Entropy of ordering in some uranium compounds

Solid State Communications,

vol. 8, pp. 1673—1676, 1970. Pergamon Press.

Printed in Great Britain

ENTROPY OF ORDERING IN SOME URANIUM COMPOUNDS

J. Grunzweig-Genossar Department of Physics Technion, Israel Institute of Technology, Haifa, Israel. (Received 5 August 1970 by J.L. Olsen)

Calorimetric data on the magnetic entropy in certain ordering UX compoundsare interpreted, using a model which assumes the presence 4~ions, with a singlet ground state. The crystal field splittings of areUcalculated and compared with splittings obtained from magnetic measurements.

IN A RECENT review article, Westrum and Lyon1 collected the results of calorimetric measurements on uranium compounds and discussed, amoung others, the problem of Sm, the change of entropy connected with the magnetic ordering transitions. In some uranium compounds, Sm is small, and thus cannot easily be interpreted using assumptions implied in the formula Sm = Nk ln(2J + 1), where J is the appropriate angular momentum (total or spin) quantum number of the magnetic uranium ion.

interpreted fairly satisfactorily, using a molecular field model. However, as we shall deal here with the difference in entropy, only the initial and final states are of importance; the difference does not depend on the ipproximations used to evaluate the complicated interactions between the moments close below the ordering temperature, TN or T~.

E

Counsell, Dell and MartinZ3 suggested that, in uranium compounds, the effect is due to the splitting of levels in the crystalline field of the neighbouring ions. We shall develop this suggestion and attempt to interpret the reported measurements of Sm. Our model assumes that the ground multiplet of the paramagnetic ion is split by the crystal field, so that the resulting lowest state is a singlet. At low temperature, under influence of the internal exchange magnetic fields, the singlet acquires a magnetic moment due to the off-diagonal matrix elements of the exchange interaction. Wang and Cooper 4have discussed in detail the magnetization M, and the magnetic specific heat Cm, of a system of interacting ions with a singlet ground state. Their results show that the detailed temperature dependence of M and of Cm varies greatly with the approximations used in calculations (molecular field, RPA, two-site-correlation approximation). Recently, Stutius5 showed that his Cm measurements on rare earth nitrides can be

1673

AE,

I

-

I

H~ b

a

Low energy levels of the paramagnetic ion as function of Hex, the exchange field (schematic). FIG-i.

1674

ENTROPY OF ORDERING IN SOME URANIUM COMPOUNDS

The entropy in the ordered state at T = 0 (state (a) in Fig. 1) is therefore S = 0. On heating, as the temperature exceeds the Curie (or Ned) temperature, the exchange field H~ collapses and the system becomes paramagnetic (state (b) in Fig. 1). At temperature Tb close above TCN the entropy of the system is Sb; hence Sm = Sb = Sb. We assume that at Tb (= ~ + c), only the two low levels with energies E E2 and degeneracies g1, g2

i

Vol.8, No.21

—

~ 3 g~.2

—

~,

respectively (g1 = 1) are appreciably occupied and the population of the higher levels can be neglected. We put x = 13(E2 E1) = ~L\E12 where ~ = 1/kT. The entropy Sb is calculated as usual. Z is the partition function. Thus

~

-

—

Sm

Sb

=

=

+

=

Nk (in Z

Nk Un (1

xg2 exp(

—

+

—

(~/Z)aZ/a~~b

g2 exp(

x)/[1

+

—

=

x)~+

g2 exp(

—

x)1

(1)

a function of g2 and of x: Sm is plotted for = 2,3,4 in Fig. 2 as a function of x. The calculations show that, for crystal field splitting, appreciably larger than the magnetic transition temperature, the change of entropy is small because the ground singlet remains mainly occupied, in spite of the disappearance of the magnetic field on passing TCN. In the model previously proposed for the 6 it UX compounds with (B 1)ion structure, was assumed that theNaCl uranium is present in the 2 3H 4 configuration. This configuration, in a crystal field of octahedral symmetry, splits with a singlet ground state, while the next higher level is a F4 state with g2 = 3. This is a system analogous to the one dealt with in this paper and hence equation (1) should be applicable for the calculation of the entropy of ordering in these UX compounds. From the magnetic entropy changes reported in literature, and the transition temperatures, we calculated the splitting between the ground singlet and the first excited state (using the above equation): t\ E~2 = E2 E1. The results are shown in Table 1, together with those obtained from rnagnetic measurements A E7~.UP has an additional transformation at 22°K when the magnetisation moment of the uranium ion changes abruptly by ~

T

10

~o’

I

0

I

2

3

4

5

6

7

8

X

2. Entropy of the system with singlet level 1 and g2 fold degenerate next higher level as function of x = (E2 E1)/kT. FIG.

—

—

14 per cent, but there is no accompanying change of magnetic order or crystal we structure. Forentropy the purpose of our calculation, added the changes at 22°K and around TN. The agreement

—

—

between L~.E~2and ~ is fair, except in the case of UN, where the two differ by a factor of about two. In the rest of the cases the L\.EI’~’ values are usually lower than those obtained from the magnetic entropy measurements. This may be due to the simplified assumptions used in the magnetic model case, which neglected the decrease of the RKKY coupling with temperature. On heating, the mean free path of the conduction electrons decreases, due both to the scattering by phonons and by the localized magnetic spins disordered on approach to the transition temperature. This usually causes a reduction of the exchange interaction between the paramagnetic moments, and hence of the local magnetic field, thus speeding up the collapse of the ordered state. The actual transition temperature is

Vol.8, No.21

ENTROPY OF ORDERING IN SOME URANIUM COMPOUNDS

1675

Table 1. The experimentally determined entropy of ordering Sm of certain magnetic uranium compounds, the crystal field splittings AE~ 6 (column 7).2as x =derived (E from Sm (column 6) and AE,?~~splittings obtained from magnetic measurements 2 El)/kTb Tmax is the temperature of the maximum of the Schottky specific heat in a non-ordering system with the same crystal splitting as in column 6. Apart from the cases indicated, all calculations aTe for g2 = 3. —

Compound

TN,C (°IQ

UN

52

(Sm)exp (cal.deg -2 g.ion’ 0.15

uS

22.5 121 180

0.17 0.45 0.74 1.19 1.63

USe U3P4

161 136.5

1.05 0.51

UP

g2

=

gg2

=

1.03

x

AE~/K AE~/K IOL~\ ~ ~ ~ z-~ 290

185

110

3

280 340

270

110

9

2.2

400

430

10 3

3.0 4.1

485 560

410-510 330

130 160 200

11

3.5 3.

480 415

330

180 120

2.4

330

130 190

2 0.12

2

5.8

545

33 203 160

0.05 1.31 1.07

2 12 13

6.8 2.6 3.0

220 530 480

13

0.16

14

5.4

70

~—UH3

171

1.18

15 2.8

475

~—UD3

168

1.19

16

USe2

(°K’

5.4 2.8

94

(UN.,3) UP2 UOTe

Tmax

5.6

2 8

2 138

U2N3 (UN1 .~)

Reference

70 180 160

580 310

23 460

160

therefore lower than the one calculated from low temperature saturation magnetization, on the assumption of constant RKKY coupling. On the other hand, in the calculation of entropy we assumed no correlation between magnetic moments above the ordering temperature, while presumably a residuary short range order exists even then,

hence an f ~ multiplet splits into 2 doublets and 5 singlets. Here too, it is likely that the ground level is a singlet and the next level above it is a doublet and a singlet close together; the remaining levels are appreciably higher. In such a case, the approximation used above would still be valid. Similar arguments can be offered in the case of the remaining compounds.

Apart from crystals with the NaCI structure, where the U site has full octahedral symmetry, we have included some other uranium compounds in Table 1. These compounds also show low entropy changes connected with the disappearance of magnetic order, but their uranium site symmetry is not octahedral. In U3 P4, for example, the point symmetry of the uranium site is S~(4),

The proposed model should show an additional specific heat term of the Schottky type, both below and above Tc T~,connected with the excitation of the paramagnetic ions to the higher crystal levels. In the absence of exchange interaction (crystal field only) and neglecting the presence of levels higher than the second, this specific heat is given by ,

1676

ENTROPY OF ORDERING IN SOME URANIUM COMPOUNDS C~

=

Nk~(a2(lnZ)/~~)

Vol. 8, No. 21

pure Schottky type peak. Wang and Cooper’ and also Stutius5 showed that, in such a case, Cm

=

Nkx ~ g 2 exp(

—

x)/(1

+

g2 exp(

—

x)

)a

and would have a maximum in C,~, at Xm 2.66, 2.84, 3.00 (kTmaz/AE12 0.376;7 0.352; 0.333) for 2,3,4ofrespectively. The maximum is ofg2the= order about 2.0 2.0 cal./deg. K gram-ion uranium (assuming g 2 = 3). These temperatures Tmz, are listed in Table 1 and are usually close to the magnetic transition temperatures where there is a peak in C~ In the case of exchange interaction below the ordering temperature, this Schottky term becomes the magnetic specific heat Cm, and its temperature variation is different from the rounded

is reduced at low temperatures, and peaks up sharply just below and at the transition temperature, in agreement with the experiment. It would be desirable (a) to have specific heat measurements on analogous thorium compounds, so as to determine the lattice contribution to the specific heat, and (b) to extend measurements on uranium compounds to high ternperatures, in the hope of settling the problem of the presence of additional Schottky peaks. Acknowledgements — The author is grateful to Drs. H.E. Flotow and D.W. Osborn and Professor E. F. Westrum for criticism of, and comments on, the first version of the manuscript.

1.

REFERENCES WESTRUM E.F. and LYON W.G., in Thermodynamics of Nuclear Materials 1967. p. 239, I.A.E.A. Vienna (1968).

2.

COUNSELL J.F., DELL R.M.. and MARTIN J.F., Trans. Faraday Soc. 62, 1736 (1966).

3.

COUNSELL J.F., DELL R.M., JUNKISON A.R. and MARTIN J.F., Trans. Faraday Soc. 63, 72 (1967).

4. WANG Y.L. and COOPER BR., Phys. Rev. 172, 539 (1968); 185, 696 (1969). 5. 6.

8.

STUTIUS W., Phys. kondens. Materie, 10, 152 (1969). GRUNZWEIG-GENOSSAR J. and KUZNIETZM., I. appl. Phys. 39, 905 (1968); GRUNZWEIGGENOSSAR J., KUZNIETZ M. and FRIEDMAN F., Phys. Rev. 173, 562 (1968). JUSTICE B.H. and WESTRUM E.F., J. Phys. Chem. 67, 339 (1963). WESTRUM E.F. and BARBER C.M., J. Chem. Phys. 4, 635 (1966).

9.

WESTRUM E.F., WALTERS R.R., FLOTOW H.E. and OSBORNED.W., I. Chem. Phys. 48, 155 (1968).

7.

10.

TAKAHASHI Y. and WESTRUM E.F., J. Phys. Chem. 69, 3618 (1965).

11. 12.

STALINSKI B., BIEGANSKI Z. and TROC R., Pkys. Status Solidi. 17, 837 (1966). STALINSKI B., BIEGANSKI Z. and TROC R., Bull. Acad. Polon. Sci. Ser. Chem. 15, 257 (1967).

13.

STALINSKI B., NIEMIEC

14.

WESTRUM E.F., in Symposium on Thermodynamics of Solids, Vol. U, p. 497 I.A.E.A. Vienna, 1966, as quoted in Reference 2.

15.

FLOTOW H.E., LOHR H.R. ABRAHAM B.M. and OSBORNE D.W., J. Am. Chem. Soc. 81, 3529 (1959).

16.

ABRAHAM B.M., OSBORNE D.W., FLOTOW H.E. and MARCUS R.B., I. Am. Chem. Soc. 82, 1064 (1960).

J.,

and BIEGANSKI Z., Bull. Acad. poE. Sd. Set. Sci. Chim. 11, 267 (1963).

Nous avons analyse résultats des mesures d’entropie magnCtique dans certains composes ferro ou antiferromagnétique d’uraniuxn UK, en utiisant modile qui singlet. supposeLes la presence den d’Cnergie ions 4~,avec us étatunfondamental differences provoquies par Ic champ cristallin sont calculées et comparCes U aux valeurs obtenues a partir des mesures magnCtiques.