on the pressure variation of resistivity of solid pr, la and ac

Z. Phys. B - Condensed Matter 58, 105-110 (1985) Condensed Matter for Physik B

�9 Springer-Verlag 1985

On the Pressure Variation of Resistivity of Solid Pr, La and Ac

K. Iyakutti* and M. Dakshinamoorthy

Department of Nuclear Physics, University of Madras, Madras, India

Received May 10, 1984; revised version August 1, 1984

The high pressure bandstructure results are used to study the pressure variation of the electrical resistivity in Praseodymium (Pr), Lanthanum (La) and Actinium (Ac). The resistivity is calculated using Huang's formula. In the case of Pr and La in the pressure range upto 200kbar the resistivity increases with pressure and approaches saturation value whereas for Ac, the resistivity increases with pressure initially and then it decreases with increase of pressure. The results are compared with the available experimental and theoretical results. The transitions reported by other investigators in Pr and La are visible in the curves of resistivity Vs pressure. We have also computed the pressure coefficients for Pr and La which indicate the probable phase transitions in them.

1. Introduction

On the pressure variation of the resistivity of metals, only a few attempts appear to have been made on the theoretical side. The earlier calculations were due to Dickey et al. [1] Kaveh and Wiser [2] and Khanna [3]. Dickey et al. used Ziman's resistivity formula [4], Kaveh and Wiser used Ziman-Baym resistivity formula [5] and have obtained good agreement with experiment. However, due to the absence of a suitable form for the electron ion interaction in non-simple metals, the above calculations have not been extended to other metals. Khanna [-3] in his work has used the ' t ' matrix approach to calculate the pressure coefficient of resistivity of solid Cu, Ag and Au. Here one has to compute the quantity S(q), the structure factor describing the dynamics of the ions [6]. For that one has to have neutron scattering measurements. In the absence of such information, one has to look for a different formula. Szabo [7] has given a formula for the isothermal resistivity which does not require these quantities. We have used this formula in our calculation to arri'~e at the pressure variation of resistivity of Pr, La and Ac. For Pr and La we have the bandstructure results for various pressures [8, 9]

* Present address: School of Physics, Madurai Kamaraj Univer- sity, Madurai, India

whereas for Ac we have bandstructure results corresponding to various V/Vo values [10]. The rare earth metals display a rich variety of transport phenomena [ l l , 12]. The transport properties were last reviewed by Legvold [13] who presented data for the resistivity, thermal conductivity and Seebeck coefficients of almost all the heavy rare earth metals. For the rare earths Pr and La, lot of experimental work has been carried out with respect to their transport properties both in solid and liquid states [14-25]. But there is no investigation on the theoretical side especially regarding the pressure dependence of resistivity. This is the first calculation in that direction. No work has been done on the transport properties of Ac either experimentally or theoretically. In Sect. 2, we give our justification for the use of Huang formula. In Sect. 3, we present the pressure dependence of the various phase shifts. In the last section, we give the results of our investigation on resistivity and discuss about the various features observed and compare with available experimental work.

2. Justification for the Use of Huang Formula

The only other formula apart from Huang formula for the calculation of resistivity is Ziman's formula

106 K. Iyakutti and M. Dakshinamoorthy: Pressure Variation of Resistivity of Solid Pr, La and Ac

(or modified) using ' t ' matrix approach of scattering theory together with the non-overlapping MT mod- el. According to this formula resistivity p is given by

12~ 1. q 3

where S(q) is the structure factor describing the dynamics of the ions and is given by

S(q) = 1 ~ S(q)dq (2)

where

S(q)=k~_ qD-l(q)q - \kBT!( 1 ~2qZ12

1( )4 . . . . + ~ qD(q)q (3)

To calculate the pressure variation of resistivity we need to know the pressure variation of structure factor S(q) and the ' t ' matrix. From equation (3) it is seen that a calculation of the pressure variation of the structure factor S(q) requires a knowledge of the corresponding variation of D(q). This can be de- termined from the neutron scattering measurements of the force constants carried out at different pressures. However for the metals considered here the neutron scattering measurements are not available. For some of the metals, S(q) are directly tabulated but they correspond to a very high temperature (liquid metals). So, for want of sufficient information on S(q) we could not use this formula. The Huang formula given by Szabo [71 for the isothermal resistivity is

gator to arrive at some other physical quantities such as the superconducting transition temperature, susceptibility etc. [8-10, 26, 27]. In the present case also this formula has given reasonably good results.

3. Pressure Variat ion of Phase Shifts

In this section we consider the pressure variation of the phase shifts. We have calculated 60, 61, 6 z and 63. The values of 64, 65 etc. are negligibly small and they are not considered. The variation of the phase shifts 60, 6~ and 6 z with pressure for Pr, La and with volume for Ac are given in Figs. 1 to 3 respec- tively. 63 is not shown in the figure because of its

0 . 6

0 . 4

:E u~

0. - 0 . 4

82

46) s'o I ,~o I ,~o I ' 2 0 0 ~ Pressure ( K bor}

0 -88 0.81 0.71 0 . 6 6 q V / VO

-0.8

6j -I.2

-I,6 ~ . ~ 8o

-2.0

Fig. 1. Variation of phase shifts with pressure for Pr

Er ~ 2(l + l)sin2(6,(EF)-6Z+l(E~)) (4) Pi--37ce2n2(2o 1

where E v is the Fermi energy, n is the valence electron density and 6~(EF) is the phase shift. This formula is similar to Ziman's formula except for the factor S(q). The phase shift term here is similar to the 't' matrix term which also involves phase shifts. The effect of change of electron-phonon interaction and the change of bandstructure enter this formula through the phase shifts, which is calculated using the M.T. potential corresponding to various pressures, Fermi energy and the atomic cell volume. Eventhough this formula does not invoke a strong Coulomb correlation beyond the one-particle picture within the frame of one electron bandstructure this gives reasonably acceptable results as Ziman's formula. Even for rare-earths the one-electron bandstructure formula has been used by many investi-

0 " 8

0 . 4

- 0 . 4

- 0 . 8

- I . 2

- I . 6 -

- 2 . 0

82

i [ ~ ) i [ 50 I00 150 2 0 0 ~ Pressure (K bor)

0.92 0 .82 0 .7 0 . 6 6 ~ V / Vo

8)

Fig. 2. Variation of phase shifts with pressure for La

K. Iyakutti and M. Dakshinamoorthy: Pressure Variation of Resistivity 107

0 . 4 - - -

- 0 . 4 -

~- - 0 -8 - L~

Z

u~ - i ' 2 -

o.

- I . 6

- 2 . 0

- 2 , 4 -

- 2 -8

o18 i i i ;

0.8 0 .7 0 .6 0'5

V/ Vo

Fig. 3. Variation of phase shifts with reduction of volume for Ac

smal l value. F o r Pr and La, 6 0 and 61 increase (negatively) with increase of pressure while 6 2 increases (posit ively) with increase of pressure. But the change in 6 2 is not so much as we have for b o and 61. W e can see tha t there is close s imi lar i ty in Figs. 1 and 2 (i.e. for Pr and La). F o r Ac (Fig. 3), 6 o and 61 increase (negatively) wi th increase of pressure (decrease of cell volume) bu t the increase is more r ap id than for Pr and La. But 6 2 decreases (posit ively) with increase of pressure (decrease of cell volume) which is ent i rely different f rom the na tu re of b 2 of Pr and La. These features are reflected in the final resist ivi ty ca lcu la t ion as we are going to see in the next section.

4. Pressure Dependence of Resistivity

Using re la t ion (4) and the c o m p u t e d phase shifts we have ca lcu la ted the resist ivi ty as a funct ion of pressure for Pr and L a and as a funct ion of V/Vo for Ac. The results are given in Tables 1 to 3 and in Figs. 4 to 6, we have given the plot, resist ivi ty Vs pressure. F o r Pr and La, the resist ivi ty increases with increase of pressure and tends towards a sa tu ra t ed value. F r o m the Tab le 1, we can see tha t in the case of Pr above 1 4 0 k b a r and up to 1 9 0 k b a r we got a lmos t same value of resistivity. The same feature is observed in the case of La also (Table 2). F o r Ac, ini t ia l ly for smal l pressure (small r educ t ion in its cell volume), the resist ivity increases and reaches a maxi- m u m value and then decreases for higher pressure (for small values of V/Vo). W e have also ca lcu la ted

of Solid Pr, La and Ac

Table 1. Variation of resistivity with Pressure for Pr

V/V o Pressure (kbar) p micro-ohm (cm)

1.00 Normal 80 0.88 45 92 0.81 100 95 0.71 140 104 0.70 150 104 0.68 160 105 0.67 170 106 0.66 180 108 0.65 190 108

Table 2. Variation of resistivity with pressure for La

V/V o Pressure (kbar) p micro-ohm (cm)

1.00 Normal 94 0.92 25 102 0.86 50 107 0.82 75 111 0.80 100 112 0.70 150 117 0.66 200 118

Table 3. Variation of resistivity with volume for Ac

V/V o p micro-ohm (cm)

1.0 114 0.9 116 0.8 117 0.7 111 0.6 98 0.5 77

"E 112 -

i

o 1 0 4 -

E

_-" 96 >

~: 88

~ f

f . ; I 8'o I ' i ' I ' 120 160 200 - - P ressu re (Kbo r )

0.88 0.81 0.71 0.66 - - V / Vo

Fig. 4. Variation of resistivity with pressure for Pr

the pressure coefficient of resist ivi ty for Pr and La. Since the resist ivi ty behaves differently at different regions of pressure, we have given the values of pressure coefficient for different pressure regions (Ta- ble 5). The effect of hydros ta t i c pressure upon the


1 1 8

i lO

102

9 4 f

I 4 0 8 0 120 160 2 0 0 ~ Pressure ( K b a r ) 0 .92 0 .82 0 .8 0.7 0 . 6 6 ~ V / Vo

Fig. 5. Variation of resistivity with pressure for La

120

E I I 0

i

E

o 1 0 0 '

u

9 0

-g

~ 80 g~

70 I I I i i

0 . 9 0 . 8 0 . 7 0"6 0 - 5

V / V 0

Fig. 6. Variation of resistivity with reduction of volume for Ac

Table 4. Comparison of the resistivity of Pr and La with experimental work at 296 ~

Metal p micro-ohm (cm)

Present work Experimental work

Ref. 17 Ref. 18

Pr 80 78 70 La 94 100 81

Table 5. Pressure coefficient of resistivity at different Pressure ranges for Pr and La

Pr La

Pressure range kba r - 1 Pressure range k b a r - 1 kbar kbar

Normal-45 0.00307 Normal -40 0.00253 45-100 0.00059 40-70 0.00144

100-150 0.00200 70-100 0.00065 150-190 0.00073 100-150 0.00105

150-200 0.00027

k b a r - 1 = 107 kg - 1 m z

resistivity of metals may be attributed to (1) a change in the interaction between the electrons and the lattice waves caused by a stiffening of the lattice (2) a change in the lattice parameter and the associated change in the bandstrncture and Fermi sur- face and (3) crystallographic modifications and the accompanying changes in the bandstructure [28].

4.t. Comparison with Experimental Results

The first experimental work on the resistivity of Pr and La was reported by James et al. [17]. Later Alstad et al. [18] used metals of higher purity and measured the resistivities of Pr and La. The effect of pressure on the electrical resistivity of Pr was investigated experimentally by Stager and Drickamer [21]. Recently Wittig [24] has investigated the resistance of Pr under high pressure. We have compared our values with the available experimental values (Table4), Stager and Drickamer [21] have measured the resistance of Pr as a function of Pres- sure upto 600 kbar (the pressures are now believed to be lower by about 30~o according to revised calibration) for different temperatures. For them, the resistance curve increases with pressure upto 300 kbar pressure with a hump at 40 kbar. After that the resistance value remains almost a constant upto 500 kbar. Above 500 kbar, there is a tendency that the resistance curve may come down with increase of pressure. The latest experimental work on resistance of Pr as a function of pressure is carried out by Wittig [24]. He has measured the resistance of Pr as a function of pressure as well as a function of temperature. For him, the resistance at room temperature increases with increase of pressure with a drop at 40 kbar. Then at 190 kbar there is a sudden drop in the resistance value. This transition pressure has been corrected to be 210kbar [29]. His obser- vations at room temperature coincide with the ob- servations of Stager and Drickamer at 77 ~ From the experimental results he predicts that the sudden changes in the resistance curves are the indication of phase transformation. He calls this as Pr III-IV transition. In his view this Pr III-IV transition is a valence transition (of the non-promotional type) with the valence of Pr changing from 3 to 5 since the conduction band gains 2 itinerant 4f electrons [29]. We could compare our results with the experimental works only qualitatively. Our result is in overall agreement with their observation upto 190 kbar. Corresponding to the dip (or hump) in the region 40 to 80 kbar, we got a sudden change in the slope of the curve (Fig. 4): The low pressure coefficient value for Pr in the pressure ranges 40 kbar to

K. Iyakutti and M. Dakshinamoorthy: Pressure Variation of Resistivity of Solid Pr, La and Ac 109

70 kbar and again in the ranges 150 kbar to 190 kbar are the indications for the experimentally observed [30] and theoretically predicted phase transfor- mations [31]. Probably, since our result is arrived assuming a single structure in the entire pressure range, it is not able to show a clear cut indication of phase transition as the experimental results have shown. For La, the pressure dependence of resistance was investigated experimentally at 30~ and 75 ~ by Bridgman [14]. He found that pressure coefficient of resistance was small and varied strongly with temperature. His value at 30~ is -1 .12 x 10 _3 kbar -1. Rapp and Sundqvist [25] reported a similar result. Their pressure coefficient value at this temperature is - 1.09 x 10 - 3 k b a r - 1. Recently Balster and Wittig [22] reported a positive pressure coefficient which is entirely different from the previous two experiments. Our result is in agreement with Balster and Wittig's results qualitatively. Their experimental work is more relevant to us than the previous two for the following reasons (i) The samples used by Bridg- man and Rapp and Sundqvist were dhcp La and (ii) Rapp and Sundqvist have measured the resistance only upto 15 kbar (1.5 GPa). Balster and Wit- tig have measured the resistance of fcc La for various pressures and temperatures. They have reported the resistance upto 140 kbar. Our result also more or less has the same trend. In the resistivity curve obtained by us, there is a sudden change in the slope of the curve corresponding to the pressure region 75 kbar to 100kbar. Also the value of the pressure coefficient attains its first minimum value (Table 5) in this pressure range. This anomaly in resistivity was observed by Balster and Wittig around 70 kbar. They pointed out that this anomaly is not associated with a change of crystal symmetry because Syassen and Holzapfel [32] found that La has the fcc structure between 30 kbar and 120 kbar at room temperature. It was hence concluded that the anomaly in resistance may indicate the existence of a iso-struc- tural phase change [33]. F rom our results it looks that there may be another phase transformation in the range 150 to 200 kbar. For Ac, there are no experimental results available for resistivity. But in general, it is reported that the resistivity of the actinide metals and metallic com- pounds behaves anomalously compared with those of the rare earth and transition metals. In the present case, the variation of resistivity with pressure is similar to the variation of resistivity with temperature in Pu and PuA12. It has been shown that the electrical resistivity in some actinide metals after having a maximum, decreases with increasing temperature due to the s - f dehybridization which sup-

presses the scattering of the s electrons [34]. Here also in our case, the behaviour of resistivity in Ac as a function of reduced volume may be due to the s - d transition and s - f dehybridisation between the unoccupied itinerant 5f bands and the 6d7s bands.

4.2. Conclusion

Since the three metals Pr, La and Ac have same valence electron configuration one may expect them to have similar transport properties. But Ac being a member of actinide series, its behaviour is entirely different from that of Pr and La which are in the rare earth series. The resistivity of Ac metal behaves anomalously when compared to rare earths. This is well brought out by our present calculation. Under the same pressure range, for Pr and La, the resistivity increases and reaches a saturation value, but for Ac, the resistivity increases, reaches a maximum and then it has decreased. Even its normal pressure resistivity is very much higher than Pr and La. From the experimental work it looks that the resistivity of Pr also decrease above 190kbar. Wittig has observed a sudden drop of resistance at 210 kbar and attributes it to a phase transformation. Will the same argument hold good for Ac which belongs to actinide series, for which we have estimated a drop in resistivity in the volume range V/Vo=0.7 to 0.5? Only experimental results can throw more light on this problem. Property wise, among the three metals, Pr is a complicated metal. At normal pressure, its resistivity should have contribution from its magnetic behaviour. But at higher pressure its character is not certain. Under high pressure, because of the delocalisation of the 4 f electrons, it may become nonmagnetic [35].

The authors are thankful to Profs. V. Devanathan and T. Nagarajan for their encouragement. Useful discussions with Dr. R. Asokamani and Mr. M. Rajagopalan are acknowledged with thanks. The financial assistance from the Department of Atomic Energy, Government of India is also acknowledged with thanks.

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K. Iyakutti* M. Dakshinamoorthy Department of Nuclear Physics University of Madras A.C. College Campus Guindy Madras 600025 India

* Present address: School of Physics Madurai Kamaraj University Madurai 621025 India

on the pressure variation of resistivity of solid pr, la and ac

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