The low-temperature specific heat of the Ce compounds is dominated by magnetic contributions, which in most cases are related by a certain degree of hybridization between the 4f state and the conduction band. The magnetic properties of a system can obviously be studied by other techniques such as magnetization and electrical resistivity, or by more sophisticated techniques like neutron scattering. In this section we shall briefly discuss some relationships between the specific heat and the tech- niques mentioned.
10.1. Magnetization and susceptibility
In a ferromagnetic system the order parameter is given by the spontaneous magnetization, Ms, which in an ideal second-order transition rises from zero at T c (see, e.g., Belov 1959). As mentioned in sect. 2, CM and M~ are related by CM ~-- -- TSM2/O T and therefore, if the temperature of the maximum of C M is taken as the thermo- dynamical definition of Tc, it corresponds to the maximum slope o f M s. Many authors define T c (T* for us) as the temperature where the maximum slope of Ms extrapolates to M~ = 0. Although this is not a thermodynamical definition, the difference:
AT c = T~ -- Tc is a measure of the fluctuations above T c as was mentioned in sect. 2.
The magnetic susceptibility of a free-electron gas (or the Pauli susceptibility) Zo, is related with the specific heat through the density of states, having a ratio of )~o/7 = 0.014 emu K z J - 1. In a Fermi liquid, where the effective mass of the electrons increases significantly, such a ratio retains its validity after renormalizing by the effective magnetic moment and the ground-state degeneracy: t:o/7 = (#eff/~k) 2 (1 + ½J), where 2 #~ff = 9 ~ # 2 J ( J + 1), gs being the 9 factor and J the total angular momentum (Wilson 1975, Newns et al. 1982). Experimentally z #~ff can be evaluated from the Curie constant, then (#eff/rck) 2 = 3C/rc2R. We can therefore evaluate the )~o/7 ratio for cubic Ce compounds as: 0.035 and 0.014 e m u K 2 J - 1, for IV ( J = 5) and H F ( J = ½) compounds, respectively. F o r lower symmetries the C value correspond- ing to the doublet ground state has to be evaluated for each case. An extensive
50 J.G. SERENI
~2
x
Ni 5
/ / / Sn3 Be ~3
/ Ni
PdK
/ / Co2/
.,., IRh-~
N Rh Ir2~i~
/ " ~ h 3 0'3 2 /
2~0
Ru 2
/ ./
At~
Cu~"0 40 60 80 100
"y(mJ / k"2Ce at. )
Fig. 28. Correlation between Z0 and 7 for CejX k compounds from table 5, labelled as jX k. The dashed line is the expected value for a Fv-CF ground state, see the text.
application of this ratio to cubic IV Ce compounds, made by Besnus et al. (1985), confirms the predicted value. The experimental values for the Zo/~' ratio are shown in table 5. In fig. 28, we show some new data for small 7- and Zo-values. The deviation from the expected values is large because of the reduced contribution of the f states to y and Zo, which becomes comparable to that of the conduction band (particularly in CeNi 5 and CeRu2). In addition, CeCu 6 and CeA13 are shown in the figure, whose Zo/7 ratio is closer to that expected for a J = ½ system. A phenomenological correlation for Z and ~, including a large number of Ce and actinide compounds has been presented by De Long (1986).
10.2. Electrical resistivity
It was found that for a variety of systems the resistivity behaves as p(T) ~_ Po + AT2 in the limit of T--* 0, where Po is the residual resistivity. The T 2 term is usually attributed to the Umklapp process of the electron-electron collision (Abrikosov 1972). In the case of the so-called concentrated K o n d o materials, the A factor has much larger values than those observed in pure TM, as happens with the 7 term. At high temperatures, the resistivity of these materials increases as - l n T with decreas- ing temperature. After a maximum, it decreases rapidly showing what is known as a coherent behaviour, which is connected with the regularity of the magnetic lattice. On the other hand, the impurities show the characteristics of incoherent K o n d o scattering. It is surprising that materials which have different ground states with different low-temperature scattering mechanisms follow the s a m e T 2 relationship in the resistivity (Kadowaki and Woods 1986).
Regardless of the large values of A and 7 in the H F compounds, the ratio A/72 is constant and equals 1 x 10-5f~ cm (J K - ~ / C e atom) -2. In fig. 29 we reproduce this
LOW-TEMPERATURE BEHAVIOUR OF CERIUM COMPOUNDS 51 universal relationship shown by Kadowaki and Woods (1986) for Ce and some U compounds, adding some other IV systems, like CeBe13, CeNi5 and CeRh. As the IV regime is reached, the
A/,/2
ratio tends to the values shown by pure TM, such as Pd, Pt and Ni[A/72~_O.9xlO-61af~cm(mJK-t/mol) -2]
where a Baber's type of scattering is expected (Rice 1968). We have to be reminded that in the case of CeRh, where 7 has very low values, the conduction band and f contributions to the density of states become comparable. Note, however, that Ce compounds with a spin fluctuation character (CeSn 3 and CeBe13 ) behave closer to the TM elements.10.3.
Neutron scattering
Although the magnetic internal energy, ~#M, is not a parameter that is much used, it becomes an important tool for recognizing systems with low dimensionality. For example,
~M(TN)/~(oO)
= 0.71 in the two-dimensional Ising model and 0.27 in the three-dimensional one (de Jongh and Miedema 1974). Specific heat and neutron scattering measurements can be related through ~M by the eqs. (4) and (5) (cf. sect. 2).Experimental results on CeA12 and CeIn 3 were compared by Peyrard (1980), as shown in fig. 30.
CeA[ 3 -/
~7 CeCu6
1
CeCu 2S~,I~
UPI~/y"
~/Ce B6
/
"CeRu2Si2 USn3/,
0
CeSitB6,y" UAI 2 (~-1 UPt2 /,Up t
=t. Ce.Pd3 /Uln3
<o /'UGa3
~-2
/ ///
~CeBe13 -3 [#CeRh///CeSn 3
ia Ce/
J e//CeRh3
-4 I i
I 2 3
Ioglo'Y (mJ/mo[e'K 2)
Fig. 29. Correlation between the T z coefficient of the resistivity, A, and 7 for HF and IV compounds. Full and dashed lines represent
A/72=
10" and 0.9 x 10 -6 [~£~ em(mJ K-a/mol)-Z] respectively, see the text. Part of the data are taken from Kadowaki and Woods (1986).52 J.G. SERENI
z.O
&
E 2
E 20
T(K)
8 12
I I i i i i |
/o'
'Jm : CMdT oO
o,, °
~I o C)
m oOO
,- oo
Celo3 T ; ° /
~o ~ i ' Ce A[ 2
0 2 4
T(K)
10
Fig. 30. Internal magnetic energy of CeI% and CeA12 as obtained from specific heat (C) ) and neutron scattering ( • and • ), after Peyard (1980).