RESISTIVITY OF LIQUID Ga BETWEEN 303 K AND 573 K FROM STRUCTURE FACTOR MEASURENENTS

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RESISTIVITY OF LIQUID Ga BETWEEN 303 K AND

573 K FROM STRUCTURE FACTOR

MEASURENENTS

M. Bellissent-Funel, R. Bellissent, G. Tourand

To cite this version:

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JOURNAL DE PHYSIQUE CoLZoque C8, suppZ6ment au nOO, Tome 41, aoct 1980, pageC8-262

R E S I S T I V I T Y O F L I Q U I D G a BETWEEN 3 0 3 K A N D 5 7 3 K F R O F STRUCTURE FACTOR MEASUREKENTS

M.C. Bellissent-Funel, R. Bellissent and G. Tourand

CEN-SacZay, IlPh-G/I'SRM, B.P. 11°2, 91190 C i f - s w - Y v e t t e , France.

1. INTRODUCTION

In order to explain the temperature dependence

of the resistivity of liquid Ga between 303 K and 573 K [I], very accurate structure factors have been determined by neutron diffraction. Using the Ziman formalism [2], the resistivity of liquid Ga has been evaluated and compared with the experimental data.

2. DETERIIINATION OF THE STRUCTURE FACTORS The experiments have been performed on the

spectrometer HI0 ( A = 1.136 A) of the reactor EL3 at

Saclay 131. The use of a 640-cell multidetector pro- vides us with a relative uncertainty on the inten-

sity I (28) scattered by the sample of about 0.4 Z.

S

The samples were contained in sealed silica cylin-

drical cells of 10 mrn diameter and 0.5 mm thickness

inside a cylindrical vanadium furnace.

The method used to determine the intensity

IS(26) has already been presented [4],[5]. After

geometrical corrections for the absorption of the furnace, container and sample, the correction within the static approximation has been made by using a Placzek type method [6]. The additional error for

the resuLting intensity is less than 0.1 ?

.

1,le also

estimated the contribution due to the multiple scattering [7]; this correction to first order is less

than 0 . 5 % : moreover, it is constant with the wave

vector modulus q which is confirmed by previous measurements [ 8 ] .

We have therefore normalized the intensity by

usin2 the relation :

Is (q)

-x

S(q) =

-

IS (rn) -X

where X represents the no-q dependent contribution

(incoherent scattering, multiple scatterin Is

(61

x

:

The structure factor for q = O is S(0) =---

Is (w)

-x

' X is deduced from the thermodynamic limit

S(0)=nkgT

xT

where kB is the Boltzmann constant ;

n the number of atoms per unit vo1ume;T the abso-

lute temperature and

xT

the isothermal compressibi-

lity. All the values of S(0) are calculated from

density and sound velocity data [ 9 ] . In addition,

the values IS(0) and IS(") are determined on the

basis of the experimental curve.

Furthermore, for each temperature, we have chec-

2 2

ked the general relation :

lm

q [S(q)-l]dq = -2n n

obtained from the general fgrm of the radial distri- bution function g(r) when r becomes 0.

The relative uncertainty AS(q)/S(q) taking

account of statistical errors and errors due to all corrections is given in the table I.

Small q

2.55 -3.30

Table I

The experimental structure factors are represented in fi~ures 1 and 2 . We observe a s l i ~ h t decrease of the first peak as the temperature increases from

303Kto 5 7 3 K , the ~osition of this first maximum

being located at 2.55

1-I.

In addition, the results

clearly show a smoothing of the shoulder located

at 3.1

1-I

without any shift versus temperature.

These results are in agreement with previous data

[lo]

and clearly confirm that the vanishing of the

shoulder with increasing temperature is a very slow phenomenon.

3 . CALCULATION OF THE RESISTIVITY

The resistivity is calculated in the Nearly

Free Electron Model

[I

I]. The electronic scattering

probability term P(u) is related to the relaxation

time T by the relation

The description of the liquid in terms of the struc-

ture factor S(q) leads to :

mk _{+ +} _-t ₂

P(U) =

2

N-I

s

(q) (<k+q

1

w

1

k>

(

(3.2)

ns3 m

From the classical relation p = ---;;-- and the

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Fi,q. l :

Ex;oemi~e.r&d n L t u m e jact0h.J doh fiq&d

303K

5

T

5

383K.

relations (3.1) and (3.2), the well known Ziman

formula [2] is obtained :

m is the electron mass ; e the electron charge ;

6

Planck's constant and Z the atomic number of the 2

element. The Fermi wave vector k = (371 nZ)'I3 is gi-

F

ven as a function of the temperature in the table 11.

+ + 3

The matrix elements I<k+qlwlk>l are the atomic form factors of the screened pseudopotential w of a single ion. The calculation of the resistivity has been made using three different form factors

of the pseudopotential :

- the model potential of Heine and Abarenkov [12] (HAP). We used the atomic form factors obtained by Animalu et a1 [ 1 3 ] .

- the optimized model potential of Shaw [I41 (OMP). We used an improved form proposed by AppapillaY et a1 [151.

- the empty core model potential (ECP) proposed by

Ashcroft [16].

In table I1 and in figure 3 the results of the calculation of the resistivity using the two first

models (HAP and 0P.P) are presented and compared

with the experimental data of Pokorny et a1

[I].

Fig.3 :

Tempenatwte dependei~ce

0 4

t h e e l e t h i c &

-

hen~!,LLv.Lty

06 Liquid Ga.

x

Expdrnentae data

0 6

Pvkohny

eR

a4?

[I]

Uiin,- ,the Heine-Abmenkov model

poteiu5a.L.

@

Uiing t h e o p t h i z e d r m d d potentiae.

These two models allow one to reproduce the slope of the curve giving the variation of the resistivi-

ty versus temperature ; a similar calculation per-

formed by Wagner

1

1 7 ) did not lead to a so good aggreement between the values of the resistivity

calculated with the HAP model and the experimental

ones of CUSACK

1

181

.

The values of the resistivity

given by the HAP model arc higher than the experi-

mental ones but we had not taken into account the I

*

electron effective mass m

.

We have the efore

evaluated the ration R =

-

4

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C8-264 JOURNAL DE PHYSIQUE

Table I1

3+

find at 303K, the value R=0.966 which corresponds to that of the ionic radius of Ga [201. The varia-

to the value of (m*/m) given to the first order by tion of R versus temperature is represented on the

Weaire 1191. The ratio R may give an idea about the fig.5.

behaviour of m* as a function of the temperature for

the HAP model and this last one is given in fig.4.

pow 1-152 x cm 24.413 25.013 25.319 25.534 25.778 26.254 26.448 26.996 27.736 28.231 HAP P

ufi

x cm 27.733 28.298 28.614 28.879 29.153 29.646 29.865 30.519 31.398 32.026

Fig.4 : T e m p e h a t ~ r e de p e n d e n c e

0 4

;the &o (m*/m)

-quid

Ga.

112 m* -

f

pexp\

T-

(PHApJ

0.966 0.964 0.966 0.967 0.971 0.970 0.974 0.980 0.982 0.988 pexP

un

x cm 25.905 26.305 26.708 27.111 27.515 27.918 28.320 29.311 30.301 31.270

While the OMF' model takes into account the effective mass and exchange correlation corrections, the calculated values of the resistivity are not in as good agreement with the experimental ones as is the case for the HAP model. This disagreement may be due to an overestimate of these additional effects. We have also calculated the relative uncertainty Ap/p due

to AS(q) /S(q) ; its value is approximately 0.9 %

.

However, this precision is not yet sufficient to put in light the maximum in dp/dT at 363K as it is

observed by Pokorny et a1

[I].

The ECP model is sa-

tisfying because the fit of the experimental resis-

tivity curve is obtained for empty core radius R

values given in table I1 and lying between 1.21 a.u. (T=303K) and 1.24 a.u. (T=573K) which are close

Rc a.u. 1.209 1.21 1 1.213 1.215 1.217 1.219 if222 1.227 1.232 1.237 T K 303 323 343 363 383 403 423 473 523 573

Fig.5 : Tempem.-tme d e p e n d e n c e

0 6

Rhe e m p i y c o h e

tLadiun

Re d o h L i q u i d

Ga.

5

A-1 1.672 1.671 1.670 1.668 1.667 1.666 1.664 1.661 1.658 1.655 4. CONCLUSION

We have calculated the resistivity of liquid Ga as a function of the temperature using our experi-

mental structure factors. A fairly good agreement is

obtained between the calculated resistivity and the experimental data of Pokorny et al. By taking into account the effective electron mass in the HAP model and the empty core radius in the ECP model, an accurate fit of experimental data can be achieved which is impossible with the OMP model. This calculation may be a way to obtain the parameters of the pseudo-

potential like m* and Rc. Nevertheless, at least,

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of the model potential. This point is consistent with the fact that the structure factor, itself, is due to a true potential which includes a pseudopotential term. This strongly suggests that, at least, for liquid Ga, the resistivity depends mainly on the structure factor.

REFERENCES

[ l ] Pokorny M. and AstrGm H.U., J.Phys.F,Metal.Phys.

6 (1976) 5 5 9 .

[ 2 ] Ziman J.M., Adv.Phys. 16 (1967) 551.

[ 3 ] L.L.B. Internal Report 1978 (Saclay).

[ 4 ] Bellissent-Funel M-C., ThSse Grenoble 1977.

[ 5 ] Paalrnan H.H. and Pings C.J., J.Appl.?hys. 33

(1962) 2635.

[ 6 ] Yarnell J.L., Katz M.J., Wenzel R.G. and Koenig

H.S., Phys.Rev.A 7 (1973) 2130.

[ 7 ] Blech I.A. and Averbach B.L., Phys.Rev.A 137

(1965) 1113.

[ 8 ] Antonini M. and Corchia ! I . , Nuovo Cimento 10B

(1972) 117.

[ 9 ] KSster V.H., Hensel F. and Franck E.U., Ber.

Buns.Gesells. 74 (1970) 43.

[ l o ] Bizid A., Defrain A., Bellissent K. and Tourand G., J.Physique 39 (1978) 554.

1111 Shimoji, 1977 Liquid Metals (London : ~cademic

Press INC.).

[ 1 2 ] Heine V. and Abarenkov I., Phil.Mag. 9 (1964)

451.

[ I 3 1 Animalu A.O.E. and Heine V., P h i l . ~ a ~ . 12 (1965)

1249.

[ 1 4 ] Shaw R.W., Phys.Rev. 174 (1968) 769.

[ I 5 1 Appa~illai M. and Williams A.R., J.Phys.F :

Metal Phys. 3 (1973) 749, 4 (1973) 772.

[ I 6 1 Ashcroft N.W., Phys.Lett. 23 (1966) 48.

1171 C .N.J. Wagner, International Conference on

Liquid Metals, 3rd, Bristol 1976.

[:18] Cusack N. and Kendall P., Proc. Phys. Soc.

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[ 1 9 1 ~ e a i r e D., Proc.Phys.Soc. 92 (1967) 956.

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