- Email: [email protected]
Bulk properties of the actinides
Journal of the Less-Common
Metals, 121 (1986)
101 - 120
101
BULK PROPERTIES OF THE ACTINIDES* J. M. FOURNIER? Centre d%tudes Nu&aires,
DRF/SPh-MDN,
85 X, 38041 Grenoble Ct!dex (France)
Summary Measurements of bulk transport and magnetic properties are basic in the study of any given compound. A short exposition of the fundamental concepts of actinide magnetic research and of the different aspects of magnetic ordering in actinide solids is followed by a discussion of experimental and interpretational aspects, in particular the difficulty always encountered of separating the different contributions. A number of examples have been chosen to illustrate the variety in the magnetic behaviour and its meaning in establishing a coherent picture of actinide solid state physics.
1. Introduction Measurements of bulk properties are essential and are the first to be made in the study of a given compound. Thus, although they do not give detailed microscopic information, such as may be obtained from spectroscopic techniques, it is not surprising that these measurements also comprise the bulk of publications made on actinide metals and simple compounds. The central feature of the actinides is the progressive filling of the incomplete 5f shell, associated with the extended nature of the 5f wavefunctions. Depending on the actinide element, the interactinide spacing and the actinide environment in a compound, we find properties ranging from that of itinerant transition-like systems to that of localized lanthanidelike systems. Thus, the magnetic behaviour of the 5f electrons concerns not only the magnetic measurements but also others such as electrical resistivity or specific heat. ‘2. Localization and magnetic properties The problem of localization of electrons belonging to an incomplete shell and its implication for the magnetic behaviour is familiar to *Paper presented at Actinides 85, Aix en Provence, September 2 - 6,1985. ‘Universit.6 Scientifique et MBdicale de Grenoble, France. 0022-5088/86/$3.50
@ Elsevier Sequoia/Printed in The Netherlands
I.02
all physicists working on tr~sition elements and, more recently, to those working on actinides. Schematically there are two extreme points of view from which to approach the problem. (i) The single-particle band point of view (Stoner theory and its developments [ 11) and (ii) the many-particle local point of view (Mott-Hubb~d theory and its developments [Z]). In the Stoner theory of magnetism, magnetic ordering is possible in a band scheme when spin polarization becomes energetically favourable. In the simpler case of ferromagnetism, a simple criterion for ordering is derived: In(E,) > 1
(1)
where I is the so-called Stoner parameter and n(E,) is the electronic density of states at the Fermi energy. It has been found [3] that I is fairly constant for different, actinide NaCl-type compounds (I a 3 eV). Thus, interestingly enough, from a band point of view the possibility of magnetic ordering in NaCl-type compounds is essentially governed by the 5f density of states at the Fermi energy. However, band magnetism is not permitted in the approach of Mott and Hubbard. An instability exists in the narrow band structure [4]; if the bandwidth decreases below a critical value, a sudden transition [5] takes place, driving the system from a non-magnetic itinerant state into a magnetic localized one. Again, in the simplest case of non-degenerate electron states, a simple criterion for magnetic ordering is derived: W<
Ueff
WJ
where W is the 5f bandwidth and U,,, the effective intra-atomic Coulomb correlation. Theoretical efforts are currently being made to bridge the gap between the fully localized and fully itinerant models, aiming at a unified theory of magnetism covering all the degrees of localization. While the band approach has been recently improved by including spin-orbit coupling [6], it still suffers the main drawback of neglecting the strong intra-atomic correlations. However, localized models are being improved to allow some degree of itinerancy to the 5f states (adding hopping energies, hybridization [7 - 91). 2.1. Pure nae tals and Mot&like transition The weight of theoretical and experimental evidence shows that in lighter actinide metals (thorium, protactinium, uranium, neptunium, plutonium) 5f electrons are truly itinerant and strongly hybridized with the 6d band [lo - 131. As an illustration, we report in Table 1 W, U and In(&) for the actinide metals which focuses on the transition that occurs between plutonium and americium. Americium behaves like a lanthanide in its physical properties with a J = 0 non-m~etic ground state (5f6 configuration). Very recently direct proof of 5f localization was obtained from photoemission measurements [14]. Since plutonium metal does not order magnetically -and in fact photoemission results show that 5f states are at the Fermi energy [ 151 -this has led Johansson [ 121 to view this sudden
103 TABLE 1 5f bandwidth W, intra-atomic Coulomb correlation 5f density of states n(E,) for light actinide metals
W (eW u (eV) I”&%)
U and Stoner parameter I times the
PLl
u
NP
Pu
5 1.5 = 0.3
4 2.3 = 0.9
3 2.6 mo.9
2 3.5 + 1 = 0.8
Am 0.1 5+1 >l
Reference 12 12 13
change as a Mott-like transition of the 5f electrons along the actinide series. Following americium, curium metal is antiferromagnetic and the value of its paramagnetic effective moment definitively supports the localized picture [16]. It should be noted that a single electron ground state from a filled band (which would be the case of americium within a band description because the 5f band is spin-orbit split onto a filled 5/2 subband and an empty 7/2 subband) is equivalent to a localized state and thus a spin-polarized band description leads to the same conclusion as a simple Mott description. 2.2. Metallic compounds and Hill plots In the 1970s Plutonium conference Hill [ 171 pointed out that uranium, neptunium and plutonium compounds could be divided into magnetic and non-magnetic subgroups, depending on the interactinide spacing dAn-**, the critical value being about 3.4 A. This systematic behaviour clearly indicates a localization process related to direct 5f-5f overlap. As noted by Hill himself, if it seems to be the essential parameter it is not the only one and the first correction comes from hybridization with d or p band states; anomalies in Hill plots reveal anomalously high hybridization (e.g. USns is non-magnetic although d,_, = 4.6 A) or on the contrary anomalously low hybridization (e.g. UNiz is magnetic although d,_, = 3.0 A showing that a pure 5f band should always lead to magnetic ordering) and we shall come back later to these most interesting classes of intermetallics. Enough theoretical and experimental work has now been accumulated [3,18,19] to show that the Hill limit does not correspond to a Mott-like transition but rather has to be looked on from the Stoner point of view.
3. Different aspects of magnetic ordering From the previous section, it is clear that the magnetic properties of actinide compounds are most difficult to interpret since we encounter the situation of non-fully localized 5f electron states and thus are faced by the very severe question: does a localized picture or an itinerant picture provide a better description of the magnetic properties? The answer is by no means unequivoqual. Crudely speaking, the itinerant picture seems more appropriate for intermetallics with small dAn+,, (e.g. Laves phases) or with strong
104
hybridization (e.g. AuCus-type structures) while the local picture seems to work rather well for most semimetallic, and of course insulating, compounds (e.g. NaCl-type, An&-type, AnXz-type compounds) and we shall try to give simple criteria for itinerant magnetism. 3.1. Localized magnetism If we assume 5f localization, it is possible to classify the different interactions on an energy scale. Similar to lanthanides, intra-atomic Coulomb correlation U is most important and of the order of 5 eV. In the presence of U alone the wavefunction is characterized by the total spin S = Zpi and the total angular momentum L = &Zi and the total momentum is given by J = L + S. The important spin-orbit coupling (about 0.3 eV) will mix LS multiplets so that only J remains a good quantum number: the RussellSaunders coupling scheme is no longer valid and an intermediate coupling scheme is more appropriate; in fact, up to the 5f4 configuration the departure from the pure Russell-Saunders ground state is small and the correction to the Land6 factor gJ is less than 3%. Only with the 5f5 configuration does the correction become significant (20%). The inclusion of the crystal field destroys the rotational symmetry of the ion and lifts the degeneracy of J levels (except of course Kramers’ degeneracy) : the only good quantum numbers will be the irreducible representations of the point-group symmetry operations. If the crystal field interaction is comparable with J-J splitting (and this may be the case in the actinides) it will also cause an admixture of different J multiplets. Finally exchange interaction (about 0.01 eV) can mix low lying crystal field energy levels. We thus see that a detailed quantitative treatment of the magnetic properties of an actinide compound within crystal field theories is by no means simple. Examples of localized magnetism are given below. 3.1.1. uoz In this semiconductor, the 5f levels lie in the 6 eV gap between an oxygen 2p derived valence and a uranium 6d derived conduction band [20]. This situation prevents hybridization ensuring good localization with U4+ ions (5f’ configuration). A first-order magnetic transition takes place at TN = 31 K and it has been recently shown that the antiferromagnetic structure is triple-k [21]. Crystal field calculations [22] lead to a r5 triplet ground state with peff = 2.9 pg. A small amount of J mixing is sufficient to obtain the observed effective and ordered moment. Specific heat data [23] reveal the first-order nature of the transition as do susceptibility data [24]. 3.1.2. PuSb This metallic plutonium ferromagnet (Tc = 67 K) is being extensively studied thanks to the recent success in growing large single crystals [25]. The localized nature of the 5f electrons has been confirmed by photoemission experiments [26]; it is corroborated by the values of the ordered and
105
effective moments which agree with those calculated in an intermediate coupling scheme (/&, = 0.74 pa, &f = 1 pa) for the 5f5 configuration with a I’s ground state. The magnetization has been found to be very anisotropic [25] with a (100) easy axis of m~etization (Fig. 1). Moreover, in a weak field an incommensurate antiferromagnetic phase develops above Tc up to TN = 85 K. The electrical resistivity has also been measured on single crystals [27] (Fig. 2); it goes through a large rn~~urn well above TN before decreasing following an exponential law. All this is reminiscent of properties of CeSb and further work is in progress on this interesting transuranium system. PuSb T=42K
I -30
-m
I' :
i
0.75
t Fig. 25).
0.
1.
0*
Magnetization
of
too TERPEAATURE
PuSb
along
(loo),
(110)
and
200
(K )
Fig. 2. Electrical resistivity of PuSb single crystal (from ref. 27).
(111)
directions
(ref.
106
At the beginning of this section we emphasized the difficulty of the choice between a localized and an itinerant picture. One may then ask whether there is any simple criterion to guide us in the identification of itinerant magnetism. In our opinion, the following ensemble of criteria is a very strong indication of itinerant magnetism: (i) a reduced ordered moment (in particular when incompatible with the effective moment in the crystal field scheme), (ii) a large electronic specific heat coefficient (r > 10 mJ mol-’ Km2)and (iii) a reduced magnetic entropy (AS, < Rln 2). It is also possible to use the pressure effects on the ordered moment and ordering temperature as criteria since for a well-localized system one expects d(ln po)/dP = 0 [ 281. Examples of itinerant magnetism are given below. 32.1. UN Uranium nitride orders antiferromagnetically at TN = 52 K. While it is possible to explain its low ordered moment (CL,= 0.75 pg) and its high effective moment (p,rr = 2.7 ,Q) assuming a 5fZ configuration with a I’, ground state and small crystal field splitting, it is not possible to explain concurrently the high electronic specific heat coefficient (y = 50 mJ mol-’ K-*) and the small entropy at the transition (0.1 Rln 2) [29]. According to our criteria, these properties strongly suggest itinerant antiferromagnetism. Strong support has been recently obtained from two kinds of measurements: the rapid decrease under pressure of both TN and &, at the same rate ]19] ; the observation of a structure in the photoemission spectrum corresponding to gap opening [ 301. These experimental results received theoretical support from band calculations [3]. It should be noted that in the case of PUN, the absence of a detectable ordered moment in neutron diffraction measurements [31] together with a high electronic specific heat coefficient (r = 64 mJ mol-i Ke2) and a small entropy at the transition (0.004 Rln 2) [32] also strongly support the assumption of itinerant antiferromagnetism which is again in agreement with band calculation results [ 31. 3.2.2. NpSn 3 In this antiferromagnet (TN = 9 K) the ordered moment is reduced to ,.&,= 0.28 ,.$. The most elegant experimental evidence for itinerant magnetism comes from specific heat measurements [ 331 (Fig. 3): the electronic specific heat fits well a B~een-Cooper-Schleffer (BCS) law predicted by the gap opening at the transition. The large change in r at the transition (y. = 88 mJ mol-l Ke2, and yp = 242 mJ mol-’ K-*) is further evidence of the itinerant nature of magnetism. 3.3. Very weak itinerant magnetism A limiting case of itinerant magnetism is very weak ferromagnetism, a well-known example of which is ZrZn, (II, = 0.15 pUB).The best example of very weak magnetism in the actinides is UN& (hexagonal Laves phase
107
O
5
10
15
20
25
30 TIKI
Fig. 3. CE/T vs. T for NpSna where CE is the electronic specific heat. The full curve is the mean-field theoretical prediction. The electronic specific heat coefficients of the paramagnetic and ordered states are designated by yP = 242 and ‘ye = 88 mJ mol-’ K-* respectively (from ref. 33). Fig. 4. Arrott plots (A@ vs. H/M) of the magnetization of UNiz (from ref. 35).
structure), with Tc = 21 K and ps = 0.06 pB, as measured on powders [34 361 (Fig. 4). Above Tc the reciprocal susceptibility is strongly curved as is usually the case for very weak ferromagnets; measurements of the specific heat yield a very large y value (y = 65 mJ mol-l K-*) [37]. The recent success in growing single crystals of UNi2 [38] allowed the discovery of very interesting properties: the magnetization is very anisotropic which is unusual for weak ferromagnets and points toward the existence of an orbital component in the magnetization [37] ; polarized neutron diffraction measurements definitively show that the moments are mainly those of uranium 5f [39]; the same experiments also definitively show that the 5f magnetization has a very large orbital moment incompatible with local magnetism in uranium but in agreement with recent band calculation conclusions, for which it is the first experimental evidence [6]. All these results make UNi2 a very interesting system displaying new and unusual magnetic properties.
4. Analysis of macroscopic properties The complex magnetic behaviour of solid actinides, described in this article, can be better understood when a set of measurements is performed together. This is why we discuss here not only magnetic measurements (susceptibility and magnetization) but also transport properties which are sensitive to the band structure of the solid and to the form of its density of states. Only bulk properties, in which thermodynamically averaged functions are measured, are presented while microscopic ones are presented
108
elsewhere at this conference. In the previous section we have already seen all the physical information which can be obtained from bulk measurements. It is thus of importance not only to be aware of the possibilities but also of the limitations presented by the analysis of the experimental data. It is also to be kept in mind that measurements on transuranium compounds are delicate and can be made in only very few laboratories. As for measurements on tr~sc~um compounds, made on the submil~~~ scale, they represent a “tour de force”; nevertheless, such pioneering work allows a coherent picture of the physical properties of the heavy actinides to emerge slowly. 4.1. Magnetic susceptibility Magnetic susceptib~ity me~~ernen~ are the cornerstone for studies of magnetic compounds; they are easily carried out on powder samples, involve a small amount of material (100 mg or less) and safety requirements for transuranium compounds may be fulfilled by working with sealed containers [ 401. As an achievement in experimental sensitivity we present in Fig. 5 measurements made on 20 pg of berkelium metal [ 411. The experimental susceptibility can be written as a sum of different contributions: Xexp =
XD + XC + XLOC
(3)
where xD is the radon core diamagnetism, xc the conduction electron contribution and xLoc the localized 5f electron contribution. The core dia~g~et~~ is due to the orbital moments of the electrons in the complete radon shell and is given (per mole) by C{r,‘> (4) e n where (rn2> are mean values of the square of the radial extension of the atomic wave~n~tions and e, N m,, c, have their usual meaning. Using accurate values for (rn2> calculated within the Dirac-Fock formalism [42], we give in Table 2 results obtained for light actinide ions, together with the diamagnetism of localized 5f electrons. While not important for concentrated magnetic systems, the diamagnetic corrections have to be taken into account when analysing the susceptibility of weak paramagnets (e.g.
XD =-
s
Fig. 5. Magnetic susceptibility of d.h.c.p. berkelium (from ref. 41).
109
TABLE 2 Radon core and localized 5f electrons diamagnetic corrections for light actinide ions
Ion
XD’”
xDs f
-38 -36 -35 -36 -34 -34 -33 -33 -31 -31 -30
-
(lo@
(lO+j e.m.u. mol-‘)
Th+4 Pa+4 Pa+’ $: NP+~ NP+~ pu+3 Pu+4 Am+3 Am+4
e.m.u. mol-‘)
-2 -5 -3 -7 -4 -a -5 -8 -6
in the case of thorium metal where the diamagnetism amounts to 40% of the experimental susceptibility [ 401. In the case of free electrons, the conduction electron susceptibility is simply XC = XLandau
+
XPauli = * ~2
NW
(5)
which is of the order of lops e.m.u. mol- ‘. In the case of compounds having narrow hybridized 5f bands, xc can be written as a sum of four terms: xc =
XL + x0
+ xs-0
+ sxs
(6)
the Landau diamagnetism, may be neglected because of the large effective mass of the electrons. x0, xs-o and xs are respectively the orbital, spinorbit and spin contributions, S is the Stoner enhancement factor. Usually xc is taken as being only Sxs (enhanced Pauli susceptibility). However, a detailed treatment of narrow bands such as was done for transition metals in the tight-binding approximation [43] indicates that the orbital term, in particular, may be large and even more important than the spin term [44,45]. Thus one should be very circumspect when using a constant susceptibility to “extract” the S factor. Thermal broadening of the density of states should also be taken into account. The general formula giving the locahked susceptibility for free ions was established by Van Vleck [46]
xL,
X=
W~J + G?JP*J(J + ~)~/~~B~IW
+ 1) exp(--Ei/&T)
Zr,(W + 1) exp(--E~/k,T)
(7)
where bd is an atomic quantity characteristic of the particular state J,p = eh/2mc, gJ is the Land6 factor. In the case where multiplet splitting is large compared with k,T, which may be assumed in the case of the actinides, only the ground multiplet intervenes and eqn. (7) reduces to the wellknown Curie law
110 x=C/T
(8)
where C = N/,+/3& and peff = pag,, {J(J + 1)}‘12. When the ion is embedded in a matrix, the crystal field splits the ground multiplet and the crystal field levels give rise to the same problem as the free ion multiplets and are solved in the way. If several crystal field levels are involved a complete treatment using eqn. (7) is needed. However, if excited states are only slightly thermally populated, they give rise to a constant susceptibility x0 and x takes the modified Curie law form x = xo + C/T
(9)
In a magnetic compound we have to consider exchange interaction; in the molecular field approximation, the Curie law takes the Curie-Weiss form x=C/T-9
(10)
P
being negative for antiferromagnetic coupling. The experimental data are often well fitted by a so-called modified Curie-Weiss law [ 471 13,
x=x$+
C*/T-Op
(11)
x0*being taken as a Pauli or Van Vleck term and C* as a Curie constant. However, it has recently been shown [48] that applying the same molecular field treatment as for eqn. (10) to the modified Curie law (9) leads to a renormalization of the Curie-Weiss constant as well as of the constant term; d and C* are different from the physical x0 and C. This applies to both the ferromagnetic and the antiferromagnetic case. As an example the results for plutonium monopnictides PUP, PuAs, PuSb are given in Table 3: while raw data would indicate a full J ground state (peff = 1 pg in the intermediate coupling), the corrected value is substantially smaller and points towards a I’s ground state in agreement with more detailed discussion (e.g. neutron diffraction results [ 491).
TABLE 3 Bare (*) and renormalized values for the constant susceptibility and the effective moment p,ff of plutonium monopnictides (from ref. 48) X8 (lo@ e.m.u. mol-‘) PUP PUAS PuSb PuBi
190 330 200 -
l-cff
x0
(PB)
(lo@ e.m.u. mol-‘)
1.06 0.98 1.01 0.8
157 213 171 -
/Jeff (PB)
0.87 0.63 0.85 0.8
111
4.2. Magnetization
Very large anisotropy seems to be the rule for actinide magnetic systems. One should thus treat with caution saturation moments obtained on ferromagnetic powders. Knowing the easy axis of magnetization it is possible to relate averaged powder saturation moment and the saturation moment along the easy axis ps. In the case of cubic symmetry, for example, one obtains: & = 1.27 vcs= l-1
E.(powder
ppowder
for (100) easy axis for (111) easy axis
When single crystals are available detailed magnetization measurements may be made as for example on US [50] (Fig. 6). An extremely large anisotropy restricts the magnetic moments to the (111) axis. The anisotropy constant K, (at T= 0 K) is about 8.5 X 10’ erg cme3 from which an anisotropy field HA = 3000 kOe is deduced. PuAs and PuSb have recently been studied; they are strongly anisotropic with a (100) easy axis of magnetization and the magnetization along the (110) and (111) directions are strictly given by the ratios l/J2 and l/,/3 of that along (100) [25] (see Fig. 1). The saturation moment in PuSb is & = 0.63 pg equal to p. [5l] indicating no conduction electron polarization effects. The coercivity field (H, = 15 kOe) is about three times that in US. Many actinide compounds have complex magnetic behaviour and the application of an external field may change or suppress antiferromagnetic structures. For example, UAs is antiferromagnetic below TN = 127 K (simple k, type I structure) and at 66 K a first-order transition occurs toward a double k, type IA structure. High field measurements reveal the existence of ferrimagnetic phases in the temperature range 2 - 66 K and in the neighbourhood of TN (Fig. 7) [52]. All these complicated magnetic phase diagrams need further study by neutron diffraction to characterize fully the different magnetic structures.
F
z 30
u 'J 25 20 15 10 5 0
T IK) Fig. 6. Temperature dependence of the magnetization u of US measured along the main crystallographic directions on cooling from the paramagnetic region in 15 kOe (from ref. 50).
Fig. 7. Magnetic phase diagram of UAs (from ref. 52).
4.3. Electrical resistivity These measurements are more limited than magnetic ones in the case of the actinides: solid samples with well-defined dimensions are needed and electrical contacts must be made to the samples, which is a rather delicate operation, in glove box on the millimetre scale. The very first information obtained is whether a given compound is a conductor, semiconductor or insulator. Such a basic answer is of importance concerning the 5f localization and the magnetic couplings but it is possible to extract more information from the analysis of experimental data. The electrical resistivity may be written as a sum of contributions: Pexp=PO+PL+PE+PM
w
Apart from the temperature independent residual resistivity po, mainly arising from lattice defects, charge carriers are scattered by the lattice (pL), by interband electron-electron scattering (~z) or by magnetic excitations (PM)-
Phonon scattering calculations require the knowledge of the phonon density of states; thus, practically, the only way is to estimate it from measurements on an isostructural compound. Interband scattering is also very difficult to calculate since it requires knowledge of the electron density of states: whereas s-d scattering may be obtained in the same way as phonon scattering from an isostructural non-magnetic compound (usually thorium based). It will be impossible to separate f-d or f-s interband scattering from pM when 5f electrons are itinerant. In this case thermal broadening of the electron density of states must be taken into consideration. Resistivity has proved to be a very sensitive probe to the possibilities of magnetism: contrary to the susceptibility which samples the spin-fluctuation propagator x(q, w) only for q = 0, the resistivity integrates over all q values ]531. One of the most famous and puzzling examples has been the electrical resistivity of plutonium metal [54]. It rises as T2 at low temperature and
113
saturates at high temperature which is a typical magnetic behaviour. The absence of magnetic ordering having been established, physical interpretation comes through spin fluctuation theory [55] : scattering by spin fluctuations gives a p dependence, as do magnons below TSf, the characteristic spin fluctuation temperature. As T increases, these fluctuations are destroyed and at high temperature, one eventually recovers a spin disorder resistivity [ 561. Curium metal is antiferromagnetic [16] and its resistivity has been measured as well as that of isostructural non-magnetic americium metal [57]. The resistivity difference is taken as the magnetic contribution and is shown in Fig. 8. It goes through a maximum at TN = 52 K and above has a In T dependence in contrast with the classical spin disorder theory [58] but in agreement with the extended Kondo formulation [ 591 (13)
P=PM
where pM is the spin disorder resistivity, 2 the number of conduction electrons per atom, J the exchange integral and EF the Fermi energy of the conduction electron. Fitting the data to eqn. (13) we obtain pM = 156 p!L?cm
t = 3ZJ/E, = 0.5 Curium being mainly in a spin only 8S,,, state, a more detailed treatment in the frame of the Ruder-man-Kittel-Kasuya-Yoshida (RKKY) model has been made [60]. It follows that Z = 1 suggesting that f-d exchange interaction should predominate over s-f interaction (contrary, for example, to gadolinium). The exchange integral is found to be about 0.13 eV. In the case of antiferromagnetism there may be additional scattering below TN caused by new Brillouin zone boundaries [61]. A spectacular example is given by the resistivity of single crystal NpAs, [62] (Fig. 9). Tetragonal NpAs, is antiferromagnetic below TN = 52 K (incommensurate structure) and becomes ferromagnetic below Tc = 18 K. The moments
temperature I K I
Fig. 8. Magnetic
contribution
to the electrical
resistivity
of curium
(from
ref. 57).
114 f200
i
E
rloo-
E 2 ;; _;; %a0 PO
NPASZ 50
Fig. 9. Electrical
100
150 200 TEPPERATURCK)
resistivity
230
lo
of NpAsz single crystal.
being along the c axis, extra scattering arising from new Brillouin zones should be observed only in the plane pe~endi~ul~ to c [61], and experimental data may be satisfactorily fitted to theory [62]. Another spectacular property of NpAsz is the huge electrical anisotropy since plc = 5OOp,, but correct measurements along the c axis are extremely difficult to perform because single crystals grow as thin platelets perpendicular to c. The series of ferromagnetic uranium monochalcogenides (US, Use, UTe) is also in~rest~g and in the p~am~etic range one observes a change of regime from a saturation in US to a well-defined In 2’ behaviour in UTe [63]. In the case of PuSb [27] Tc corresponds to a clear anomaly while a broad maximum is observed well above TN = 85 K, and a In T behaviour is followed above the maximum (Fig. 2). This behaviour is similar to that of CeSb and makes PuSb a possible Kondo system of great interest. 4.4. Hulleffect Although a first principles calculation of the Hall constant RH is almost impossible, very crude two-band models may be sufficient to interpret experimental results qualitatively. In low fields, we obtain: (14) where N, (Nh) is the number of electrons (holes) per unit volume and pu, (,u,J is the electron (hole) mobility. Thus the Hall effect is a very sensitive probe to determine the dominant charge carriers and to detect changes in condensation and/or mobility of carriers. In magnetic compounds the Hall resistivity may be written
,,H=RoB+R,M
(15)
where R. is the ordinary Hall constant and R, is the extraordinary magnetic contribution, Hall effect measurements are difficult in magnetic systems and very few have been reported. In the case of We [ 641 the separation into normal and magnetic parts (Fig. 10) leads to a carrier concentration of 0.45 electrons ( f .u.)-” .
115
5-
6.182K
.H-20kOe
L.LE
I 0
1
L
5
10h-0;~K4~
Fig. 10. The total ref. 64).
Hall resistivity
of USe single crystal
as a function
of (T - &,)-l
(from
In this conference the first results of Hall effect measurements on a transuranium single crystal NpAs, are presented [65]. The analysis of the results in the paramagnetic region leads to a carrier concentration of about 0.44 electron (f-u.)-‘, while it is 0.22 electron (f.u.)-* in the ferromagnetic region. In the ~tife~oma~etic range fitting to eqn. (15) using the experimental susceptibility leads to a linear temperature dependence of R,. 4.5. Specific heat As in the case of electrical resistivity, the experimental specific heat may be written as a sum of several separate contributions, leading to the same d~ficulties in qu~titatively in~rpret~g the data [ 661.
c, = c, + c, + CL
+ CE + CM + C&-H
(16)
C, term includes the dilatation contribution (C, - C,) and the contribution due to anharmonic forces; at high temperature it may reach several per cent of C,. The nuclear term C, is usually negligible above 1 K. The lattice term CL is normally predom~~t above 10 K. At low enough temperature (7’ < B&O, where 8, is the Debye temperature) the Debye model should be followed well leading to a T3 temperature dependence. The conduction electron term CE = yT, the electronic specific heat coefficient y being proportional to the density of electronic states at the Fermi energy. Thus plotting C/T uersus TZ at low temperature (typically below 10 K) will lead to the determination of Y, the importance of which has been already stressed; y is enhanced over the bare density of states value by a factor 1 f &,,, + A,* where Xel_-phand XSf are the electron-phonon and spin-fluctuation enhancement factors. When the density of states is high, The
116
thermal broadening has again to be taken onto account. The magnetic term Clllris due to changes in internal energy related to magnetic transitions. Its thermal dependence leads to the determination of the magnetic entropy A&. Whereas in the case of localized 5f states the magnetic entropy is Rln(m), where m is the multiplicity of the ground state, in the case of itinerant magnetism AS, may be as low as 0. Thus, as stated in the previous section, it is an important criterion for itinerant magnetism. Structural transitions may also lead to anomalies similar in shape to magnetic anomalies. The Schottky term CscH comes from excitations of crystal field energy levels in compounds with localized 5f electrons. Apart from the difficulties associated with data analysis, specific heat measurements are severely affected by self-heating problems and sample size (about 1 g) so that the development of microcalorimeters [67] represents a great advance in the field of the actinides. For non-magnetic actinide metals specific heat data have been employed very successfully to corroborate 5f localization starting with americium, as indicated by the sudden drop in y values [68] (rpu = 12 mJ mol-’ K-‘, YAm = 2 mJ mol-’ KP2). It has also allowed the discovery of superconductivity in protactinium and americium metals [ 68, 691. As presented in Section 3 (Fig. 3), specific heat measurements allowed the characterization of itinerant magnetism in, for example, NpSns. Going to more localized systems, the case of UAs is particularly interesting and has been studied in detail [70, 711 leading to a surprisingly high y value of 53 mJ mol-l K-*. Temperature dependence of the magnetic entropy is shown in Fig. 11. Its maximum value of 0.8 Rln 4 at 250 K is between rs (Rln 4) and I’, (Rhr 2) crystal field level values leading to a rs ground state with a I’, first excited level in agreement with magnetic susceptibility analysis [ 721. Measurements under magnetic fields up to 80 kOe [71]
Fig. 11. Temperature
dependence
of the magnetic
entropy
&+,J of UAs (from
ref. 70).
117
temperature
(K)
Fig. 12. Temperature dependence of the specific heat of UAs for constant magnetic field applied along the (100) axis (from ref. 71).
applied along the (100) direction of a single crystal are reported in Fig. 12. In higher fields the peak splitting around TN corroborates the occurrence of the ferrimagnetic state (see Fig. 7). Last, but not least, specific heat measurements have permitted the discovery of a new and very interesting class of exotic superconductors, the so-called “heavy-fermion” superconductors which are characterized by a huge electronic specific heat term y x 1 J mol-’ Kp2, the first being cubic UBels and others such as UPts have now been added to the list. One may speculate whether UNi2 might become a heavy-fern-non superconductor under pressure when compensation between the spin and orbital moment will occur. These most interesting materials will be discussed elsewhere in this conference.
5. Conclusion At this point, we hope that the reader has been able to appreciate the richness and the variety of the macroscopic properties of the actinide systems. The importance of bulk measurements for the study of the actinides is thus clearly shown and will not decrease in the future. On the contrary, progress in being made to develop experimental techniques adapted to the specificity of the actinides, the key words being work with the smallest quantity possible. Advances have been made in magnetic measurements using the SQUID magnetometer [ 411, specific heat measurements using microcalorimeters [67], neutron diffraction measurements using single crystals and high flux reactors [73], and electrical measurements on transuranium compounds using single crystals [ 271. Measurements under high pressure are also being developed, following large magnetovolume effects which have already been both calculated [3] and observed [28,74].
118
Among systems displaying new and promising physical properties, one should note the following: itinerant magnets with large orbital moments, the first being UN&, heavy-fermion superconduc~~, the first being UBe13 and highly anisotropic quasi-localized systems like aetinide monopnictides.
References 1 2 3 4 5 6 7 8 9
10
11 12 13
14 15 16 17 18 19 20 21
22 23 24 25 26 27 28 29
E. C. Stoner, Proc. R. Sot. London, 165 (1938) 372. J. Hubbard, Proc. R. Sot. London, Ser. A, 277 (1964) 237. M. S. S. Brooks and D. Glotzel, PhysicaB, 102 (1980) 51. R. E. Peierls, Quantum Theory of Solids, Oxford University Press, Oxford, 1955. N. F. Mott, Philos. &Zag., 6 (1961) 287. M. S. S. Brooks and P. J. Kelly, Phys. Ret?. Let&, 51 (1983) 1708. B. R. Cooper, J. Magn. Magn. Mater., 29 (1982) 230. H. A. Razafimandimby and P. Erdos, 2. Phys. B, 46 (1982) 193. K. Takegahara, Y. Yanase and T. Kasuya, in R. P. Guertin, W. Suski and Z. Zolnierek (eds.), Proc. 4th Znt. Conf. on Crystal Field and Structural Effects in f-Electron Systems, Plenum, New York, 1982, p. 533. A. J. Freeman and D. 13. Koelling, in A. J. Freeman and J. B. Darby, Jr., (eds.), The Act&tides, Electronic Structure and Related Properties, Vol. 1, Academic Press, New York, 1974, p. 5. R. Jullien and B. Coqblin, Phys. Rev. B, 8 (1973) 5263. B. Johansson, Phys. Rev. B, 11 (1975) 2740. M. S. S. Brooks, B. Johansson and Skriver, in A. J. Freeman and G. H. Lander (eds.), Handbook on the Physics and Chemistry of the Actinides, Vol. 1, North-Holland, Amsterdam, 1984, p. 153. J. R. Naegele, L. Manes, J. C. Spirlet and W, Miiller, Phys. Rev. Lett., 52 (1984) 1834. D. Courteix, J. Chayrouse, L. Heinz and R. Baptist, Solid State Commun., 39 (1981) 209. J. M. Fournier, A. Blaise, W. Miiller and J. C. Spirlet, Physica B, 86 - 88 (1977) 30. H. H. Hill, in W. N. Miner (ed.), Plutonium 1970 and other Actinides, Metal Society AIME, New York, 1970, p. 2. Y. Baer, Physica B, 102 (1980) 104. J. M. Fournier, J. Beille, A. Boeuf, C. Vettier and A. Wedgwood, Physica B, 102 (1980) 282. B. W. Veal and D. J. Lam, Phys. Rev. B, 10 (1974) 4902. J. Rossat-Mignod, G. H. Lander and P. Burlet, in A. J. Freeman and G. H. Lander (eds.), Handbook on the Physics and Chemistry of the Actinides, Vol. 1, NorthHolland, Amsterdam, 1984, p. 478. H. U. Rahman and W. A. Runciman, J. Phys. Chem. Solids, 27 (1966) 1833. J. J. Huntzicker and E. F. Westrum, Jr., J. Chem. Therm., 3 (1971) 61. M. J. M. Leask, L. E. Roberts, A. J. Walter and W. P. Wolf, J. Chem. SOC., 1963 (1963) 4788. B. R. Cooper, P. Thayamballi, J. C. Spirlet, W. Miiller and 0. Vogt, Phys. Rev. Lett., 51 (1983) 2418. R. Baptist, J. Chayrouse, D. Courteix, L. Heinz, D. Damien and A. Wojakowski, J, Phys. (Parer, 44 (1982) 241. A. Blaise, J. M. Fournier, J. M. Collard, J. C. Spirlet, J. Rebizant and 0. Vogt, in Proc. 138mes dour&es des Actinides, Nuclear Research Center, Negev, Israel, 1983. J. M. Fournier, Physica B, 130 (1985) 268. C. H. De Novion, Rep. CEA No 4113 (1970).
119 30 B. Reihl, G. Hollinger and F. J. Himpsel, J. Mugn. Magn. Mater., 29 (1982) 141. 31 A. Boeuf, J. M. Fournier, L. Manes, J. Rehizant and F. Rusticelli, Solid State Commun., 52 (1984) 451. 32 D. J. Martin, R. D. A. Hall, J. A. Lee, M. J. Mortimer and P. W. Sutcliffe, AERE Rep. No 7612599 (1976) (Harwell, Didcot). 33 R. J. Trainor, M. B. Brodsky, B. D. Dunlap and G: K. Shenoy, Phys. Rev. Lett., 37 (1976) 1511. 34 V. Sechovsky, Z. Smetana, G. Hilscher, E. Gratz and H. Sassik, Physica B, 102 (1980) 277. 35 J. M. Fournier, One Day Meet. on Magnetism of Actinides, Bedford College, London, unpublished (1980). 36 J. J. M. Franse, P. H. Frings, F. R. De Boer and A. Menovsky, in J. S. Schilling and R. N. Shelton (eds.), Physics of Solids under High Pressure, North-Holland, Amsterdam, 1981, p. 181. 37 P. H. Frings, J. J. Franse, A. Menovsky, S. Zeroual and B. Barbara, ZCM 1985, J. Magn. Magn. Mater., 54 - 57 (1986) 541. 38 A. Menovsky and J. J. M. Franse, J. Cryst. Growth, 65 (1983) 286. 39 J. M. Fournier, A. Boeuf, M. Bonnet, J. K. Boucherle, A. Delapalme, P. Frings and A. Menovsky,J. Less-Common Met., 121 (1986). 40 J. M. Fournier, Thesis, University of Grenoble; CNRS Rep. No. A.O. 10568, 1975. 41 S. E. Nave, P. G. Huray and R. G. Haire, in J. F. Crowe, R. P. Guertin and T. W. Mihalisin (eds.), Crystalline Electric FieM and Structural Effects in f-Electron Systems, Plenum, New York, 1980, p. 269. 42 J. P. Desclaux, At. Data Nucl. Data Tables, 12 (1973) 312. 43 R. Kubo and Y. Obata, J. Phys. Sot. Jpn., 11 (1956) 547. 44 C. M. Place and P. Rhodes, Phys. Status Solidi, 47 (1971) 475. 45 N. Mori, J. Phys. Sot. Jpn., 31 (1971) 359. 46 J. H. Van Vleck, Electric and Magnetic Susceptibilities, Oxford University Press, London, 1952. 47 M. B. Brodsky, Rep. Prog. Phys., 41 (1978) 103. 48 G. Amoretti and J. M. Fournier, J. Magn. Magn. Mater., 43 (1984) L217. 49 G. H. Lander, A. Delapalme, P. J. Brown, J. C. Spirlet, J. Rebizant and 0. Vogt, Phys. Rev. Lett., 53 (1984) 2262. 50 D. K. Tillwick and P. Du Plessis, J. Magn. Magn. Mater., 3 (1976) 329. 51 P. Burlet, S. Quezel, J. Rossat-Mignod, J. C. Spirlet, R. Rebizant, W. Miiller and 0. Vogt, Phys. Rev. B, 30 (1984) 6660. 52 G. Busch, 0. Vogt and H. Bartholin, J. Phys. (Paris), 40 (1979) C4-64. 53 J. M. Lawrence, P. S. Riseborough and R. D. Parks, Rep. Prog. Phys., 44 (1981) 1. 54 G. T. Meaden,F’roc. R. Sot. London, Ser. A, 276 (1963) 553. 55 A. B. Kaiser and S. Doniach, Znt. J. Magn., 1 (1970) 11. 56 R. Jullien, M. T. Beal-Monod and B. Coqblin, Phys. Rev. B, 9 (1974) 1441. 57 R. Schenkel, Solid State Commun., 23 (1977) 389. 58 P. G. De Gennes and J. Friedel, J. Phys. Chem. Solids, 41 (1958) 71. 59 J. Kondo, Prog. Theor. Phys., 32 (1964) 37. 60 G. Amoretti and V. Dallacasa, in Proc. 1 Odmes Journees des Actinides, University of Stockholm, Stockholm, 1980, p. 57. 61 R. J. Elliott and F. A. Wedgwood, Proc. Phys. Sot., 81 (1963) 846. 62 P. G. Therond, A. Blaise, J. M. Fournier, J. Chiapusio, J. P. Charvillat and A. Wojakowski, Physica B, 130 (1985) 102. 63 J. Schoenes, B. Frick and 0. Vogt, Phys. Rev. B, 30 (1984) 6578. 64 J. Schoenes and 0. Vogt, in Proc. 12emes Journees des Actinides, Institut de Physique Nucleaire, Orsay, 1982, p. 46. 65 P. G. Therond, A. Blaise, J. M. Fournier, A. Wojakowski and J. Chiapusio, J. LessCommon Met., 121 (1986) 66 A. Blaise, J. Phys. (Paris), 40 (1979) C4-49. 67 G. R. Stewart, Cryogenics, 18 (1978) 120.
120 68 J. L. Smith, G. R. Stewart, C. Y. Huang and R. G. Haire, J. Phys. (Paris), 40 (1979) c4-138. 69 G. R. Stewart, J. L. Smith, J. C. Spirlet and W. Miiller, in H. Suhl and M. B. Maple (eds.), Proc. 3rd Co& on Superconductivity in d- and f-Band Metals, La Jofla, CA, 1979, Academic Press, New York, 1980. 70 A. Blaise, R. Trot, R. Lagnier and M. J. Mortimer, J. Low Temp. Phys., 38 (1980) 79. 71 J. Rossat-Mignot, P. Burlet, H. Bartholin, R. Tchapoutian, 0. Vogt, C. Vettier and R. Lagnier, Physica& 102 (1980) 177. 72 R. Trot and D. J. Lam, Phys. Siatus Solidi, 65 (1974) 317. 73 A. Delapalme, M. Forte, J. M. Fournier, J. Rebizant and J. C. Spirlet, Physica B, 102 (1980) 171. 74 J. J. M. Franse, P. H. Frings, A. Menovsky and A. de Visser, Physica B, 130 (1986) 180.
Metals, 121 (1986)
101 - 120
101
BULK PROPERTIES OF THE ACTINIDES* J. M. FOURNIER? Centre d%tudes Nu&aires,
DRF/SPh-MDN,
85 X, 38041 Grenoble Ct!dex (France)
Summary Measurements of bulk transport and magnetic properties are basic in the study of any given compound. A short exposition of the fundamental concepts of actinide magnetic research and of the different aspects of magnetic ordering in actinide solids is followed by a discussion of experimental and interpretational aspects, in particular the difficulty always encountered of separating the different contributions. A number of examples have been chosen to illustrate the variety in the magnetic behaviour and its meaning in establishing a coherent picture of actinide solid state physics.
1. Introduction Measurements of bulk properties are essential and are the first to be made in the study of a given compound. Thus, although they do not give detailed microscopic information, such as may be obtained from spectroscopic techniques, it is not surprising that these measurements also comprise the bulk of publications made on actinide metals and simple compounds. The central feature of the actinides is the progressive filling of the incomplete 5f shell, associated with the extended nature of the 5f wavefunctions. Depending on the actinide element, the interactinide spacing and the actinide environment in a compound, we find properties ranging from that of itinerant transition-like systems to that of localized lanthanidelike systems. Thus, the magnetic behaviour of the 5f electrons concerns not only the magnetic measurements but also others such as electrical resistivity or specific heat. ‘2. Localization and magnetic properties The problem of localization of electrons belonging to an incomplete shell and its implication for the magnetic behaviour is familiar to *Paper presented at Actinides 85, Aix en Provence, September 2 - 6,1985. ‘Universit.6 Scientifique et MBdicale de Grenoble, France. 0022-5088/86/$3.50
@ Elsevier Sequoia/Printed in The Netherlands
I.02
all physicists working on tr~sition elements and, more recently, to those working on actinides. Schematically there are two extreme points of view from which to approach the problem. (i) The single-particle band point of view (Stoner theory and its developments [ 11) and (ii) the many-particle local point of view (Mott-Hubb~d theory and its developments [Z]). In the Stoner theory of magnetism, magnetic ordering is possible in a band scheme when spin polarization becomes energetically favourable. In the simpler case of ferromagnetism, a simple criterion for ordering is derived: In(E,) > 1
(1)
where I is the so-called Stoner parameter and n(E,) is the electronic density of states at the Fermi energy. It has been found [3] that I is fairly constant for different, actinide NaCl-type compounds (I a 3 eV). Thus, interestingly enough, from a band point of view the possibility of magnetic ordering in NaCl-type compounds is essentially governed by the 5f density of states at the Fermi energy. However, band magnetism is not permitted in the approach of Mott and Hubbard. An instability exists in the narrow band structure [4]; if the bandwidth decreases below a critical value, a sudden transition [5] takes place, driving the system from a non-magnetic itinerant state into a magnetic localized one. Again, in the simplest case of non-degenerate electron states, a simple criterion for magnetic ordering is derived: W<
Ueff
WJ
where W is the 5f bandwidth and U,,, the effective intra-atomic Coulomb correlation. Theoretical efforts are currently being made to bridge the gap between the fully localized and fully itinerant models, aiming at a unified theory of magnetism covering all the degrees of localization. While the band approach has been recently improved by including spin-orbit coupling [6], it still suffers the main drawback of neglecting the strong intra-atomic correlations. However, localized models are being improved to allow some degree of itinerancy to the 5f states (adding hopping energies, hybridization [7 - 91). 2.1. Pure nae tals and Mot&like transition The weight of theoretical and experimental evidence shows that in lighter actinide metals (thorium, protactinium, uranium, neptunium, plutonium) 5f electrons are truly itinerant and strongly hybridized with the 6d band [lo - 131. As an illustration, we report in Table 1 W, U and In(&) for the actinide metals which focuses on the transition that occurs between plutonium and americium. Americium behaves like a lanthanide in its physical properties with a J = 0 non-m~etic ground state (5f6 configuration). Very recently direct proof of 5f localization was obtained from photoemission measurements [14]. Since plutonium metal does not order magnetically -and in fact photoemission results show that 5f states are at the Fermi energy [ 151 -this has led Johansson [ 121 to view this sudden
103 TABLE 1 5f bandwidth W, intra-atomic Coulomb correlation 5f density of states n(E,) for light actinide metals
W (eW u (eV) I”&%)
U and Stoner parameter I times the
PLl
u
NP
Pu
5 1.5 = 0.3
4 2.3 = 0.9
3 2.6 mo.9
2 3.5 + 1 = 0.8
Am 0.1 5+1 >l
Reference 12 12 13
change as a Mott-like transition of the 5f electrons along the actinide series. Following americium, curium metal is antiferromagnetic and the value of its paramagnetic effective moment definitively supports the localized picture [16]. It should be noted that a single electron ground state from a filled band (which would be the case of americium within a band description because the 5f band is spin-orbit split onto a filled 5/2 subband and an empty 7/2 subband) is equivalent to a localized state and thus a spin-polarized band description leads to the same conclusion as a simple Mott description. 2.2. Metallic compounds and Hill plots In the 1970s Plutonium conference Hill [ 171 pointed out that uranium, neptunium and plutonium compounds could be divided into magnetic and non-magnetic subgroups, depending on the interactinide spacing dAn-**, the critical value being about 3.4 A. This systematic behaviour clearly indicates a localization process related to direct 5f-5f overlap. As noted by Hill himself, if it seems to be the essential parameter it is not the only one and the first correction comes from hybridization with d or p band states; anomalies in Hill plots reveal anomalously high hybridization (e.g. USns is non-magnetic although d,_, = 4.6 A) or on the contrary anomalously low hybridization (e.g. UNiz is magnetic although d,_, = 3.0 A showing that a pure 5f band should always lead to magnetic ordering) and we shall come back later to these most interesting classes of intermetallics. Enough theoretical and experimental work has now been accumulated [3,18,19] to show that the Hill limit does not correspond to a Mott-like transition but rather has to be looked on from the Stoner point of view.
3. Different aspects of magnetic ordering From the previous section, it is clear that the magnetic properties of actinide compounds are most difficult to interpret since we encounter the situation of non-fully localized 5f electron states and thus are faced by the very severe question: does a localized picture or an itinerant picture provide a better description of the magnetic properties? The answer is by no means unequivoqual. Crudely speaking, the itinerant picture seems more appropriate for intermetallics with small dAn+,, (e.g. Laves phases) or with strong
104
hybridization (e.g. AuCus-type structures) while the local picture seems to work rather well for most semimetallic, and of course insulating, compounds (e.g. NaCl-type, An&-type, AnXz-type compounds) and we shall try to give simple criteria for itinerant magnetism. 3.1. Localized magnetism If we assume 5f localization, it is possible to classify the different interactions on an energy scale. Similar to lanthanides, intra-atomic Coulomb correlation U is most important and of the order of 5 eV. In the presence of U alone the wavefunction is characterized by the total spin S = Zpi and the total angular momentum L = &Zi and the total momentum is given by J = L + S. The important spin-orbit coupling (about 0.3 eV) will mix LS multiplets so that only J remains a good quantum number: the RussellSaunders coupling scheme is no longer valid and an intermediate coupling scheme is more appropriate; in fact, up to the 5f4 configuration the departure from the pure Russell-Saunders ground state is small and the correction to the Land6 factor gJ is less than 3%. Only with the 5f5 configuration does the correction become significant (20%). The inclusion of the crystal field destroys the rotational symmetry of the ion and lifts the degeneracy of J levels (except of course Kramers’ degeneracy) : the only good quantum numbers will be the irreducible representations of the point-group symmetry operations. If the crystal field interaction is comparable with J-J splitting (and this may be the case in the actinides) it will also cause an admixture of different J multiplets. Finally exchange interaction (about 0.01 eV) can mix low lying crystal field energy levels. We thus see that a detailed quantitative treatment of the magnetic properties of an actinide compound within crystal field theories is by no means simple. Examples of localized magnetism are given below. 3.1.1. uoz In this semiconductor, the 5f levels lie in the 6 eV gap between an oxygen 2p derived valence and a uranium 6d derived conduction band [20]. This situation prevents hybridization ensuring good localization with U4+ ions (5f’ configuration). A first-order magnetic transition takes place at TN = 31 K and it has been recently shown that the antiferromagnetic structure is triple-k [21]. Crystal field calculations [22] lead to a r5 triplet ground state with peff = 2.9 pg. A small amount of J mixing is sufficient to obtain the observed effective and ordered moment. Specific heat data [23] reveal the first-order nature of the transition as do susceptibility data [24]. 3.1.2. PuSb This metallic plutonium ferromagnet (Tc = 67 K) is being extensively studied thanks to the recent success in growing large single crystals [25]. The localized nature of the 5f electrons has been confirmed by photoemission experiments [26]; it is corroborated by the values of the ordered and
105
effective moments which agree with those calculated in an intermediate coupling scheme (/&, = 0.74 pa, &f = 1 pa) for the 5f5 configuration with a I’s ground state. The magnetization has been found to be very anisotropic [25] with a (100) easy axis of m~etization (Fig. 1). Moreover, in a weak field an incommensurate antiferromagnetic phase develops above Tc up to TN = 85 K. The electrical resistivity has also been measured on single crystals [27] (Fig. 2); it goes through a large rn~~urn well above TN before decreasing following an exponential law. All this is reminiscent of properties of CeSb and further work is in progress on this interesting transuranium system. PuSb T=42K
I -30
-m
I' :
i
0.75
t Fig. 25).
0.
1.
0*
Magnetization
of
too TERPEAATURE
PuSb
along
(loo),
(110)
and
200
(K )
Fig. 2. Electrical resistivity of PuSb single crystal (from ref. 27).
(111)
directions
(ref.
106
At the beginning of this section we emphasized the difficulty of the choice between a localized and an itinerant picture. One may then ask whether there is any simple criterion to guide us in the identification of itinerant magnetism. In our opinion, the following ensemble of criteria is a very strong indication of itinerant magnetism: (i) a reduced ordered moment (in particular when incompatible with the effective moment in the crystal field scheme), (ii) a large electronic specific heat coefficient (r > 10 mJ mol-’ Km2)and (iii) a reduced magnetic entropy (AS, < Rln 2). It is also possible to use the pressure effects on the ordered moment and ordering temperature as criteria since for a well-localized system one expects d(ln po)/dP = 0 [ 281. Examples of itinerant magnetism are given below. 32.1. UN Uranium nitride orders antiferromagnetically at TN = 52 K. While it is possible to explain its low ordered moment (CL,= 0.75 pg) and its high effective moment (p,rr = 2.7 ,Q) assuming a 5fZ configuration with a I’, ground state and small crystal field splitting, it is not possible to explain concurrently the high electronic specific heat coefficient (y = 50 mJ mol-’ K-*) and the small entropy at the transition (0.1 Rln 2) [29]. According to our criteria, these properties strongly suggest itinerant antiferromagnetism. Strong support has been recently obtained from two kinds of measurements: the rapid decrease under pressure of both TN and &, at the same rate ]19] ; the observation of a structure in the photoemission spectrum corresponding to gap opening [ 301. These experimental results received theoretical support from band calculations [3]. It should be noted that in the case of PUN, the absence of a detectable ordered moment in neutron diffraction measurements [31] together with a high electronic specific heat coefficient (r = 64 mJ mol-i Ke2) and a small entropy at the transition (0.004 Rln 2) [32] also strongly support the assumption of itinerant antiferromagnetism which is again in agreement with band calculation results [ 31. 3.2.2. NpSn 3 In this antiferromagnet (TN = 9 K) the ordered moment is reduced to ,.&,= 0.28 ,.$. The most elegant experimental evidence for itinerant magnetism comes from specific heat measurements [ 331 (Fig. 3): the electronic specific heat fits well a B~een-Cooper-Schleffer (BCS) law predicted by the gap opening at the transition. The large change in r at the transition (y. = 88 mJ mol-l Ke2, and yp = 242 mJ mol-’ K-*) is further evidence of the itinerant nature of magnetism. 3.3. Very weak itinerant magnetism A limiting case of itinerant magnetism is very weak ferromagnetism, a well-known example of which is ZrZn, (II, = 0.15 pUB).The best example of very weak magnetism in the actinides is UN& (hexagonal Laves phase
107
O
5
10
15
20
25
30 TIKI
Fig. 3. CE/T vs. T for NpSna where CE is the electronic specific heat. The full curve is the mean-field theoretical prediction. The electronic specific heat coefficients of the paramagnetic and ordered states are designated by yP = 242 and ‘ye = 88 mJ mol-’ K-* respectively (from ref. 33). Fig. 4. Arrott plots (A@ vs. H/M) of the magnetization of UNiz (from ref. 35).
structure), with Tc = 21 K and ps = 0.06 pB, as measured on powders [34 361 (Fig. 4). Above Tc the reciprocal susceptibility is strongly curved as is usually the case for very weak ferromagnets; measurements of the specific heat yield a very large y value (y = 65 mJ mol-l K-*) [37]. The recent success in growing single crystals of UNi2 [38] allowed the discovery of very interesting properties: the magnetization is very anisotropic which is unusual for weak ferromagnets and points toward the existence of an orbital component in the magnetization [37] ; polarized neutron diffraction measurements definitively show that the moments are mainly those of uranium 5f [39]; the same experiments also definitively show that the 5f magnetization has a very large orbital moment incompatible with local magnetism in uranium but in agreement with recent band calculation conclusions, for which it is the first experimental evidence [6]. All these results make UNi2 a very interesting system displaying new and unusual magnetic properties.
4. Analysis of macroscopic properties The complex magnetic behaviour of solid actinides, described in this article, can be better understood when a set of measurements is performed together. This is why we discuss here not only magnetic measurements (susceptibility and magnetization) but also transport properties which are sensitive to the band structure of the solid and to the form of its density of states. Only bulk properties, in which thermodynamically averaged functions are measured, are presented while microscopic ones are presented
108
elsewhere at this conference. In the previous section we have already seen all the physical information which can be obtained from bulk measurements. It is thus of importance not only to be aware of the possibilities but also of the limitations presented by the analysis of the experimental data. It is also to be kept in mind that measurements on transuranium compounds are delicate and can be made in only very few laboratories. As for measurements on tr~sc~um compounds, made on the submil~~~ scale, they represent a “tour de force”; nevertheless, such pioneering work allows a coherent picture of the physical properties of the heavy actinides to emerge slowly. 4.1. Magnetic susceptibility Magnetic susceptib~ity me~~ernen~ are the cornerstone for studies of magnetic compounds; they are easily carried out on powder samples, involve a small amount of material (100 mg or less) and safety requirements for transuranium compounds may be fulfilled by working with sealed containers [ 401. As an achievement in experimental sensitivity we present in Fig. 5 measurements made on 20 pg of berkelium metal [ 411. The experimental susceptibility can be written as a sum of different contributions: Xexp =
XD + XC + XLOC
(3)
where xD is the radon core diamagnetism, xc the conduction electron contribution and xLoc the localized 5f electron contribution. The core dia~g~et~~ is due to the orbital moments of the electrons in the complete radon shell and is given (per mole) by C{r,‘> (4) e n where (rn2> are mean values of the square of the radial extension of the atomic wave~n~tions and e, N m,, c, have their usual meaning. Using accurate values for (rn2> calculated within the Dirac-Fock formalism [42], we give in Table 2 results obtained for light actinide ions, together with the diamagnetism of localized 5f electrons. While not important for concentrated magnetic systems, the diamagnetic corrections have to be taken into account when analysing the susceptibility of weak paramagnets (e.g.
XD =-
s
Fig. 5. Magnetic susceptibility of d.h.c.p. berkelium (from ref. 41).
109
TABLE 2 Radon core and localized 5f electrons diamagnetic corrections for light actinide ions
Ion
XD’”
xDs f
-38 -36 -35 -36 -34 -34 -33 -33 -31 -31 -30
-
(lo@
(lO+j e.m.u. mol-‘)
Th+4 Pa+4 Pa+’ $: NP+~ NP+~ pu+3 Pu+4 Am+3 Am+4
e.m.u. mol-‘)
-2 -5 -3 -7 -4 -a -5 -8 -6
in the case of thorium metal where the diamagnetism amounts to 40% of the experimental susceptibility [ 401. In the case of free electrons, the conduction electron susceptibility is simply XC = XLandau
+
XPauli = * ~2
NW
(5)
which is of the order of lops e.m.u. mol- ‘. In the case of compounds having narrow hybridized 5f bands, xc can be written as a sum of four terms: xc =
XL + x0
+ xs-0
+ sxs
(6)
the Landau diamagnetism, may be neglected because of the large effective mass of the electrons. x0, xs-o and xs are respectively the orbital, spinorbit and spin contributions, S is the Stoner enhancement factor. Usually xc is taken as being only Sxs (enhanced Pauli susceptibility). However, a detailed treatment of narrow bands such as was done for transition metals in the tight-binding approximation [43] indicates that the orbital term, in particular, may be large and even more important than the spin term [44,45]. Thus one should be very circumspect when using a constant susceptibility to “extract” the S factor. Thermal broadening of the density of states should also be taken into account. The general formula giving the locahked susceptibility for free ions was established by Van Vleck [46]
xL,
X=
W~J + G?JP*J(J + ~)~/~~B~IW
+ 1) exp(--Ei/&T)
Zr,(W + 1) exp(--E~/k,T)
(7)
where bd is an atomic quantity characteristic of the particular state J,p = eh/2mc, gJ is the Land6 factor. In the case where multiplet splitting is large compared with k,T, which may be assumed in the case of the actinides, only the ground multiplet intervenes and eqn. (7) reduces to the wellknown Curie law
110 x=C/T
(8)
where C = N/,+/3& and peff = pag,, {J(J + 1)}‘12. When the ion is embedded in a matrix, the crystal field splits the ground multiplet and the crystal field levels give rise to the same problem as the free ion multiplets and are solved in the way. If several crystal field levels are involved a complete treatment using eqn. (7) is needed. However, if excited states are only slightly thermally populated, they give rise to a constant susceptibility x0 and x takes the modified Curie law form x = xo + C/T
(9)
In a magnetic compound we have to consider exchange interaction; in the molecular field approximation, the Curie law takes the Curie-Weiss form x=C/T-9
(10)
P
being negative for antiferromagnetic coupling. The experimental data are often well fitted by a so-called modified Curie-Weiss law [ 471 13,
x=x$+
C*/T-Op
(11)
x0*being taken as a Pauli or Van Vleck term and C* as a Curie constant. However, it has recently been shown [48] that applying the same molecular field treatment as for eqn. (10) to the modified Curie law (9) leads to a renormalization of the Curie-Weiss constant as well as of the constant term; d and C* are different from the physical x0 and C. This applies to both the ferromagnetic and the antiferromagnetic case. As an example the results for plutonium monopnictides PUP, PuAs, PuSb are given in Table 3: while raw data would indicate a full J ground state (peff = 1 pg in the intermediate coupling), the corrected value is substantially smaller and points towards a I’s ground state in agreement with more detailed discussion (e.g. neutron diffraction results [ 491).
TABLE 3 Bare (*) and renormalized values for the constant susceptibility and the effective moment p,ff of plutonium monopnictides (from ref. 48) X8 (lo@ e.m.u. mol-‘) PUP PUAS PuSb PuBi
190 330 200 -
l-cff
x0
(PB)
(lo@ e.m.u. mol-‘)
1.06 0.98 1.01 0.8
157 213 171 -
/Jeff (PB)
0.87 0.63 0.85 0.8
111
4.2. Magnetization
Very large anisotropy seems to be the rule for actinide magnetic systems. One should thus treat with caution saturation moments obtained on ferromagnetic powders. Knowing the easy axis of magnetization it is possible to relate averaged powder saturation moment and the saturation moment along the easy axis ps. In the case of cubic symmetry, for example, one obtains: & = 1.27 vcs= l-1
E.(powder
ppowder
for (100) easy axis for (111) easy axis
When single crystals are available detailed magnetization measurements may be made as for example on US [50] (Fig. 6). An extremely large anisotropy restricts the magnetic moments to the (111) axis. The anisotropy constant K, (at T= 0 K) is about 8.5 X 10’ erg cme3 from which an anisotropy field HA = 3000 kOe is deduced. PuAs and PuSb have recently been studied; they are strongly anisotropic with a (100) easy axis of magnetization and the magnetization along the (110) and (111) directions are strictly given by the ratios l/J2 and l/,/3 of that along (100) [25] (see Fig. 1). The saturation moment in PuSb is & = 0.63 pg equal to p. [5l] indicating no conduction electron polarization effects. The coercivity field (H, = 15 kOe) is about three times that in US. Many actinide compounds have complex magnetic behaviour and the application of an external field may change or suppress antiferromagnetic structures. For example, UAs is antiferromagnetic below TN = 127 K (simple k, type I structure) and at 66 K a first-order transition occurs toward a double k, type IA structure. High field measurements reveal the existence of ferrimagnetic phases in the temperature range 2 - 66 K and in the neighbourhood of TN (Fig. 7) [52]. All these complicated magnetic phase diagrams need further study by neutron diffraction to characterize fully the different magnetic structures.
F
z 30
u 'J 25 20 15 10 5 0
T IK) Fig. 6. Temperature dependence of the magnetization u of US measured along the main crystallographic directions on cooling from the paramagnetic region in 15 kOe (from ref. 50).
Fig. 7. Magnetic phase diagram of UAs (from ref. 52).
4.3. Electrical resistivity These measurements are more limited than magnetic ones in the case of the actinides: solid samples with well-defined dimensions are needed and electrical contacts must be made to the samples, which is a rather delicate operation, in glove box on the millimetre scale. The very first information obtained is whether a given compound is a conductor, semiconductor or insulator. Such a basic answer is of importance concerning the 5f localization and the magnetic couplings but it is possible to extract more information from the analysis of experimental data. The electrical resistivity may be written as a sum of contributions: Pexp=PO+PL+PE+PM
w
Apart from the temperature independent residual resistivity po, mainly arising from lattice defects, charge carriers are scattered by the lattice (pL), by interband electron-electron scattering (~z) or by magnetic excitations (PM)-
Phonon scattering calculations require the knowledge of the phonon density of states; thus, practically, the only way is to estimate it from measurements on an isostructural compound. Interband scattering is also very difficult to calculate since it requires knowledge of the electron density of states: whereas s-d scattering may be obtained in the same way as phonon scattering from an isostructural non-magnetic compound (usually thorium based). It will be impossible to separate f-d or f-s interband scattering from pM when 5f electrons are itinerant. In this case thermal broadening of the electron density of states must be taken into consideration. Resistivity has proved to be a very sensitive probe to the possibilities of magnetism: contrary to the susceptibility which samples the spin-fluctuation propagator x(q, w) only for q = 0, the resistivity integrates over all q values ]531. One of the most famous and puzzling examples has been the electrical resistivity of plutonium metal [54]. It rises as T2 at low temperature and
113
saturates at high temperature which is a typical magnetic behaviour. The absence of magnetic ordering having been established, physical interpretation comes through spin fluctuation theory [55] : scattering by spin fluctuations gives a p dependence, as do magnons below TSf, the characteristic spin fluctuation temperature. As T increases, these fluctuations are destroyed and at high temperature, one eventually recovers a spin disorder resistivity [ 561. Curium metal is antiferromagnetic [16] and its resistivity has been measured as well as that of isostructural non-magnetic americium metal [57]. The resistivity difference is taken as the magnetic contribution and is shown in Fig. 8. It goes through a maximum at TN = 52 K and above has a In T dependence in contrast with the classical spin disorder theory [58] but in agreement with the extended Kondo formulation [ 591 (13)
P=PM
where pM is the spin disorder resistivity, 2 the number of conduction electrons per atom, J the exchange integral and EF the Fermi energy of the conduction electron. Fitting the data to eqn. (13) we obtain pM = 156 p!L?cm
t = 3ZJ/E, = 0.5 Curium being mainly in a spin only 8S,,, state, a more detailed treatment in the frame of the Ruder-man-Kittel-Kasuya-Yoshida (RKKY) model has been made [60]. It follows that Z = 1 suggesting that f-d exchange interaction should predominate over s-f interaction (contrary, for example, to gadolinium). The exchange integral is found to be about 0.13 eV. In the case of antiferromagnetism there may be additional scattering below TN caused by new Brillouin zone boundaries [61]. A spectacular example is given by the resistivity of single crystal NpAs, [62] (Fig. 9). Tetragonal NpAs, is antiferromagnetic below TN = 52 K (incommensurate structure) and becomes ferromagnetic below Tc = 18 K. The moments
temperature I K I
Fig. 8. Magnetic
contribution
to the electrical
resistivity
of curium
(from
ref. 57).
114 f200
i
E
rloo-
E 2 ;; _;; %a0 PO
NPASZ 50
Fig. 9. Electrical
100
150 200 TEPPERATURCK)
resistivity
230
lo
of NpAsz single crystal.
being along the c axis, extra scattering arising from new Brillouin zones should be observed only in the plane pe~endi~ul~ to c [61], and experimental data may be satisfactorily fitted to theory [62]. Another spectacular property of NpAsz is the huge electrical anisotropy since plc = 5OOp,, but correct measurements along the c axis are extremely difficult to perform because single crystals grow as thin platelets perpendicular to c. The series of ferromagnetic uranium monochalcogenides (US, Use, UTe) is also in~rest~g and in the p~am~etic range one observes a change of regime from a saturation in US to a well-defined In 2’ behaviour in UTe [63]. In the case of PuSb [27] Tc corresponds to a clear anomaly while a broad maximum is observed well above TN = 85 K, and a In T behaviour is followed above the maximum (Fig. 2). This behaviour is similar to that of CeSb and makes PuSb a possible Kondo system of great interest. 4.4. Hulleffect Although a first principles calculation of the Hall constant RH is almost impossible, very crude two-band models may be sufficient to interpret experimental results qualitatively. In low fields, we obtain: (14) where N, (Nh) is the number of electrons (holes) per unit volume and pu, (,u,J is the electron (hole) mobility. Thus the Hall effect is a very sensitive probe to determine the dominant charge carriers and to detect changes in condensation and/or mobility of carriers. In magnetic compounds the Hall resistivity may be written
,,H=RoB+R,M
(15)
where R. is the ordinary Hall constant and R, is the extraordinary magnetic contribution, Hall effect measurements are difficult in magnetic systems and very few have been reported. In the case of We [ 641 the separation into normal and magnetic parts (Fig. 10) leads to a carrier concentration of 0.45 electrons ( f .u.)-” .
115
5-
6.182K
.H-20kOe
L.LE
I 0
1
L
5
10h-0;~K4~
Fig. 10. The total ref. 64).
Hall resistivity
of USe single crystal
as a function
of (T - &,)-l
(from
In this conference the first results of Hall effect measurements on a transuranium single crystal NpAs, are presented [65]. The analysis of the results in the paramagnetic region leads to a carrier concentration of about 0.44 electron (f-u.)-‘, while it is 0.22 electron (f.u.)-* in the ferromagnetic region. In the ~tife~oma~etic range fitting to eqn. (15) using the experimental susceptibility leads to a linear temperature dependence of R,. 4.5. Specific heat As in the case of electrical resistivity, the experimental specific heat may be written as a sum of several separate contributions, leading to the same d~ficulties in qu~titatively in~rpret~g the data [ 661.
c, = c, + c, + CL
+ CE + CM + C&-H
(16)
C, term includes the dilatation contribution (C, - C,) and the contribution due to anharmonic forces; at high temperature it may reach several per cent of C,. The nuclear term C, is usually negligible above 1 K. The lattice term CL is normally predom~~t above 10 K. At low enough temperature (7’ < B&O, where 8, is the Debye temperature) the Debye model should be followed well leading to a T3 temperature dependence. The conduction electron term CE = yT, the electronic specific heat coefficient y being proportional to the density of electronic states at the Fermi energy. Thus plotting C/T uersus TZ at low temperature (typically below 10 K) will lead to the determination of Y, the importance of which has been already stressed; y is enhanced over the bare density of states value by a factor 1 f &,,, + A,* where Xel_-phand XSf are the electron-phonon and spin-fluctuation enhancement factors. When the density of states is high, The
116
thermal broadening has again to be taken onto account. The magnetic term Clllris due to changes in internal energy related to magnetic transitions. Its thermal dependence leads to the determination of the magnetic entropy A&. Whereas in the case of localized 5f states the magnetic entropy is Rln(m), where m is the multiplicity of the ground state, in the case of itinerant magnetism AS, may be as low as 0. Thus, as stated in the previous section, it is an important criterion for itinerant magnetism. Structural transitions may also lead to anomalies similar in shape to magnetic anomalies. The Schottky term CscH comes from excitations of crystal field energy levels in compounds with localized 5f electrons. Apart from the difficulties associated with data analysis, specific heat measurements are severely affected by self-heating problems and sample size (about 1 g) so that the development of microcalorimeters [67] represents a great advance in the field of the actinides. For non-magnetic actinide metals specific heat data have been employed very successfully to corroborate 5f localization starting with americium, as indicated by the sudden drop in y values [68] (rpu = 12 mJ mol-’ K-‘, YAm = 2 mJ mol-’ KP2). It has also allowed the discovery of superconductivity in protactinium and americium metals [ 68, 691. As presented in Section 3 (Fig. 3), specific heat measurements allowed the characterization of itinerant magnetism in, for example, NpSns. Going to more localized systems, the case of UAs is particularly interesting and has been studied in detail [70, 711 leading to a surprisingly high y value of 53 mJ mol-l K-*. Temperature dependence of the magnetic entropy is shown in Fig. 11. Its maximum value of 0.8 Rln 4 at 250 K is between rs (Rln 4) and I’, (Rhr 2) crystal field level values leading to a rs ground state with a I’, first excited level in agreement with magnetic susceptibility analysis [ 721. Measurements under magnetic fields up to 80 kOe [71]
Fig. 11. Temperature
dependence
of the magnetic
entropy
&+,J of UAs (from
ref. 70).
117
temperature
(K)
Fig. 12. Temperature dependence of the specific heat of UAs for constant magnetic field applied along the (100) axis (from ref. 71).
applied along the (100) direction of a single crystal are reported in Fig. 12. In higher fields the peak splitting around TN corroborates the occurrence of the ferrimagnetic state (see Fig. 7). Last, but not least, specific heat measurements have permitted the discovery of a new and very interesting class of exotic superconductors, the so-called “heavy-fermion” superconductors which are characterized by a huge electronic specific heat term y x 1 J mol-’ Kp2, the first being cubic UBels and others such as UPts have now been added to the list. One may speculate whether UNi2 might become a heavy-fern-non superconductor under pressure when compensation between the spin and orbital moment will occur. These most interesting materials will be discussed elsewhere in this conference.
5. Conclusion At this point, we hope that the reader has been able to appreciate the richness and the variety of the macroscopic properties of the actinide systems. The importance of bulk measurements for the study of the actinides is thus clearly shown and will not decrease in the future. On the contrary, progress in being made to develop experimental techniques adapted to the specificity of the actinides, the key words being work with the smallest quantity possible. Advances have been made in magnetic measurements using the SQUID magnetometer [ 411, specific heat measurements using microcalorimeters [67], neutron diffraction measurements using single crystals and high flux reactors [73], and electrical measurements on transuranium compounds using single crystals [ 271. Measurements under high pressure are also being developed, following large magnetovolume effects which have already been both calculated [3] and observed [28,74].
118
Among systems displaying new and promising physical properties, one should note the following: itinerant magnets with large orbital moments, the first being UN&, heavy-fermion superconduc~~, the first being UBe13 and highly anisotropic quasi-localized systems like aetinide monopnictides.
References 1 2 3 4 5 6 7 8 9
10
11 12 13
14 15 16 17 18 19 20 21
22 23 24 25 26 27 28 29
E. C. Stoner, Proc. R. Sot. London, 165 (1938) 372. J. Hubbard, Proc. R. Sot. London, Ser. A, 277 (1964) 237. M. S. S. Brooks and D. Glotzel, PhysicaB, 102 (1980) 51. R. E. Peierls, Quantum Theory of Solids, Oxford University Press, Oxford, 1955. N. F. Mott, Philos. &Zag., 6 (1961) 287. M. S. S. Brooks and P. J. Kelly, Phys. Ret?. Let&, 51 (1983) 1708. B. R. Cooper, J. Magn. Magn. Mater., 29 (1982) 230. H. A. Razafimandimby and P. Erdos, 2. Phys. B, 46 (1982) 193. K. Takegahara, Y. Yanase and T. Kasuya, in R. P. Guertin, W. Suski and Z. Zolnierek (eds.), Proc. 4th Znt. Conf. on Crystal Field and Structural Effects in f-Electron Systems, Plenum, New York, 1982, p. 533. A. J. Freeman and D. 13. Koelling, in A. J. Freeman and J. B. Darby, Jr., (eds.), The Act&tides, Electronic Structure and Related Properties, Vol. 1, Academic Press, New York, 1974, p. 5. R. Jullien and B. Coqblin, Phys. Rev. B, 8 (1973) 5263. B. Johansson, Phys. Rev. B, 11 (1975) 2740. M. S. S. Brooks, B. Johansson and Skriver, in A. J. Freeman and G. H. Lander (eds.), Handbook on the Physics and Chemistry of the Actinides, Vol. 1, North-Holland, Amsterdam, 1984, p. 153. J. R. Naegele, L. Manes, J. C. Spirlet and W, Miiller, Phys. Rev. Lett., 52 (1984) 1834. D. Courteix, J. Chayrouse, L. Heinz and R. Baptist, Solid State Commun., 39 (1981) 209. J. M. Fournier, A. Blaise, W. Miiller and J. C. Spirlet, Physica B, 86 - 88 (1977) 30. H. H. Hill, in W. N. Miner (ed.), Plutonium 1970 and other Actinides, Metal Society AIME, New York, 1970, p. 2. Y. Baer, Physica B, 102 (1980) 104. J. M. Fournier, J. Beille, A. Boeuf, C. Vettier and A. Wedgwood, Physica B, 102 (1980) 282. B. W. Veal and D. J. Lam, Phys. Rev. B, 10 (1974) 4902. J. Rossat-Mignod, G. H. Lander and P. Burlet, in A. J. Freeman and G. H. Lander (eds.), Handbook on the Physics and Chemistry of the Actinides, Vol. 1, NorthHolland, Amsterdam, 1984, p. 478. H. U. Rahman and W. A. Runciman, J. Phys. Chem. Solids, 27 (1966) 1833. J. J. Huntzicker and E. F. Westrum, Jr., J. Chem. Therm., 3 (1971) 61. M. J. M. Leask, L. E. Roberts, A. J. Walter and W. P. Wolf, J. Chem. SOC., 1963 (1963) 4788. B. R. Cooper, P. Thayamballi, J. C. Spirlet, W. Miiller and 0. Vogt, Phys. Rev. Lett., 51 (1983) 2418. R. Baptist, J. Chayrouse, D. Courteix, L. Heinz, D. Damien and A. Wojakowski, J, Phys. (Parer, 44 (1982) 241. A. Blaise, J. M. Fournier, J. M. Collard, J. C. Spirlet, J. Rebizant and 0. Vogt, in Proc. 138mes dour&es des Actinides, Nuclear Research Center, Negev, Israel, 1983. J. M. Fournier, Physica B, 130 (1985) 268. C. H. De Novion, Rep. CEA No 4113 (1970).
119 30 B. Reihl, G. Hollinger and F. J. Himpsel, J. Mugn. Magn. Mater., 29 (1982) 141. 31 A. Boeuf, J. M. Fournier, L. Manes, J. Rehizant and F. Rusticelli, Solid State Commun., 52 (1984) 451. 32 D. J. Martin, R. D. A. Hall, J. A. Lee, M. J. Mortimer and P. W. Sutcliffe, AERE Rep. No 7612599 (1976) (Harwell, Didcot). 33 R. J. Trainor, M. B. Brodsky, B. D. Dunlap and G: K. Shenoy, Phys. Rev. Lett., 37 (1976) 1511. 34 V. Sechovsky, Z. Smetana, G. Hilscher, E. Gratz and H. Sassik, Physica B, 102 (1980) 277. 35 J. M. Fournier, One Day Meet. on Magnetism of Actinides, Bedford College, London, unpublished (1980). 36 J. J. M. Franse, P. H. Frings, F. R. De Boer and A. Menovsky, in J. S. Schilling and R. N. Shelton (eds.), Physics of Solids under High Pressure, North-Holland, Amsterdam, 1981, p. 181. 37 P. H. Frings, J. J. Franse, A. Menovsky, S. Zeroual and B. Barbara, ZCM 1985, J. Magn. Magn. Mater., 54 - 57 (1986) 541. 38 A. Menovsky and J. J. M. Franse, J. Cryst. Growth, 65 (1983) 286. 39 J. M. Fournier, A. Boeuf, M. Bonnet, J. K. Boucherle, A. Delapalme, P. Frings and A. Menovsky,J. Less-Common Met., 121 (1986). 40 J. M. Fournier, Thesis, University of Grenoble; CNRS Rep. No. A.O. 10568, 1975. 41 S. E. Nave, P. G. Huray and R. G. Haire, in J. F. Crowe, R. P. Guertin and T. W. Mihalisin (eds.), Crystalline Electric FieM and Structural Effects in f-Electron Systems, Plenum, New York, 1980, p. 269. 42 J. P. Desclaux, At. Data Nucl. Data Tables, 12 (1973) 312. 43 R. Kubo and Y. Obata, J. Phys. Sot. Jpn., 11 (1956) 547. 44 C. M. Place and P. Rhodes, Phys. Status Solidi, 47 (1971) 475. 45 N. Mori, J. Phys. Sot. Jpn., 31 (1971) 359. 46 J. H. Van Vleck, Electric and Magnetic Susceptibilities, Oxford University Press, London, 1952. 47 M. B. Brodsky, Rep. Prog. Phys., 41 (1978) 103. 48 G. Amoretti and J. M. Fournier, J. Magn. Magn. Mater., 43 (1984) L217. 49 G. H. Lander, A. Delapalme, P. J. Brown, J. C. Spirlet, J. Rebizant and 0. Vogt, Phys. Rev. Lett., 53 (1984) 2262. 50 D. K. Tillwick and P. Du Plessis, J. Magn. Magn. Mater., 3 (1976) 329. 51 P. Burlet, S. Quezel, J. Rossat-Mignod, J. C. Spirlet, R. Rebizant, W. Miiller and 0. Vogt, Phys. Rev. B, 30 (1984) 6660. 52 G. Busch, 0. Vogt and H. Bartholin, J. Phys. (Paris), 40 (1979) C4-64. 53 J. M. Lawrence, P. S. Riseborough and R. D. Parks, Rep. Prog. Phys., 44 (1981) 1. 54 G. T. Meaden,F’roc. R. Sot. London, Ser. A, 276 (1963) 553. 55 A. B. Kaiser and S. Doniach, Znt. J. Magn., 1 (1970) 11. 56 R. Jullien, M. T. Beal-Monod and B. Coqblin, Phys. Rev. B, 9 (1974) 1441. 57 R. Schenkel, Solid State Commun., 23 (1977) 389. 58 P. G. De Gennes and J. Friedel, J. Phys. Chem. Solids, 41 (1958) 71. 59 J. Kondo, Prog. Theor. Phys., 32 (1964) 37. 60 G. Amoretti and V. Dallacasa, in Proc. 1 Odmes Journees des Actinides, University of Stockholm, Stockholm, 1980, p. 57. 61 R. J. Elliott and F. A. Wedgwood, Proc. Phys. Sot., 81 (1963) 846. 62 P. G. Therond, A. Blaise, J. M. Fournier, J. Chiapusio, J. P. Charvillat and A. Wojakowski, Physica B, 130 (1985) 102. 63 J. Schoenes, B. Frick and 0. Vogt, Phys. Rev. B, 30 (1984) 6578. 64 J. Schoenes and 0. Vogt, in Proc. 12emes Journees des Actinides, Institut de Physique Nucleaire, Orsay, 1982, p. 46. 65 P. G. Therond, A. Blaise, J. M. Fournier, A. Wojakowski and J. Chiapusio, J. LessCommon Met., 121 (1986) 66 A. Blaise, J. Phys. (Paris), 40 (1979) C4-49. 67 G. R. Stewart, Cryogenics, 18 (1978) 120.
120 68 J. L. Smith, G. R. Stewart, C. Y. Huang and R. G. Haire, J. Phys. (Paris), 40 (1979) c4-138. 69 G. R. Stewart, J. L. Smith, J. C. Spirlet and W. Miiller, in H. Suhl and M. B. Maple (eds.), Proc. 3rd Co& on Superconductivity in d- and f-Band Metals, La Jofla, CA, 1979, Academic Press, New York, 1980. 70 A. Blaise, R. Trot, R. Lagnier and M. J. Mortimer, J. Low Temp. Phys., 38 (1980) 79. 71 J. Rossat-Mignot, P. Burlet, H. Bartholin, R. Tchapoutian, 0. Vogt, C. Vettier and R. Lagnier, Physica& 102 (1980) 177. 72 R. Trot and D. J. Lam, Phys. Siatus Solidi, 65 (1974) 317. 73 A. Delapalme, M. Forte, J. M. Fournier, J. Rebizant and J. C. Spirlet, Physica B, 102 (1980) 171. 74 J. J. M. Franse, P. H. Frings, A. Menovsky and A. de Visser, Physica B, 130 (1986) 180.