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Spin-phonon interactions in classical spin systems
Vol. 7, pp. 1271—1273, 1969.
Solid State Communications,
Pergamon Press.
Printed in Great Britain
SPIN—PHONON INTERACTIONS IN CLASSICAL SPIN SYSTEMS N.W. Dalton Theoretical Physics Division, Atomic Energy Research Establishment, Harwell, England and
D.W. Wood Mathematics Department, University of Nottingham, Nottingham, England (Received 16 June 1969 by C.W. McCombie)
A paramagnetic lattice system is discussed where spin—phonon interactions are present, and where the spin operators are treated classically. It is found that the spin—phonon coupling gives rise to an effective exchange coupling between spins.
1of cerium THE SPECIFIC heat measurements and praseodymium ethyl sulphates have revealed unexpectedly large Schottky anomalies, which • of the standard cannot be explained on the basis theory of Schottky specific heats. The underlying assumptions of this theory are:
=
_msh*~H°~S~,
(3)
and a
a
_1L~L.~l~j -u R ~p — 2 R ~ \ R
}(
[a~(PYZ*~
+
~
(4)
(a) that the Hamiltonian of the system can ke expressed as a sum of a lattice Hamiltonian V which describes the thermal vibrations, and a spin Hamiltonian which covers the Zeeman interaction of the spins with the applied magnetic field H, and
Here ~R’ and UR are the a-components (a = x, y, z) of the momentum and displacement operators of the spin at the lattice site R; S~ is the a-component of the spin operator at R, Ha is the a.component of the external magnetic field, m ~ is the magnetic moment per spin and h* = h • • • if the spins are finite and is zero in the limit of
1~,
(b) that the exchange interactions between the spins are negligible.
•
It has therefore been suggested2 that a form of spin-lattice coupling may be significant and that a term representing such an effect should be included in the total Hamiltonian; and several authors 2—4 have studied Ham iltonians of the form =
L
S +
with this the case ~y(p) = 0, and spin ½. ~ith simplification the effects of the Even term
where
LS
HL
=
\‘f 1 \~a~a •~)~_•)PR ‘~R
+
ic’v
a
(P)
Va —
R~pJ
R
upon the thermodynamic properties of the
system very with complications arising is from theinvolved spin commutation relations. The purpose of this note is to show that if the spin operators are replaced by classical spin vectors (which will be a good approximation
R R
•
Previous work has been mainly concerned
(1)
LS
•
infinite spin (classical spins). The p are the lattice vectors; Ea~(P), and ~y(~) are components of the spin-lattice coupling tensors which vanish in zero magnetic field, and Ka(p) is the a-component of the elastic force constant.
‘2~
R+pJ~ ~
1271
1272
SPIN—PHONON INTERACTIONS IN CLASSICAL SPIN SYSTEMS
when the total spin quanturr~,number is 1arge)~ then the principle effect of ~LS is to add an effective spin exchange HamiltonianH~ to J{
with the result that
3.
+
k +
=
+~)+H~(13)
k
3{~,
(14)
where
H~
~
=
(15)
•
The effective classical spin Hamiltonian can be written in the more transparent form
It is convenient to rewrite HL and ~ LS in terms of the phonon creation and annihilation operators , and respectively, which satisfy the boson commutation relations
(R
,
~
=
R ‘)S~S~,
—
—
R R’
—
(h2/8m)~~J~ (R R ‘)S~S~,SR~’s—
~a+1
—
j
~4k,Ak’ La
— —
LS
The effect is analagous to an RKY 5coupling of The large spins by spin—electron Schottky anomalies now interactions. become understandable since the specific heat of an exchange coupled spin system in a weak magnetic field can be anomalously large for temperatures in the region of a critical temperature T~(see below).
Vol. 7, No. 17
(5)
=
RR’
—
The result is
(~2/8m)~~J~)(R- R’)S~S~S~,S~, ~~a+A~a
=
+
~),
(6)
(16)
RR’
—
where for example and =
,
-
LS
(7)
J~ (R
R’)
-
=
Ic
where (wIc) a2 =
~(K~
=
m
~i(
K~), K~
-
~ I~
~
=
(0)
~ Ka(p)eik.P; (8) p
~a$ (k)
=
(0)- ~
(k)ll
(0)
~
Ea~ (-k)~e(R~’)
(17)
(k)}s~ f
~
with similar expressions for J,~. (R
2mw~/ +
E~
k
{~‘c~~ (0)— ~
(k)~S~7] (9)
c~. (k)
—
~
—
R
‘), and
R’) (which are zero when ~~a~p)=
0).
In the linear coupling approximation where
(p)et~P
~a$
J~2~ (R
—
(p)eP
, (10)
the ~~~(p) are zero, the effect of is to add a Heisenberg spin—spin coupling term H ‘~ to Hç, where the coupling constant is approxi-
-
mately proportional to the square of the tensor
se
=
N1~2
Y~s~ ~
If J~ (R — R’) = J(R — R’) (say) is positive then from the simple molecular field ~a~3(P).
~
=
RSRe
;
(11)
and where S~ is the c-number (component of classical s~invector) which replaces the operator h*S~.The term linear in , and Ak is readily removed by a canonical transformation to the new boson operators B~, and defined by
A~a k
=
~a Bk
+
a /ha~, g_~
(12)
theory of the Heisenberg ferromagnet6 in the presence of a small magnetic field we would expect a large excess magnetic specific heat in the neighbourhood of the temperature T~ where
h2
2J (R R kT~ —
)
1/z
and z is the number of neighbours interacting with a given spin.
(18)
Vol. 7, No. 17
SPIN—PHONON INTERACTIONS IN CLASSICAL SPIN SYSTEMS
1273
REFERENCES 1.
MEYER H. and SMITH P.L., Physics Chem. Solids 9, 285 (1959).
2.
PERSICO F. and STEVENS K.W.H., Proc. phys. Soc. 82, 855 (1963).
3.
ELLIOT R.J. and PARKINSON J.B., Proc. phys. Soc. 92, 1024 (1967).
4.
STEVENS K.W.H. and VAN EEKELEN H.A.M., Proc. phys. Soc. 92, 680 (1967).
5.
RUDERMANN K. and KITTEL C., Phys. Rev. 96, 99 (1954); KASUYA T., Prog. theor. Phys. (Kyoto), 16, 45 (1956); YOSIDA K., Phys. Rev. 106, 93 (1957).
6.
DOMB C.,Adv. Phys. 9, 149 (1960).
On a discuté un système paramagnétique oü des interactions spin— phonon se trouvent, et o~ion a traité de façon dassique les operateurs de spin. On a trouvé que le couplage de spin—phonon engendre un couplage d’echange entre les spins qui est effectif.
Solid State Communications,
Pergamon Press.
Printed in Great Britain
SPIN—PHONON INTERACTIONS IN CLASSICAL SPIN SYSTEMS N.W. Dalton Theoretical Physics Division, Atomic Energy Research Establishment, Harwell, England and
D.W. Wood Mathematics Department, University of Nottingham, Nottingham, England (Received 16 June 1969 by C.W. McCombie)
A paramagnetic lattice system is discussed where spin—phonon interactions are present, and where the spin operators are treated classically. It is found that the spin—phonon coupling gives rise to an effective exchange coupling between spins.
1of cerium THE SPECIFIC heat measurements and praseodymium ethyl sulphates have revealed unexpectedly large Schottky anomalies, which • of the standard cannot be explained on the basis theory of Schottky specific heats. The underlying assumptions of this theory are:
=
_msh*~H°~S~,
(3)
and a
a
_1L~L.~l~j -u R ~p — 2 R ~ \ R
}(
[a~(PYZ*~
+
~
(4)
(a) that the Hamiltonian of the system can ke expressed as a sum of a lattice Hamiltonian V which describes the thermal vibrations, and a spin Hamiltonian which covers the Zeeman interaction of the spins with the applied magnetic field H, and
Here ~R’ and UR are the a-components (a = x, y, z) of the momentum and displacement operators of the spin at the lattice site R; S~ is the a-component of the spin operator at R, Ha is the a.component of the external magnetic field, m ~ is the magnetic moment per spin and h* = h • • • if the spins are finite and is zero in the limit of
1~,
(b) that the exchange interactions between the spins are negligible.
•
It has therefore been suggested2 that a form of spin-lattice coupling may be significant and that a term representing such an effect should be included in the total Hamiltonian; and several authors 2—4 have studied Ham iltonians of the form =
L
S +
with this the case ~y(p) = 0, and spin ½. ~ith simplification the effects of the Even term
where
LS
HL
=
\‘f 1 \~a~a •~)~_•)PR ‘~R
+
ic’v
a
(P)
Va —
R~pJ
R
upon the thermodynamic properties of the
system very with complications arising is from theinvolved spin commutation relations. The purpose of this note is to show that if the spin operators are replaced by classical spin vectors (which will be a good approximation
R R
•
Previous work has been mainly concerned
(1)
LS
•
infinite spin (classical spins). The p are the lattice vectors; Ea~(P), and ~y(~) are components of the spin-lattice coupling tensors which vanish in zero magnetic field, and Ka(p) is the a-component of the elastic force constant.
‘2~
R+pJ~ ~
1271
1272
SPIN—PHONON INTERACTIONS IN CLASSICAL SPIN SYSTEMS
when the total spin quanturr~,number is 1arge)~ then the principle effect of ~LS is to add an effective spin exchange HamiltonianH~ to J{
with the result that
3.
+
k +
=
+~)+H~(13)
k
3{~,
(14)
where
H~
~
=
(15)
•
The effective classical spin Hamiltonian can be written in the more transparent form
It is convenient to rewrite HL and ~ LS in terms of the phonon creation and annihilation operators , and respectively, which satisfy the boson commutation relations
(R
,
~
=
R ‘)S~S~,
—
—
R R’
—
(h2/8m)~~J~ (R R ‘)S~S~,SR~’s—
~a+1
—
j
~4k,Ak’ La
— —
LS
The effect is analagous to an RKY 5coupling of The large spins by spin—electron Schottky anomalies now interactions. become understandable since the specific heat of an exchange coupled spin system in a weak magnetic field can be anomalously large for temperatures in the region of a critical temperature T~(see below).
Vol. 7, No. 17
(5)
=
RR’
—
The result is
(~2/8m)~~J~)(R- R’)S~S~S~,S~, ~~a+A~a
=
+
~),
(6)
(16)
RR’
—
where for example and =
,
-
LS
(7)
J~ (R
R’)
-
=
Ic
where (wIc) a2 =
~(K~
=
m
~i(
K~), K~
-
~ I~
~
=
(0)
~ Ka(p)eik.P; (8) p
~a$ (k)
=
(0)- ~
(k)ll
(0)
~
Ea~ (-k)~e(R~’)
(17)
(k)}s~ f
~
with similar expressions for J,~. (R
2mw~/ +
E~
k
{~‘c~~ (0)— ~
(k)~S~7] (9)
c~. (k)
—
~
—
R
‘), and
R’) (which are zero when ~~a~p)=
0).
In the linear coupling approximation where
(p)et~P
~a$
J~2~ (R
—
(p)eP
, (10)
the ~~~(p) are zero, the effect of is to add a Heisenberg spin—spin coupling term H ‘~ to Hç, where the coupling constant is approxi-
-
mately proportional to the square of the tensor
se
=
N1~2
Y~s~ ~
If J~ (R — R’) = J(R — R’) (say) is positive then from the simple molecular field ~a~3(P).
~
=
RSRe
;
(11)
and where S~ is the c-number (component of classical s~invector) which replaces the operator h*S~.The term linear in , and Ak is readily removed by a canonical transformation to the new boson operators B~, and defined by
A~a k
=
~a Bk
+
a /ha~, g_~
(12)
theory of the Heisenberg ferromagnet6 in the presence of a small magnetic field we would expect a large excess magnetic specific heat in the neighbourhood of the temperature T~ where
h2
2J (R R kT~ —
)
1/z
and z is the number of neighbours interacting with a given spin.
(18)
Vol. 7, No. 17
SPIN—PHONON INTERACTIONS IN CLASSICAL SPIN SYSTEMS
1273
REFERENCES 1.
MEYER H. and SMITH P.L., Physics Chem. Solids 9, 285 (1959).
2.
PERSICO F. and STEVENS K.W.H., Proc. phys. Soc. 82, 855 (1963).
3.
ELLIOT R.J. and PARKINSON J.B., Proc. phys. Soc. 92, 1024 (1967).
4.
STEVENS K.W.H. and VAN EEKELEN H.A.M., Proc. phys. Soc. 92, 680 (1967).
5.
RUDERMANN K. and KITTEL C., Phys. Rev. 96, 99 (1954); KASUYA T., Prog. theor. Phys. (Kyoto), 16, 45 (1956); YOSIDA K., Phys. Rev. 106, 93 (1957).
6.
DOMB C.,Adv. Phys. 9, 149 (1960).
On a discuté un système paramagnétique oü des interactions spin— phonon se trouvent, et o~ion a traité de façon dassique les operateurs de spin. On a trouvé que le couplage de spin—phonon engendre un couplage d’echange entre les spins qui est effectif.