration, term, spin—orbit and crystal-field splittings. In this communication we shall use this latter approach, as it would seem more physically appropriate than the former, to discuss the magnetic susceptibilities of Zn: Mn and Zn: Cr. We shall not attempt a full discussion of this general approach and fully realise that the ionic model for the case of ions in metals has not been justified. Nevertheless is is of interest to see if such an approach can provide a useful physical interpretation of the experimental data.
c-axis throughout and Hmt has been written with J
11 * J1 so that the anisotropic nature of the host is reflected in the symmetry of ~ 1618 We assume that the spin—orbit coupling is isotropic. To calculate the magnetic susceptibility we have used the Kubo method used by Kondo19 and Borchi and Gennaro2°using the exchange Hmt and the spin— orbit coupling as perturbations on the hexagonal crystal field states. The subsequent analysis applies to Zn: Cr with an orbital singlet lowest (this seems likely since the anisotropy in the susceptibility is small) and also to Zn : Mn with L = 0. The parallel and perpendicular susceptibilities may be written
f
2
x~ (~—~—) =
dA (e~’(L~ + 2S~)e~”(L~ + 2Sf))
and 2
Xi
k-i—) f dX (e~’(L~+ 2S~)e~H(Lx + 2S~))
=
where H is the total Hamiltonian which we write as H = A [3L~ —L(L + 1)1 + B[35L~ 30L(L + 1)L~ —
+ 25L~
—
6L(L + 1) + 3L2 (L + 1)2 1 + HF
+ XL~S~ =
An interpretation in which the interaction with conduction electrons is neglected, retaining spin—orbit coupling and using spin and orbital angular momenta appropriate for the free ion, has been found to be completely inadequate to describe the results for Zn : Cr. In order to explain the low temperature experimental results, it was found to be sufficient only to have a ground ionic state which was almost degenerate (~1 cm’) with two other states. For spin—orbit
Ho+XL-S+H~~.
The terms in A and B represent the hexagonal crystal field in a manifold with orbital angular momentum L, HF describes the Fermi sea of conduction electrons
and the third term is the spin—orbit coupling. Using Fermion annihilation and creation operators for the Fermi sea we may re-write Hmt as ~
=
Nk,k’ [Ji 1SZ(~tC~’t
coupling parameters of the same order magnitude 2~,Aof—60 cm1) it as those found into insulators(for Cr situation unless was not possible produce the above the crystal field was an order of magnitude greater than one would expect using a point-charge-like-model. Therefore to make our model more physically realistic we have introduced the interaction Hmt with the conduction electrons, which we have taken for simplicity to be of Heisenberg type, and write Hmt
=
Vol. 16, No. 12
—
C4C~’~)
+ JiS+Ck~CJ~t + .JjS_C~tC~’j where N is a constant. To second order in ~ + XL- S
one fmds the separable form XII = X + x~ + x~°~ + x~ (1) and similarly for x~, where ~ is the zeroth order contribution to the susceptibility from the orbital angular momentum, ~ is the free spin susceptibility,
J 11S2c,~+ ~J1(S÷a_+ S_o+)
where S and a are the ion and conduction—electron spins respectively. The z-axis will be taken to be the
x~°~ is due to the spin—orbit coupling alone and x~ the susceptibility from ~ alone. There are no terms in first order from exchange although spin—orbit coupling contributes a small term for Zn : Cr if the
Vol. 16, No.12
ANISOTROPIC EXCHANGE IN HEXAGONAL CLOSE PACKED METALS
orbital state is a singlet. (If the lowest orbital state is degenerate such a term could be large and the perturbation series may be non-converging). The reason for
this is that
2 S~S+1) 3kT
N~ F_4p2
—
P0H~F,,H~Po +... — E0
~
~
(4)
0
where H0 = H0 — HF, P0 is a projection operator for the FermiseaandP~thatforthe presence of elementary excitations. Fermiseain The third term the
1~—J~)S~ + ~(J~ —/D)
_~p2~(log~ 4) 7S(S + 1)J~.
(5)
(ZMB)
=
x (0.432_lose [~~})_~~kT then
(~i~
=
+ N2 2 ~ZIzB)2 k2T2
[s(s+ 1
15
1)
p2D~1oge4)~J~f ~Th ______
.
(3S~+ 3S
—
1)—
(s(s+
=
E
1))2]
(2) and
The second (5) produces only a constant energy shiftterm in theineffective Hamiltonian and as this this term. Our effective Hamiltonian can therefore will not affect the magnetic susceptibility, we drop be written 11 = 110 + XLS+ES~ with
2
+ ~(J~ —
XFC
perturbation and use the degenerate perturbation formalism of Bates et al.22 We find 11 = H+?L•S+P 0H~F0
in (4)~.p2D(1og~ vanishes and the fourth becomes 4)(J
~-~—j)
and
1303
22 (Si.zB) k2T2 2D (loge 4)(j p 1t — J~2) Xf + N ~S(S+i) X ~ 30 (2S2 + 2S + l) (s(s+ l))2] _____
~p2D(log~
=
4XJ~—J.?).
(6)
IfJ 11 >Jj, E < 0 and its effect will be to compensate terms from second order spin—orbit coupling in S~ the ML = ±1 crystal field states are ~ cm’ higher in and powers of S~from higher orders (for Zn: Cr, if energy than the singlet ground state such a term is 3X2S~/L~~). To agree with experimentE can be adjusted until the three states lowest in energy are degenerate in which case, at low temperatures, if we use an ef—
(3) where MB is the Bohr magneton, k is Boltzmann’s constant, p is the assumed constant density of states over the band width D.
fective spin Hamiltonian E ‘S2 with S’ = I (for Zn: Cr), E’ will be small, a complicated function of A and E and a high temperature approximation for the susceptibility can be used. To first order in AL• S + ESL the term in E contributes x~ and x’ to the susceptibility, given by
Unfortunately one cannot add ~ + + x~°. to (2) to obtain the total susceptibility because A i~ not small enough (if A 60 cm’), compared with the unperturbed energy differences ofH 0, (forusing esti- a Xi mation purposes these can be those calculated point charge model) so the perturbation expansion for x should go to a higher order in A before the necessary convergence is obtained. Expressions for xff, for /ll = Jj have been given by Yosida and Okiji21 using infinite order ~rturbation theory. To find an effective Hamiltonian, H, for the ion we treat HMt only as a
2132E {~s(s+ 1X3S2 + 3S — 1) =
—
— and
xl =
(gl.LB)
IS(S + l)\21 ~ )
,~
— .g~B2~2E fS(s+ IS(S+ l)\2) -~
3
(7) 1)(~2 +
2S + 1)
1304
ANISOTROPIC EXCHANGE IN HEXAGONAL CLOSE PACKED METALS
Vol. 16, No. 12
Comparing equations (7) with (2) and (3) we see that contributions to x~,apart from the terms are the same as those using H, to first order, so for Zn : Cr we
For Zn : Mn, assuming the Kondo temperature TK = 0.25 K, and the free electron band width, we obtain an anisotropy of 0.0038 per cent, with J11 > J~.In the
can approximate and include higher order terms in A by diagonalising H in (6) by computer and then add x~’~ afterwards,
case of Zn: Cr the situation is more complicated because of the presence of spin—orbit coupling and the possibility of low-lying crystal field states. The ground orbital state of Zn—Cr we believe to be ML = 0) since if either the IML = ±1) or ML = ±2) state were lowest a much higher degree of anisotropy would be observed. The value deduced from experiment for E depends on the value taken for the spin—orbit coupling and also on the disposition of higher levels. However, we find that E is between 15 and 45 cm~and hence the anisotropy in J is between 2.4 and 7.2 per cent. These values for the anisotropy in the exchange seem reasonable in view of the anisotropic nature of the host. Further details of the crystal fields in Zn—Cr will be presented elsewhere.
We believe that the origin of the Si-like term in both the case of Zn—Mn and Zn—Cr is the anisotropic exchange interaction rather than a crystal-field mcclianism suggested by Hedgcock et a!. Our coeffIcient E is not to be equated with the A of the latter authors since they have used a phenomenological Hamiltoman with an effective spin S” = 2 for Mn and S” = 4 for Cr. However, we expect this to be proportional in the case of Mn but not in the case of Cr because of additional contributions from non-vanishing spin—orbit coupling,
REFERENCES 1.
HEDGCOCK FT., LI P.L., MUIR W.B. and STROM-OLSEN J.O., Phys. Rev. Lett.. 31, 29(1973).
2. 3.
HEDGCOCK F.T., LENIS S., LI P.L., STROM-OLSEN J.O. and WASSERMANN E.F., Can. J. Phys., 52, 1759 (1974). CAMWELL P.L., Ph.D. Thesis University of Warwick (1974).
4.
GRIFFITH J.S., The Theory of Transition Metal Ion. Cambridge University Press (1964).
5.
CHATTERJEE R., NEWMAN Di. and TAYLOR C.D., J. Phys. C6, 706 (1973).
6.
LULEK T.,Acta. Phys. Polon. A36, 551 (1969).
7. 8.
LACROIX R. and EMCH G.,Helv. Phys. Acta 35, 592 (1962). BUCKMASTER H.A., CHATTERJEE R., SHING Y.H., Can. J. Phys. 50, 78(1972).
9.
HAN P.S., DAS T.P. and RETTIG M.F.,J. Chem. Phys. 49, 4665 (1968).
10. 11. 12. 13.
ANDERSON P.W.,Phys. Rev. 124, 41 (1961). FRIEDELJ.,Nuovo CinentoSuppl. 7,287(1958); Can. J. Phys. 34,1190(1956); BLANDIN A. and FRIEDEL J.,J. Phys. Radium 19, 573 (1958). YOSIDA K., OKIJI A. and CHICKAZUMI S., Frog. Theor. Phys. 33, 559 (1965). COQBLIN B. and BLANDIN A.,Adv. Phys. 17, 281 (1968).
14.
SCHRIEFFER J.R.,J. App!. Phys. 38, 1143(1967).
15.
FIIRST L.L.,Phys. Cond. Mar. 11,255(1970).
16.
RIMMER D.E. and JOHNSTON D.F., Proc. Phys. Soc. 89, 943 (1966).
17. 18.
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