Susceptibility maximum and logarithmic temperature dependence of magnetocaloric effect: Application to U2C3

Solid State Communications,Vol. 16, Pp. 1215—1218, 1975.

Pergamon Press.

Printed in Great Britain

SUSCEPTIBIUTY MAXIMUM AND LOGARITHMIC TEMPERATURE DEPENDENCE OF MAGNETOCALORIC EFFECT: APPLICATION TO U2C3 S. Misawa Department of Physics, College of Science and Engineering, Nihon University, Kanda-Sdrugadai, Tokyo, Japan (Received 16 December 1974; in revised form 27 January 1975 by P. G. de Gennes)

It is shown that the Fermi liquid model of magnetis~,when appliçd to U2 C3, can explain 2 In Tthe with susceptibility respect to temperature. maximum at Itabout is proposed 65 K through that a more a term precise of the proof form Ton the existence of this term may be found by observing the Tin T dependence of the magnetocaioric effect at low temperatures.

IN RECENT years efforts have been stimulated to find such substances for which there appears a maximum in the magnetic susceptibility as a function of temperature, yet the maximum is neither associated with an antiferromagnetic transition nor with a particular structure of the density-of-states curve. As is typical in Pd, the usual band theory based on the Stoner model has failed to explain the anomalous peakthat, of the biity at about 85°K.’It is noteworthy forsusceptiPd, any of recent results2 on band calculation which have been reported as to possibly account for the anomalous susceptibility behaviour can never explain theDensityexperi3 of the electronic specific heat. mental data of-states curves which give a peak in the susceptibility curve necessarily result in predicting, within the band model, a corresponding peak in the electronic specific heat coefficient 7(T),1 whereas no such peak has been observed experimentally.3 (Here 7(T) is defined as the electronic specific heat divided by temperature T). An alternative explanation on the phenomena of the susceptibility maximum has been given by the 4 in the Fermi liquid model of magnetism. present authora system of strongly correlating electrons If one regards as a Fermi liquid, the temperature dependence of the susceptibility x(T) is shown to be given by a logarithmic formula x(T)

=

a

—

bT2 hi (T/T*) (b > 0)

(1)

because of interactions and the existence of a sharp Fermi surface, where a, b and T* are constants. This relation predicts the existence of a maximum at a temperature Tm~,= T*/s,/e. We have already stated that, for Pd,5 a-Mn6 and CeSn 3 7 x follows this formula. We have also pointed out that peaks of the susceptibility appearing in 8’9Pt, Ta, a-U and a-Pu may be of similar nature. Here, in this note, we shall describe the case of uranium sesquicarbide U 2C3. Earlier a maximum of the susceptibility at about 65°Kwas thought to be attributed to an antiferromagnetic transition.10 However, since no anomaly was found in the specific heat and transport properties in the temperature range concerned, nor evidence of a magnetic ordering was observed in neutron diffraction and 13C Knight shift experiments, the antiferromagnetic model of U2C3 11 should have been abandoned. Now we shall examine whether the peak of the susceptibility in U 2C3 the canlogarithmic be explained by the(1). Fermi liquid model through formula First, it is to be noted that, if the theoretical prediction is right, T* should be about 105°K,since T* = .~.JeTm~and Tm~is already known to be about 65°K. In fact, the experimental data11 below about 75°K fit formula (1) very well107°K.Figure with any value1 of T* inthe thethe range between 103 and shows

1215

1216

MAGNETOCALORIC EFFECT: APPLICATION TO U2C3

/~1

24

~

/

a2

/ //

20 /
T

FIG. 1. Magnetic susceptibility of U 21n (lOS/T). Experimental data areasdue 2 C3 plotted a to Boutard function of and T de Novion.11 Solid straight line shows

that a fit of the data to the T21n (105/T) law is excellent below about 75°K.Solid and open circles denote respectively the experimental points below and above Tm~, 64°K. situation for T* = lOS°K.Above about 75°K the susceptibility begins to deviate from the T21n (T/T*) law. However, it is noteworthy that the deviation can be accurately expressed by form next order term in the logarithmic formula of the T41n (T/Tt), which has initially omitted (1). This reflects thatbeen the description by theinlogarithmic formulatheis fact not fortuitous but consistent and physically plausible. Thus, below about l00°K, the susceptibility data are precisely reproducible through the formula

~(T)

1.91 x l0~

=

+

F Ii

( ~)

2

—

-~

79.5

ln

T 105

+

1 ~-~--~J

T T ~-i-i-~-) ln

/

N

/

N

N

N

/ ~

—

8p~ 70~

3 2 l~(105/T}

I?

Vol. 16, Nos. 10/Il

emu/mole.

(2)

This relation is plotted in Fig. 2 together with the experimental data for comparison.

A word is prepared here concerning the plausibility of the the appearance of the logarithmic in susceptiU2 C3. From data on the specific heat12 terms and the bility, it is found that the electronic specific heat coefficient at T = 0, ‘ye, is anomalously large, 40 mu mole deg2, and the Stoner (or Landau) enhancement factor of the susceptibility, L, is fairly large, 3.65. Since the parameter b in (1) is, besides other factors,

~

0

-

25

50

FIG. 2. Magnetic susceptibility as a function of ternperature. Circles show the same experimental data as in Fig. 1. Solid curve shows a theoretical fit to the data, (2), predicted from the Fermi liquid model in the text. proportional to the square of y~and L,4’7 the logarithmic terms seem to appear in U 2C3 even much more preferably2 than in a-Mn for which mJ/ emu~ and L = 2.6. In fact, h y~ = 4= x 14 lO~~ moleS deg deg2/gr for a-Mn,6 while b for U 2C3 given in (2) 2/gr: this relative size may be is under6stood x l0~b0emu deg very roughly as to arise from the difference of these two Yo values. We have just seen that the susceptibility is really described by the logarithmic formula, nevertheless such a view is not wholly denied that the peculiar temperature dependence of x can be explained within the band model. This point may be ascertained, as

mentioned before, by examining whether or not there

exists a corresponding peak in y(T). There appears. however, one difficulty in this regard. Since the simple Debye model seems not to be valid for U 2 C3 ,12 separation of the observed specific heat into the electronic and the lattice parts is very difficult unless the phonon 3 spectrum is directly measured, as really done scattering. in Pd. through the experiment of inelastic neutron

We henceforth propose an alternative check as to whether the peak of x actually arises from the Fernil liquid effect or is simply explained by the band model. Since we are concerned with the magnetic susceptibility

Vol. 16, Nos. 10/il

MAGNETOCALORIC EFFECT: APPLICATION TO U2C3

it is natural to consider thermodynamic properties in the presence of magnetic field. First one might of observing thea specific heat in the presence of athink magnetic field H; the low temperature specific heat is given, in addition to the usual T.linear term, by tIC

where T

=

—

bH2 T ln (T/T),

=

Tmaic/e. This term predominates

=

e_3/2T*

(3)

over the usual term, because of the Tln Tbehaviour, at the lowest temperatures. However, tIC practically contributes only a tiny portion to the total specific heat even at very low temperatures and at very high fields; using the data of U2C3 and for H = 100 kG, one has

varies as ln (T/Tmax). On the contrary, according to 2, LInT the usual band theory on which x varies as T should be constant with T. Since in (T/Tm~)falls down more steeply as T 0 in marked contrast to the constant behaviour, one can in principle determine -+

whether theperforms data fit the ln T law at or moderately fit the constant law. If one experiment low temperatures, to avoid very low temperatures where the effect of magnetic impurities becomes appreciable, one should take the deviation of c(T) from Yo T into consideration; we henceforth define LITmod = LIT’ c(T)/yoT. Using the data for U 2C3,with H = 20 kG, one has LITmod T

~

1217

~ X

T n64.

Yo which is too small to be observed with our experimental accuracy. Apparently a more powerful method is to use the magnetocaloric effect; one measures the change of the temperature of the system, LIT, when a magnetic field is adiabatically applied from zero to a certain value H. This change is proportional to (3X/aT)H and is given, as far as equation (1) is valid, by

This order of a relative temperature change can be very easily detected in present-day experimental techniques. It is to be noted that, even Pd, which 5’13 theforrelative change has aHrelatively (for = 20 kG)small b/yo value, LIT T mod 3 x l0~In— T 87 can be measured within our experimental accuracy.

bT2H2 1T~ ln

In conclusion, it is desirable that the experiment along this line is performed presently for any substances exhibiting the susceptibility maximum.

LIT

=

C~ )

T

i~i~— ,

(4)

max

where c(T) is the specific heat per mole (if b is measured per mole) at constant field. The specific heat being proportional to T at very low temperatures LIT/T

1. 2. 3.

4. 5.

Acknowledgement The author is much indebted to Professor T. Ogasawara for suggesting that the magnetocaloric effect is experimentally superior to the specific heat measurements. —

REFERENCES THOMPSON E.D.,J. Phys. Chem. Solids 30, 1181(1969); SHIMIZU M., TAKAHASHI T. and KATSUKI A., I Phys. Soc. Japan 18, 240 (1963). VAN DAM J.E. and ANDERSEN O.K., Solid State Commun. 14, 645 (1974); LIPTON D. and JACOBS R.L., J. Phys. C: Metal Phys. Suppi. 3,S389 (1970). MILLER A.P. and BROCKHOUSE B.N., Phys. Rev. Lett. 20, 798 (1968). Precisely speaking, y(T) given in this reference exhibits a small local maximum at about 1 50°K.In the present article, however, a “peak” means exclusively an absolute maximum (not a local maximum) and is defined, moreover, as to satisfy the requirement that d 7(T)/dT> 0, or dX(T)/dT> 0, for 0< T< Tmax, where Tmax is the temperature of the maximum in y(T) or MISAWA S., Phys. Lett. 32A, 153 (1970); in Proc. 12th mt. Conf on Low Temp. Phys. Kyoto, 197U(Edited by KANDA E.) p. 151, Keigaku, Tokyo (1971). MISAWA S.,Phys. Lett. 32A, 541 (1970);Phys. Rev. Lett. 26, 1632 (1971); JAMIESON H.C. and MANCHESTER F.D., J. Phys. F.’ Metal Phys. 3, 323 (1972).

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MAGNETOCALORIC EFFECT: APPLICATION TO U2C3

Vol. 16, Nos. 10/li

6.

MISAWA S.,Phys. Lett. 44A, 333 (1973).

7. 8.

MISAWA S., Solid State Commun. 15, 507 (1974). MISAWA S.,Proc. mt. Conf Magnetism, Moscow, 1973, Vol. III, p. 349, Nauka, Moscow (1974). 21n Tbehaviour; cf. reference 4. In contrast to the case of x(T), however, Corresponding wheretothe x(T), coefficient y(T) also of the shows T2ln a TT term is negative definite because of thermodynamic stability, here the coefficient is generally positive and hence this contribution to y(T) exhibits no maximum. It should be noted, moreover, that this term is generally not observable because of being masked by the lattice specific heat, as seen from the fact that the relative magnitude to the lattice term is of the order of (~~/Tp~)3, where 0D and T are the Debye and the effective Fermi temperatures. DE NOVION C.H., COSTA P. and DEAN G.,Phys. Lett. 19,445(1965).

9

10. 11. 12.

BOUTARD J.L. and DE NOVION C.H., Solid State Commun. 14, 181(1974). ANDON R.J.L., COUNSELL J.F., MARTIN J.F. and HEDGER H.J., Trans. Faraday Soc. 60, 1030 (1964); FARR J.D., WITTEMAN W.G., STONE P.L. and WESTRUM E.F., Jr.,Advances in Thermophysical Properties at Extreme Temperatures and Pressures, pp. 162—166, Conference ASME, New York (1965).

13.

VEAL B.W. and RAYNE J.A.,Phys. Rev. 135, A442 (1964).