Indi an Journal of Pure & Applied Physics Vol. 40, Janu ary 2002, pp. 32-4 1

Evaluation of partial and total structure factors, Bhatia-Thornton correlation functions, compressibility and diffusion coefficient of

Pb-Pd alloy at different compositions and temperatures Sumita Bandyopadhyay, R Yenkates h & R V Gopala Rao

Department of Chemistry. Banaras Hindu Un iversity, Yaranasi 22 1 005

Received 8 February 200 I; revised I November 200 I; accepted 2 November 200 I

The part ial and total structure factors of Pd-Pb alloy have been computed using Lebowitz solut ion of hard spheres with a sq uare well attractive tail. The calcu lated values of the total structure factors and also that of the reduced radial distribution function are in excellent agreement wi th experiment. It is found that the partial structure factors show peculiarities due to the fo rmation of a compou nd Pb2Pd as reported by Chri sti an et al.[J NonCJ)•St Solids , 156( 1993)38]. Bhati a-Thornton correlat ion fu nctions are also calcul ated, namely, SNN(k). which show sirong correlation in k-space wh il e Scc(k) and SNc(k) do not show such correlation. The partial and total radial distribution functions (RDF) are also calcu lated from which first coordin ation numbers are obtained. The RDFs show for all composi tions the first peaks at 3.35 A. Exactl y the same value has been obtained by Christian eta/. This value is in no way nearer to the firs t nearest neighbour distances in pure Pb or Pd . Thus, the small va lue of 3.35 A, which corresponds to the first peak positi on, has been attributed to the formation of a compound Pb2Pd, which in crystall ine form shows exactly the same distance between unlike neighbouring atoms. The va lues of compressibility have also been calcu lated , which show a bigger va lue of 41 .6 x I o-12 cm1/dyne at 60 atomic percent of Pb. This also indicates the formation of a compound. The diffusion coefficients have been calculated through Helfand 's prescription. No observable sudden changes are seen in the case of diffu sion coefficient. As suggested by Chris tian et a l., the authors also calculated the total sum of the partial structure factor S,(k) , namely that of SPh-rh(k) and Srh-ru(k) and a strong right hand shoulder at 60 atomic percent of Pb has been found and this perhaps is a characteristic of compound formation and the shoulder is obtained at 2.8 A whi le in Pb2Pd an exact distance is found in its crys talline form between unlike atoms. It may be pointed out Pd-Pb distance in amorphous Pb2Pd compound is exactly 2.83 A whi le the present shoulder is obtained at 2.8 A. Hence it is concluded that Lebowitz solution wi th a square well att ractive tai l reproduced several characteristics shown in the experiment and also clearly indicate the formati on of a compound.

1 Introduction

It is found that certain alloys like 50 atomic percent of Sn wi th Au u exhibit a split in the first peak·\ Several amorphous all oys like CuSn and AuSn also exhibit a sp li t in the first peak3

•

Amorphous alloys prepared by liquid quenching are supposed to retain an atomic arrangement simi lar to that observed in the liquid state. Christian et a/.4

investigated Pb-Pd alloys at different compositions and temperatures by neutron diffraction method and obtained the nature of the total structure factor and also the total reduced distribution function G,(r ) which is the Fourier transformation of G,(k) = k[S(k)- I] . They did neither observe any split of the firs t peak in the total structure factor S(k) of the eutectic alloy Pbli(]Pd40 nor a shoulder in S(k) as was fou nd out by Rao & Satpathy5·6 in the case of Cu-Bi all oy' and in certain peculi ar metal s6 like Sb, Bi , Ga and Sn. In the present calcu lations it was thought

best to study Pb-Pd all oy theoretically. Hence it was fe lt that it is interesting to obtain Ashcroft Langreth structure factors of Pb-Pd alloy at different compositions and temperatures7

·K. In these invest igations the au thors studied part ial and total structure factors of the all oy, the partial and total radial distribution functions , the values of compressibility as a function of composition and also the reduced total distribution function , the Bhatia-Thornton corre lation functions, the coord ination numbers and the diffusion coefficients th rough the use of Helfand 's linear trajectory methocl9

•

2 Theory

The well-known solut ion of Percus Yevick 's equat ion for hard sphere mixture obtained by Lebowitz 10 and Baxter 11 was used. The solut ions they obtained for the direct correlation function

BANDYOPADHY A Y eta/. :PARTIAL AND TOTAL STRUCTURE FACTORS 33

(DCF) are suitable and convenient for numerica l calculations.

Within the framework of mean spherical approx imat ions (MSMA) the DCF can be written as:

C1u( r), 0 < r < cru

Cu(r) = £;jK8T, CJu< r<A ucru

0 ,

... (I)

Here, the conditi on cr2 > cr , must be fulfilled so as to ap ply Lebowitz so luti on. Further, C1u is the DCF for a mixture of hard spheres obtained by Lebowitz for the Percus Yevicks equation 10. Here

CJ;;, £;;and Au are the diameter, potential energy depth and breath respectively, of the square well attractive tail added to the hard sphere solution of the ith species. The mixed parameters are determined through the use of Lorentz Be rthelot rulesL'. Thus:

(J ii + (J" jj (J" .. = ----"'-

II 2

£ .. = (£ £)1 /2 IJ I j

Acr + A .. cr .. A .. = II II }} }}

I.J 2

... (2)

The C;/ s obtained by Lebowitz have been di scussed a lreadyR·12

• They have been used in the evaluat ion of total correlation function hu and is re lated to C;1 as:

... (3)

As is well-known , one can have:

... (4)

Rao & Murthy"-' 4, a lso give the Fourier

transforms of the DCFs in great detail. The partia l structures are finally obtained as :

Sll(k)= l-piCII (k ) -plp2CI22(k)

1- PzCzz(k)

S22

(k)= {I-p,C11 (k)}S 11 (k)

J-pzC22 (k)

... (5)

... (6)

( ) "2 c k sl 2(k) = P1P2 12( )SI2(k )

1-p2C22 (k) . .. (7)

The C;/k) are the Fourier transforms of C;j(r) and , S,J(k), S22(k) and S 12(k) refers to partial structure factors of Pd-Pd , Pb-Pb and Pd-Pb respectively. The tota l structure factors have been calculated through the following equation:

S(k) = ± I(CC) 11 2 .IJ; S(k) . . I } c .(." 2 c 2 I) t = l J= l iJ I + 2j2

... (8)

Here,_{;' s are the neutron scatteri ng factors. The

necessary parameters CJ;, £; and A; are taken from literature'6

· '7 and the neutron scattering factors data

are a lso taken from literature1x. It is found that the

parameters cru' s are nearer to Pauling's va lues. From Cu(k), the authors obtained the partial structure fac tors namely S,,(k), Sdk) and S 12(k) and from these partial structure factors the total structure factors were calculated through Eq. (8), while the partial structure factors are computed from Eqs (5)­(7).

2.1 Bhatia-Thornton (BT) correlation functions

Bhatia & Thornton 19·21 have introduced three correlation function s in terms of fluctuation s in (i) particle density, SNN(k) (ii) in local concentration, Scc(k) and (iii ) in terms of number and composition SNc(k). These functions are very important because they are related to partial structure factors linearly. Further, they are re lated to several thermodynamic properties as we!F0

. The linear re lationships between BT function s and partial structure factors are as follows :

sNN(k)=C,S11 (k)+C2S22 (k)+ ~C1 C2 S 1 2 (k) ... (9)

Scc(k) = C,C2[C2S,,(k) + C,S22(k) - 2 ~C,C2 S 12(k)]

... ( I 0)

... ( I I )

2.2 Calculation of compressibility of the alloy at different compositions

The Fourier transformations of the DCFs in r space to k space are important. Thus, C;j(k) can be

34 INDIAN 1 PURE & APPL PHYS, VOL 40, JANUARY 2002

used to compute the compressibility of the alloy at different compositions by evaluating the same in the long wave length limit. Thus, the values of compressibility can be obtained from Kirkwood Buff's equation, namely:

pK 8T -- =[I- C1p1CII(O)- C2p2C22(0)

Kr

... (12)

Here KT is the bulk modulus. p; the number density of the ith constituent, p the total number density and Cu(O) the value of Cu(k) at k = 0. Hence:

I [ fa bO" ceo>=-;;; -247J;rt+~!._+

dO"?} )+[81J;E;;(A;/ -I)] 6 KBT

... (13)

-4Jr0" 3{ b(O"I + 2b2) + MO"l (30"1 +50"2) I ]2 10

+d0"2 (20"1 +30"2) } I 30

... (14)

The symbols a;, b;, d;, T);, etc. have already been explained5·w . The Cu(O) are called the DCFs in the long wave limit.

2.3 Partial and total radial distribution functions and evaluation of first coordination number

The partial and total radial distribution functions can be obtained from the corresponding partial structure factors. Thus, one can get:

.. . ( 15)

and hence the first coordination numbers 'l'u can be obtained from the following equation:

rmm 2 \jl;i = 4npu f r gu(r)dr

0

... (16)

Here, rmin corresponds to the first minimum in the corresponding g;lr) and so also in gT.,,a1(r). The

latter is obtained from the Fourier transformation of the total structure factor.

2.4 Evaluation of diffusion coefficient of the alloy at different concentrations

The partial structure factors can be lucratively used to evaluate the diffusion coefficient through the use of Helfand's linear trajectory principJeY. The authors understand that diffusion coefficient of ith component can be obtained from the Einstein ' s equation, namely:

D = KBT I ~j

... (17)

where ,; is the friction constant and can be written as the sum of hard sphere contribution, soft part contribution and cross effect contribution. Thus, ~; is given by7·x· 13 :

where

J:.S _ 8 2 P; 7rJ1u (

2 )1 /2 <:. · ---I--

I 3 j=l 3 KBT

I ~J 1 , d -( \2 k·uij(k)hu(k) k

27r J 0

f dk(kO"u coskO"u - sin kO"J u;~ (k) 0

... (18)

... ( 19)

... (20)

... (2 1)

Here, Jl;j is the reduced mass, hu(k) and uu(k) are the Fourier transforms of the total correlation function hu(r) = [gu(r)- I] and the atiractive part of the potential function . Thus:

and

s 47rE u. (k) = __ u [AkO" coskA O" -sin 1\.kO"

I} k) IJ I} I) I} I) If

BANDYOPADHY A Y et al. :PARTIAL AND TOTAL STRUCTURE FACTORS 35

Here, m; and mi are the atomic masses of the ith

and jth components of the alloy and ()ii is the Kronecker de lta.

sr----------------------

Temp. 01o Pb

49

.....

.X ... ,;;

1

Fig. I- Partia l structure facto rs of S11 (k)

3 Results and Discussion

The parameters fitted to obtain the parti a l and total structure facto rs are g iven in Table I (A and B). T he fo ll ow ing compos it ions of the Pb-Pd a ll oys have been in vestigated. T hey arc 49, 55 , 60 and 90 atomic percent of lead at d iffere nt temperatures. The structure fac tors of each of these compositi ons have been computed th rough Eq. (8). The parti a l structure factors are shown in Figs 1-3, while total structure factors are shown in Figs 4a and 4b. In Tab le 2 the first peak pos it ion of the reduced radia l di stribution functi on and a lso that of the total structure fac tor ST(k) are given (v ide Figs 4a and 4b) as obtained by the authors along w ith that obta ined

by Chri stian et a/. 4 in the ir diffracti on studies. In Fig. 5 is given the sum of parti al structure fac tors of Sr~rrh(k) and Sr~rrd (k).

-.X

N N

l/)

sr---------------------0foPb

Fi g. 2 - Parti al structu re factors of S22(k)

The importance of the sum of Srh-Ph( k) and Srh­r~Ck) has been di scussed . The Bhatia-Thorn ton corre lation functi ons for a ll these concentrati ons are shown in Figs 6a-6c. The values of compress ibili ty have been calculated and presented in Table 3. Th is table also presents the parti a l diffu sion coeffic ients . In Table 4 the partia l and tota l coordination numbers have been presented. Here, it is reminded that I stands fo r Pd and 2 stands for Pb. The factor

T) 11 stands for the number o f nearest ne ighbours of the same kind , T) 12 gives the number of nearest

ne ighbours of the second kind while llT stands for the tota l number. It may be pointed out that in Table 3, D1 refers to the d iffusion coefficient of Pd while D2 that of Pb. In Table 5 nearest ne ighbour di stances have been presented .

36 INDIAN 1 PURE & APPL PHYS , VOL 40, JANUARY 2002

4 Ttmp. 0/oPb

800°C 49

3

Fig. 3- Partial structure factors of S12(k)

It can be observed from Figs 1-3 that the partial structure factors are not normal. Thus, S 11 (k) shows a second big peak for all concentrations except for 90 atomic percent of Pb for which the second peak is small. Thus, at lower concentrations of Pd and hence higher concentrations of Pb it is possible that a compound-like substance is forming and is giving a second peak in S 11 (k). This possibility has also been pointed by Christi an et a/. 4 Bhatia also gives the same reason 21 that such satel lite peaks appear when interaction between the constituents of the alloy are strong. Thi s information is also corroborated by the fact that the authors get the first peak in radial di stribution function (RDF) at 3.4 A and thi s corresponds to the di stance in the crysta ll ine compound Pb2Pd as pointed by Christian et a/. 4 At 90 atomic percent of Pb, a very small satellite peak is obtained, while it becomes conspicuous as the concentration of Pb decreases. Srh-rik) also reveal s simil ar characte ristics of compound formation of the alloy. Similar peculiarities are a lso noti ced in S2i k) due to the

-~ ~ 2

(/)

1

2

-- Th11or. o oo Expt .

Temp. 0 /o•P b

500°C 60

8

Fig. 4a-Total structure factors ST(k) at 500 oc and 600 oc

-~ ~ 2

(/)

1

2

650°C

--ThCEor. o oo Expt .

6

01ePb

49

55

8

Fig. 4b- Total structure factors ST(k) at 650 oc and 800 oc

compound formation. The total structure factors ST(f.:) obtained through Eq. (8) agrees very we ll with

BANDYOPADHY A Yet al. :PARTIAL AND TOTAL STRUCTURE FACTORS 37

-,:,t. C" N

~

3~--------------------·

Ta mp. •!e Pb

500°C 60

0~----~----~----~----~~ 0 2 6

Fig. 5- Sum of the structure factors S,(k) of

Srb-Pb(k) and Srb-Pd(k)

8

Table I A- Parameters used in the calculation of ST(k)

Metal

Pb Pd

cr;j(A>

2.970 2.400

70.00 245.49

1.40 1.50

Table I B - Temperatures, parti al number densities and scattering lengths

Partial number densities of Scattering constituents I and 2 in A-3 length

(I 0' 13 em)

Temp PI P2 Pb Pd (oC)

600 0.00607 0.02710 9.4 6.0 500 0.02430 0.01864 650 0.02728 0.01710 800 0.03085 0.01523

that of experiment (vide Figs 4a-4b). Surprisingly the total structure factor does not reveal the details of the compound formation . In Fig. 5, the authors present the sum of the partial structure factors Srt.-rb(k) and Srbrd(k) at 60 atamic percent of Pb. In Figs 7a-7d are presented the partial and total radial distribution functions, respectively. It is observed from Fig. 6a that in the case of gPd-Pb(r), the first peak position does not change while in grh-rh( r) there

is a shift of the first peak to the right with increasing concentration of Pb. No such shift is observed in the other two graphs, namely grd-rh( r ) and the total radial distribution function. The presently computed total reduced radial distribution function G,(r) is shown in Fig. 8. The agreement is excellent in all the cases, thus proving that the present method is suitable for the calcu lation of structure factors of an alloy containing transition metals like Pd.

In Figs 6a-6c the Bhatia-Thornton correlation functions are presented. The SNN(k) shows a great resemblance to the total structure factor (vide Figs 4a and 4b). The SNN(k) shows a conspicuous first peak and a small second peak and so on. This shows the strong number-number correlations in momentum space. Scc(k) shows no oscillations. It may be seen that the peak height decreases with increasing concentration of Pd. Further Scc(k) shows just only one small peak and becomes constant.

z z

(/)

5

4

3

2

,

2 4 6 8

k ( .A-1)

Fig. 6a- The SNN(k) of Bhatia-Thornton correlation function

38 INDIAN J PURE & APPL PHYS, VOL 40, JANUARY 2002

This shows Scc(k ) is on ly short ranged correlation function in thi s alloy. The cross correlation

3 T~mp. •r. Pb

('--800°C 49

2 ....... .X

u ~50°C u 55 (/)

1

2 4 6 8

k c.A1 >

Fig. 6b- The Scc(k) of Bhati a-Thornton correlat ion fu nction

.:::t. ....... u z

(f)

500°C 1

600°C 0

0/oPb 49

55

60

90

-1 L---~----~----~----~

0 2 4 6 0

k (A )

Fi g. 6c- The SNc(k) of Bhatia-Thornton correlation functi on

functions are not conspicuous and just shows only one peak, while in the rest of the momentum space it is nearly zero. The peak height in S ,c(k ) is almost a constant showing only weak correlations .

Table 2 - Composition of Pb-Pd alloy and characteristics of the reduced RDF of G,(r) and total structure factors ST(k) along with experimental Ya lues

Comp- I st peak Christ- First peak Christ-osi tion position ian position ian (atomic of total va lue4 from total value4

% Pb) reduced SF (k 1)

RDF,A

90 3.40 3.35± 2.30 2.24± 0.02 0.02

60 3.20 3.35± 2.42 2.38± 0.02 0.04

55 3. 1 3.35± 2.50 2.48± 0.05 0.04

49 3.1 3.35 ± 2.50 2.57± 0.05 0.04

Table 3- Values of compressibility and di ffu sion coe ffici ents of Pb and Pd at different composit ions

Temp at. % ~T X o,x 105 0 2 X 105

(a C) ofPb 1012 (Pd ) (Pb) cm2/dyn cm2/sec crr.2/sec

600 90 11.83 13.23 7.43 500 60 4 1.63 11.78 7.67 650 55 12.40 9.40 8.68

Table 4 - Partial and total coordination numbers

Temp at% r cA-Jl 11 11 11 22 11 12 11 T (a C) of

Pb

600 CJO 0.046 1 2.0 9.0 3.3 9.8 500 00 0.0444 6.3 6.1 4.1 5.8 650 55 0.0429 9.0 4.0 4.8 5.8 800 49 0.0340 9.0 3.6 7.8 5.7

The values of compressibility are given in Table 3 and the order of magnitude is good. There are no experimental values. However, the authors would like to point out in the above investi gations4

, both temperature and composition have been varied and also a compound formation Pb2Pd is there, which

BANDYOPADHY A Y et al. :PARTIAL AND TOTAL STRUCTURE FACTORS 39

makes the in terpretation difficult. T he compressibi lity as obtained from Kirkwood Buffs

Table 5- Nearest neighbour distances

Temp at. % of p (k' ) Nearest neighbour (oC) Pb distance

Present Litera-value ture

value 600 90 0.0461 3.4 3.35±

0.02 500 60 0.0444 3.2 3.35±

0.02 650 55 0.0429 3.2 3.35±

0.05 800 49 0.0340 3. 1 3.35±

0.05

o~--~~--~----~---L_J

0 2 4 6 8 0

r (A)

Fig. 7a- Partial radia l distribution funct ion g 11 (r)

-'-N N

01

Tczmp. 0/o P b

0 r ( A)

Fig. 7b - Partial radial distribution fun ction gn( r)

equation at 60 atomic percent of Pb which is the eutectic composition very c learly indicates the formation of a compound. However, it may be

noticed that at 800 °C no observable split can be noticed even in the experiments of Christian et al.4

as also in total structure factor.

-'-. ._..,. .

6 T~Zmp . •r. Pb

Fig. 7c- Partial radial distribution function gdr)

8.---------------------~

Tczmp. 0/o Pb

55

60

2 6 8 0

r ( A)

Fig. 7d - Total radial distribution function g(r)

The self-diffusion coefficient of Pb ts decreasing with its increas ing concentration while D2, the self-diffusion coefficient of Pb increases with it s decreasing concentration. Unfortunately the authors could not get any information from the

40 INDIAN J PURE & APPL PHYS, VOL40, JANUARY 2002

ca lcul ated diffusion coeffic ients regarding the formation of compound (vide Table 3).

....

--Theor. o o o Exp't.

Fig. 8-Total reduced radial d istribution function Gs(r}

The change in the number of nearest neighbours is given in Table 4 and it is not smooth with increasing concentration of Pb. This is probabl y due to the formation of a compound. Pure Pb and Pd have fcc structures. The nearest neighbour di stance in the crystalline pure Pb is 4.94 A while in Pd it is 3.9 A. Neither the present in ves ti gat ion nor that of Chri st ian er a/.4 indicate any distance like this. The va lues of first max imum in both theory and experiment coincide and the distance of 3.3 A is found from the first peak of the RDF and is attributed to the format ion of Pb2Pb compound. With 90 atomic percent of Pb, the number of nearest neighbours is found to be about 10 in our calcu lati ons at 600 oc and is nearer to the fcc

structure val ue even at a high temperature of 600 °C (the melting point of Pb is 327.5 °C). The formation of compound is c lear from the fact that with increasing concentration of Pd . the number of nearest neighbours suddenly changes to 5.8 at 60 atomic percent of Pb from about I 0 obtai ned at 90 atom ic percent of Pb.

The authors get only a shoulder for the sum of partial structure factors Srh-rh(k) and Srh-rd(k) (v ide Fig. 5). The reasons are not clear for not gett ing a sp lit as obtained by Christian et a/4

• However, in their case too , the split disappears fas t beyond 60 atomic percent of Pb. Perhaps the other peaks are not that much consp icuous. Further. 60 atomic percent Pb corresponds to the eutecti c compos iti on

of this alloy and the authors got a ha lf width (L\Q) = 1.0 A while Christian et al. get around 0.8 A. The sum shows a shou lder at 2.8 A. which Christian et a!. also got exactly at the same pos ition . This c learly shows as pointed out by them, is due to the forma ti on of a compound Pb2Pd, which shows a si mil ar distance between unlike atoms in the crysta l! i ne state.

Acknowledgements

Two of the authors (SB and RV) are th ankfu l to the Department of Science and Technology, New Delhi and the Council of Scientific and Industria l Research, New Delhi , for financia l aJsistance, and RVGR is gratefu l to the Indian National Science Academy and the Council of Scientific and Industria l Research for f inanc ia l assistance.

References

Henders H. Z Natwforsch , 2a ( 1947) 505.

2 Kaplow R & Averbach B L. Local atomic arrangements. studied by X-ray difji"action (Gordon and Breach. ew York) 1966. p. 159.

3 Le itz H, Z Phys B. 49 ( 1980) 65.

4 Chris tian N J. Wagner C N J & Boldrick M S. 1 Non cryst So lids, 156 (1993) 38 .

5 Gopala Rao R V & Satpath y A, Phys Rev B, 4 I ( I 993) 38 .

6 Gopala Rao R V & Satpathy A, Physica Sta t Solidi (b), I 58 ( I 990) I I.

7 Gopala Rao R V & Yenkatesh R, Phys Rev B, 39 ( 1987) 3563.

8 Gopala Rao R V & Yenkatesh R. Phys Rev B, 39 (1989) 8 142 .

9 Heli fand E , Fluids . 4 ( 196 I ) 68 I.

10 LcbowitzJL.Phys RevA, 133(1964)895.

I I Baxter R J. 1 Chem Phys, 52 ( I 970) 4559.

I 2 Gopala Rao & Das Ratna, Phys Rev B , 36 ( i 987) 6325 .

I 3 Gopala Rao R V & Murthy A K. Indian 1 Phys. 50 ( 1976) 856.

BANDYOPADHY A Y et al. :PARTIAL AND TOTAL STRUCTURE FACTORS 4 1

14 Gopala Rao R V & Dasgupta B. Phys Rev B, 32 ( 1985) 6429.

15 Yenkatesh R & Gopala Rao R V, Th eochem. 36 1 ( 1966) 283.

16 Gopala Rao R V & Murthy A K. Z Natwforsch , 30a (1976) 61.

17 Gopala Rao R V & Sen D, Indian J Phys, 51 A ( 1977) 382.

18 Wills B T M, Chemical application .\· of thermal neutron scattering, (William C lowes, London ), 1973. p. 269.

19 Bhatia A B & Thornton DE, Phys Rev B. 2 ( 1972) 472 .

20 Knoll W & Steeb S, Z Nauuforch. 339 ( 1972) 472 .

21 Bhatia A B, Conference series no. Vol. 30 (Institute of Physics , London), 1977, p. 35.