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Magnon-phonon interaction in the itinerant electron model of ferromagnetism
TECHNICAL
Therefore,
The value of (76) can now be evaluated from the experimental values of B, at any two temperatures, and, in turn, is used to calculate the bulk modulii at various temperatures with a proper arbitrary parameter B,. A comparison of the estimated and experimental values are given in Fig. 3. In all the cases examined, a good agreement is found between the calculated and experimental values; and, therefore, Wachtman’s equation, as modified by Anderson, can very well be used for estimating the bulk modulii even at temperatures at which the experiments have not been carried out. Generally, it is observed that the estimation is quite near to the experimental points when the room temperature value of y and S = 2y are used. We thus confirm our earlier conclusion[ I] that the Wachtman’s equation can safely be used for calculating the temperature dependence of bulk modulus in oxide and nonoxide :.olids. Acknowledgemenfs - The author is grateful to Professor G. S. Verma and Dr. M. M. Joshi for their interest in the work. Financial assistance from the Department of Atomic Energy, India is thankfully acknowledged. S. RAJAGOPALAN Ultrasonic andSolid State Research Physics Department. Alluhahad Universiry. Alluhahud. India
1647
NOTES
6. DUGDA1.E J. S. and MCDONALD D. K. C., Phys. Rev. 89,832 (1953). 7. CHANG Y. A., J. Phys. Chem. Solids 28,697 (1%7). 8. GERLICH D. and FISHER E. S., J. Phys. Chem. Solids 30, I I97 ( I969 ). 9. WONG C. and SCHLELE D. E., J. Phys. Chem. Solids 29, I309 ( 1968). R. Q. and SCHUELE D. E., J. Phys. IO. FUGATE Chem. Solids 27,493 ( 1966). Theory of I I. BORN M. and HUANG K., Dynamical Crysral Lurrices. Clarendon Press, Oxford (I 954). 12. MATHUR V. K. and SINGH S. P., J. Phvs. Chem. Solids 29, Y59 (1968). T. and TRIVISONNO J.. J. Phps. 13. SLOTWINSKI Chem. Solids 30. 1276 (I 969). R. A. and SCHUELE D. E.. J. Phys. 14. BARTELS Chem. Solids 26,537 (1965).
Vol. 3 I,
J. Phys. Chem. Solids
pp. 1647..
1649.
Magnon-phonon interaction in the itinerant electron model of ferromagnetism (Rrceired I I September I Y69; in recisedf~wm I4 Norember 1969) WANT to present here a derivation of the bilinear magnon-phonon coupling using as a starting point the electron-phonon coupling and the itinerant electron model of ferromagnetism. The consequences of this coupling are well known and have been discussed by Kittel[ I] from a phenomenological point of view. The model Hamiltonian for a metallic ferromagnet in the presence of a magnetic held H is[2] WE
L&oratory.
+;
c C,+,,Ck,cq,Ck~,
(1)
kk’q
REFERENCES I. RAJAGOPALAN S., Solid Stale Commun. 7, 1649 (1969). 2. WACHTMAN J. B.. TEFFT W. E., LAM D. G. Jr. and APSTEIN C. S..Phys. Rev. 122, 1754(1961). 3. ANDERSON 0. I.., Phys. Rev. 144, 553 (1966). 4. MORSE G. E. and I.AWSON A. W.,J. Phys. Chem. Solids 28,93Y ( 1967). 5. American Institute of Physics Handbook. Chap. 7. Table 4. McGraw-Hill, New York (1957).
with EL”
=
lk-upH
(2)
where lk is the energy of the Bloch state Ikv) and CLis the Bohr magneton. C,‘, is the creation operator of an electron in the state )ka). U is the Coulomb integra1.o = ( f , & ).
TECHNICAl.
1638 Following
Rajagopal
the electron-phonon
The
parameter
of the coupling.
NOTES
and Joshi [3] we take
interaction
in the form
g,“,,,,(q) measures the strength bqh is the destruction operator
+&
for a phonon of momentum q and polarization A. From this interaction Hamiltonian we consider only those terms which give direct magnon-phonon coupling. i.e. we shall keep as interaction
F
;b+ = tE kq
A.
kfql
an
isotropic
model.
angular
(‘I
-Ekr)B:q
k
since we shall be interested
For
%
equation (6) becomes:
only in the qualitative aspects of the interaction we shall drop in what follows the polarization index
&Q \/,‘,/ T
= oFh,++
within the RPA
where
Furthermore,
(6)
and ih,+ =\H.b,‘I
(4)
,(‘k?
- C‘;_, _c,’ eq , lb;
Hamiltonian
c g;,cq)B& be + 1l.C. Hmwh = / ’ L N llqh
‘Oq’)K-:,,_,.
(8)
-,-,+; where
momentum conservation implies that only circularly polarized phonons would be coupled to the magnons. The total Hamiltonian of the system is then: with
H = H,, + Hr,t,+ H,,,.,,
(0
4
Ilk,, =
= ;
F (CLC,,>.
where Fourier
//“h = 2’ of;“h;hq a
transforming
and eliminating
is the free phonon Hamiltonian. Following the equation of motion method [4] we try now to hnd simultaneous eigenfrequencies within the random-phase approximation (RPA) corresponding created by b,’ tphonons) and
qq
= E,: q.
uk can be determined cigenfrequencies. we lind:
and (8)
-E,-)B;q+~(n,,ql-“kt)
later. in solving for the a simple
amplitudes
B;,tIZ).
Multiplying equation (9) by (Y . and summing over k we are led to a set of homogeneous
tmagnons)
After
(7)
to the excitation for the Fourier
Si = C tr,R:,
equations
b: we obtain
calculation
equations involving the I?;,. The independence of the B;, implies a new set of equations for the cykfrom which we obtain the following
TECHNICAL
NOTES
eigenvalue equations[4]
[
REFERENCES
cl-g@$I = 1.
(10)
9
Assuming a spherical Fermi surface for evaluating the sum in equation (IO) we obtain to second order in q
I. KITTELC.,Phys. Reti. 110,836(1958). 2. HEKRING C., Exchange Inreracrions among Ilinerclnr Electrons (Edited by W. G. Rado and H. Suhl), Vol. 4. Academic Press, New York (I 966). 3. RAJAGOPAI. A. K. and JOSHI S. K.. Phys. Lert. 24A, 95 ( 1967). 4. See for instance: BLANDIN A., Thcwp of (‘ondenwd Marfar. IAEA, Vienna ( 1966).
J. Phys. Chem. Solids
~yJ(TL[]_~(,_~~)]
”
I649
(Received
1 (11)
where LI = CI- urn - ZpH, m =.; (/ll, 7 - /lk, ) is the net moment, and Ed, IS the Fermi energy for the up and down bands respectively. Taking now p-H -+Um and using the magnon frequency
the roots of equation ( 1 1) can be written as: J’H
n=
+
q2
&/
q -+[
1”:;3)1 +,p(q),!2m]l/2. (12)
We see from ( 12) that the gap at resonance
is directly coupling.
related
to the electron-phonon
BI.AS AIASCIO ARTURO 1.t)PF.Z Crnrro Aromiw Ruriloclw, Comision Notional dc Encrgkr lnsfiiuto de Fisiccr.
“Dr
Josh
A. Balseiro”.
Unioersidud Nrrcionul de Cuyo, San C‘arlos de Bariloche, Rio Negro -Argentitw
Aldmicu.
pp. 1649- 1650.
Field emission in high magnetic fields
[
-~[,+~(I+;~)]]=
Vol. 3 I,
I7 September
1969)
BEHAVIOR of a field emission current in a magnetic field has been of interest for some time [ I, 21. Blatt [3J was the first to theoretically investigate such behavior for a gas of free electrons at the absolute zero of temperature. His results showed that the current should exhibit oscillations which are periodic in H-‘. Independently, and very shortly after Blatt’s calculation, Gogadze, Itskovich, and Kulik[41 considered the more general case of an arbitrary Fermi surface and non-zero temperature, with the restriction that the electron orbits around the extremal areas are closed and non-self-intersecting. Both papers consider the case for which the Fermi energy is independent of magnetic field, and for which the Fermi energy oscillates with magnetic field. Gogadze et ~I.[41 estimated that observable oscillations in both cases are a possibility only when the current originates from small pockets of electrons with low effective masses. This was essentially the same conclusion reached by Blatt. Numerical estimates of the amplitude of the oscillations from both calculations indicate effects on the order of 2 or 3 per cent at 1 tesla for an electron pocket with an effective mass of - 10 z m,,. At 3 teslas the amplitudes should be nearly 10 per cent of the emission current in zero magnetic field. In addition to the oscillatory terms, Blatt [3] THE
Therefore,
The value of (76) can now be evaluated from the experimental values of B, at any two temperatures, and, in turn, is used to calculate the bulk modulii at various temperatures with a proper arbitrary parameter B,. A comparison of the estimated and experimental values are given in Fig. 3. In all the cases examined, a good agreement is found between the calculated and experimental values; and, therefore, Wachtman’s equation, as modified by Anderson, can very well be used for estimating the bulk modulii even at temperatures at which the experiments have not been carried out. Generally, it is observed that the estimation is quite near to the experimental points when the room temperature value of y and S = 2y are used. We thus confirm our earlier conclusion[ I] that the Wachtman’s equation can safely be used for calculating the temperature dependence of bulk modulus in oxide and nonoxide :.olids. Acknowledgemenfs - The author is grateful to Professor G. S. Verma and Dr. M. M. Joshi for their interest in the work. Financial assistance from the Department of Atomic Energy, India is thankfully acknowledged. S. RAJAGOPALAN Ultrasonic andSolid State Research Physics Department. Alluhahad Universiry. Alluhahud. India
1647
NOTES
6. DUGDA1.E J. S. and MCDONALD D. K. C., Phys. Rev. 89,832 (1953). 7. CHANG Y. A., J. Phys. Chem. Solids 28,697 (1%7). 8. GERLICH D. and FISHER E. S., J. Phys. Chem. Solids 30, I I97 ( I969 ). 9. WONG C. and SCHLELE D. E., J. Phys. Chem. Solids 29, I309 ( 1968). R. Q. and SCHUELE D. E., J. Phys. IO. FUGATE Chem. Solids 27,493 ( 1966). Theory of I I. BORN M. and HUANG K., Dynamical Crysral Lurrices. Clarendon Press, Oxford (I 954). 12. MATHUR V. K. and SINGH S. P., J. Phvs. Chem. Solids 29, Y59 (1968). T. and TRIVISONNO J.. J. Phps. 13. SLOTWINSKI Chem. Solids 30. 1276 (I 969). R. A. and SCHUELE D. E.. J. Phys. 14. BARTELS Chem. Solids 26,537 (1965).
Vol. 3 I,
J. Phys. Chem. Solids
pp. 1647..
1649.
Magnon-phonon interaction in the itinerant electron model of ferromagnetism (Rrceired I I September I Y69; in recisedf~wm I4 Norember 1969) WANT to present here a derivation of the bilinear magnon-phonon coupling using as a starting point the electron-phonon coupling and the itinerant electron model of ferromagnetism. The consequences of this coupling are well known and have been discussed by Kittel[ I] from a phenomenological point of view. The model Hamiltonian for a metallic ferromagnet in the presence of a magnetic held H is[2] WE
L&oratory.
+;
c C,+,,Ck,cq,Ck~,
(1)
kk’q
REFERENCES I. RAJAGOPALAN S., Solid Stale Commun. 7, 1649 (1969). 2. WACHTMAN J. B.. TEFFT W. E., LAM D. G. Jr. and APSTEIN C. S..Phys. Rev. 122, 1754(1961). 3. ANDERSON 0. I.., Phys. Rev. 144, 553 (1966). 4. MORSE G. E. and I.AWSON A. W.,J. Phys. Chem. Solids 28,93Y ( 1967). 5. American Institute of Physics Handbook. Chap. 7. Table 4. McGraw-Hill, New York (1957).
with EL”
=
lk-upH
(2)
where lk is the energy of the Bloch state Ikv) and CLis the Bohr magneton. C,‘, is the creation operator of an electron in the state )ka). U is the Coulomb integra1.o = ( f , & ).
TECHNICAl.
1638 Following
Rajagopal
the electron-phonon
The
parameter
of the coupling.
NOTES
and Joshi [3] we take
interaction
in the form
g,“,,,,(q) measures the strength bqh is the destruction operator
+&
for a phonon of momentum q and polarization A. From this interaction Hamiltonian we consider only those terms which give direct magnon-phonon coupling. i.e. we shall keep as interaction
F
;b+ = tE kq
A.
kfql
an
isotropic
model.
angular
(‘I
-Ekr)B:q
k
since we shall be interested
For
%
equation (6) becomes:
only in the qualitative aspects of the interaction we shall drop in what follows the polarization index
&Q \/,‘,/ T
= oFh,++
within the RPA
where
Furthermore,
(6)
and ih,+ =\H.b,‘I
(4)
,(‘k?
- C‘;_, _c,’ eq , lb;
Hamiltonian
c g;,cq)B& be + 1l.C. Hmwh = / ’ L N llqh
‘Oq’)K-:,,_,.
(8)
-,-,+; where
momentum conservation implies that only circularly polarized phonons would be coupled to the magnons. The total Hamiltonian of the system is then: with
H = H,, + Hr,t,+ H,,,.,,
(0
4
Ilk,, =
= ;
F (CLC,,>.
where Fourier
//“h = 2’ of;“h;hq a
transforming
and eliminating
is the free phonon Hamiltonian. Following the equation of motion method [4] we try now to hnd simultaneous eigenfrequencies within the random-phase approximation (RPA) corresponding created by b,’ tphonons) and
= E,: q.
uk can be determined cigenfrequencies. we lind:
and (8)
-E,-)B;q+~(n,,ql-“kt)
later. in solving for the a simple
amplitudes
B;,tIZ).
Multiplying equation (9) by (Y . and summing over k we are led to a set of homogeneous
tmagnons)
After
(7)
to the excitation for the Fourier
Si = C tr,R:,
equations
b: we obtain
calculation
equations involving the I?;,. The independence of the B;, implies a new set of equations for the cykfrom which we obtain the following
TECHNICAL
NOTES
eigenvalue equations[4]
[
REFERENCES
cl-g@$I = 1.
(10)
9
Assuming a spherical Fermi surface for evaluating the sum in equation (IO) we obtain to second order in q
I. KITTELC.,Phys. Reti. 110,836(1958). 2. HEKRING C., Exchange Inreracrions among Ilinerclnr Electrons (Edited by W. G. Rado and H. Suhl), Vol. 4. Academic Press, New York (I 966). 3. RAJAGOPAI. A. K. and JOSHI S. K.. Phys. Lert. 24A, 95 ( 1967). 4. See for instance: BLANDIN A., Thcwp of (‘ondenwd Marfar. IAEA, Vienna ( 1966).
J. Phys. Chem. Solids
~yJ(TL[]_~(,_~~)]
”
I649
(Received
1 (11)
where LI = CI- urn - ZpH, m =.; (/ll, 7 - /lk, ) is the net moment, and Ed, IS the Fermi energy for the up and down bands respectively. Taking now p-H -+Um and using the magnon frequency
the roots of equation ( 1 1) can be written as: J’H
n=
+
q2
&/
q -+[
1”:;3)1 +,p(q),!2m]l/2. (12)
We see from ( 12) that the gap at resonance
is directly coupling.
related
to the electron-phonon
BI.AS AIASCIO ARTURO 1.t)PF.Z Crnrro Aromiw Ruriloclw, Comision Notional dc Encrgkr lnsfiiuto de Fisiccr.
“Dr
Josh
A. Balseiro”.
Unioersidud Nrrcionul de Cuyo, San C‘arlos de Bariloche, Rio Negro -Argentitw
Aldmicu.
pp. 1649- 1650.
Field emission in high magnetic fields
[
-~[,+~(I+;~)]]=
Vol. 3 I,
I7 September
1969)
BEHAVIOR of a field emission current in a magnetic field has been of interest for some time [ I, 21. Blatt [3J was the first to theoretically investigate such behavior for a gas of free electrons at the absolute zero of temperature. His results showed that the current should exhibit oscillations which are periodic in H-‘. Independently, and very shortly after Blatt’s calculation, Gogadze, Itskovich, and Kulik[41 considered the more general case of an arbitrary Fermi surface and non-zero temperature, with the restriction that the electron orbits around the extremal areas are closed and non-self-intersecting. Both papers consider the case for which the Fermi energy is independent of magnetic field, and for which the Fermi energy oscillates with magnetic field. Gogadze et ~I.[41 estimated that observable oscillations in both cases are a possibility only when the current originates from small pockets of electrons with low effective masses. This was essentially the same conclusion reached by Blatt. Numerical estimates of the amplitude of the oscillations from both calculations indicate effects on the order of 2 or 3 per cent at 1 tesla for an electron pocket with an effective mass of - 10 z m,,. At 3 teslas the amplitudes should be nearly 10 per cent of the emission current in zero magnetic field. In addition to the oscillatory terms, Blatt [3] THE