2. STRUCTURAL STUDIES OF RARE EARTH COMPOUNDS USING DIFFRACTION TECHNIQUES UNDER HIGH
2.1 X-ray diffraction .1 GdCu
It is well known that Murnaghan’s equation is applicable for a wide range of materials, and the elastic constants are sensitive to the electronic state. As example of rare earth compound, we show first the data of non-Kondo GdCu. GdCu show a structural phase transformation from the cubic CsCl- type to the FeB-type structure at low temperature (Blanco et al., 1999; Van Dongen et al., 1983). This transformation is known as a diffusionless mar- tensitic transition, in which a large thermal hysteresis is observed with large volume and electrical resistivity anomalies (Ohashi et al., 2003b). The struc- tural study was carried out by using high-pressure X-ray diffraction, which was performed byRodrı´guez et al. (2007)at room temperature up to 11 GPa.
It was found that the CsCl structure is stable in the whole range of pressures.
The evolution of the cell volume with pressure is plotted inFigure 1. It is seen that the change of volume is smooth against pressure. In order to estimate the bulk modulus, a least-square fit of the data to the following Murnaghan’s equation of state was employed:
V V0¼ B00
B0 Pþ1 1=B00
: (1)
1 GdCu
0.95
0.9
0.850 2 4 6 8 10 12 14
P (GPa) VIV0
Experimental Fitting
FIGURE 1 Relative volume of GdCu as a function of pressure (Rodrı´guez et al., 2007).
Here,B0is the bulk modulus andB00is its pressure derivative. The solid line inFigure 1shows the result of fitting, in which the observed data are well reproduced. The obtained values areV0¼42.68 A˚3,B0¼69.7 GPa, and B00 ¼3.97. It is interesting to note that the initial compressibility is very close to the average compressibility of Gd and Cu elements. The authors suggested possible crystal structural transformation above 11 GPa to a low-symmetry structure. This result confirms the hypothesis of Degtyareva et al. (1997) in which the existence of an orthorhombic structure with the AuCd-type at high pressure has been proposed.
2.1.2 Structural transitions in CeXmKondo compounds
These compounds show interesting electronic properties depending onm.
First, we consider the Kondo compound with m¼1 that has the CsCl- type structure CeZn, and then we show the case ofm¼6 and a HF Kondo compound CeCu6.
2.1.2.1 CeZn Kondo compound CeZn is an antiferromagnet with TN ¼30 K, where a cubic to tetragonal phase transition occurs (Pierre et al., 1981).Kadomatsu et al. (1986)reported a pronounced enhancement of the Kondo anomaly, an antiferromagnetic transition and a structural phase transition by applying high pressure. The magnetic and crystallo- graphic structures of CeZn single crystal were studied byShigeoka et al.
(1990)using neutron diffraction (ND) under pressure up to 1.2 GPa and below 60 K. X-ray diffraction study of CeZn at high pressure was carried out byUwatoko et al. (1992b). The pressure dependence of the (110) peak position is shown in Figure 2. Bragg angle increases smoothly with increasing pressure, which indicates a decrease in lattice constant.
Above 3 GPa, however, the (110) peak splits into two peaks (001) and (110), whereas the (001) and (002) peaks do not. This result indicates that the crystal structure changes from cubic to rhombohedral structure above 3 GPa as was found in previous ND measurement (Kadomatsu et al., 1986; Shigeoka et al., 1990).
The pressure variation of the axial anglebof the rhombohedral cell is shown inFigure 3(Uwatoko et al., 1992b). The authors could not deter- mine whether the cell expands or shrinks along the [111] direction. Here, they assumed that the cell expands. From the results ofFigures 2 and 3, the structural phase transition takes place atP¼2.6 GPa at room temper- ature. This value is in good agreement with that estimated on the basis of previous phase diagram which is shown in Figure 4: the boundary between the cubic and rhombohedral structures is extrapolated to room temperature, and they gotP¼2.6 GPa as a transition pressure.
Figure 5shows the change of unit cell volumeVof CeZn as a function of pressure at room temperature up to 10 GPa (Uwatoko et al. 1992b). The volume Vis calculated by the following relationV¼a3(13cos2bþ2
cos3b)1/3, where aand b are the lattice constants. Vdecreases approxi- mately linearly with increasing pressure. A discontinuity change is not observed within the experimental error in the pressure–volume relation due to first-order structural transition. The volume compressibility is estimated to bek¼ (1/V)(@V/@P)¼1.71102GPa1, which is com- parable with those of other magnetic Kondo compounds, for example, CeAg (1.98102 GPa1; Takke et al., 1980) and CeIn3
(1.30102GPa1;Oomi et al., 1986). Considering these facts and that P (GPa)
8q (deg.)
0 5 10
65 70
(011)
(011)
(110)
CeZn RT
FIGURE 2 The pressure dependence of the (110) peak position (Bragg angle is given as 8y) for CeZn at room temperature (Uwatoko et al., 1992b).
CeZn
TM
0 85 90
b (deg.)
P (GPa)
5 10
RT
FIGURE 3 The pressure dependence of the axial anglebof the rhombohedral cell (Uwatoko et al., 1992b).
00
1 2
Tc Ferro
1.5 1
0.5 0
40
20 60
T (K) TN
TM
P (GPa) Para
Kondo state (Rhombohedral) Kondo state
(Cubic)
−Δa/a (%)
(Tetra. Ι) (Tetra. ΙΙ) (c/a>1) (c/a<1)
T + R AF + F Antiferro
FIGURE 4 Magnetic and structural pressure–temperature (P–T) phase diagram for CeZn. The solid circles represent the data reported byShigeoka et al. (1990).
CeZn
P (GPa) V (Å3)
45
0 5 10
50
RT
FIGURE 5 Pressure dependence of the unit cell volumeVof CeZn at room temperature (Uwatoko et al., 1992b).
there is no trace of volume anomaly in the compression curve, like g–a transition in Ce, the valence of Ce in CeZn may be 3þat least up to 10 GPa at room temperature. In other words, no valence transition, which has been observed in several Kondo compounds, is induced by high pressure up to 10 GPa at room temperature.
2.1.2.2 CeCu6 CeCu6 has an orthorhombic crystal structure at ambient pressure and is a typical HF material that has a very large effective electron mass, which is about 1000 times greater than that of free electron (Stewart, 1984). It has been reported that the crystal structure changes from orthorhombic to monoclinic around 200 K (¼Tm) at ambient pres- sure (Asano et al., 1986; Gratz et al., 1987). Although this phase transition does not appear in the temperature dependence of electrical resistance, it is observed in the measurements of elastic constants (Suzuki et al., 1985), neutron scattering (Noda et al., 1985), and lattice constants (Bauer et al., 1987).Goto et al. (1988)revealed that the elastic constant C66showed a complete softening around Tm; further, Tm decreases with increasing pressure up to 0.4 GPa. The X-ray diffraction study was carried out both at room temperature and at liquid nitrogen temperature (Oomi et al., 1988a; Shibata et al., 1986).Figure 6shows the lattice constantsa,b, and cas a function of pressure up to about 10 GPa at room temperature. It is found thatais almost constant against pressure, whilebandcdecrease smoothly with increasing pressure, indicating a large anisotropy in lattice compression. The linear compressibilityki(i¼a,b, orc) is estimated by using the compression curves.kais extremely small and the values ofkb
andkcare 4.8103and 4.2103GPa1, respectively. In other words, theb-axis is more compressible than thec-axis. It is concluded that there is
0 0.95 1.00
2 4 6 8 10 12
P (GPa) a/ao
b/bo
c/co
CeCu6 RT
FIGURE 6 Lattice constants of CeCu6as a function of pressure at room temperature (Oomi et al., 1988a).
no crystal structure change and no volume anomaly at room temperature under high pressure at least up to 10 GPa.
The crystal structure of CeCu6has been reported to transform from orthorhombic to monoclinic around 200 K at ambient pressure, which cannot be observed by electrical resistivity measurement (Shibata et al., 1986).Figure 7shows the pressure dependence ofa,b, andcat 77 K. In the course of the data analysis at 77 K, the orthorhombic structure was assumed because the angle betweenaandcaxes,b, is nearly 90 (Asano et al., 1986). The value ofais almost independent of pressure, which is similar to room temperature data. The pressure dependences ofb-andc- axes can be approximated by two straight lines within experimental errors, with a break of slope around 4.5 GPa. This fact indicates that a phase transition takes place near 4.5 GPa at 77 K. Theb-andc-axes below 4.5 GPa are more compressible than above 4.5 GPa. FromFigure 7,kbat 77 K is larger thankc. Further, the values ofkbandkcbelow 4.5 GPa are found to be larger than those above 4.5 GPa, which indicates that the high- pressure phase (hpp) above 4.5 GPa is harder than the low-pressure phase (lpp) below 4.5 GPa. The bulk moduli were roughly estimated to be B (hpp)¼200 GPa andB(lpp)¼100 GPa.
We now consider the anomaly in the lattice compression observed at 77 K around 4.5 GPa. TheTm of CeCu6 was reported to decrease with increasing pressure with a rate of@Tm/@P¼ 20 K GPa1(Suzuki et al., 1985). Considering this result and Tm¼168 K at ambient pressure, we obtain a speculative P–T phase diagram, which is shown in Figure 8 (Oomi et al., 1988a). It is surprising that the point (T¼77 K, P¼4.5 GPa) falls on the line which is extrapolated from the data men- tioned above. In other word, the anomaly is considered to correspond to
0 2 4 6 8 10 12
P (GPa) 0.95
1.00
T= 77 K CeCu6 b/bo
c/co
a/ao
FIGURE 7 Lattice constants of CeCu6as a function of pressure at 77 K (Oomi et al., 1988a; Shibata et al., 1986).
the monoclinic–orthorhombic phase transition induced by pressure at 77 K. Further, the present result is confirmed to be consistent with that obtained by alloying with La (Ōnuki et al., 1985).
Finally, we consider the origin of the large bulk modulus observed in hpp at 77 K. There have been several investigations of elastic anomalies of rare earth alloys and compounds (Lavagna et al., 1983; Neumann et al., 1982, 1985). It was shown that the magnitude of longitudinal mode C11of CeCu6 decreases around Tm. The difference in the magnitude of C11
between the two phases is about 0.1%. Since the bulk modulus of hpp is larger than that of lpp by several tens of percent, the structural change is not the main origin of the large bulk modulus in hpp.Neumann et al.
(1982)showed that there is a linear relationship between bulk modulus and the valence. According to their empirical relation, a valence change of 0.3 is estimated to correspond to an increase of several tens of gigapascal in the magnitude ofB. Further, it should be noted that the lattice constants of a-Ce, which is stable above 5 GPa at room temperature is almost independent of pressure below 9 GPa (Franceschi and Olcese, 1969).
P (GPa) 5 100
200
Suzuki et al. (1985) Oomi et al. (1988)
77 K
Orthorhombic
Monoclinic T (K)
CeCu6
00
FIGURE 8 Speculated pressure–temperature phase diagram of CeCu6(Oomi et al., 1988a).
This fact implies that the bulk modulus ofa-Ce is extremely large. Taking these facts into account, the Ce atom in the hpp of CeCu6may be in the same electronic state as ina-Ce which is considered as the mixed-valence state material. Thus the large bulk modulus may be originated mainly from a valence change of Ce atom induced by pressure. This consider- ation is supported by the pressure-induced electronic crossover as was observed in the electrical resistance measurements (Oomi et al., 1993b).
2.1.3 Continuous valence transition under pressure
In this section, we show an interesting electronic state observed in some Ce compounds, that is, intermediate (or mixed) valence state, which is induced by applying pressure. This electronic transition or electronic crossover may be related to Kondo states that have very high TK. In Section 4, we will show that TKof Kondo compounds is strongly influ- enced by applying pressure since the magnitude of hybridization between conduction electrons and 4f electrons changes significantly at high pressure (Oomi et al., 1993b). In other words, we can induce a valence transition by controlling the magnitude of TK. It should be noted that this type of transition is usually accompanied by a continuous change in volume.
2.1.3.1 CeAl3 CeAl3 has the Ni3Sn-type hexagonal structure and is a prototype HF compound, in which a large value of specific heat coeffi- cientgand an extremely large value ofT2term in the electrical resistivity have been found (Andres et al., 1975). These properties are affected significantly by applying pressure. At high pressure, this compound shows a crossover from HF (small TK) to intermediate valence (IV) (largeTK) states through the increasingTK. The details about this cross- over will be mentioned inSection 4.1.1.
Figure 9shows the fractional change of lattice constantsaandcalong with the unit cell volume V¼ ffiffiffi
p3
a2c=2 as a function of pressure (Kagayama and Oomi, 1995). The lattice constants were determined mainly from the reflections (110), (101), and (201) of X-ray diffraction measurement at high pressure. Since no extra diffraction lines were observed, it was assumed that the hexagonal structure is stable at room temperature up to 17 GPa. A discontinuous change in the lattice constant, like for ag–atransition in Ce metal, is not observed within experimental error. Thus the crossover from concentrated Kondo (CK) to IV state observed in the electrical resistivity of CeAl3 (Kagayama and Oomi 1996; and see Section 4.1.1) occurs gradually without volume anomaly.
To estimate the bulk modulus, a least-squares fit of the volume data to Eq. (1) was attempted. We obtained the following values, B0¼54 GPa andB00 ¼3.0. If Eq. (1)is extended to the negative pressure region, the chemical pressure associated with the substitution of Ce by La is
estimated to be 1.7 GPa by using the volume of LaA13. The pressure coefficient of electrical resistivityrat ambient pressure,@lnr/@P¼@lnR/
@P1/3B0, is estimated to be 0.186 GPa1(Kagayama and Oomi, 1995).
The volume dependence ofr,@lnr/@PB0, is 9.97, which is nearly the same as that of CeInCu2 (Kagayama et al., 1991a). B00 is commonly observed to be 3–4 in HF systems. Since B00 is written as B00 ¼ @ln (BV)/@ln V|P¼0 and BV is considered as cohesive energy per unit cell volume (B has dimension of [energy]/[volume]), B00 is regarded as a Gru¨neisen parameter of cohesive energy. The compression is strongly anisotropic reflecting a noncubic crystal structure. To examine the anisot- ropy in the compression, the axial ratio c/a is plotted in Figure 10as a function of pressure (Kagayama and Oomi, 1995). It increases with increasing pressure. At ambient pressure c/a0.7 which is approxi- mately half of the ideal close-packed hexagonal value (¼ ffiffiffiffiffiffiffiffi
p8=3
), but it tends to show higher symmetry under pressure compared to that at ambient pressure. The solid line is obtained by least-squares fit to the equation
a/a0
c/c0
V/V0
at room temperature
P (GPa) 0
0.80 0.85 0.90 0.95
a/a0,c/c0,V/V0
1.00
5 10 15
CeAl3
FIGURE 9 Relative changes of the lattice constantsa,c, and the unit cell volumeVof CeAl3as a function of pressure (Kagayama and Oomi, 1995).
c a¼c0
a0
V0 V
g1
; (2)
which is assumed tentatively. We obtainedg1¼0.24. It was found that the magnitude ofB0(¼53.6 GPa) is much smaller than those of other HF materials (e.g.,B0¼100 GPa for CeCu6and CeInCu2), which may suggest an unstable electronic state of CeAl3 under the influence of external forces.
In order to examine the relation between the electronic state and the lattice compression, we will show the temperature-dependent electrical resistivity under high pressure because it reflects the electronic state more sensitively than lattice compression. The temperature dependence of the electrical resistivityrof CeAl3at various pressures up to 8 GPa androf LaAl3 at ambient pressure are shown in Figure 76 in Section 4.1.1 (Kagayama and Oomi, 1996). Ther of LaAl3is similar to the ordinary nonmagnetic metal; it varies linearly with temperature above 100 K with- out any anomaly. While, at ambient pressure,rof CeAl3increases loga- rithmically with decreasing temperature until it reaches a maximum at 35 K and has a shoulder near 6 K. This behavior is due to the Kondo scattering on a thermally populated level split by crystalline electric field (CEF; (Cornut and Coqblin, 1972). With increasing pressure, the peak and the shoulder are merged into one peak, which is shifted toward higher temperatures. Therat 8 GPa becomes similar to that of LaAl3. This result is interpreted as a pressure-induced crossover in the electronic state of CeAl3 from small TK HF state to large TK IV state associated with an increase in the hybridization between conduction electrons and 4f electrons. More details can be found inSection 4.1.1.
at room temperature 0.74
0.73 0.72 0.71 0.70
0 5 10
P (GPa)
c/a
15 CeAl3
FIGURE 10 The axial ratioc/aof CeAl3as a function of pressure (Kagayama and Oomi, 1995).
Figure 11 is isothermalr–Pcurve of CeAl3at various temperatures.
Ther at room temperature shows a maximum near 4 GPa(¼Pc), which corresponds to the crossover in the electronic state. Above 4 GPa, CeAl3is in the IV state. It should be noted that there is no anomaly in the pressure dependence of volume or lattice constants as is seen fromFigure 9. This result indicates that the electronic crossover in CeAl3occurs without any volume anomaly. The result in Figure 11 shows thatPc increases with increasing temperature, which is consistent with the pressure depen- dence of TK as will be mentioned below. In order to get the tempera- ture-dependent 4f magnetic contributionrmag, therof LaAl3is assumed to be pressure-independent phonon part of CeAl3 and subtracted from the r data of CeAl3 at various pressures, rmag¼r(CeAl3)r(LaAl3) (Kagayama and Oomi, 1996). InSection 4.1.1,Tmax is found to increase with increasing pressure. SinceTmaxis roughly proportional to theTK, it may be inferred thatTKalso increases with pressure. This result is the same as mentioned above.
2.1.3.2 CeRh2Si2and CeAu2Si2 In this section, we illustrate some exam- ples of pressure-induced valence transitions in rare earth compounds exhibiting magnetic order. The compounds, CeRh2Si2and CeAu2Si2, are selected as typical examples which are antiferromagnetic at low tempera- ture. In CeRh2Si2,TNdecreases with increasing pressure, and antiferro- magnetism (AFM) disappears at high pressure as will be also mentioned inSection 4.2.1. However,TNof CeAu2Si2increases with pressure, which is in sharp contrast with CeRh2Si2. Considering these facts, we suggest that CeRh2Si2is on the right hand side of the peak of Doniach’s phase diagram (1977), but CeAu2Si2is on the left hand side of that the same
00 100 150
50 200
2 0 K
10 K 50 K
100 K Pc
200 K 300 K
4 P (GPa)
r(mΩcm)
6 8
CeAl3
FIGURE 11 Pressure dependence ofrof CeAl3at various temperatures (Kagayama, 1995).
diagram. Hence, it is interesting to compare the electronic and lattice properties of these two compounds under high pressure.
CeRh2Si2with the ThCr2Si2-type tetragonal structure orders antiferro- magnetically below the Ne´el temperatureTN1¼36 K. BelowTN2¼24 K, there occurs a change of the magnetic structure, from the antiferromag- netic state with the propagation vectorq1¼(0.5, 0.5, 0) to the 4q structure (Kawarazaki et al., 2000). CeRh2Si2 is considered to be a standard 4f-localized system.
In order to investigate pressure dependence of lattice parameters, X-ray diffraction measurement was carried out (Ohashi et al., 2003a). At ambient pressure, the crystal structure is tetragonal with the lattice para- metersa¼4.070 A˚ andc ¼10.156 A˚ , which are consistent with previous experiments (Settai et al., 1997). Figure 12 shows the pressure depen- dences of the relative lattice parametersa/a0,c/c0, and the relative volume V/V0as a function of pressure at room temperature, wherea0,c0, andV0
are the values at the ambient pressure. The tetragonal structure is stable up to 13 GPa at room temperature. Bothaandcdecrease with increasing pressure; no discontinuous changes have been observed within experi- mental error. Linear compressibilities, ki¼(1/i)(@i/@P), i¼a or c, of a- and c-axes are 2.2103 and 3.0103 GPa1. Thec-axis is more compressible than thea-axis. We attempted a least-squares fit of the data of V/V0 to the first-order Murnaghan’s equation of state Eq. (1). The results are shown in Figure 12as a solid line for V/V0. The agreement between the observed points and the calculated ones is satisfactory.
B0andB0are estimated to be 139 GPa and 2.2, respectively.
The temperature dependence of the electrical resistivity along the a-axis is shown in Figure 105 in Section 4.2.1 in a wide temperature
0
CeRh2Si2
P (GPa) a/a0, c/c0, V/V0
1.00 0.98 0.96 0.94 0.92
0.90 2 4 6 8 10 12 14
-c/c0
V/V0
-a/a0
FIGURE 12 Pressure dependence ofa/a0,c/c0, andV/V0of CeRh2Si2. The solid line for the pressure dependence ofV/V0shows the result of the least-square fitting using Murnaghan’s equation. The dashed lines fora/a0andc/c0are guides to the eye (Ohashi et al., 2003a).