IACS Institutional Repository

Bhadauriya, Preeti

Indian J. Phys. 81 (5 & 6), 573-584 (2007) 
Equations of state and thermoplastic properties of the 
Earth's lower mantle 
Preeti Bhadauriya 
Department of Physics, Institute of Basic Sciences, 
Khandan, Agra-282 002, Uttar Pradesh, India 
Email preeti jbs(a>sity com 
Received 22 November 2006, accepted 12 June 2007 
Abstract The thermoetastic properties of the lower mantle of the Earth were studied using different 
equations of state The values of bulk modulus and its pressure derivative, thermal expansivity, Gruneisen 
parameter y and its volume derivatives, q and A have been calculated for the entire depth of the lower mantle 
ranging from 670 km to 2891 km The results obtained in the present study are found to compare well with the 
corresponding values based on the seismological data reported by Stacey and Davis [Phys Earth Planet Inter 
142 137 (2004)]. 
Keywords Thermal expansivity, Gruneisen parameter, seismological data, bulk modulus. 
PACS Nos. 64 70 Dv, 62 50 + p, 65 40 -b 
1. Introduction 
The lower mantle of the Earth ranges from 670km to 2891km in depth and the 
corresponding pressure ranges 24GPa to 136GPa according to the Preliminary Reference 
Earth Model [1] (PREM), and the corresponding temperature [2] is estimated to be, for 
example, from 1980K to 2940K. Geochemical and geophysical evidences [3-5] indicate 
that the composition of Earth's lower mantle is primarily (Mg, Fe) Si03 perovskite with a 
little dissolved Al203, (Mg, Fe) O magnesiowustite, and a few percent CaSi03 perovskite. 
This composition is just simple enough to make a useful comparison of its properties 
with seismological data. 
©2007IACS 
574 Preeti Bhadauriya 
For understanding the thermoelastic properties of Earth's lower mantle we need equations 
of state (EOS) for minerals applicable at high pressures and high temperatures. The 
applicability of an EOS can be judged by comparing the calculated P-V results with the 
experimental data. However, the measured laboratory data are subject to uncertainties 
arising from calibration errors related to the use of pressure scales [6,7]. On the other 
hand, seismological data are free from calibration errors, and offer much more effective 
tests of high pressure equations of state than do laboratory data and can be used to 
recalibrate laboratory pressure scales. Geophysics offers two wide pressure ranges, one 
for the lower mantle and other for the core, over which there are reliable values not just of 
pressure, but also of bulk modulus at different densities. 
In the present paper, thermoelastic properties of the lower mantle are studied using 
equations of state, such as, (i) Birch-Mumaghan EOS [8] (ii) Rydberg-Vinet EOS [9-11], 
and (iii) Shanker EOS [12]. The results are obtained for pressure P, bulk modulus Kand 
its pressure derivative K' for the entire depth of the lower mantle, and compared with the 
seismological data [7] derived from the Preliminary Reference Earth model (PREM) [1] 
using the Stacey relationships for P, K and K1. The thermoelastic properties such as the 
Gruneisen parameter y and its volume derivatives q and A, and thermal expansivity a 
have been calculated and compared with the seismological data [7]. 
2. Method of analysis 
The following three equations of state are considered 
/. Birch-Mumaghan EOS: 
This EOS has been derived from the Eulerian finite strain theory [8]. Expressions for 
pressure P, bulk modulus K and its pressure derivative K = dK/dP are given below 
P=3-K0(x-7 / 3-x-5 / 3) 1 + | (K 0 -4) (x- 2 / 3 -1) | (1) 
K = l/C0(7x-7/3 -5x~5/3) + ~K0(K0 -4)(9x-3 -14x~7/3 +5x'5 / 3 ) (2) 
2 8 
*•.&[<«- 4)(81x"3 - 98x~7'3 + 25x~5'3) + - (49x~7/3 - 25x"5/3) (3) 
where x = V/VQ , KQ and KQ are the values of K and K" at P • 0. V is the volume at 
pressure P and V0 is the volume at zero pressure. Also V/V0 - p0/p where Po is the 
density p at P = 0. 
Equations of state and thermoelastic properties of the Earth's lower mantle 575 
//. Rydberg -Vinet EOS: 
Vinet et al [11] have obtained an EOS which is based on the potential energy function 
due to Rydberg [10] 
E(r)=E(a) 1-6| 1 - - expfa i - £ (4) 
Here, E{r) is the potential energy expressed a$ a function of the interatomic distance 
r. For a given solid, a and b are constants, 'a' is the equilibrium value of interatomic 
distance r, and 'tf is the hardness parameter in the potential function [10]. At r = a, we 
have E (i) = E(a). The EOS derived from Eq. (4) is known as the Rydberg - Vinet EOS. 
The expressions for P, K, and K" obtained from this EOS are given below [9,13] 
P = 3K0x-2/3(l-x1/3)exp[»7(1-x1/3)] 
K = KQx-2/3 [1 + {r,xU3 +1}(1 - x1/3 )]exp[rj(1 - x1/3)] 
(5) 
(6) 
3 1 + (rjx1/3+1)(1-x1/3) (7) 
V 3 
where x = - - and »? = (KQ -1) . 
///. Shanker EOS: 
Shanker et a/[12] have obtained an EOS using the volume dependence of the interatomic 
force constant for interatomic potentials. The force constant is defined as follows [14]. 
A = -
3 
lfo*2E 2c/E 
dr2 r dr (8) 
where E is the lattice potential energy and r is the interatomic separation. The expressions 
based on the Shanker EOS are given below [12,15] 
,-4/3 
P = /C„ [f 1 - ) + 1 ]{exp(fy) -1}+ y f l + y - ^lexp(fy) j (9) 
/C = K0x-1/3(1 + y + y2) exp(ry) + ^P 
4 f„ 4 P , 1 f, (1 + 2y) . . K' = -+ 1 x +xlt + — —5-M 3 I 3K 1 3 I (H-y + y 2 ) ] ] 
(10) 
(11) 
576 Preeti Bhadauhya 
V . {V ] . . . ,v 8 
where x = — , y = 1 -
v^ o 
and t as KQ - -3 
We make use of equations (1-3), (5-7) and (9-11) to calculate Pt / (and /<" for the lower 
mantle. The Gruneisen parameter y is an important quantity which relates thermal and 
elastic properties of materials. We have the following expression for y [16] 
r = Y?Kr_ = VccKs ( 1 2 ) 
Cy Cp 
It is dimensionless combination of four familiar quantities, volume thermal expansion 
coefficient, a ; bulk modulus, K; Volume, V; specific heat, C; with subscripts T, Vt S, 
P, indicating constant temperature, volume, entropy and pressure. These alternative 
definitions are linked by the identities 
KKT = % = ' + YaT • ( 1 3 ) 
The values of y at different compressions can be calculated using the generalized 
formula for the Gruneisen parameter 
= (1/2)K'-(1/6)-(f/3)(1-(1/3)(P//C)) 
7
 1-(2/3)f(P//C) ( 1 4 ) 
with different values of f. The simplest of these is Slater's equation [17], for which f = 0. 
The Dugdale - MacDonald formula appears with f =1, and the free - volume formula [18] 
with f as 2. Values of y, calculated from Eq. (14) with different values of f, are not well 
appropriate for the lower mantle. Stacey and Davis have suggested that the general form 
of the free volume equation (Eq. (14)) could be used with a suitably adjusted value of f. 
By using the acoustic value of y at P = 0, we can calculate the value of f which is f = 
1.436 [7]. 
To proceed further we examine the behaviour of q and A for the lower mantle, which 
are the volume derivatives of Gruneisen parameter such as 
v
 [BYnV) (15) 
and 
JT 
(16) 
Equations of state and thermoelastic properties of the Earth's lower mantle 577 
The fact that q->0 at V - » 0 suggested to Stacey and Isaak [19] that the next 
derivative 
A = 8lncjf 
dlnl/ 
might be constant, at least to a useful approximatioii With constant A, eq. (16) integrates 
to 
Q = <to Vn (17) 
and In y = °0 
A 
vT 
-1 (18) 
As p _• oo, v -> 0 . and y -> y„, and therefore eq. (18) yields 
A = Qo 
ln(y0/y») 
(19) 
However, we have no fundamental reason for believing that A is constant, we know 
that it is a much better assumption than constant q, as has often been assumed in 
mineral physics [16,20]. In the present study, on the basis of generalized free volume 
theory, it is found that q as well as A, both depend on pressure or volume. On differentiating 
eq. (14), we obtain the expressions for q and A, given as follows 
(20) 
A - * 44 (21) 
where A- I
 KJP\K» 2fZpV fPK' K+2f* (22) 
Aj = 
2 6 ~ 3 l 3/Cj 
2 fP 
3 K (23) 
578 Preeti Bhadauhya 
and 
As^^iKK+KK )| (K' - 1 | - * L * _ + X \ - 9PK*K -6PKKKH -3K* { 3 ] 4 18L 
+ 3 W _3 / c 2 / c«. + 6PKK* _3/CK/C" + 6PK'fC" + 3PKK" -2/CfC" + K* 
A 
AC + ^ 3/c* - -
3
- ^ - + 4PKK + 2PKK"' + P 2 ** '" + 2 ^ ^ -
9 AC /v 
-p . * * - K
 + «L + PK- - M - °?KK~ • ?£. HW + < " * " 
AC AC 8 1 AC 
ooi/1 OD2I/" 9P2K"fC" SP2** P3*" 4P3AC'K" P V „ 
-3PAC -3P*K - — + — + - - + - -—+2 
K K2 K K2 Kz 
4f4 
243 
3PAC ZP^K" 3P2K P*K P*K" 2P
 0 
- + —— - + - - - —^ - - + 3 AC AC Kd ACJ K< AC (24) 
where AC" = (PKIdP2 and AT' = cPK/dP* are respectively the second and third order 
derivatives of bulk modulus. 
The behaviour of higher derivatives is important for understanding the thermoelastic 
properties. However, the traditional equations of state (such as the Birch - Murnaghan 
EOS (1-3), the Rydberg-Vinet EOS (5-7), and the Shanker EOS (9-11)) lead to quite 
complicated expressions for the higher derivatives of bulk modulus (AC" and AC"') and 
therefore less convenient to use. On the other hand, the Stacey EOS which is based on 
the reciprocal AC-primed equation gives a slight advantage and that is in manipulating 
higher derivatives. Expressions for Stacey EOS are given below 
AC AC, 
1- p 
AC (25) 
where AC„ is the value of AC' in the limit P -»°° . 
Eq. (25) integrates to expressions for ACAC^  and p/p0 in terms of P/K 
K^-K-i< 
-Ko'lt 
(26) 
Equations of state and thermoelastic properties of the Earth's lower mantle 579 
In ' P
N 
[Po 
K0 AP 
[Z )K (27) 
where p is density of the material under study, p0 is the value of p at P = 0. 
By differentiating eq. (25) we can express t ie next two derivatives, at arbitrary 
compression, in terms of K , KQ and K l with convenient simplicity 
K%2 KK =-—r(K -K„)
 ( 2 8 ) 
and K K =
 2-(K -K^SK -2 /C + KQ) ^9) 
As p -* 00, K -* K' such that 
p - 1 
^ •
r
- : t (30) 
By fitting the data on p-P-K for the lower mantle in the above expressions (25-
29), Stacey and Davis7 have found K0 = 206GPa, K0 =4.2 and K^= 2.4. We have used 
these values as input in the present study. 
3. Results and Discussions 
First, we have calculated P, bulk modulus K and K% = dK/dP as a function of density 
p0/p-V/V0 for the lower mantle at different values of r, the distance from the centre of 
Earth. Thus the range of /t5701-3480km) for the lower mantle corresponds to the depth 
670-2891 km. We have used equations (1-3), (5-7) and (9-11) based on the Birch-Murnaghan 
EOS, the Rydberg-Vinet EOS and the Shanker EOS. The input data used in the present 
study are the zero pressure values, KQ = 206 GPa, K%0 = 4.2 and Ki =2.4 for the 
lower mantle reported by Stacey and Davis [7]. The results are given in Tables 1-3. The 
results for pressure P, bulk modulus K and pressure derivative K obtained from different 
equations are compared with the seismological data [7]. The comparison reveals that the 
Rydberg-Vinet EOS and the Shanker EOS yield very similar results which are close to 
the seismological data for the entire depth of the lower mantle. On the other hand, the 
Birch-Murnaghan EOS yields significant deviations. 
We have also calculated thermoelastic properties such as Gruneisen parameter r , 
thermal expansivity a , and volume derivatives of Gruneisen parameter q and A for the 
580 Preeti Bhadaunya 
Table 1. Values of P(GPa) calculated from (a) Birch - Murnaghan EOS, (b) Rydberg-Vinet EOS and (c) 
Shanker EOS, compared with the seismological data [7] for the lower mantle 
r 
(km) 
3480 
3600 
3630 
3800 
4000 
4200 
4400 
4600 
4800 
5000 
5200 
5400 
5600 
5701 
P 
(kgm3) 
5567 
5508 
5493 
5409 
5310 
5210 
5108 
5005 
4899 
4790 
4677 
4560 
4438 
4374 
(H>1P = V/V0 
0 7144 
0 7220 
0 7240 
0 7353 
0 7490 
0 7633 
0 7786 
0 7946 
08118 
0 8303 
0 8503 
0 8721 
0 8961 
0 9092 
P(GPa) 
Seismological 
data 
135 75 
128 71 
126 97 
117 35 
106 39 
95 76 
85 43 
75 36 
65 52 
55 90 
46 46 
37 29 
28 29 
23 83 
Calculated values of P(GPa) 
(a) 
141 02 
133 54 
131 62 
121 24 
109 57 
98 36 
87 37 
76 88 
66 62 
56 65 
46 97 
37 57 
28 44 
23 93 
(b) 
136 26 
129 30 
127 51 
117 80 
106 81 
96 19 
85 71 
75 64 
65 73 
56 05 
46 59 
37 35 
28 33 
23 87 
(c) 
136 41 
129 44 
127 66 
11794 
106 94 
96 30 
85 80 
75 71 
65 78 
56 07 
46 60 
37 35 
28 32 
23 85 
Table 2. Values of K(GPa) calculated from (a) Birch - Mumaghan EOS, (b) Rydberg-Vinet EOS and (c) 
Shanker EOS, compared with the seismological data [7] for the lower mantle 
r 
(km) 
3480 
3600 
3630 
3800 
4000 
4200 
4400 
4600 
4800 
5000 
5200 
5400 
5600 
5701 
P 
(kgm3) 
5567 
5508 
5493 
5409 
5310 
5210 
5108 
5005 
4899 
4790 
4677 
4560 
4438 
4374 
Po'P^VIVo 
0 7144 
0 7220 
0 7240 
0 7353 
0 7490 
0 7633 
0 7786 
0 7946 
08118 
0 8303 
0 8503 
0 8721 
0.8961 
0 9092 
K(GPa) 
Seismological 
data 
667 
645 
640 
610 
576 
542 
509 
476 
444 
411 
379 
347 
315 
299 
Calculated values of K(GPa) 
(a) 
721 
695 
689 
653 
613 
574 
535 
498 
461 
425 
389 
354 
320 
302 
(b) 
669 
648 
643 
613 
579 
546 
512 
479 
447 
414 
382 
349 
316 
300 
(c) 
670 
649 
643 
614 
580 
547 
514 
481 
448 
415 
383 
350 
317 
300 
Equations of state and thermoelastic properties of the Earth's tower mantie 581 
Table 3. Values of K calculated from (a) Birch - Mufnaghan EOS, (b) Rydberg-Vmet EOS and 
(c) Shanker EOS, compared with the seismological data [7} for the lower mantle. 
r 
(km) 
3480 
3600 
3630 
3800 
4000 
4200 
4400 
4600 
4800 
5000 
5200 
5400 
5600 
5701 
(kgm-3) 
5567 
5508 
5493 
5409 
5310 
5210 
5108 
5005 
4899 
4790 
4677 
4560 
4438 
4374 
PO'P = V/V0 
0.7144 
0.7220 
0.7240 
0.7353 
0.7490 
0.7633 
0.7786 
0.7946 
0.8118 
0.8303 
0.8503 
0.8721 
0.8961 
0.9092 
K 
Seismological 
data 
3.08 
3.10 
3.10 
3.13 
3.16 
3.19 
3.23 
3.27 
3.32 
3.38 
3.45 
3.52 
3.62 
3.68 
— 4 
* (a) 
—4> 
3.40 
t 3.41 
3.41 
3.44 
3.46 
3.49 
3.53 
3 56 
3.60 
3.65 
3.70 
3.76 
3.83 
3.87 
Calculated values of 
(b) 
3.01 
3.03 
3.04 
3.07 
3.12 
3.16 
3.21 
3.27 
3.33 
3.40 
3.47 
3.56 
3.66 
3.72 
K 
(c) 
2.97 
3.00 
3.00 
3.04 
3.09 
3.15 
3.21 
3.27 
3 33 
3.41 
3.49 
3.58 
3.69 
3.75 
lower mantle at different values of r. The results for Gruneisen parameter y and thermal 
expansivity a are reported in Tables 4 and 5 and compared with seismological values. 
The results obtained from the Rydberg-Vinet EOS and the Shanker EOS are in good 
agreement with the seismological values. On the other hand, the results obtained from 
the Birch-Murnaghan EOS are nearly ten percent higher than the seismological values. 
For estimating the value of q and A, we need the higher derivatives /.e., KK and K2K%. 
We have calculated the values of KK and K2K using equations (28) and (29) 
respectively, and then the values of q and A with the help of equations (20)-(24). The 
calculated values of q and A are reported in Table 6, and compared with the values 
based on seismological data [7]. It is found that the calculated values of q are in good 
agreement with the seismological values while the calculated values of A are only in fair 
agreement with seismological values. 
It should be mentioned that y, q and A are determined respectively from the first, 
second, and third pressure derivatives of bulk modulus, which are in turn related to the 
second, third, and fourth derivatives of pressure-volume relationships expressed in different 
forms of equation of state. Thus y, q and A can be expressed in terms of third-, 
fourth-, and fifth derivatives of potential energy function respectively. The calculation of A 
is therefore most sensitive to the form of the potential function or the equation of state 
used as it depends on the third pressure derivative of bulk modulus. 
582 Preeti Bhadauriya 
Table 4. Values of /calculated from modified free volume formula [eq. (14)] with f = 1.436 using different 
EOS (a) Birch-Murnaghan EOS, (b) Rydberg-Vinet EOS, (c) Shanker EOS and (d) Seismological data [7]. 
r(km) (a) (b) (c) (d) 
3480 
3600 
3630 
3800 
4000 
4200 
4400 
4600 
4800 
5000 
5200 
5400 
5600 
5701 
1.3360 
1.3359 
1.3340 
1.3430 
1.3428 
1.3482 
1.3579 
1.3603 
1.3667 
1.3765 
1.3839 
1.3930 
1.4032 
1.4090 
1.1082 
1.1142 
1.1186 
1.1278 
1.1465 
1.1581 
1 1745 
1.1955 
1.2140 
1.2365 
1.2557 
1.2831 
1.3115 
1.3285 
1.0832 
1.0957 
1.0942 
1.1092 
1.1282 
1.1519 
1.1738 
1.1947 
1.2136 
1.2417 
1.2667 
1.2938 
1.3273 
1.3446 
Seismological data 
1.1412 
1.1447 
1.1454 
1.1515 
1.1591 
1.1676 
1.1757 
1.1881 
1.2008 
1.2154 
1.2335 
1.2548 
1.2815 
1.2972 
Table 5. Values of thermal expansivity aOO^K'1) from eq. (12) corresponding to the results for y given 
in Table 4. 
r(km) (a) (b) (c) (d) 
3480 
3600 
3630 
3800 
4000 
4200 
4400 
4600 
4800 
5000 
5200 
' 5400 
5600 
5701 
12.6 
12.7 
12.8 
13.4 
14.0 
14.7 
15.6 
16.5 
17.6 
18.8 
20.2 
21.8 
23.8 
25.0 
11.0 
11.2 
11.3 
11.8 
12.5 
13.1 
14.0 
14.9 
15.9 
17.2 
18.5 
20.2 
22.3 
23.5 
10.8 
11.0 
11.1 
11.6 
12.3 
13.0 
13.9 
14.8 
15.9 
17.2 
18.6 
20.4 
22.5 
23.8 
Seismological data 
11.3 
11.6 
11.7 
12.1 
12.7 
13,4 
14.2 
15.0 
15.9 
17.0 
18.3 
19.8 
21.8 
23.0 
Equations of state and thermoelastic properties of the Earth's lower mantle 583 
Table 6. Values of q and A; (a) calculated from eq. (20) and eq. (21) with f = 1.436, and comparison 
with (b) the values based on seismological data [7]. 
r(km) 
3480 
3600 
3630 
3800 
4000 
4200 
4400 
4600 
4800 
5000 
5200 
5400 
5600 
5701 
Q ( a ) 
0.3058 
0.3176 
0.3196 
0.3395 
0.3641 
0.3935 
0.4252 
0.4634 
0.5064 
0.5453 
0.6187 
0.7013 
0.7992 
0.8592 
A (a) 
3.72 
3.91 
3.77 
4.01 
4.01 
3.90 
4.05 
4.03 
4.19 
4.69 
4.78 
4.71 
5.11 
5.37 
. Q (b) 
\ 0.2770 
' 0.2894 
0.2926 
; 0.3117 
0.3365 
0.3644 
0.3919 
0.4324 
0.4748 
0.5245 
0.5860 
0.6602 
0.7549 
0.8136 
A(b) 
3.36 
3.37 
3.37 
3.39 
3.41 
3.43 
3.45 
3.50 
3.54 
3.60 
3.67 
3.76 
3.88 
3.94 
Acknowledgments 
Author is thankful to Professor Jai Shanker, Institute of Basic Sciences, Agra for his 
valuable guidance and useful discussions. Thanks are also due to the reviewer for his 
valuable comments. 
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584 PreetiBhadauriya 
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English

Equations of State and Thermoelastic Properties of the Earth's Lower Mantle

Advisory Board and Editors

Rafique, S M

Kumar, Ashwani

Indian J. Phys. 81(1) 35-39(2007) 
w 
Thermodynamic properties and alloying behaviour of liquid 
binary alloy :CdZn 
S M Rafique* and Ashwani Kumar 
University Department of Physics, T M Bhagalpur University, Bhagalpur-812 007, Bihar, India 
E-mail sm_rafq(a>yahoo com 
Abstract : The complex formation model first proposed by Bhatia and Hargrove [1] assumes the existence of chemical 
complexes or pseudomolecules in liquid binary alloys On the basis of this concept Singh and his coworkers [2-5] have studied 
various thermodynamical properties of different binary alloys We have also studied different alloys within this framework [6-8] 
The present paper envisages the study of some thermodynamic properties viz, Qibbs free energy of mixing (GM), enthalpy of 
mixing (HM) and entropy of mixing (SM) of CdZn binary alloy The results are In reasonable agreement with expenment and throw 
light on the ionic interactions of the constituent atoms leading to the alloying behavior of the alloys under investigation 
Keywords : Complex formation model, entropy of mixing, CdZn alloy, liquid alloys, thermodynamic model 
PACS Nos. : 64 70 Fx, 65 20 +w. 61 25 Mv, 65 20 +m, 71 25 Cz 
1. Introduction 
The concept of formation of complexes or pseudomolecules in liquid binary alloys 
proposed by Bhatia and Hargrove [1] is being widely used for the study of thermodynamic 
properties and alloying behaviour of binary alloys [2-5]. In our earlier papers [6-8], we 
have studied thermodynamic, structural and surface properties of some binary alloys 
within this framework. 
In this paper, we present the computation and results of free energy of mixing 
(GM), heat of mixing (HM) and entropy of mixing (SM) of liquid CdZn alloy. Comparison 
has also been made with the available experimental data. The results of the computed 
properties and parameters throw light on the nature and behavior of this alloy. 
2. Formalism and computation 
The formalism of the present work has been presented in our earlier papers [6-8]. Here 
we confine with the final expressions of the computed properties. In the binary alloy, 
we assume that chemical complexes or pseudomolecules are present. They are 
represented as /iA + vB » AM By. Here // and v are integers. 
Let the binary alloy contains nA atoms of A, nB atoms of B where nA = cAN 
'Corresponding Author ® 2 0 0 7 I A C S 
36 S M Rafique and Ashwani Kumar 
and nB = cBN where N is the Avogadro's number and cA and cB are the concentrations 
of the two species. Then it is assumed to consist of n,N free atoms of A n2N free 
atoms of B and n3/V complexes AMBV acting as independent scattering centers in the 
alloy. From the conservation of atoms : 
n = ny + ty + nz. 
The number of complexes n3 is obtained through the condition 
(aGM/d^)r>PfC = o, 
(D 
(2) 
where GM is the Gibb's free energy of mixing and Tf P, c are the temperature, pressure 
and concentration respectively. The conformal solution approximation enables us to 
express GM as 
Gu = -fW + * r E n i ( l n n i - | n n ) + £ £ r^i 
/</ /7 
W„ (3) 
where IV,, are the interaction energies which are independent of concentration, g is the 
formation energy, R is the gas constant. From eqs. (2) and (3), we get 
= e-»,rn exp(y, + y2 + y3), (4) 
where 
y( RT «—^H^)M 
y2 
~ AT 
7 3
 RT 
(/i+V-1) 'fvo £&Lrai 
n JJ 
K^j-ftH*)] (5) 
The equilibrium value of n3 is obtained by solving numerically eq. (4) and then it is 
used to obtain G„ through eq. (3). The interaction energies W„ and formation energy 
(fir) are determined through the Bhatia-Hargrove technique [1J. 
The heat of mixing HM and entropy of mixing SM are given by 
H, 
'M 
• * - l ' ^ JP.CN 
5 f ^ L l ^ l l 
(6) 
(7) 
Thermodynamic properties and alloying behaviour of liquid binary alloy: CdZn 37 
The computation of the above mentioned parameters has been done through eqs. ( 1 -
7) and the computed results are presented in Figures 1 ancjl 2 paragraph alongwith the 
experimental data for comparison against Zn concentration cZn. 
f 
00 
o3 -02 
—•-
- 0 -
A 
•"» 
G« 
GM(E>qX) 
2 
CO 
3 
X 
,fi 
^4,-< 
0.0 02 
—1 • 1 — 
04 0.6 10 12 
Q8 
07A 
05H 
Q4-J 
0.3-I 
02 
OH 
Q0-
" • - H M 
• HM (B0.) 
* SM(Bqpt) 
/ 
/ 
/ " \ 
' \ \ 
1 
QO 02 —,— Q4 0.6 
— 1 — 
08 1.0 
Figure 1. Number of complexes and energy of 
mixing of CdZn liquid alloy (800 K), against Zn 
concentration cZn. 
C * 
Figure 2. Heat and entropy of mixing of CdZn 
liquid alloy (800 K), against Zn concentration c2n. 
3. Observation and inferences 
The computation of the energy of mixing (GM), heat of mixing (HM) and entropy of 
mixing (SM) was carried out for CdZn alloy at different concentrations at 800 K. It was 
observed that the numbers of complexes are very small. This indicates that CdZn form 
simple binary alloy. This seems reasonable because both of the constituents have the 
same valency and lie in the same group I IB of the periodic table. The physical and 
chemical properties are also similar in various aspects. Inspection of Figure 1 reveals 
that the number of complexes n3 is quite small and the computed value of GM (in 
terms of RT) agrees well with the experimental results of Hultgren et ai [9]. Hence, 
simple binary alloy CdZn has been considered onwards with ju = v = 1 and the 
equilibrium values of n3 have been arrived at. However, if we apply the equilibrium 
condition (4) no reasonable agreement is obtained with experimental data of GM [9] 
which excludes the possibility of complex formation. 
The computed values of HM and SM are also in good agreement with experimental 
result (vide Figure 2). The interaction energies are computed on the basis of Bhatia 
and Hargrove technique. The equilibrium values of chemical complexes n3 have been 
obtained through numerically solving eq. [2]. The interaction energies Wy are adjusted 
until a good fit for GM is obtained. Once the energy parameter is selected, they remain 
the same for all concentrations. The computed values are g = 1.359, W12 = 1.29, 
W13 * -1.0, W23 a 1.8 (in terms of RT). The derivatives needed for the computation 
38 S M Rafique and Ashwani Kumar 
of HM and SM are found to be dgldt = 0.99485, dW.Jdt = 0, dW.Jdt = 0.3, dW22l 
dt = 1.32. As the complex formation energy g/RT is 1.359 which is appreciably 
smaller than 3.55, the CdZn alloy may be assumed to be a weakly interacting system 
like CuSn, AgAI, MgSn etc. For strongly interacting systems, g/RT has much larger 
value e.g., MgBi (16.7), TITe (10.84), HgK (9.965), HgNa (8.294). Also we find that W12 
and W23 are positive i.e., repulsive in nature while W13 is attractive which also justifies 
the weakly interacting nature. 
An inspection of Figure 1 reveals that the Gibbs free energy of mixing (GM) of 
CdZn binary alloy is in reasonably good agreement with experimental observations and 
is symmetric about c = 0.5 (Hultgren et al [9]). From Figure 2 we observe that the 
heat of mixing HM and entropy of mixing SM are also in good agreement with 
experiment [9] and they also show symmetric behaviour. However, it is to be noted that 
HM of CdZn alloy is positive for all concentrations like CdTI, AIGa, AlPb, AlZn, CdGa, 
Cdln, GaZn, InTI, InZn, KNa etc. alloys [9]. In general, the heat of mixing is found to 
be negative for many alloys. This gives a special character to this alloy. CdZn forms 
a simple eutectic phase diagram like GaZn, NaRb, AlBe, CdPb etc. It is to be 
mentioned that solutions whose components exhibit negative deviations from Rouit's 
law, form exothermically i.e., HM < 0. In contrast the solution whose components 
exhibit positive deviations from Rouit's law will form endothermically i.e., HM > 0. This 
is the case for CdZn alloy. 
4. Summary and conclusion 
In the present work, the CdZn alloy has been studied within the framework of complex 
formation technique proposed by Bhatia and Hargrove [1] in respect of its thermodynamical 
properties viz., GM, HM and SM. The following important conclusions may be drawn from 
the present investigation : 
(i) As CdZn presents an eutectic phase diagram and both the constituents belong 
to the same group II B of the periodic table, the probability of complex formation 
is meagre which is indicated by the very small value of n3 at all concentrations. 
The previous experimental and theoretical workers have also not suggested any 
other complex in the liquid or solid phase. 
(ii) All the computed properties are \n fairly good agreement with experiment if 
simple CdZn alloy is considered taking //
 8 v = 1. Other values of // and v do 
not provide experimental agreement excluding the possibility of formation of 
complexes. 
(Hi) All the computed properties show symmetrical behavior about equiatomic 
concentration. 
<M The heat of mixing of CdZn is positive for all concentrations which places it in 
the category of CdTI, CdGa, Cdln etc. in contrast to CdPb, GaZn, AlBe etc. This 
throws light on the alloying behavior of this alloy. 
Thermodynamic properties and alloying behaviour of liquid binary alloy; CdZn 39 
Acknowledgments 
The authors are thankful to Dr. R N Singh, Professor of Physics, College of Sciences, 
Sultan Qaboos University, Muscat, Oman for valuable suggestions and providing reprints 
of his research work. We would also like to thank Dr. B B Sahay, Professor of 
Physics, Bhagalpur College of Engineering for his valuably cooperation and guidance. 
References 
[1] A B Bhatia and W H Hargrove Lett. Nuovo Cim. 8 1025 (1973) 
[2] R N Singh Thermodynamics of Alloy Formation (eds) Y A Chang aAd F Sommer (The Minerals, Metals 
and Materials Society) p229 (1997) > 
[3] 0 Akinlade, R N Singh and F Sommer J. Alloys and Compounds 291163 (2000) 
[4] R N Singh and IK Mishra Advances in Condensed Matter Physics (New Delhi: Willey Eastern Ltd.) p403 
(1990) 
[5] F Sommer, R N Singh and V Witusiewiez J. Alloys and Compounds 325 118 (2001) 
[6] Ashwani Kumar, S M Rafique, N Jha and A K Mishra Physica B 357 445 (2005) 
[7] N Jha, S M Rafique, A K Mishra, R N Singh and V N Singh Indian J. Phys. 74A 461 (2000) 
[8] N Jha, A K Mishra, S M Rafique and A N Singh Indian J. Phys. 75A 519 (2001) 
[9] R Hultgren, P D Desai, T D Hawkins, M Gleiser and K K Kelley Selected Values of the Thermodynamic 
Properties of Binary Alloys (Metals Park, Ohio, ASM) 


Thermodynamic Properties and Alloying Behaviour of Liquid Binary Alloy : CdZn

Bishoyi, K C

Rout, G C

Behera, S N

Indian J. Phys. 81 (1) 47-53 (2007) 
Study of magnetism and hybridization through ultrasonic 
attenuation in cuprate systems 
K C Bishoyi1*, G C Rout2 and S N Bebera3 
lP G Department of Physics, F M College (Autonomous), Balasote-756 001, Orissa, India 
2Condensed Matter Physics Group, Govt Science College, Chatrapur-761 020, Orissa, India 
^Physics Enclave, HIG-23/1, Housing Board Phase-1, Chandrasekharpur, Bhubaneswar-751 016, Onssa, India 
E-mail bishoyi(<?)iopb res in 
Abstract : In the present communication, an attempt is made to present a microscopic theoretical model for the cuprate 
systems in presence of antiferromagnetism (AFM), the position of impurity f-level, hybridization between Mevel and copper 
^electrons as well as phonon interaction to the hybridization in harmonic phonon vibration approximation When the high 
frequency ultrasonic wave travels in the solid, it deforms the lattice In our theoretical model calculation, the phonon Green's 
function is evaluated by Zubarev's technique The imaginary part of the phonon self-energy gives the ultrasonic attenuation of 
the system and displays the interplay of AFM and hybridization in it The effect of model parameters like staggered magnetic field 
(/?), position of (-level (<-,), hybridization (V), and electron-phonon coupling constant (g) on the temperature variation of ultrasonic 
attenuation is studied and the results are reported 
Keywords : High- Tecuprate superconductors, electron-phonon interaction 
PACSNos. : 74 72-h, 63 20 Kr 
1. Introduction 
Ultrasonic methods are extremely valuable in elucidating the electron-phonon mechanism 
in strongly correlated electron systems. The ultrasonic peak and minimum in velocity 
are found in non-superconducting sample of NCCO at 260 K and in the superconducting 
sample at 200 K [1]. At low frequences and temperatures below 200 K, the attenuation 
increases with decreasing temperature and frequency and possibly arises due to 
interaction with the motion of domain wall. The attenuation arises possibly due to 
structural phase transition. A low tem-perature transformation from the orthorhombic to 
the tetragonal low temperature phase is observed with partial lattice stiffening and a 
small attenuation peak at T < Tc is observed in LSCO [2]. Again .attenuation 
measurements on LSCO showed two peaks, one close to the superconducting 
transition (Tc = 37.9 K) possibly combined with same structural instabilities and another 
at TN s 27 K possibly of magnetic origin. Though some ultrasonic measurements are 
reported for NCCO and LSCO compounds, further experimental details are needed. The 
measurements do not show the effects of the hybridisation, antiferromagnetism. The 
Corresponding Author © 2 ° 0 7 l A C S 
48 KCBishoyi, GC Rout and SN Behera 
microscopic theory of ultrasonic attenuation has not yet been reported. We propose a 
microscopic theoretical model to describe ultrasonic attenuation of high-7"c systems in 
normal phase in presence of antiferromagnetism (AFM), the position of f-level, 
hybridization between f-level and Cu d-electrons as well as phonon interaction to the 
hybridization in harmonic phonon vibration approximation. Behera and co-workers have 
investigated the velocity of sound [3] on this type of model. The under-doped systems 
exhibit AFM with low magnetic moments of the Cu2* ion in Cu02 plane. When the high 
frequency ultrasonic wave travels in the solid, it deforms the lattice and sets up lattice 
vibration. In our theoretical model calculation, the phonon Green's function is evaluated 
by Zubarev's technique [4]. 
2. Formalism 
The Hamiltonian in /c-space for the cuprate system is taken as 
«» = « * + « . + «* + * / . (1) 
where 
^•Z^oWKA+H, (2) 
Ha^{hl2)Jia(4,aBk^blabkt0)t (3) 
^^Z( f lLW^Vi,^H, (4) 
^ *
 ef lLf\k,ofyxo . (5) 
k.a v ' 
Hdt H8t Hv and Ht are conduction electron, staggered field, hybridization interaction 
and /-electron Hamiltonian, respectively. The Fourier transformed electron-phonon interaction 
Hamiltonian is 
H
-P - jE l 4 ( C ^ + 4UAk*) +**]**
 (6) 
with Aq « bq + blq where bq (tfq) are annihilation (creation) operators for phonon with 
wave vector q and f(q) is the electron-phonon coupling constant. The free phonon 
Hamiltonian with phonon energy mq is written as Wp = X 9 V > J * V Hence the total 
Hamiltonian of the system is described by [3] 
3. Calculation of renormalized phonon energy 
The double time phono Green's function of Zubarev [4] type is defined as 
- -»<f-0<[VftVOj>. (8) 
Study of magnetism and hybridization through ultrasonic attenuation in cuprate systems 49 
Applying Dyson approximation, the phonon Green's function can be written as 
D „ > ) = (coq ln)[w* - o>* - I,(a>)]-1, (9) 
where phonon self energy is given by 
£g(<a) = AxPl-qtotZquio), (10) 
XvfW = I ^ - l r , + r2 + r3 + rA]. ( n ) 
r/(/c,/f/,Q,c7/lfi)),s(/ = l to4) represent the electron response functions. We make some 
approximations, while calculating the electron response function, keeping the essential 
physics intact. We use certain decoupling schemes folr the higher order Green's 
functions such that all the physical parameters like f-1 eve I, staggered magnetic field, 
hybridization term (-V2) and phonon coupling term are retained. The limit of zero wave 
vector (q —> 0) and finite temperature are considered in evaluating the response 
function. 
4. Expression for ultrasonic attenuation coefficient (a) 
The high frequency ultrasonic sound wave, as it travels through the cuprate system, the 
lattice gets distorted giving rise to a new vibrational spectrum (phonon). The ultrasound 
is absorbed by the system corresponding to some appropriate energy of the sub-atomic 
system. This gives an idea about the electron configuration of the sub-atomic system. 
The ultrasonic attenuation coefficient a(o),T) for sound waves of frequency co and at 
temperature T of the system is given by the imaginary part of the phonon self-energy. 
Mathematically it is written as 
a(coJ) = -—\n)x(q = 0,(D), (12) 
v 
where z(Q = 0,a>) is the phonon response function for long wave length longitudinal 
phonons (q -» 0). The renormalised velocity of sound v is given by 
v = ^ + 4A2(co) 
and the ultrasonic attenuation co-efficient a is written as 
a.-JSS-. (13) 
where the functional dependence of A2 and Bz on frequency and the model parameters 
has been calculated in detail. Different parameters used in the eq. (13) are made 
dimensionless by dividing them by hopping integral 2^, where the width of the 
conduction band is W = 8^. Those are : 
g = f2(0)N(0)/co0\ d = e,/2f0; c = o>l2t0\ 
v = V/2t0\ e~rj/2t0-t p = a)0/2t0; a)=»c/p; 
x0^€0(k)/2t0; M = h/2t0; f«k f l7/2^; 
where kB is the Boltzmann constant. 
50 K C Bishoyi, G C Rout andSN Behera 
5. Results and discussion 
Figure 1 shows the temperature dependence of the attenuation when the electron-
phonon coupling strength (g) varies representing different high-Tc superconducting 
compounds containing R-atoms When ultra sound interacts stongly with the electronic 
sub-system of the solid, it strongly modulates the f-electron and the conduction 
electron orbitals This causes the increase in hybridization (v) and hence increases the 
electron-phonon coupling strength (g) We investigate here the effect of the electron-
phonon coupling on the attenuation co-efficient (a) It is observed from the Figure 1 that 
ultrasonic attenuation increases, as the electron-phonon coupling increases from g = 
0 0180 to 0 0255 Attenuation peaks at temperature t - 0.10 corresponding to the fixed 
position of the f-level at d = 006 which is the middle of the hybridization gap. 
6* -05 
< 
o 
r——' "T ' 
i 
i 
i 
; i 
;i: 
n , ,, 
g «- 0 0180 
g * 0 02^0 
1 p *• 0 0250 
» g J 0 0252 
1
 0 - 0 0?5«S 
1 
\ \ \ 
< 
i 
Be ob 
6e 05 
4* OS 
2e-05 
' A 
/" K 
/o 
cf*4^ 
v - 0 060 < 
v * 0 050 
v « 0 0 1 5 
^ " ~ — . 
. . . , . I 
0 01 0^ 03 04 05 
reduced temperature (t) 
Figure 1. The plot of ultrasonic attenuation vs 
reduced temperature for fixed values of e = 0 018 
p= 1 0, d= 0 06, v= 0 015, M = 0 195 f = 0 09, w= 0 1 
and for different values of g = 0 0180, 0 0220, 0 025, 
0 0252, 0 0255 
0 01 02 
reduced temperature (t) 
Figure 2 The plot of ultrasonic attenuation vs 
reduced temperature for fixed values of e = 0 018 
p = 1 0, 0= 0 025, c/= 0 06, A)1 = 0 195, m^ 0 1 and 
for different values of v = 0 07, 0 05, 0 015 
Figure 2 shows the temperature dependence of the attenuation due to the 
hybndization (v) between the f-electrons and the conduction electrons. Though small, 
the high-Tc systems possess different magnitudes of hybridization strength. When 
ultrasonic sound interacts with the electrons, it deforms the lattice and enhances the 
hybridization This process causes damping of the ultrasonic wave and hence producing 
ultrasonic absorption It is worthwhile to mention here that the attenuation peak 
becomes sharper for stronger strength of the hybridization parameter 
Figure 3 shows the temperature dependence of ultrasonic attenuation (a) for 
different ultrasonic frequencies The high frequency waves distort the lattice more, 
thereby increase the hybridization of the f-electrons and the conduction electrons. The 
attenuation increases with the increase of hybridization and also with increase in 
frequency (o>) and becomes sharper at higher frequencies. The attenuation decreases 
with decrease of temperature and becomes zero at temperature t a 0; but the 
Study of magnetism and hybridization through ultrasonic attenuation in cuprate systems 51 
attenuation decreases more slowly and remains nearly constant at higher temperature. 
In studying the temperature variation of ultrasonic attenuation for the system, 
from Figure 1 to Figure 3, we have considered a constant staggered magnetic field M. 
But the staggered field varies with temperature. This tempeteture variation of M can be 
shown by a simple BCS type expression (Not shown heije). Taking into account this 
type of temperature variation of frt, the temperature variation of 'or* is shown in 
Figure 4. 
S 1*-os 
I / 
• / / / /, 
/' 
I' 
I\ 
M 
t \ 
1 1 
i I 
1 \ 
1 » 
J \ 
i \ 
i \ 
1 \ 
N 
^ 
"~-^ * : - - -
"-
(0=0.01 ! 
<o=0.05 
to=0.08 
OBO.IO 
"' • - -
-
, » _ , j 
CO 
D 
C 
<D 
» ?e-06 
le-06 
, i
 h 1 ( 0 ) = 0 420, V, * 0 118 
/ i h1(0) x, 0 3S5 V- - 0 100 
/ l . h1(0) ^ 0 151, IN * 0 040 
"0 01 02 03 04 
reduced temperature (t) 
Figure 3. The plot of ultrasonic attenuation vs. F^ure 4. The plot of ultrasonic attenuation vs. reduced 
reduced temperature for fixed values of e = 0.018, p temperature for fixed values of the N6el temperature 
= 1.0, g= 0.025, d= 0.06, v = 0.015, fi1 =0.195 and for *N = 0.018, 0.100, 0.040 with corresponding values of 
different values of <w= 0.01, 0.05, 0.08, 0.10. ^ ( 0 ) and h\ value as per BCS type variation i.e. /?1 = 
/71(0)/?1 - />1(0)[1 - (t/tN)2Ju and for fixed values of e = 
0.018, p = 1 0, g = 0 025, d = 0.06, v = 0.015. 
The Figure 4 shows the ultrasonic absorption peaks, one at Neel temperature 
tN and another at hybridization gap at F with tN > F. It is also observed that at tN < F, 
only the absorption peak of staggered field gap appears but the hybridization gap 
disappears altogather. It is further noticed that ultrasonic peak shows asymmetry in its 
peak in magnetic phase for t < tN. The ultrasonic absorption decreases rapidly in non-
magnetic low-7* phase, while absorption decreases more slowly for temperature t > tN. 
This asymmetry in ultrasonic peak is a clear indication of the presence of two different 
phases in the system. Below the characteristic temperature F the hybridization effect 
is suppressed, so that the system containing localised f-level exhibits heavy fermion 
behaviour even in presence of antiferromagnetism. The weak hybridization between the 
lattice of 4f-ions of Nd with the strongly correlated electrons of Cu-O plane is observed 
experimentally [5J. 
The Figure 5 shows the effect of the position f-level (d) on the ultrasonic 
absorption peaks in which we observe two peaks. The high temperature peak is due 
to the ab-sorption of sound energy through the AFM gap, while the low temperature 
gap is due to absorption through hybridization gap. When f-level approaches the Fermi 
52 K C Bishoyi, G C Rout and S N Behera 
3e-06 
s 
'£ 2e-06 
3 
C 
<D 
< 
.9 
c 
o 1e-06 
/ / 
. . -wf l^T 1 . .,,...., 
/ ' ' 
* . bj, 
\ d*0.060 ^ 
\ d = 0.065 
\ - - - d = 0.070 
* ~~^'^*>»**J 
0.05 Ot 015 0.2 
reduced temperature (t) 0.25 
Figure 5. The plot of a vs. t for Neel temperature tN = 0.1 with corresponding value of M (0) = 0.335 and M value 
as per BCS type variation i.e. M = /rt(0)[l - (f / f J ? | ' 2 and for fixed values of e = 0.018, p = 1.0, g = 0.025, w = 
0.15, and with the different values of d. 
level (from d = 0.070 to 0.060), the two peaks become sharp with decreasing line width 
of both the peaks. This happens because of the strong hybridization. When the f-level 
approaches the Fermi level, phonon coupling to the hybridization between the f- and 
oelectrons become stronger leading to increasing absorption of sound energy. In other 
words, when f-level moves away from the Fermi level, the hybridization becomes 
weaker and hence the two peaks become flat and ultimately merge to a single flat 
peak. 
6. Conclusion 
We calculated the temperature dependence of ultrasonic absorption coefficient (a) for 
the cuprate system in normal phase. The phonon coupling to the hybridization 
between f-electrons and conduction electrons is considered in the formalism of the 
model m presence of an anti-ferromagnetic (AFM) ground state. Introducing a simple 
BCS type temperature variation of the stagged field (M). we observed two peaks in the 
a vs. t plot (Figure 5), one at tN - 0.118 and shifted hybridization gap at r - 0 107 
1 * T K 9 ^ " 6 r e d f i e , d M ( 0 ) * °420- S i m i , a r P e a k s « * • staggered field is 
observed by Zhang « a/ [6] for the ultrasonic measurements in LSCO system. 
Acknowledgments 
The authors would like to thank the UGC New Delhi for nm»Mim «~ • . 
«».«.»: F-Psca^ e-^ o, < ! * "££ pc09Jrr.cZr: 
I . -~~.ch to**. „„*«, * »8 l n 8 t l t u t e „ ^ BhlLsw r^ O ^ T t ^ 
Rsfsrsness 
m He Yuaheno and Sun Xiangzhong Physica B 165/166 1291(1890) 
Study of magnetism and hybridization through ultrasonic attenuation in cuprate systems 53 
[2] Tetsuo Fukase et al Physica B165/166 (1990) 
[3] K C Bishoyi, G C Rout and S N Behera Indian J. Phys. 76A 87 (2002) 
[4] D N Zubarev Sov. Phys. Usp. 95 71 (1960) 
[5] P Pulde J Low Temp. Phys. 95 45 (1994) 
[6] H Zhang, M J McKenna, Carsten Hucho, B K Sarma Moises Levy, T Kimura, K Kisho and K Kitazawa 
Physica B223 554 (1996) 


Study of Magnetism and Hybridization Through Ultrasonic Attenuation in Cuprate Systems

Ghosh, P K

Ahmed, Sk F

Maiti, U N

Chattopadhyay, K K

Indian J. Phys. 81 (1). 69-73 (2007) 
Structural and optical properties of ZnS nanostructured 
films synthesized via RF-magnetron sputtering technique 
P K Ghosh, Sk F Ahmed, U N Maiti and K K Chatiopadhyay* 
Thin Film & Nanosciencc Laboratory, Department of Physics, 
Jadavpur University, Kolkata-700 032, India 
E-mail kalyan_chattopadhyay(atyahoo
 c o r n 
Abstract : Nanostructured ZnS thin films have been synthesized by radio frequency magnetron sputtering technique X-ray 
diffraction and selected area electron diffraction studies confirmed the formation of nanocrystattine cubic phase of ZnS in the 
films The particle size calculated from the XRD patterns of the thin films was found in the range 2 06-4 86 nm TEM micrographs 
of the thin films revealed the manifestation of ZnS nanoparticles with sizes lying in the range 3 00-5 83 nm UV-Vis-NIR 
spectrophotometric measurements showed that the films are highly transparent (-90 %) in the wavelength range 500-2600 nm 
1 he absorption edge showed a blue shift The direct allowed bandgaps was found in the range 3 89-4 44 eV The room temperature 
photolummescence spectra of the films showed two peaks centered around 315 nm and 450 nm We assigned the first peak due 
to bandgap transittons while the latter due to sulfur vacancy in the films The composition analysis by energy dispersive X-rays 
also supported the existence of sulfur deficiency in the films The thicknesses of the films were in the range 260-350 nm 
depending on the deposition time -
Keywords : Nanostructured films, RF-sputtermg, TEM, photolummescence 
PACS Nos. : 61 14 Lj, 61 46 +w. 78 55 -m 
1. Introduction 
In recent years nanostructured materials have gained special interest due to their novel 
properties and possibilities of many applications. Previously, one-dimensional ZnS 
nanostructures have been reported, including ZnS nanowires by liquid crystal template, 
micelle template, thermal evaporation using Au catalyst etc. Recently, ZnS nanobelts 
synthesis by chemical bath deposition (CBD) within the self-organized pores of polyvinyl 
alcohol (PVA) was reported by us [1]. ZnS nanoparticles have also been reported [2] 
via chemical routes without capping agent, using thioglycerol as capping agent [3] and 
using sodium dioctyl sulfosuccinate as a surfactant [4]. But to our knowledge there is 
no report on the synthesis of ZnS nanoparticles in thin film form via radio frequency 
magnetron sputtering without using any matrix and its detail studies of optical 
properties. In this paper we report the size dependent optical properties of ZnS 
nanoparticles synthesized via RF-magnetron sputtering without using any capping 
agents or matrix. 
'Corresponding Author © 2 0 0 7 , A C S 
70 P K Ghosh, Sk F Ahmed, U N Maiti and K K Chattopadhyay 
2. Experimental 
Zinc sulfide (ZnS) target was fabricated by taking a suitable aluminium holder (5 cm 
dia.) and compacting the ZnS polycrystalline powder (99.99%, Aldrich) by applying 
suitable hydrostatic pressure (-100 kg/cm2). Detail of experimental procedure can be 
found elsewhere [5]. The films were synthesized at room temperature (298 K) on glass 
and Si substrates for 10-30 min. The sputtering parameters for deposition of thin films 
were : effective RF-power : 170 Watt, electrode distance : 3 cm, sputtering gas : Ar, 
gas pressure : 0.1 mbar. 
3. Results and discussion 
3.1. Nanostructural studies : 
The nanostructures of the films were studied at room temperature by a using a 
transmission electron microscope (TEM, Hitachi-H600). The micrographs of ZnS 
nanoparticles have been shown in Figure 1(a) and (b). Presence of ZnS nanoparticles 
is clearly visible in TEM bright field images and corresponding diffraction patterns of the 
film consists of several concentric rings. From the diameter of the rings, we calculated 
the inter-planer spacing (dhw) values, which correspond to reflection from (111), (200), 
(220) and (311) planes of cubic ZnS with a0 = 5.406 A. The average particles sizes 
obtained from TEM micrograph were in the range of 3.0-5.83 nm. 
S S T 1 " J E M mlcro9raPhs of Z n S for deferent deposition time (a) 10 min, (b) 20 min and Inset: corresponding 
oAcP patterns. 
Figure 2 shows the X-ray diffraction (XRD) patterns of ZnS thin films. One peak 
of ZnS has been obtained due to diffraction form (111) plane of cubic ZnS. The particle 
size (Z.) of the deposited films have been obtained from standared Scherrer relations. 
The interplanner spacing (cW corresponding to XRD peaks, TEM measurement and 
Joint committee for powder diffraction studies (JCPDS) card [6] matched well. 
The composition of the films was determined by energy dispersive X-ray analysis 
(EDX, JEOL JSM 6300 Oxford-7582)). The Zn : S in the target material was - 1 : 1 . 
The final sulfur concentration in the films decreases with the increase in deposition 
Structural and optical properties of ZnSnanostructured films synthesized etc. 71 
time. The final composition inside the film for different deposition time is as shown in 
Table 1. 
100 
E 
W 
C 
CO 
40 50 
26 (deg.) 
Figure 2. XRD patterns of thin ZnS films deposited 
on glass substrate for different deposition time 
(a) 10 min, (b) 15 min, (c) 20 min and (d) 25 min. 
500 1000 1500 2000 
Wavelength (nm) 
2500 
Figure 3. Transmittance spectra for different 
deposition time and inset: corresponding (ahv)2 vs. 
/?v plots. 
Table 1. The direct bandgap {Eg), particle size (L) from shift of Eg and Zn/S ratios obtained from EDX for different 
deposition time (dt): 
Sample name 
(a) 
(b) 
(c) 
(d) 
(e) 
dt (min) 
10 
15 
20 
25 
30 
E0 (eV) 
4.44 
4.26 
4.09 
3.98 
3.89 
L(nm) 
3.23 
3.72 
4.36 
4.92 
5.6 
Zn/S 
1.09 
1.12 
1.15 
1.19 
1.26 
3.2. Optical studies : 
The transmission vs. wavelength traces for different ZnS thin films have been recorded 
as shown in Figure 3. The direct bandgap values lie in the range 3.89 eV to 4.44 eV 
calculated from (ahv)2 vs. hv plots as shown in inset of Figure 3. The direct bandgap 
values of the films are higher than that of bulk value of ZnS (3.6 eV) because of 
quantum confinement of carriers in ZnS nanocrystals. 
The shift of band gap might also be utilized in determining the crystal radius (r) 
using reported standard relation [7]. The values of direct bandgap, and particle size 
from shift of direct bandgap for different deposition time are shown in Table 2. It is 
clear from the table that the direct bandgap decreases with the increase of deposition 
time and the particle size increases with the increase of deposition time. It can also 
be noticed that that there is some mismatch of particle size determined from XRD, 
TEM and optical method. Particle size determined from XRD is always found to be 
smaller than that measured by TEM or optical method because from XRD we get 
length of coherence in the grains which do not include grain boundaries. 
72 P K Ghosh, SkF Ahmed, U N Maiti and K K Chattopadhyay 
Table 2. The comparison of particle size obtained from TEM (LTEM). XRD (*-XRD) and shift of direct optical bandgap 
(^ optto«i) and corresponding film thickness (F,) for different deposition time (dt) 
F,(nm) 
260 
330 
410 
490 
530 
i-TEM(nm) 
3 00 
3 52 
4 60 
5 31 
5 83 
*-XRD (nm) 
2 06 
2 65 
3 26 
4 07 
4 86 
Logical (nm) 
3 23 
3 72 
4 36 
4 92 
56 
3 3 Photoluminescence studies 
Photoluminescence (PL) spectra, measured at room temperature (300 K) of the 
nanocrystalline ZnS thin films deposited on glass substrates are shown in Figure 4. All 
the plots contain two peaks centered at 315 nm and at 450 nm The excitation 
wavelength was 200 nm (xenon lamp, 100 W) It is evident from Figure 4 that the 
position of 2nd peak is not shifted with the variation of particle sizes Hence, the 
photoluminescence in this region is due to the presence of sulfur vacancies in the 
lattice, which is also previously reported by Sooklal et al [8] The sulfur deficiency in 
the synthesized nanocrystalline ZnS films was also confirmed from EDX measurements 
of the samples. Also it was observed that with the increase of deposition time, the 
intensity of the PL peak increased 
w 
c 
55 
C 
B 
0. 
T • — — i . — - rr » —i « 1 • — i 
300 320 440 460 480 
Wavelength (nm) 
Figure 4. PL-spectra of nano-ZnS thin films for different deposition time (a) 10 mln, (b) 20 mm and (c) 30 mm 
This is due to the fact that with the increase of deposition time sulfur deficiency 
increases which is supported by the EDX results. The other PL peak appeared at -315 
nm and from Figure 4 it is clear that the position of this peaks changes with the 
deposition time which indicate the peak position depends on the particle size. This 
peak is assigned due to the band-to-band transition of nanocrystalline ZnS. As the 
bandgap energy depends on the particle size, the position of the peak changes 
accordingly. In the present study trap related emission was not observed, most 
Sample name 
W 
(b) 
(c) 
(d) 
(e) 
d,(min) 
10 
15 
20 
25 
30 
Structural and optical properties of ZnS nanostructured films synthesized etc 73 
probably this emission is obstructed by the grain boundaries owing to compactness of 
nanograms. 
4. Conclusion 
Nanoparticles of ZnS have been successfully synthesized by rf-magnetron sputtering 
technique. TEM micrographs of the films revealed the manifestation of ZnS nanoparticles 
with sizes lying in the range 3.0-5.83 nm. From the transjnittance and reflectance 
spectra of the films the direct bandgap values have been calcinated and they lie in the 
range 3.89-4.44 eV. The room temperature photoluminescence spectra of the films 
showed two peaks centered around 315 nm and 450 ran. EDX measurements 
confirmed the sulfur vacancy in the films. 
Acknowledgment 
The authors wish to thank Defence Research & Development Organization (DRDO), 
Government of India, for financial supports during the execution of the work. 
References 
[1] P K Ghosh, R Maity and K K Chattopadhyay J. Nanoscience & Nanotechnology 5 301 (2005) 
[2] W A Chen, G Joly, J O Malm and J O Bovin J. Appl. Phys. 95 667 (2004) 
[3] J Nanda, S Saora, D D Sharma, N Chandrasekharan and G Hodes Chem. Mater. 12 1018 (2000) 
[4] A D Dinsmore, D S Hsu, S B Quadri, J O Cross, T A Kennedy, H F Gray and B R Ratna J. Appl. Phys. 88 
4985 (2000) 
[5] P K Ghosh, U N Maiti, Sk F Ahmed and K K Chattopadhyay Sol. Energy Mater. & Sol. Cells in press (2006) 
[6] J.CP.D.S. File Card 05*566 
[7] Y S Yuang, F Y Chen, Y Y Lee and C L Liu Jpn. J. Appl. Phys. 76 3041 (1994) 
[8] K Sooklal, B S Cullum, S M Angel and C J Murphy J. Phys. Chem. 100 4551 (1996) 


Structural and Optical Properties of ZnS Nanostructured Films Synthesized Via RF-Magnetron Sputtering Technique

Das, U

Indian J. Phys. 81(1) 155-159(2007) 
Transition metal doped ZnS nanoparticle systems and their 
response to swift heavy ion irradiation 
} 
U Das1, D Mohanta1*, F Singh2, A Tripathi2, D K Avastrt2 and A Choudhury1 
department of Physics, Tezpur University, Napaam, Assam-784 028, Assam, India 
2Inter University Accelerator Centre, Aruna Asaf Ali Marg, New Delhi-110 067, India 
E-mail : best@tezu.emet.in 
Abstract : Transition metal (Cr) doped ZnS nanoparticles were fabricated by adopting a solution growth-
chemical route. They were irradiated by 150-MeV titanium ion beams using tandem pelletron accelerator available 
at IUAC, New Delhi. Irradiation led manifestations in luminescence patterns and morphology has been discussed 
in analogy with modification in magnetic domains. The present study would find promising application in magnetic 
sensors and spintronics devices. 
Keywords : Nanostructures, luminescence, ion irradiation. 
PACS Nos. : 81.20.Ka, 81.05.Dz, 82.35.Np 
1. Introduction 
Semiconductor nanostructures, which use the principle of three dimensional carrier 
confinement, have emerged as important technological assets for their dimensional 
dependent optoelectronic properties. The nanoscaled systems exhibit quantum size 
effect thus, inducing quantization of the bulk electronic bands into discrete electronic 
states. This leads to size dependent electronic and optical properties of nanoparticles 
[1-4]. The fact that the band gap of the material varies with the crystallite size which 
makes them an interesting category of materials for potential application in novel 
devices. Among binary semiconductor systems, ZnS is the most widely investigated 
system having a direct band gap of 3.7 eV at room temperature. It is an important 
phosphorescent material both in doped and un-doped form. It has been established that 
when doped with some magnetic impurities, these nanostructures develop a unique 
class of material called diluted magnetic semiconductors (DMS) which are promising 
candidates for magnetic memories, sensors and other spin-based devices [5,6]. Such 
semimagnetic and semiconducting structures, where carrier and spin confinement is 
possible provides a matchless system for spin manipulation and spin transportation [7]. 
Again, swift heavy ion irradiation is a useful tool for the development of nanostructures 
'Corresponding Author © 2 ® ° 7 I A C S 
156 U Das, D Mohanta, F Singh, A Tripathi, D KAvasthi and A Choudhury 
through ion implantation, ion beam mixing and template synthesis etc, [8-12]. In this 
particular report, we highlight impact of 150-MeV Ti1 u ions on the photoluminescence 
and morphological changes in polymer encapsulated ZnS:Cr nanostructures. Further, 
significant modification in magnetic domains has also been discussed. 
2. Experimental 
First, 5% (w/v) transparent polyvinyl alcohol (PVOH, 1 mw) matrix was prepared under 
moderate stirring (-200 rpm) and heating (65°C) for 3 hours. Next, 0.15 M 2nCI2 was 
added to the PVOH matrix under stirring environment and the precursor was kept at 
ice cold temperature. Then, 0.0066 M Cr203 solution at 100°C was quenched into the 
as-prepared precursor. The whole solution was left undisturbed overnight. Next, H2S gas 
was allowed to diffuse through the solution for maximum absorption and result Cr 
doped ZnS nanocrystals. Nanocrystalline films are casted on 1 x 1 cm2 laboratory 
slides for the irradiation experiments. 
The irradiation was carried out on four identical samples. The nanoparticle 
samples were mounted on a vacuum shielded vertical sliding ladder having four 
rectangular faces. One sample was reserved as virgin to compare with the irradiated 
ones. They were irradiated at high vacuum (1.6 x 1C6 torr) by 150-MeV T i m beam with 
approximated beam current 1.0 pnA (particle nano ampere). The fluence was varied in 
the range 1010 to 1012 ion/cm2. In order to expose the whole target area, the beam was 
scanned vertically over the sample plane. The energy of the ion beam was chosen in 
such a way that the projectile range of the incident ion (21.3 urn, as computed by 
SRIM program [13]) exceeds thickness of the sample films -0.5 jim). Again, the ion 
beam energy and thickness of the target was selected so that modification due to 
electronic energy loss (Se) affects the sample. Energy dependent electronic energy loss 
(S#) and nuclear energy loss (Sn) for Ti-ions is shown in Table 1. 
Table 1. Energy loss vs. projectile range [13]. 
Energy 
MeV 
icr* 
10-' 
1 
10 
100 
150 
Electronic energy 
\oss S0*(dE/dx)B 
eV/A 
1.53 
5.03 
25.47 
109.75 
512,37 
587.29 
Nuclear energy 
\ossSn*(dE/dx)n 
eV/A 
18.99 
20.59 
8.59 
1.76 
0.23 
0.17 
Projectile range 
.0419 
.2913 
.2820 
1.446 
15.26 
21.32 
3. Results and discussion 
The X-ray diffraction pattern of ZnS nanoparticles is shown in Figure 1. It depicts cubic 
crystalline structure corresponding to three diffraction peaks (111), (220) and (311) at 
27.5°, 46.2° and 56°, respectively. The broadening in the diffraction peak corresponds 
to the formation of nanoparticles. The average particle size estimated is -10 nm, 
Transition metal dopedZnS nanoparticle systems and their response to swift heavy ion irradiation 157 
(111) 
— ! • I 1 * 1 
20 30 40 50 60 70 
2 e (degrees) 
Figure 1. XRD pattern of ZnS nanoparticle. 
obtained by measuring full-width-at half maxima (FWHM) and using Scherrer's formula 
of s 0.9/i/wcos 6. 
Normally, the photoluminescence study provides information relating to different 
energy states available between valence band and conduction band responsible for 
radiative recombination. It is well known that when emission peak energies are less 
than the band gap energy of the material, these bands ascribe to transition involving 
donors, acceptors, free electrons and holes. The appearance of the PL peak with 
energy close to the band gap energy suggests band edge emission [14]. It has been 
studied that the photoluminescence efficiency of coated ZnS:Mn nanoparticle is higher 
than the powder sample due to passivation of surfaces [15]. In addition to Mn+2 yellow-
orange emission, irradiation led surface emission enhancement in ZnS:Mn nanoparticles 
has been reported by our group very recently [16]. A Cr-related near-infrared 
photoluminescence (PL) band in ZnSe with a main no-phonon line at 10702.2 cm"1 has 
previously been reported [17]. Crystal field calculations demonstrate that the observed 
no-phonon lines and the corresponding luminescence band can reasonably be interpreted 
in terms of a 3TL(H) -» ST2(D) transition within the d* configuration of Cr2* [16]. 
Similarly, Luo have extensively studied the effect of incorporation of Cr into epilayers 
of ZnTe, developed by molecular beam epitaxial method [18]. They observed that Cr*"2 
ions produce a broad emission peak in the IR-regime 2-3 fxm. The temperature 
dependent luminescence studies have also been performed by them to determine 
thermal quenching activation energies. 
Figure 2 represents PL response of pristine and ion irradiated ZnS:Cr 
nanostructures. It depicts three distinct bands around 475 nm, 540 nm and 700 nm. 
We ascribe the emission peak -475 nm as donor-acceptor (D-A) pair transition and the 
band at -700 nm corresponds to surface state fluorescence activation due to nano-ZnS, 
in consistency with our recent work in ZnSrMn systems [19]. However, the blue-green 
emission peak at -540 nm is expected due to ^electron transfer of Cr ions into the 
<5 
158 UDas, D Mohanta, F Singh, A Tripathi, D KAvasthiandA Choudhury 
1 1 1 i 
300 400 500 600 700 800 900 
Wavelength in nm 
Figure 2. PL emission spectra of (a) unirradiated and irradiated, (b) 5 x 1010, (c) 2 x 101 \ (d) 8 x 1011, (e) 3.2 x 
1012 ions/cm2 ZnS: Cr nanoparticles. 
ZnS host. Strong Cr-emission dominates the fluorescence activation as well as 
emission due to D-A pair transitions to the extent that the heavily exposed sample also 
possesses significant emission band - 5 4 0 nm. The PL peaks decrease in intensity 
with the increase in ion dose, possibly due to suppression in recombination emission 
owing to grain growth phenomena under ion irradiation. 
It has been found that V or Cr doped ZnS, ZnSe, ZnTe are ferromagnetic, 
without p-type or n-type doping treatment. However, Mn, Fe, Co, Ni substituted ZnS, 
ZnTe etc. are found to be in spin glass states [20]. To exploit magnetic properties, we 
have carried out magnetic force microscopic studies on irradiated ZnS:Cr nanosystems, 
represented in Figure 3. The figures are basically phase images of the magnetic 
impurity (Cr) doped ZnS nanostructures. Figure 3a is the phase image of the pristine 
Phase Image Phase Image 
I 500 i 2 0° 500 
250 
1.0° 
I 0 0° 
Digital Instruments Nanoscope 
Scansize 500 Onm 
Scan rate 3005H? 
No ofSampiM 512 
Data scale 2 000* 
2.0° 
1.0* 
0.0° 
Digital Instruments Nanoscope 
Scan sua 500.0 wn 
Scan rate 1502 Hz 
NoofSampNM 256 
Data seek 2000* 
250 
•2 0-
I to0 
•oo° 
Digital Instruments Nanoscope 
Scansize 500 Onm 
Scan rata 3005 Hz 
No of Samples 256 
Data scale 2000" 
500 
250 I 
2.0° 
1.0° 
0.0° 
Digital instrument! Nenoeoope 
500.0IW 
1.507 Hz 
Figure 3. MFM images (200 x 200 nm) of (a) unirradiated and irradiated, (b) 5 x 1010. (c) 2 x 1 0 \ (d) 8 x 101,k>ns/ 
cm2 ZnS:Cr nanoparticies. 
Transition metal doped ZnS nanopartic/e systems and their response to swift heavy ion irradiation 159 
ZnS.Cr, which does not show appreciable results on the magnetic domains owing to 
matrix encapsulation, which prevents from detecting magnetic domains by the tip of the 
MFM. Figure 3b is the sample irradiated at 2 x 1011 ions/cm2 where spherical single 
domains (-30 nm) are visible which correspond to ZnScCr nanoparticles distributed 
uniformly in the matrix. Irradiation with fluence 4 times leads to distorted domains, 
shown in Figure 3c. Surprisingly, sample exposed at 3.2 x 1012 ions/cm2 results in the 
aligned magnetic domains normal to ion trajectories. Thesjje aligned domains in which 
elongation upto -200-250 nm has been observed would Jfind application in magnetic 
tapes, magnetic recording devices and other spin-based devices. 
5. Conclusion 
In conclusion, we have produced Cr doped ZnS nanostructures by simple and 
inexpensive chemical route. Photoluminescence shows a new emission peak around 
-540 nm due to Cr+2. The low ion fluence recovers the nano-ZnS:Cr from matrix 
encapsulation and affect the shape of the magnetic domains. Spherically distributed 
domains turn into elongated domains upon ion impacts possibly, due to pressure led 
development of structural elongation normal to the ion trajectories. 
Acknowledgment 
The authors thank pelletron group, IUAC, New Delhi for quality beam and material 
science group for providing PL data and taking MFM images. 
References 
[1] A P Alivisatos J. Phys. Chem. 100 13226 (1996) 
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[3] Y Kayanuma Phys. Rev. B38 9797 (1998) 
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[16] D Mohanta, S Nath, N C Mishra and A Choudhury Bull. Mater. Sci. 26 289 (2003) 
[17] MU Lehr, B Litzenburger, J Kreissl, U W Pohl, H R Selber, H-J Schulz, A Klimakow and L Worscheoh 
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[18] M Luo, B L Vanmil, R P Tompkins, Y Cul, T Mounts, U N Roy, A Burger, T H Myers and N C Giles J. Electron. 
Mater. 32 7 (2003) 
[19] D Mohanta, G A Ahmed, A Choudhury, F Singh and D K Avasthi J. Nanopart. Res. (In press, 2005) 
[20] J KFurdyna J. Appl. Prtys. 64 R29 (1988) 


Transition Metal Doped ZnS Nanoparticle Systems and their Response to Swift Heavy Ion Irradiation

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