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The spin-wave contribution to the specific heat of NiF2
Volume 28A, number 8
THE
PHYSICS
SPIN-WAVE
LETTERS
CONTRIBUTION
TO THE
27 January 1969
SPECIFIC
HEAT
OF NiF2
S. J. JOSHUA and A. P. CRACKNELL Department
of Physics,
University
of Essex,
Wivenhoe Park,
Colchester,
Essex,
U.K.
Received 13 December 1968
The spin-wave contribution to the specific heat of NiF over the range 0.36oK to 500K has been calculated using the dispersion relations given by Moriya; t;i,e results agree well with the experimental results of Stout and Catalano.
A theoretical derivation of expressions for the magnon frequencies in the canted antiferromagnetic state of NiF2 was given by Moriya [l] for magnons near the centre of the Brillouin zone. The theory was later extended to cover all wave vectors in the Brillouin zone [2]. By using the experimental values of the canting angle and the exchange constants, determined from magnetic susceptibility and antiferromagnetic resonance experiments [3-41, we have previously shown [5] that the expressions for the two magnon frequencies v+(k) and v_(k) can now be evaluated all over the Brillouin zone :
u,(k) = 125.0
[I
1.01569 -
(1
Table 1 Comparison of the experimental [6] and calculated magnetic specific heat of NiF2. T
31.1 - (173k)J3
I
where rIk, y2k and Y3k are certain functions of k used by Moriya [2]. At any given temperature T we may use the Bose-Einstein distribution to determine the total energy UM of the spin waves that are excited in the crystal,
(2) The contribution CM to the specific heat of the crystal due to the excitation of spin waves is therefore given by aUM/aT, so that
where x = (hv/kT). We have used the expression given above for v+(k) to calculate the magnon energies at 8 x 106 points in the Brillouin zone and thence the magnon density of states curve g(v). This density of
CM (calculated)
cM (61
(OK) (J-ldeg-lmole-1)
(J-ldeg-lmole-l) a!=1
2 62.2 -y2&2
- {ylk f 0.01534}2 1’ cm-l
562
states curve was then used to perform a numerical integration of the above expression for CM at a large number of temperatures from 0.36oK to 500K. The results are given in table 1, together with the experimental values of Stout and Catalan0 [6].
0.36 2.5 5.0 7.5 10.0 12.5 15 20 25 30 35 40 45 50
0.184 0.431 0.917 1.632 2.553 3.599 4.813 6.194
8X10-10 0.0002 0.002 0.006 0.022 0.073 0.188 0.663 1.416 2.257 3.080 3.814 4.435 4.959 __.~
a! =O.l 8X10-lo 0.0002 0.002 0.007 0.024 0.079 0.229 0.693 1.450 2.286 3.100 3.828 4.441 4.962
o! = 10 2x1O-g 0.0002 0.002 0.007 0.028 0.095 0.230 0.758 1.524 2.353 3.155 3.874 4.480 4.997
The general agreement with the experimental results is quite good but the following points should be noted. (i) Stout and Catalan0 [6] have obtained CM by
subtracting from the total specific heat of antiferromagnetic NiF2 the total specific heat of (non-magnetic) ZnF2, weighted appropriately for the differences in mass and lattice spacing. That is, the phonon dispersion relations of NiF2 and ZnF2 have been assumed to obey a “law of corres ponding states”. While this is clearly a good
PHYSICS
Volume 28A, number 8
27 January 1969
LETTERS
(iv) By plotting log CM against log T we find that first approximation it is unlikely to be completely the temperature variation of CM is quite close correct. to being a T3 variation, namely CM = AT3a4 be(ii) Moriya’s model is not valid once the degree low lOoK, which agrees well with figure 6 of Stout of order becomes significantly reduced below 1 and Catalan0 [6]. so that we have stopped our calculation well below the N6el point (TN = ‘73.2oK). (iii) The separate values of J2 and 53 are not actually known but it is known [3] that 2J2 + 4J3 = References 1. T.Moriya, whys. Rev. 117 (1960) 635. = (28.5 f 2.1) cm-l. We have written J2 =crJ3 and 2. T. Moriya, J. Phys. Sot. Japan 21 (1966) 926. satisfying the above condition on (2J2 + 453) have 3. R. J. Joenk and R. M. Bozorth, Proc. Int. Conf. on calculated CM for three different sets of values Magnetism, Nottingham(1964) 493. of (Y, see columns 3, 4 and 5 of table 1. The dif4. P. L. Richards. Phvs. Rev. 136 (1965) A1769. ference between the values of CM obtained with 5. S. J. Joshua and A.?‘. Cracknell, ‘J. Pdys. C (hoc. these three different sets of values can be seen Phys. Sot.) 2 (1969) 24. 6. J. W. Stout and E. Catalano, J. Chem. Phys. 23 to very small varying from 0 at 5oK to 1% at (1955) 2013. 500K but being somewhat larger in between, namely 15% at 15’K. ***1(
NOUVEAUX
PROCESSUS PAR
IMPACTS
D’IONISATION
DU XENON
D’ELECTRONS
J. PERESSE, F. TUFFIN and G. LE COZ Facult6
des Sciences Received
de Brest,
29 N - France
15 July 1968
Because of the large separation $etween the 2P3 and 2P~ levels of Xe+, we have observed several autoionisation processes between these two levels. ‘We can e2mphasize that in the threshold region a great number of ions are created through autoionisation processes.
La skparation entre les niveaux 2P3 et 2P+ du x&on est d’environ 1.30 eV. C’est po&quoi cette region a BtB systematiquement Btudie’e par plusieurs auteurs. D’apr& les r&ultats obtenus par ces auteurs la courbe d’ionisation peut se d&omposer en differentes parties linkaires. Fox et al. [l] obtinrent une cassure P 1.27 eV au-dessus du seuil; ce changement de pente corresponyt vraisemblablement ri l’ionisation au niveau Pi. Cloutier et Schiff [2] mirent en Bvidence le seuil d’apparation de ce processus Q 1.31 eV. Clarke [3] obtint deux cassures P 0.82 eV et 1.93 eV qui ne peuvent @tre reli6es & aucun processus connu. Foner et Nall [4] mire& en 6vidence l’apparition de nouveaux processus 5 0.70 eV, 1.33 eV et 2.05 eV au-dessus du seuil d’ionisation. RBcemment des Etudes plus pr&ises furent entreprises par Burns [5] 5 l’aide de la m&bode
R.P.D. et par Morrison [6] P l’aide d’un &lecteur e’lectrostatique cylindrique. Ces derniers auteurs ont montrd que la courbe d’ionisation entre les etats 2P1 et 2Pl n’&ait pas 1inGaire mais pr&entait uze stru&re plus compliqu~e due P des processus d’autoionisation. On remarquera done que l’accord n’est pas bon entre les differents auteurs. Il existe un d&accord d’une part SUF la forme g&&ale de la courbe d’ionisation et d’autre part sur les potentiels d’apparition des differents processus. On notera Bgalement, comme dans le cas de l’argon, que la structure obtenue par les auteurs ayant utilise la m&hode du selecteur Blectrostatique semble plus riche que celle obtenue par la m& thode R.P.D. Nous avons utilise pour cette &ude un &lecteur cylindrique de faible dispersion en e’nergie [8]. La courbe, que nous avons obtenue, de%ute 563
THE
PHYSICS
SPIN-WAVE
LETTERS
CONTRIBUTION
TO THE
27 January 1969
SPECIFIC
HEAT
OF NiF2
S. J. JOSHUA and A. P. CRACKNELL Department
of Physics,
University
of Essex,
Wivenhoe Park,
Colchester,
Essex,
U.K.
Received 13 December 1968
The spin-wave contribution to the specific heat of NiF over the range 0.36oK to 500K has been calculated using the dispersion relations given by Moriya; t;i,e results agree well with the experimental results of Stout and Catalano.
A theoretical derivation of expressions for the magnon frequencies in the canted antiferromagnetic state of NiF2 was given by Moriya [l] for magnons near the centre of the Brillouin zone. The theory was later extended to cover all wave vectors in the Brillouin zone [2]. By using the experimental values of the canting angle and the exchange constants, determined from magnetic susceptibility and antiferromagnetic resonance experiments [3-41, we have previously shown [5] that the expressions for the two magnon frequencies v+(k) and v_(k) can now be evaluated all over the Brillouin zone :
u,(k) = 125.0
[I
1.01569 -
(1
Table 1 Comparison of the experimental [6] and calculated magnetic specific heat of NiF2. T
31.1 - (173k)J3
I
where rIk, y2k and Y3k are certain functions of k used by Moriya [2]. At any given temperature T we may use the Bose-Einstein distribution to determine the total energy UM of the spin waves that are excited in the crystal,
(2) The contribution CM to the specific heat of the crystal due to the excitation of spin waves is therefore given by aUM/aT, so that
where x = (hv/kT). We have used the expression given above for v+(k) to calculate the magnon energies at 8 x 106 points in the Brillouin zone and thence the magnon density of states curve g(v). This density of
CM (calculated)
cM (61
(OK) (J-ldeg-lmole-1)
(J-ldeg-lmole-l) a!=1
2 62.2 -y2&2
- {ylk f 0.01534}2 1’ cm-l
562
states curve was then used to perform a numerical integration of the above expression for CM at a large number of temperatures from 0.36oK to 500K. The results are given in table 1, together with the experimental values of Stout and Catalan0 [6].
0.36 2.5 5.0 7.5 10.0 12.5 15 20 25 30 35 40 45 50
0.184 0.431 0.917 1.632 2.553 3.599 4.813 6.194
8X10-10 0.0002 0.002 0.006 0.022 0.073 0.188 0.663 1.416 2.257 3.080 3.814 4.435 4.959 __.~
a! =O.l 8X10-lo 0.0002 0.002 0.007 0.024 0.079 0.229 0.693 1.450 2.286 3.100 3.828 4.441 4.962
o! = 10 2x1O-g 0.0002 0.002 0.007 0.028 0.095 0.230 0.758 1.524 2.353 3.155 3.874 4.480 4.997
The general agreement with the experimental results is quite good but the following points should be noted. (i) Stout and Catalan0 [6] have obtained CM by
subtracting from the total specific heat of antiferromagnetic NiF2 the total specific heat of (non-magnetic) ZnF2, weighted appropriately for the differences in mass and lattice spacing. That is, the phonon dispersion relations of NiF2 and ZnF2 have been assumed to obey a “law of corres ponding states”. While this is clearly a good
PHYSICS
Volume 28A, number 8
27 January 1969
LETTERS
(iv) By plotting log CM against log T we find that first approximation it is unlikely to be completely the temperature variation of CM is quite close correct. to being a T3 variation, namely CM = AT3a4 be(ii) Moriya’s model is not valid once the degree low lOoK, which agrees well with figure 6 of Stout of order becomes significantly reduced below 1 and Catalan0 [6]. so that we have stopped our calculation well below the N6el point (TN = ‘73.2oK). (iii) The separate values of J2 and 53 are not actually known but it is known [3] that 2J2 + 4J3 = References 1. T.Moriya, whys. Rev. 117 (1960) 635. = (28.5 f 2.1) cm-l. We have written J2 =crJ3 and 2. T. Moriya, J. Phys. Sot. Japan 21 (1966) 926. satisfying the above condition on (2J2 + 453) have 3. R. J. Joenk and R. M. Bozorth, Proc. Int. Conf. on calculated CM for three different sets of values Magnetism, Nottingham(1964) 493. of (Y, see columns 3, 4 and 5 of table 1. The dif4. P. L. Richards. Phvs. Rev. 136 (1965) A1769. ference between the values of CM obtained with 5. S. J. Joshua and A.?‘. Cracknell, ‘J. Pdys. C (hoc. these three different sets of values can be seen Phys. Sot.) 2 (1969) 24. 6. J. W. Stout and E. Catalano, J. Chem. Phys. 23 to very small varying from 0 at 5oK to 1% at (1955) 2013. 500K but being somewhat larger in between, namely 15% at 15’K. ***1(
NOUVEAUX
PROCESSUS PAR
IMPACTS
D’IONISATION
DU XENON
D’ELECTRONS
J. PERESSE, F. TUFFIN and G. LE COZ Facult6
des Sciences Received
de Brest,
29 N - France
15 July 1968
Because of the large separation $etween the 2P3 and 2P~ levels of Xe+, we have observed several autoionisation processes between these two levels. ‘We can e2mphasize that in the threshold region a great number of ions are created through autoionisation processes.
La skparation entre les niveaux 2P3 et 2P+ du x&on est d’environ 1.30 eV. C’est po&quoi cette region a BtB systematiquement Btudie’e par plusieurs auteurs. D’apr& les r&ultats obtenus par ces auteurs la courbe d’ionisation peut se d&omposer en differentes parties linkaires. Fox et al. [l] obtinrent une cassure P 1.27 eV au-dessus du seuil; ce changement de pente corresponyt vraisemblablement ri l’ionisation au niveau Pi. Cloutier et Schiff [2] mirent en Bvidence le seuil d’apparation de ce processus Q 1.31 eV. Clarke [3] obtint deux cassures P 0.82 eV et 1.93 eV qui ne peuvent @tre reli6es & aucun processus connu. Foner et Nall [4] mire& en 6vidence l’apparition de nouveaux processus 5 0.70 eV, 1.33 eV et 2.05 eV au-dessus du seuil d’ionisation. RBcemment des Etudes plus pr&ises furent entreprises par Burns [5] 5 l’aide de la m&bode
R.P.D. et par Morrison [6] P l’aide d’un &lecteur e’lectrostatique cylindrique. Ces derniers auteurs ont montrd que la courbe d’ionisation entre les etats 2P1 et 2Pl n’&ait pas 1inGaire mais pr&entait uze stru&re plus compliqu~e due P des processus d’autoionisation. On remarquera done que l’accord n’est pas bon entre les differents auteurs. Il existe un d&accord d’une part SUF la forme g&&ale de la courbe d’ionisation et d’autre part sur les potentiels d’apparition des differents processus. On notera Bgalement, comme dans le cas de l’argon, que la structure obtenue par les auteurs ayant utilise la m&hode du selecteur Blectrostatique semble plus riche que celle obtenue par la m& thode R.P.D. Nous avons utilise pour cette &ude un &lecteur cylindrique de faible dispersion en e’nergie [8]. La courbe, que nous avons obtenue, de%ute 563