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Derivation of spin hamiltonians by tensor decomposition
J. Phys. Chem. Solids
Pergamon
DERIVATION
Press 1964. Vol. 25, pp. 635-639.
OF SPIN
Printed in Great Britain,
HAMILTONIANS
BY TENSOR
DECOMPOSITION* WALTER Department
J. C. GRANTt#
and
M. W. P. STRANDBERG
of Physics, Research Laboratory of Electronics, Cambridge, Massachusetts
Massachusetts
Institute of Technology,
(Received 3 December 1963)
Abstract-The spin Hamiltonian of a paramagnetic system is derived from the decomposition of spherical tensors having appropriate symmetry properties. Although no explicit use of group theory is involved, the Hamiltonian obtained is similar to that derived from purely group theoretical methods. Since perturbation theory is not invoked, the derivation applies regardless of the relative magnitude of the interactions, regardless of whether the lowest state of the free ion is an S-state, and regardless of whether the interaction of interest is linear, quadratic or of higher order.
THE
interaction of a paramagnetic ion with a crystal field and with an external magnetic field is usually represented by a spin Hamiltonian.(l-‘n In constructing a spin Hamiltonian one presupposes that the crystal field has removed the degeneracy of the free ion orbital levels. The energy structure of the lowest orbital state is then derived by a second order perturbation calculation on the orbital operators in the original Hamiltonian, i.e. on the orbital operators appearing in terms like gflH*(L+2S) and L-S. The perturbation sums over higher orbital states are lumped into appropriate parameters. The effects of the crystal field, which acts directly only on the orbitals, and the effects of spin-orbit coupling are then implicit in the g-factors and the ground-state splitting parameters which appear in the spin Hamiltonian. The actual values of these parameters are usually determined from experimental spectra. The relative magnitude of the interactions is crucial, since it determines the order of perturbation theory. Three orders of magnitude are involved, the crystal field interaction being con* This work was supported in part by the U.S. Army Signal Corps, the Air Force Office of Scientific Research, and the Office of Naval Research. t National Science Foundation Predoctoral Fellow, 1958-1960. $ Present address: Bell Telephone Laboratories, Inc., Murray Hill, New Jersey.
s*
sidered intermediate between the fine structure and multiplet structure. The method fails if the lowest orbital state is degenerate. It also fails for S-state ions which cannot directly interact with the crystal field at all. Very high orders of perturbation theory are required in this case.@s) Effects of covalent bonding cannot be accounted for. Excluded also are possible quadratic and higher-order effects in the applied fields. We propose in this paper a derivation of the spin Hamiltonian, which does not directly rest on perturbation theory and which consequently overcomes the limitations imposed by the perturbation approach. Our method rests on the symmetry properties of the crystalline environment. In this it is allied to the method of KOSTER and STATZ.‘~) Unlike these authors, however, we do not derive our results through formal group theory, but by a simple application of tensor algebra. We begin with a brief statement of those properties of spherical tensors that will be relevant to our discussion. Our notation is that of ROSE.@) An irreducible set of tensor operators of rank L is a set of 2L+ 1 functions TLM, -L 5 M 5 L, which transforms under the 2L+ 1 dimensional representation of the rotation group: RTLMR~
= C D&sMT~~* M’
635
(1)
636
WALTER
J.
C.
GRANT
and
M.
where R is the rotation operator. A tensor of rank L can be constructed from two tensors of rank 11 and 1s provided that the “triangle rule” (]Z1-Zs[ < L I 11+1~) and provided ml+mz
X TZ,mi( ~l>Tl,m,(
vZ>,
T&Vl,
Vs . ..) = TI,M(OVl,
OV, . ..).
(3)
that is, the functional form is unchanged when the operators Vt are expressed in the new coordinate system. Tensors of zero rank are invariant under all rotations. We consider an atom at a lattice site with a symmetry that is described by a group of operations Oi. The Hamiltonian of this atom must then be invariant under the operations Oi. If we write the Hamiltonian in the most general possible form H = 2
~.wTI,M(H,S),
(4)
I‘M
then the invariance requirement places restrictions on the coefficients ULM within a given L manifold. The constraints on the &!LM can be found in a number of ways, a general method having been given by BETHE. For certain symmetries, very simple considerations will yield the allowed coefficients. Cubic symmetry, for instance, requires that changes in 8 and + by ~12 leave the expressions invariant. Since YLM is proportional to pLM(cos 8)exp(iM+), only those M values are allowed that are integer multiples of 4. For L < 8, equations of the form fiL0 Y:(R
4) + aL4[ Y;(e, +) + Y;V,
T~I
STRANDBERG
(
1 + (e+S,++; )I
= M:
where C is a Clebsch-Gordan coefficient. For instance if we identify TILm,(Vl) with the spherical harmonic Y~,~,(rl) on the unit sphere, and TJ&V~) with the spherical harmonic Ylzmt(rs), then for II = Ia, equation (2) reduces to the wellknown additional theorem. It is true in general that the spherical harmonics YLM fulfill the condition (1). An invariant under an operation 0 is a combination of operators obeying
P.
+aL4
holds
(2)
W.
[
Y;
e+:,+++
Y;"
determine the ULM completely. For metry, the terms in the Hamiltonian L = 6, are as follows:
.*
= d’oofnd[Tk~+
+ae[T~n+
cubic sym(4), up to
(;)1’2(T,,+
Td-a)]
(;)“‘(Tc,+
Ts-,)]
(5)
Terms with odd L have odd parity and are excluded because they fail to meet the requirement of invariance and under time reversal. In the next stage of the calculation, we decompose the tensors of equations (4) and (5) into products of operators, according to the rule of equation (2). The factors in these products are functions of the magnetic field only or of the spin only. The decomposition of equation (2) is not unique, since for any given L, an infinite number of II, 1s combinations is possible, as long as the triangle rule is obeyed. The following physical restrictions limit the allowable modes of decomposition: (1) Odd parity products can be neglected, for the reasons previously given. (2) Spin tensors of rank greater than 2s can be neglected, since their matrix elements necessarily vanish. This again is an immediate consequence of the “triangle rule”. (3) Field tensors of rank Zimply terms in the field of order HI. Usually only the linear terms are of interest, although situations do occur in which second-(10) and third-orderur) terms are observed. We list below those decompositions which are possible for S = 5/2. We include field tensors only up to rank 1. (The inclusion of nonlinear interactions would considerably complicate the resulting Hamiltonian, without shedding more light on the method itself.) Terms in S only: 5”40 +
Y;(S)
T44+ T4-4 --f Y;(S)+
Yi4(S).
DERIVATION
Terms linear in
OF
SPIN
HAMILTONIANS
+
Y;l(H)Y;(S) + YRH) qm
J(
TENSOR
H:
&Y:(H)Y;l(S)+
T40 --f D$, =
BY
i3j KY:(H)Y&V )
DECOMPOSITION
637
J(A>[yil(H) y55(s> +
Y;(H)Y,5(S)].
Spin tensors of rank > 5 will give matrix elements that vanish identically for S = 5/2. It is for this reason that further terms are unnecessary in equation (5). Similarly the matrix elements of expressions Or,), Dzi, D6o and 064 would vanish for S < 512. The terms in S only give vanishing results for S < 2, which immediately indicates that for S < 2, no zero field splitting is possible in a cubic field. According to equation (5), we may now write to first order in H
+ Y;w)
y&m
+ Y3H>
y:(s)1 Z = ~o[Y~(S)+(;)~';Y;(S)+ Y;4(S)]+-b~Doo
+ T40 + Df; =
+ bz[D&j+ (;)“2D;;] 7-60
-+
D60
=
+ ba[Df;+ +
TW
qm + b4[D6o+ (;)““64].
+ T44+ T4-4 + D;i
(;)1’2Dt;]
= Y;(H)Y;(S)+
(2) _ T44-1T4-4 -+ D,, -
Y;l(H)Yi3(S) The reduction
of the field and spin tensors to functions of the components of the field and the spin is called “polarization”. The procedure is given by Rose and tables up to 1 = 5 are given by Koster and Statz. Written in terms of the field and + Y;1(H)Y,-3(S)] spin components, the Hamiltonian (6) becomes
+
Z
= Co
+ Y:(H)yY5(S)l
+ cl
qv-ws4(~)I T64-kT6-4 +- D64 =
[
$H+s-+ H-S+) + HISS 1
+c2-&H+s.+
H-S+)
[
+
Y;1(H)Y,-3(S)]
x
(
3+2+gs~+3
)
638
WALTER
J.
C.
GRANT
and
I +C3
f
~(~+S-+ILs+)(-12-~4+8ss-t-
14~2s~
+14S2S;-42S,-63S,2-42S,3+
Hz( 12Sz - 5OP& -I-1.5~4~~
-21S;) + -
7osw,3
f 105SZ3f 63s;) +~(H+s~+H-s~)(-24+S~-27Sz-9s,2) +I-St+H+S?)
M.
W.
P.
STRANDBERG
be required to interpret data, the derivation of additional terms in the spin Hamiltonian would require very ponderous perturbation calculations. (3) The presence or absence of zero-field splitting emerges with great clarity at a primary stage of the derivation. (4) The general method is applicable also to nonlinear effects. For H = Hz, the following matrix elements are obtained from the Hamiltonian (7) :
*II
= 3Cof
22s
= -9Co+
5
-C1+3’$+3OC3+6OC4 (2 +I-
>
H
~Ca-~5~Cs-3OOC4
H
! +;Hz(S4,+@)
233
(2+Sz) I
= 6Co+
;Cr-
) ;Ca+300CS+600C4
-i-c4~H+*-+H_s+~(-~~-s~+~s~
X44
[
= 6Co+
j - +I+
;Cs-3OOCs-6004
H
t
>
+14S?&+l4S%S,Z-42S,-43s;
3 --cr+-cs 2 c
JP33 = -9co+
-425,3-215;) +ZHz(12&--5OSWt-15S4&-7OS2S;
21 5 + lSOCs+ 3OOC4 H
+ lOSS,3+63S,5)
)
- ;(H+Sf+H_Sz)(-24+Sz - 27Sr95;)-7(H_S:+H+S!) + 35(H&
H
t
1
+ HzS4)(2 + Sz) .
(7)
The arbitrary constants Ci are not identical to the corresponding b’s of equation (6), because awkward numerical coefficients have been absorbed in them. The conventional spin Hamiltonian for S = 512 in a cubic field is identical to the CO and Cl terms of the Hamiltonian we have derived. The Hamiltonian derived by the method of Koster and Statz differs from ours in the arrangement of terms, so that their five constants are linear combinations of our five constants. The question naturally suggests itself: What is gained by deriving a Hamiltonian with five parameters instead of only two? The answers are several: (1) Five is an absolute limit on the number of parameters that might be needed to describe linear interactions for this system. This reassurance is not derivable from perturbation theoretic reasoning. (2) If more than two parameters should
-&3Ca-3oCs-6flC~
X66
= 3co+
x13
=
-*zs
= e%@@= ~(5~[3Co-(l8C~-140C4)~].
851
( =
1
H
2/(5)[3(70+(18C3+140C4)H]
At low fields, CO is large. To first order, the diagonalizing transformation is the one which diagonalizes the “CO” matrix. This transformation turns out to be:
5
Tll = - T55 = T26 = Ti32 =
J 6
Tzz = -To,3
= T13 = Tsl =
T33 = T44 = 1. The resulting levels are a doublet at - 12Ce and a quartet at 6Cs. The slopes of the levels at - 12Ca are )
1
+4Cz-100Cs+2~4
1.
DERIVATION
OF SPIN
HAMILTONIANS
The slopes of the levels at 6 C’s are 1000 rf +,, [
~Ca+SOCs+-CJ
rf: ;Cl-
++300Cs+600C4
3
and
1 . I
At high fields, C’s is small, and the levels are given to first order by the diagonal elements y%;r through #es, as listed above. At either high field or low field, five independent quantities can be measured, and hence the five parameters can be determined. These results, again, are in agreement with results obtained previously.(7) To summarize: It is possible to derive a spin Hamiltonian by a method which short-circuits both the machinery of perturbation theory and the machinery of group theory. The method has three steps: (1) The Hamiltonian is considered as a sum of tensors that have the invariance properties appropriate the crystalline environment. (2) These tensors are decomposed into products of spin and field operators. The decomposition is not unique, but the modes of decomposition are severely
BY TENSOR
DECOMPOSITION
639
limited by the magnitude of the spin, by the order of interaction considered, and by parity requirements. (3) Finally, these operators can be expressed in terms of spin and field components, yielding a generalized spin Hamiltonian. This Hamiltonian will contain the maximum number of free parameters consistent with the physical requirements. REFERENCES 1. PRYCE M. H. L., Proc. Phys. Sot. Lond. A63, 25 (1950). 2. ABRAGAMA. and PRYCE M. H. L., Proc. Roy. Sot.
Lond. A205, 135 (1951). 3. BLEANEY B. and STEVENS K. W. H.. Reb. Pr0.w. Phys. 16, 108 (1953).
4. BOWERS K. D. and OWEN J., Rep. Progr. Phys. 18, 304 (1955).
5. VAN VLECK J. H., Phil. 1Mup. 17. 961 (1934). 6. HUTCHINSONC. A., JLTDDB: R. and P&E D. F. D., Proc. Phvs. Sot. Lond. B70. 514 09571. 7. KOSTER G: F. and STATZ M, Phyi. Rev. 113, 445 (1959).
8. ROSE M. E., Elementary Theory of Angular Momentum. John Wiley, New York (1957). 9. BETHE H., Phys. Rev. 71, 612 (1947). 10. BAKERJ. M. and BLEANEY B., Proc. Roy. Sot., Lond. A245, 156 (1958). 11. GRAMBERGG., Z. Phys. 159, 125 (1960).
Pergamon
DERIVATION
Press 1964. Vol. 25, pp. 635-639.
OF SPIN
Printed in Great Britain,
HAMILTONIANS
BY TENSOR
DECOMPOSITION* WALTER Department
J. C. GRANTt#
and
M. W. P. STRANDBERG
of Physics, Research Laboratory of Electronics, Cambridge, Massachusetts
Massachusetts
Institute of Technology,
(Received 3 December 1963)
Abstract-The spin Hamiltonian of a paramagnetic system is derived from the decomposition of spherical tensors having appropriate symmetry properties. Although no explicit use of group theory is involved, the Hamiltonian obtained is similar to that derived from purely group theoretical methods. Since perturbation theory is not invoked, the derivation applies regardless of the relative magnitude of the interactions, regardless of whether the lowest state of the free ion is an S-state, and regardless of whether the interaction of interest is linear, quadratic or of higher order.
THE
interaction of a paramagnetic ion with a crystal field and with an external magnetic field is usually represented by a spin Hamiltonian.(l-‘n In constructing a spin Hamiltonian one presupposes that the crystal field has removed the degeneracy of the free ion orbital levels. The energy structure of the lowest orbital state is then derived by a second order perturbation calculation on the orbital operators in the original Hamiltonian, i.e. on the orbital operators appearing in terms like gflH*(L+2S) and L-S. The perturbation sums over higher orbital states are lumped into appropriate parameters. The effects of the crystal field, which acts directly only on the orbitals, and the effects of spin-orbit coupling are then implicit in the g-factors and the ground-state splitting parameters which appear in the spin Hamiltonian. The actual values of these parameters are usually determined from experimental spectra. The relative magnitude of the interactions is crucial, since it determines the order of perturbation theory. Three orders of magnitude are involved, the crystal field interaction being con* This work was supported in part by the U.S. Army Signal Corps, the Air Force Office of Scientific Research, and the Office of Naval Research. t National Science Foundation Predoctoral Fellow, 1958-1960. $ Present address: Bell Telephone Laboratories, Inc., Murray Hill, New Jersey.
s*
sidered intermediate between the fine structure and multiplet structure. The method fails if the lowest orbital state is degenerate. It also fails for S-state ions which cannot directly interact with the crystal field at all. Very high orders of perturbation theory are required in this case.@s) Effects of covalent bonding cannot be accounted for. Excluded also are possible quadratic and higher-order effects in the applied fields. We propose in this paper a derivation of the spin Hamiltonian, which does not directly rest on perturbation theory and which consequently overcomes the limitations imposed by the perturbation approach. Our method rests on the symmetry properties of the crystalline environment. In this it is allied to the method of KOSTER and STATZ.‘~) Unlike these authors, however, we do not derive our results through formal group theory, but by a simple application of tensor algebra. We begin with a brief statement of those properties of spherical tensors that will be relevant to our discussion. Our notation is that of ROSE.@) An irreducible set of tensor operators of rank L is a set of 2L+ 1 functions TLM, -L 5 M 5 L, which transforms under the 2L+ 1 dimensional representation of the rotation group: RTLMR~
= C D&sMT~~* M’
635
(1)
636
WALTER
J.
C.
GRANT
and
M.
where R is the rotation operator. A tensor of rank L can be constructed from two tensors of rank 11 and 1s provided that the “triangle rule” (]Z1-Zs[ < L I 11+1~) and provided ml+mz
X TZ,mi( ~l>Tl,m,(
vZ>,
T&Vl,
Vs . ..) = TI,M(OVl,
OV, . ..).
(3)
that is, the functional form is unchanged when the operators Vt are expressed in the new coordinate system. Tensors of zero rank are invariant under all rotations. We consider an atom at a lattice site with a symmetry that is described by a group of operations Oi. The Hamiltonian of this atom must then be invariant under the operations Oi. If we write the Hamiltonian in the most general possible form H = 2
~.wTI,M(H,S),
(4)
I‘M
then the invariance requirement places restrictions on the coefficients ULM within a given L manifold. The constraints on the &!LM can be found in a number of ways, a general method having been given by BETHE. For certain symmetries, very simple considerations will yield the allowed coefficients. Cubic symmetry, for instance, requires that changes in 8 and + by ~12 leave the expressions invariant. Since YLM is proportional to pLM(cos 8)exp(iM+), only those M values are allowed that are integer multiples of 4. For L < 8, equations of the form fiL0 Y:(R
4) + aL4[ Y;(e, +) + Y;V,
T~I
STRANDBERG
(
1 + (e+S,++; )I
= M:
where C is a Clebsch-Gordan coefficient. For instance if we identify TILm,(Vl) with the spherical harmonic Y~,~,(rl) on the unit sphere, and TJ&V~) with the spherical harmonic Ylzmt(rs), then for II = Ia, equation (2) reduces to the wellknown additional theorem. It is true in general that the spherical harmonics YLM fulfill the condition (1). An invariant under an operation 0 is a combination of operators obeying
P.
+aL4
holds
(2)
W.
[
Y;
e+:,+++
Y;"
determine the ULM completely. For metry, the terms in the Hamiltonian L = 6, are as follows:
.*
= d’oofnd[Tk~+
+ae[T~n+
cubic sym(4), up to
(;)1’2(T,,+
Td-a)]
(;)“‘(Tc,+
Ts-,)]
(5)
Terms with odd L have odd parity and are excluded because they fail to meet the requirement of invariance and under time reversal. In the next stage of the calculation, we decompose the tensors of equations (4) and (5) into products of operators, according to the rule of equation (2). The factors in these products are functions of the magnetic field only or of the spin only. The decomposition of equation (2) is not unique, since for any given L, an infinite number of II, 1s combinations is possible, as long as the triangle rule is obeyed. The following physical restrictions limit the allowable modes of decomposition: (1) Odd parity products can be neglected, for the reasons previously given. (2) Spin tensors of rank greater than 2s can be neglected, since their matrix elements necessarily vanish. This again is an immediate consequence of the “triangle rule”. (3) Field tensors of rank Zimply terms in the field of order HI. Usually only the linear terms are of interest, although situations do occur in which second-(10) and third-orderur) terms are observed. We list below those decompositions which are possible for S = 5/2. We include field tensors only up to rank 1. (The inclusion of nonlinear interactions would considerably complicate the resulting Hamiltonian, without shedding more light on the method itself.) Terms in S only: 5”40 +
Y;(S)
T44+ T4-4 --f Y;(S)+
Yi4(S).
DERIVATION
Terms linear in
OF
SPIN
HAMILTONIANS
+
Y;l(H)Y;(S) + YRH) qm
J(
TENSOR
H:
&Y:(H)Y;l(S)+
T40 --f D$, =
BY
i3j KY:(H)Y&V )
DECOMPOSITION
637
J(A>[yil(H) y55(s> +
Y;(H)Y,5(S)].
Spin tensors of rank > 5 will give matrix elements that vanish identically for S = 5/2. It is for this reason that further terms are unnecessary in equation (5). Similarly the matrix elements of expressions Or,), Dzi, D6o and 064 would vanish for S < 512. The terms in S only give vanishing results for S < 2, which immediately indicates that for S < 2, no zero field splitting is possible in a cubic field. According to equation (5), we may now write to first order in H
+ Y;w)
y&m
+ Y3H>
y:(s)1 Z = ~o[Y~(S)+(;)~';Y;(S)+ Y;4(S)]+-b~Doo
+ T40 + Df; =
+ bz[D&j+ (;)“2D;;] 7-60
-+
D60
=
+ ba[Df;+ +
TW
qm + b4[D6o+ (;)““64].
+ T44+ T4-4 + D;i
(;)1’2Dt;]
= Y;(H)Y;(S)+
(2) _ T44-1T4-4 -+ D,, -
Y;l(H)Yi3(S) The reduction
of the field and spin tensors to functions of the components of the field and the spin is called “polarization”. The procedure is given by Rose and tables up to 1 = 5 are given by Koster and Statz. Written in terms of the field and + Y;1(H)Y,-3(S)] spin components, the Hamiltonian (6) becomes
+
Z
= Co
+ Y:(H)yY5(S)l
+ cl
qv-ws4(~)I T64-kT6-4 +- D64 =
[
$H+s-+ H-S+) + HISS 1
+c2-&H+s.+
H-S+)
[
+
Y;1(H)Y,-3(S)]
x
(
3+2+gs~+3
)
638
WALTER
J.
C.
GRANT
and
I +C3
f
~(~+S-+ILs+)(-12-~4+8ss-t-
14~2s~
+14S2S;-42S,-63S,2-42S,3+
Hz( 12Sz - 5OP& -I-1.5~4~~
-21S;) + -
7osw,3
f 105SZ3f 63s;) +~(H+s~+H-s~)(-24+S~-27Sz-9s,2) +I-St+H+S?)
M.
W.
P.
STRANDBERG
be required to interpret data, the derivation of additional terms in the spin Hamiltonian would require very ponderous perturbation calculations. (3) The presence or absence of zero-field splitting emerges with great clarity at a primary stage of the derivation. (4) The general method is applicable also to nonlinear effects. For H = Hz, the following matrix elements are obtained from the Hamiltonian (7) :
*II
= 3Cof
22s
= -9Co+
5
-C1+3’$+3OC3+6OC4 (2 +I-
>
H
~Ca-~5~Cs-3OOC4
H
! +;Hz(S4,+@)
233
(2+Sz) I
= 6Co+
;Cr-
) ;Ca+300CS+600C4
-i-c4~H+*-+H_s+~(-~~-s~+~s~
X44
[
= 6Co+
j - +I+
;Cs-3OOCs-6004
H
t
>
+14S?&+l4S%S,Z-42S,-43s;
3 --cr+-cs 2 c
JP33 = -9co+
-425,3-215;) +ZHz(12&--5OSWt-15S4&-7OS2S;
21 5 + lSOCs+ 3OOC4 H
+ lOSS,3+63S,5)
)
- ;(H+Sf+H_Sz)(-24+Sz - 27Sr95;)-7(H_S:+H+S!) + 35(H&
H
t
1
+ HzS4)(2 + Sz) .
(7)
The arbitrary constants Ci are not identical to the corresponding b’s of equation (6), because awkward numerical coefficients have been absorbed in them. The conventional spin Hamiltonian for S = 512 in a cubic field is identical to the CO and Cl terms of the Hamiltonian we have derived. The Hamiltonian derived by the method of Koster and Statz differs from ours in the arrangement of terms, so that their five constants are linear combinations of our five constants. The question naturally suggests itself: What is gained by deriving a Hamiltonian with five parameters instead of only two? The answers are several: (1) Five is an absolute limit on the number of parameters that might be needed to describe linear interactions for this system. This reassurance is not derivable from perturbation theoretic reasoning. (2) If more than two parameters should
-&3Ca-3oCs-6flC~
X66
= 3co+
x13
=
-*zs
= e%@@= ~(5~[3Co-(l8C~-140C4)~].
851
( =
1
H
2/(5)[3(70+(18C3+140C4)H]
At low fields, CO is large. To first order, the diagonalizing transformation is the one which diagonalizes the “CO” matrix. This transformation turns out to be:
5
Tll = - T55 = T26 = Ti32 =
J 6
Tzz = -To,3
= T13 = Tsl =
T33 = T44 = 1. The resulting levels are a doublet at - 12Ce and a quartet at 6Cs. The slopes of the levels at - 12Ca are )
1
+4Cz-100Cs+2~4
1.
DERIVATION
OF SPIN
HAMILTONIANS
The slopes of the levels at 6 C’s are 1000 rf +,, [
~Ca+SOCs+-CJ
rf: ;Cl-
++300Cs+600C4
3
and
1 . I
At high fields, C’s is small, and the levels are given to first order by the diagonal elements y%;r through #es, as listed above. At either high field or low field, five independent quantities can be measured, and hence the five parameters can be determined. These results, again, are in agreement with results obtained previously.(7) To summarize: It is possible to derive a spin Hamiltonian by a method which short-circuits both the machinery of perturbation theory and the machinery of group theory. The method has three steps: (1) The Hamiltonian is considered as a sum of tensors that have the invariance properties appropriate the crystalline environment. (2) These tensors are decomposed into products of spin and field operators. The decomposition is not unique, but the modes of decomposition are severely
BY TENSOR
DECOMPOSITION
639
limited by the magnitude of the spin, by the order of interaction considered, and by parity requirements. (3) Finally, these operators can be expressed in terms of spin and field components, yielding a generalized spin Hamiltonian. This Hamiltonian will contain the maximum number of free parameters consistent with the physical requirements. REFERENCES 1. PRYCE M. H. L., Proc. Phys. Sot. Lond. A63, 25 (1950). 2. ABRAGAMA. and PRYCE M. H. L., Proc. Roy. Sot.
Lond. A205, 135 (1951). 3. BLEANEY B. and STEVENS K. W. H.. Reb. Pr0.w. Phys. 16, 108 (1953).
4. BOWERS K. D. and OWEN J., Rep. Progr. Phys. 18, 304 (1955).
5. VAN VLECK J. H., Phil. 1Mup. 17. 961 (1934). 6. HUTCHINSONC. A., JLTDDB: R. and P&E D. F. D., Proc. Phvs. Sot. Lond. B70. 514 09571. 7. KOSTER G: F. and STATZ M, Phyi. Rev. 113, 445 (1959).
8. ROSE M. E., Elementary Theory of Angular Momentum. John Wiley, New York (1957). 9. BETHE H., Phys. Rev. 71, 612 (1947). 10. BAKERJ. M. and BLEANEY B., Proc. Roy. Sot., Lond. A245, 156 (1958). 11. GRAMBERGG., Z. Phys. 159, 125 (1960).