- Email: [email protected]
A theory of magnetism for semi-metallic uranium compounds
J. Phys.Chem.Solids. 1973,Vol.34, pp. 1537-1542. Pergamon Press. Printedin GreatBritain
A THEORY
OF MAGNETISM FOR SEMI-METALLIC URANIUM COMPOUNDS
H. ADACHI and S. IMOTO Department of Nuclear Engineering, Faculty of Engineering, Osaka University, Suita-shi, Osaka, Japan (Received 10 August 1970; in revised form 18 March 1971)
A l ~ t r a e t - A theoretical study is carded out to explain the magnetic properties of uranium compounds of NaCI, anti-Cu~Sb and ThaP~ type. The electronic model used is a modified free electron model along the lines of the band structure for NaCI type uranium compounds previously reported. An exchange interaction via the conduction electrons is assumed for the mechanism of magnetic ordering. The molecular field approximation is used to calculate 0, TN and the exchange interaction constants. The stability range of various magnetic ordering states and the variation of 0, TN and exchange interaction constants are obtained as functions of lattice parameter. These results well explain the experimental data.
1. INTRODUCTION
IT HAS been known that uranium compounds have many interesting properties with respect to magnetism. Some of these properties have been reported in the literature, particularly by the Polish group under Trzebiatowski. The magnetic moments of these, which probably originate primarily in partially filled 5f shell, are distributed in the range of about 2 - 4 pB for paramagnetic state and of about 1 - 2/.tB for the ferro- or antiferromagnetic states. This fact makes it difficult to assign a definite elec' tronic configuration or valence to the uranium ion. The positive value of the paramagnetic Curie temperature is found for many antiferromagnetic compounds. It implies that there are ferromagnetic exchange interactions in addition to antiferromagnetic ones in these compounds, and this is confirmed from the values of the nearest neighbor exchange interaction constant (o/1) and the second nearest f~f2) calculated from molecular field theory. However, for some antiferromagnetic compounds such as the NaCI type (UX) and the anti-Cu2Sb type (UX2) uranium compounds with elements of group V, the relation between the exchange interaction constant and interatomic distance is difficult to explain by either the direct or superexchange mechanism.
These compounds have been found to show metallic conductivity, so that exchange interaction via conduction electrons, i.e. the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction[l], seems to operate and to be the origin of the magnetic ordering. For the NaC1 type uranium compounds the stability range of magnetic ordering for various types has been calculated by Grunzweig-Genossar et al. [2] using the R K K Y theory. Their result reasonably accounts for the occurence of the observed ferromagnetic ordering for compounds with the VI group and the antiferromagnetic (the first kind) for those with the V group, but hardly explains the variations of 0, TN and the exchaiage interaction constants in passing from one compound to another. The 5f electrons are shown to have itinerant character in early actinide metals [3] and even in paramagnetic uranium compounds with NaC1 type symmetry according to the calculations[4, 5]. In the latter compounds the Fermi levels intersect hybridized f - d bands, which give a high density of states at the Fermi level and account for the observed very high electronic specific heat coefficients (e.g. 208 • i0 -4 cal/mole deg z for USe[6]). Due to the intra-atomic exchange interaction the spinup and spin-down states of the 5f band may
1537
1538
H. A D A C H I and S. I M O T O
have different energies and this can bring about the spin polarization. Furthermore, polarization of the conduction electrons of the 7s band which overlaps the 5f band may be produced by two effects, as Mott[7] describes for transition metals; that is, exchange coupling between f and s electrons and s-f hybridization. However, the 5f band is much narrower than the d electron band in the transition metals, and it is likely that the 5f electrons are intermediate in character between itinerant and localized, having rather small spatial extension of the 5f wave function. But we do not have a theory to treat this complicated problem satisfactorily. In this article, therefore, we have assumed that the 5f electrons in uranium compounds have somewhat itinerant character, which leads to a very narrow 5f band with a Fermi surface, but when magnetic coupling between the conduction electron and the 5f electron is taken into account, the latter is treated as a localized magnetic carrier and the R K K Y interaction, ignoring the s-f hybridization, is assumed to operate for reasons described above. The present work was carried out on the basis of the above model to explain the magnetic properties of some uranium compounds with typical crystal structure such as NaC1, anti-Cu2Sb and Th3P4 type, using an extended molecular field approximation with the exchange interactions deduced from the R K K Y mechanism.
--2Jnm(Sn'Sm)
(2)
where J,,~ is the exchange integral between spins S, and Sin. Using the molecular field theory, one obtains T~ (the Curie temperature) or TN (the Neel temperature) and 0 from J , ' s . If the exchange interaction is due to the R K K Y mechanism, it is necessary that the molecular field approximation be extended for long-range interactions in order to take into account the interaction between spins for sufficiently large interatomic distances. Thus, Tc, TN, 0 and J , can be written as
Tc=O= 2 S 3k ( S + 1 ) ( i ~ + Joi++ ~, Joi-) i--
TN = 2S(S+3k 1)(i~+ Joi+-~i_ Joi-
J~-+ =
(3)
2 S ( S + I) jo,_+ 3k
where i• refers to the lattice site of +_ spin. Using equation (I) and (2), equations (3) are rewritten as
To=O= MS(S+I)
2. CALCULATIONS AND RESULTS
The second order exchange interaction via the conduction electrons between spin Sn localized at Rn and spin Sm at Rm is given by
H.m = MF (2K~Rnm) (S.'Sm)] M = 9zrZ2G2/4v2 EF F(x) = (xcosx--sinx)/x 4 J
the conduction electrons and v is atomic volume. Since this exchange interaction is due to a spin polarization of the conduction electrons and decreases in an oscillatory manner with distance R,m, the effect extends to long range and causes both ferro- or antiferromagnetic coupling at different distances. The exchange interaction can also be written as
(1)
where G is the exchange interaction between 5f and conduction electrons, z is valence of
3k
[~ F(2KFRoi+)+ ~ F(2KFRoi-)] TN= MS(S+ 1) 3k [~ F(2KFRo,+)-~ F(2KrRo,-)] j~+ = MS(S+3k 1) F(2KFRo~+)"
(4)
A THEORY OF MAGNETISM FOR SEMI-METALLIC URANIUM COMPOUNDS
A.s F(2KFRoO'S are modified to F~(2KFa) = F(Roda • 2KFa), then T~, 0, TN and ,,r can be represented as functions of 2KFa. I f the energy spectrum of the conduction electrons is approximated by that o f free electrons, the Fermi energy EF is given by h2 2 h2 2 /3 h2 2 213 EF=f-~m KF --2-mm(3zr n)2 =2-mm(3~" Z/v) where n is the concentration of conduction electrons. T h e n , as v increases, n and K r 2 decrease when z is constant. H o w e v e r , according to the model described in the introduction, the 5 f electrons co-transfer into the conduction band or the conduction electrons co-transfer to the 5f band to keep the Fermi energy of the two bands the same. Therefore, the 5 f band with a high density of states acts as a 'buffer', which keeps the conduction electron concentration and the F e r m i level almost constant by easily absorbing or emitting conduction electrons. Thus, 2KFa can be regarded as proportional to the lattice constant a, since KF is almost invariable. (a) NaCl type compound F o u r types have been known, at least, for the antiferromagnetic ordering of the f.c.c. lattice[8], which is formed by uranium ions in the NaC1 type uranium compounds. T h e exchange interaction constants f i and J~' obtained from the molecular field theory are given by
1539
and can also be calculated theoretically from equation (4). Figure 1 shows the stable range of each magnetic ordering and the variation of 0, TN, J ~ and J'2 with 2Kea in the respective ranges, found by the theoretical calculations thus performed. F o r the NaCI type uranium compounds with V group all showing antiferromagnetic ordering at low temperatures, the experimental values 0 and TN, and J l ,t J 2 t, which were calculated from the former values with equations (5) are plotted in Fig. 2 as functions of the lattice constant a. H e r e S was evaluated from the magnetic moment of the ordered state, where the orbital momentum seems to be quenched, as expected from the calculated band structure. As 2KFa is proportional to a, the variations in Fig. 2 are found to agree with those around 2Kra ~ 10 (or Z ~ 1) in Fig. 1, except for J ' 2" T h e departure for J ' 2 can be interpreted as an effect o f a negative 180 ~ c a t i o n - a n i o n cation superexchange interaction between the second nearest neighbors which lowers the value of J'2- By comparing these two figures, we evaluated the magnitude of the R K K Y
0.05
\\
\ ",,~\
9~.. . . . . . . . = - ~ . . . . . ) ~ . .
~tl=O-T?N' , J'2 = 0 2 T N (for the first kind)
_ 0 + TN 12 '
_
_ 0-- 3 TN 24 '
_ 0 + 3 TN (for the third 12 kind)
9-.
TN (for the second 6 kind)
(5)
_ 0 + 3 TN 12 '
2KO
= ----~ (for the fourth - kind)
(J'~ is distinguished from J . )
-0.05
.,.
Fig. 1. Stability range of various magnetic ordering and variation of e, TN, J~ and J'2 for Ec.c. lattice.
O/MS(S + 1), - .... ; Ts/MS(S + 1), - . . . . . . ; J~/MS(S + 1) x 10, - . . . . ; J~/MS(S+ 1) x 10. F; ferromagnetic ordering, I; antiferromagnetic ordering of the first kind, II; the second kind, III; the third kind.
1540
H. A D A C H I and S. IMOTO
Ts, J ' l and J ~ from the theoretical calculation and the experimental results are illustrated in Figs. 3 a n d 4, respectively. In these types the exchange interaction constants were given by
'K 40(3 200 .-... ,
"--,:.
/
-200
/../
-400 - 600
UN
LiP UAs
U5b
Fig. 2. Plots of 0, Ts, J ' l and J'~ for NaC1 type compounds from experimental data. - - ; O/S(S+I), - . . . . ; TN/
s(s+]), S(S+
,4;=
§
0 {f
- ...... ; j~/s(s+l)xlo, - . . . . . . ; f;l
1) • 10. All compounds represented belong to the first kind.
interaction as M ='. 1 eV. As 2Kea increases from about 10.5 the antiferromagnetic ordering of the first kind becomes unstable and antiferromagnetic order of the second kind and then ferromagnetic order will appear successively. The compounds with the VI group possess one more valence electron, which slightly raises the Fermi level and increases the value of 2Kea for the same lattice parameter. This increases the tendency to ferromagnetism. Within the stability range of the second kind antiferromagnetic ordering the exchange interaction becomes small. Thus it is possible that the spin direction may be easily affected by other factors, such as external field and magnetic anisotropy, and some compounds which fall in this range may show metamagnetic behavior. (b) Anti-Cu2Sb type compound (UX2) All of the compounds of this type also become antiferromagnetic at low temperatures. Three types of antiferromagnetic order in body centered tetragonal structure are found in these compounds [9]. The variations of 0,
0 + TN 8'
O-- T s
J.'=
8
It is found that the curves in Fig. 4 accord with those in Fig. 3 in the range of 2KFa from 8 to 11. The change of the ordering type from type III (UP2, UAs~ and USb2) to type I (UBi2) with increasing lattice parameter a can be also explained very well from the theoretical result (see Fig. 4). We evaluated MS (S + 1) - 2 eV in this case. (C) Th3P4 type compound (UaX4) Compounds of this type show ferromagnetism. Since no antiferromagnetic structure is known, the calculation was done only for 0. The result is shown in Fig. 5 and experimental data are plotted in Fig. 6. It is found that 0 becomes positive within the range of 15.722.5 for 2KFa. Even in this range, however, some nearest neighbor exchange interactions are negative and it is possible that the spin 0.02]
0-01
/%
,\
2Kro t,
:j
-o.o
~. \
,, ,.
/
/
/ "~j.
Fig. 3. Stability range of various magnetic ordering and variation of 0, TM, J'~ and J'2 for body centered tetragonal lattice. Symbols are similar to Fig. 1.
A THEORY OF MAGNETISM d
FOR SEMI-METALLIC URANIUM
"K
400
COMPOUNDS
1541
~
A
uaA.~ \
! ',
200
200 U3P4
100
~oU3Bi4 U3Te4
i i. -200
I
.~. Fig. 6. Plots of 0 for ThaP4 type compounds from experimental data. I
I
I
I
UP2 UA.s2 USb2UBi2 (m)(nl) (1) (1)
Fig. 4. Plots of 0, TN, J ~ and J'2 for UX2 type compounds from experimental data. ; 0, - . . . . ; T N , - . . . . . . ; J l • 10, - . . . . . . ; J'2 X 10. T y p e o f magnetic ordering is shown in ().
0'02
-0.02
- 0.04
Fig. 5. Variation of 0 for ThaP4 type compound from theoretical calculation.
directions are affected by factors other than the R K K Y interaction, as is the case for NaCl type compounds with VI group. 3. DISCUSSION
It is found that the proposed model using the R K K Y theory can well account for the
JPCS VoL 34 No. 9 - F
change of 0, TN and the exchange interaction constants as functions of the lattice constant for typical uranium compounds. The change of magnetic structure and its stability ranges are also accounted for theoretically, though the assumption used here is somewhat arbitrary one, taking account of both the itinerant and localized character of the 5f electron in a convenient way. However, there still remain several problems in this work, one of which is the estimation of the spin S. This has been determined for NaCl type compounds from the magnetic moment of the ordered state, resulting in good agreement between theory and experiment, whereas for UX2 compounds the theoretical result accords with experimental data if S is taken to be constant. Recently, another type of antiferromagnetic ordering for the f.c.c, lattice, namely I - A type, has been observed in UAs [10] and U P US solid solutions of some compositions [11]. A calculation carried out along the lines described above predicts that the I - A type ordering is more unstable than ordering of the first or second kind, though there may be a small possibility of reversal of stability in the narrow range around 2KFa -- 10-5, if the effect
1542
H. A D A C H I and S. IMOTO
of the other interactions is taken into account more exactly. According to Knight shift measurement for UP2 reported by Easwaran et al. [12], the s - f exchange energy in UP2 is rather smaller than in U3P4 and in UP, though in present work the value of the s - f exchange integral seems to be of same order for all three types of compounds. These problems need further examination from both the experimental and theoretical viewpoints. REFERENCES 1. R U D E R M A N M. A. and K I T T E L C., Phys. Rev. 96.99 (1954). K A S U Y A T., In Magnetism (Edited by G. T. Rado and H. Suhl), Vol. l i B p. 215. Academic Press, New York (1966); Y O S H I D A K., Phys. Rev. 106, 893 (1957). 2. G R U N Z W E I G - G E N O S S A R J., K U Z N I E T Z M. and F R I E D M A N F., Phys. Rev. 173, 562 (1968).
3. K O E L L I N G D. D., F R E E M A N A. J. and ARBMAN G. O., Nuclear Metallurgy, Vol. 17, Plutonium 1970 and other actinides, p. 194 (1970); KMETKO E. A. and H I L L H. H., Nucl. Met. 17, 233 (1970). 4. D A V I S H. L., Nucl. Met. 17, 209 (1970). 5. A D A C H I H. and I M O T O S., J: Nucl. Sci. TechnoL 6, 371 (1969). 6. T A K A H A S H I Y. and W E S T R U M E. F., Jr, J. Phys. Chem. 69, 3618 (1965). 7. MOTT N. F.,Adv. Phys. 13,325 (1964). 8. G O O D E N O U G H J. B., In Magnetism and the Chemical Bond. Wiley, New York (1963). 9. PRZYSTAWA J. and SUSKI W., Phys. Status Solidi 20, 451 (1967). 10. LECIEJEWICS J., M U R A S I K A. and T R O C R., Phys. Status Solidi30, 157 (1968). 11. L A N D E R G. H., K U Z N I E T Z M. and BASKIN Y., SolidState Commun. 6, 887 (1968); K U Z N 1 E T Z M., L A N D E R G. H. and BASKIN Y., J. appl. Phys. 40, 1130 (1969). 12. E A S W A R A N K. R. K., R A O V. U. S., VIJAYARAG H A V A N R. and R A O U. R. K., Phys. Lett. 25A, 683 (1967).
A THEORY
OF MAGNETISM FOR SEMI-METALLIC URANIUM COMPOUNDS
H. ADACHI and S. IMOTO Department of Nuclear Engineering, Faculty of Engineering, Osaka University, Suita-shi, Osaka, Japan (Received 10 August 1970; in revised form 18 March 1971)
A l ~ t r a e t - A theoretical study is carded out to explain the magnetic properties of uranium compounds of NaCI, anti-Cu~Sb and ThaP~ type. The electronic model used is a modified free electron model along the lines of the band structure for NaCI type uranium compounds previously reported. An exchange interaction via the conduction electrons is assumed for the mechanism of magnetic ordering. The molecular field approximation is used to calculate 0, TN and the exchange interaction constants. The stability range of various magnetic ordering states and the variation of 0, TN and exchange interaction constants are obtained as functions of lattice parameter. These results well explain the experimental data.
1. INTRODUCTION
IT HAS been known that uranium compounds have many interesting properties with respect to magnetism. Some of these properties have been reported in the literature, particularly by the Polish group under Trzebiatowski. The magnetic moments of these, which probably originate primarily in partially filled 5f shell, are distributed in the range of about 2 - 4 pB for paramagnetic state and of about 1 - 2/.tB for the ferro- or antiferromagnetic states. This fact makes it difficult to assign a definite elec' tronic configuration or valence to the uranium ion. The positive value of the paramagnetic Curie temperature is found for many antiferromagnetic compounds. It implies that there are ferromagnetic exchange interactions in addition to antiferromagnetic ones in these compounds, and this is confirmed from the values of the nearest neighbor exchange interaction constant (o/1) and the second nearest f~f2) calculated from molecular field theory. However, for some antiferromagnetic compounds such as the NaCI type (UX) and the anti-Cu2Sb type (UX2) uranium compounds with elements of group V, the relation between the exchange interaction constant and interatomic distance is difficult to explain by either the direct or superexchange mechanism.
These compounds have been found to show metallic conductivity, so that exchange interaction via conduction electrons, i.e. the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction[l], seems to operate and to be the origin of the magnetic ordering. For the NaC1 type uranium compounds the stability range of magnetic ordering for various types has been calculated by Grunzweig-Genossar et al. [2] using the R K K Y theory. Their result reasonably accounts for the occurence of the observed ferromagnetic ordering for compounds with the VI group and the antiferromagnetic (the first kind) for those with the V group, but hardly explains the variations of 0, TN and the exchaiage interaction constants in passing from one compound to another. The 5f electrons are shown to have itinerant character in early actinide metals [3] and even in paramagnetic uranium compounds with NaC1 type symmetry according to the calculations[4, 5]. In the latter compounds the Fermi levels intersect hybridized f - d bands, which give a high density of states at the Fermi level and account for the observed very high electronic specific heat coefficients (e.g. 208 • i0 -4 cal/mole deg z for USe[6]). Due to the intra-atomic exchange interaction the spinup and spin-down states of the 5f band may
1537
1538
H. A D A C H I and S. I M O T O
have different energies and this can bring about the spin polarization. Furthermore, polarization of the conduction electrons of the 7s band which overlaps the 5f band may be produced by two effects, as Mott[7] describes for transition metals; that is, exchange coupling between f and s electrons and s-f hybridization. However, the 5f band is much narrower than the d electron band in the transition metals, and it is likely that the 5f electrons are intermediate in character between itinerant and localized, having rather small spatial extension of the 5f wave function. But we do not have a theory to treat this complicated problem satisfactorily. In this article, therefore, we have assumed that the 5f electrons in uranium compounds have somewhat itinerant character, which leads to a very narrow 5f band with a Fermi surface, but when magnetic coupling between the conduction electron and the 5f electron is taken into account, the latter is treated as a localized magnetic carrier and the R K K Y interaction, ignoring the s-f hybridization, is assumed to operate for reasons described above. The present work was carried out on the basis of the above model to explain the magnetic properties of some uranium compounds with typical crystal structure such as NaC1, anti-Cu2Sb and Th3P4 type, using an extended molecular field approximation with the exchange interactions deduced from the R K K Y mechanism.
--2Jnm(Sn'Sm)
(2)
where J,,~ is the exchange integral between spins S, and Sin. Using the molecular field theory, one obtains T~ (the Curie temperature) or TN (the Neel temperature) and 0 from J , ' s . If the exchange interaction is due to the R K K Y mechanism, it is necessary that the molecular field approximation be extended for long-range interactions in order to take into account the interaction between spins for sufficiently large interatomic distances. Thus, Tc, TN, 0 and J , can be written as
Tc=O= 2 S 3k ( S + 1 ) ( i ~ + Joi++ ~, Joi-) i--
TN = 2S(S+3k 1)(i~+ Joi+-~i_ Joi-
J~-+ =
(3)
2 S ( S + I) jo,_+ 3k
where i• refers to the lattice site of +_ spin. Using equation (I) and (2), equations (3) are rewritten as
To=O= MS(S+I)
2. CALCULATIONS AND RESULTS
The second order exchange interaction via the conduction electrons between spin Sn localized at Rn and spin Sm at Rm is given by
H.m = MF (2K~Rnm) (S.'Sm)] M = 9zrZ2G2/4v2 EF F(x) = (xcosx--sinx)/x 4 J
the conduction electrons and v is atomic volume. Since this exchange interaction is due to a spin polarization of the conduction electrons and decreases in an oscillatory manner with distance R,m, the effect extends to long range and causes both ferro- or antiferromagnetic coupling at different distances. The exchange interaction can also be written as
(1)
where G is the exchange interaction between 5f and conduction electrons, z is valence of
3k
[~ F(2KFRoi+)+ ~ F(2KFRoi-)] TN= MS(S+ 1) 3k [~ F(2KFRo,+)-~ F(2KrRo,-)] j~+ = MS(S+3k 1) F(2KFRo~+)"
(4)
A THEORY OF MAGNETISM FOR SEMI-METALLIC URANIUM COMPOUNDS
A.s F(2KFRoO'S are modified to F~(2KFa) = F(Roda • 2KFa), then T~, 0, TN and ,,r can be represented as functions of 2KFa. I f the energy spectrum of the conduction electrons is approximated by that o f free electrons, the Fermi energy EF is given by h2 2 h2 2 /3 h2 2 213 EF=f-~m KF --2-mm(3zr n)2 =2-mm(3~" Z/v) where n is the concentration of conduction electrons. T h e n , as v increases, n and K r 2 decrease when z is constant. H o w e v e r , according to the model described in the introduction, the 5 f electrons co-transfer into the conduction band or the conduction electrons co-transfer to the 5f band to keep the Fermi energy of the two bands the same. Therefore, the 5 f band with a high density of states acts as a 'buffer', which keeps the conduction electron concentration and the F e r m i level almost constant by easily absorbing or emitting conduction electrons. Thus, 2KFa can be regarded as proportional to the lattice constant a, since KF is almost invariable. (a) NaCl type compound F o u r types have been known, at least, for the antiferromagnetic ordering of the f.c.c. lattice[8], which is formed by uranium ions in the NaC1 type uranium compounds. T h e exchange interaction constants f i and J~' obtained from the molecular field theory are given by
1539
and can also be calculated theoretically from equation (4). Figure 1 shows the stable range of each magnetic ordering and the variation of 0, TN, J ~ and J'2 with 2Kea in the respective ranges, found by the theoretical calculations thus performed. F o r the NaCI type uranium compounds with V group all showing antiferromagnetic ordering at low temperatures, the experimental values 0 and TN, and J l ,t J 2 t, which were calculated from the former values with equations (5) are plotted in Fig. 2 as functions of the lattice constant a. H e r e S was evaluated from the magnetic moment of the ordered state, where the orbital momentum seems to be quenched, as expected from the calculated band structure. As 2KFa is proportional to a, the variations in Fig. 2 are found to agree with those around 2Kra ~ 10 (or Z ~ 1) in Fig. 1, except for J ' 2" T h e departure for J ' 2 can be interpreted as an effect o f a negative 180 ~ c a t i o n - a n i o n cation superexchange interaction between the second nearest neighbors which lowers the value of J'2- By comparing these two figures, we evaluated the magnitude of the R K K Y
0.05
\\
\ ",,~\
9~.. . . . . . . . = - ~ . . . . . ) ~ . .
~tl=O-T?N' , J'2 = 0 2 T N (for the first kind)
_ 0 + TN 12 '
_
_ 0-- 3 TN 24 '
_ 0 + 3 TN (for the third 12 kind)
9-.
TN (for the second 6 kind)
(5)
_ 0 + 3 TN 12 '
2KO
= ----~ (for the fourth - kind)
(J'~ is distinguished from J . )
-0.05
.,.
Fig. 1. Stability range of various magnetic ordering and variation of e, TN, J~ and J'2 for Ec.c. lattice.
O/MS(S + 1), - .... ; Ts/MS(S + 1), - . . . . . . ; J~/MS(S + 1) x 10, - . . . . ; J~/MS(S+ 1) x 10. F; ferromagnetic ordering, I; antiferromagnetic ordering of the first kind, II; the second kind, III; the third kind.
1540
H. A D A C H I and S. IMOTO
Ts, J ' l and J ~ from the theoretical calculation and the experimental results are illustrated in Figs. 3 a n d 4, respectively. In these types the exchange interaction constants were given by
'K 40(3 200 .-... ,
"--,:.
/
-200
/../
-400 - 600
UN
LiP UAs
U5b
Fig. 2. Plots of 0, Ts, J ' l and J'~ for NaC1 type compounds from experimental data. - - ; O/S(S+I), - . . . . ; TN/
s(s+]), S(S+
,4;=
§
0 {f
- ...... ; j~/s(s+l)xlo, - . . . . . . ; f;l
1) • 10. All compounds represented belong to the first kind.
interaction as M ='. 1 eV. As 2Kea increases from about 10.5 the antiferromagnetic ordering of the first kind becomes unstable and antiferromagnetic order of the second kind and then ferromagnetic order will appear successively. The compounds with the VI group possess one more valence electron, which slightly raises the Fermi level and increases the value of 2Kea for the same lattice parameter. This increases the tendency to ferromagnetism. Within the stability range of the second kind antiferromagnetic ordering the exchange interaction becomes small. Thus it is possible that the spin direction may be easily affected by other factors, such as external field and magnetic anisotropy, and some compounds which fall in this range may show metamagnetic behavior. (b) Anti-Cu2Sb type compound (UX2) All of the compounds of this type also become antiferromagnetic at low temperatures. Three types of antiferromagnetic order in body centered tetragonal structure are found in these compounds [9]. The variations of 0,
0 + TN 8'
O-- T s
J.'=
8
It is found that the curves in Fig. 4 accord with those in Fig. 3 in the range of 2KFa from 8 to 11. The change of the ordering type from type III (UP2, UAs~ and USb2) to type I (UBi2) with increasing lattice parameter a can be also explained very well from the theoretical result (see Fig. 4). We evaluated MS (S + 1) - 2 eV in this case. (C) Th3P4 type compound (UaX4) Compounds of this type show ferromagnetism. Since no antiferromagnetic structure is known, the calculation was done only for 0. The result is shown in Fig. 5 and experimental data are plotted in Fig. 6. It is found that 0 becomes positive within the range of 15.722.5 for 2KFa. Even in this range, however, some nearest neighbor exchange interactions are negative and it is possible that the spin 0.02]
0-01
/%
,\
2Kro t,
:j
-o.o
~. \
,, ,.
/
/
/ "~j.
Fig. 3. Stability range of various magnetic ordering and variation of 0, TM, J'~ and J'2 for body centered tetragonal lattice. Symbols are similar to Fig. 1.
A THEORY OF MAGNETISM d
FOR SEMI-METALLIC URANIUM
"K
400
COMPOUNDS
1541
~
A
uaA.~ \
! ',
200
200 U3P4
100
~oU3Bi4 U3Te4
i i. -200
I
.~. Fig. 6. Plots of 0 for ThaP4 type compounds from experimental data. I
I
I
I
UP2 UA.s2 USb2UBi2 (m)(nl) (1) (1)
Fig. 4. Plots of 0, TN, J ~ and J'2 for UX2 type compounds from experimental data. ; 0, - . . . . ; T N , - . . . . . . ; J l • 10, - . . . . . . ; J'2 X 10. T y p e o f magnetic ordering is shown in ().
0'02
-0.02
- 0.04
Fig. 5. Variation of 0 for ThaP4 type compound from theoretical calculation.
directions are affected by factors other than the R K K Y interaction, as is the case for NaCl type compounds with VI group. 3. DISCUSSION
It is found that the proposed model using the R K K Y theory can well account for the
JPCS VoL 34 No. 9 - F
change of 0, TN and the exchange interaction constants as functions of the lattice constant for typical uranium compounds. The change of magnetic structure and its stability ranges are also accounted for theoretically, though the assumption used here is somewhat arbitrary one, taking account of both the itinerant and localized character of the 5f electron in a convenient way. However, there still remain several problems in this work, one of which is the estimation of the spin S. This has been determined for NaCl type compounds from the magnetic moment of the ordered state, resulting in good agreement between theory and experiment, whereas for UX2 compounds the theoretical result accords with experimental data if S is taken to be constant. Recently, another type of antiferromagnetic ordering for the f.c.c, lattice, namely I - A type, has been observed in UAs [10] and U P US solid solutions of some compositions [11]. A calculation carried out along the lines described above predicts that the I - A type ordering is more unstable than ordering of the first or second kind, though there may be a small possibility of reversal of stability in the narrow range around 2KFa -- 10-5, if the effect
1542
H. A D A C H I and S. IMOTO
of the other interactions is taken into account more exactly. According to Knight shift measurement for UP2 reported by Easwaran et al. [12], the s - f exchange energy in UP2 is rather smaller than in U3P4 and in UP, though in present work the value of the s - f exchange integral seems to be of same order for all three types of compounds. These problems need further examination from both the experimental and theoretical viewpoints. REFERENCES 1. R U D E R M A N M. A. and K I T T E L C., Phys. Rev. 96.99 (1954). K A S U Y A T., In Magnetism (Edited by G. T. Rado and H. Suhl), Vol. l i B p. 215. Academic Press, New York (1966); Y O S H I D A K., Phys. Rev. 106, 893 (1957). 2. G R U N Z W E I G - G E N O S S A R J., K U Z N I E T Z M. and F R I E D M A N F., Phys. Rev. 173, 562 (1968).
3. K O E L L I N G D. D., F R E E M A N A. J. and ARBMAN G. O., Nuclear Metallurgy, Vol. 17, Plutonium 1970 and other actinides, p. 194 (1970); KMETKO E. A. and H I L L H. H., Nucl. Met. 17, 233 (1970). 4. D A V I S H. L., Nucl. Met. 17, 209 (1970). 5. A D A C H I H. and I M O T O S., J: Nucl. Sci. TechnoL 6, 371 (1969). 6. T A K A H A S H I Y. and W E S T R U M E. F., Jr, J. Phys. Chem. 69, 3618 (1965). 7. MOTT N. F.,Adv. Phys. 13,325 (1964). 8. G O O D E N O U G H J. B., In Magnetism and the Chemical Bond. Wiley, New York (1963). 9. PRZYSTAWA J. and SUSKI W., Phys. Status Solidi 20, 451 (1967). 10. LECIEJEWICS J., M U R A S I K A. and T R O C R., Phys. Status Solidi30, 157 (1968). 11. L A N D E R G. H., K U Z N I E T Z M. and BASKIN Y., SolidState Commun. 6, 887 (1968); K U Z N 1 E T Z M., L A N D E R G. H. and BASKIN Y., J. appl. Phys. 40, 1130 (1969). 12. E A S W A R A N K. R. K., R A O V. U. S., VIJAYARAG H A V A N R. and R A O U. R. K., Phys. Lett. 25A, 683 (1967).