Estimation of Rossiter parameters in CuAu

Intermetallics 8 (2000) 831±833

Short communication

Estimation of Rossiter parameters in CuAu B. SprusÏ il *, B. Chalupa Department of Metal Physics, Charles University, Ke Karlovu 5, CZ-121 16 Praha 2, Czech Republic Received 28 February 2000; accepted 10 March 2000

Abstract Parameters in the Rossiter formula, which connects the resistivity of a given material with its degree of long-range order, were determined for the ®rst time for CuAu I. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: A. Intermetallics, miscellaneous; B. Order/disorder transformations; B. Electrical resistance and other electrical properties; B. Phase transformations

1. Introduction It is well known that the electrical resistivity r sensitively reacts to changes in the degree of order in an alloy. Some time ago Rossiter [1] proposed the following formula describing the dependence of the resistivity on the absolute temperature T and the degree of longrange order S ÿ
  1 ÿ S2 B 1 ‡
T: …1†
OR …S; T† ˆ …0†
2 …1 ÿ A
S † n0 …1 ÿ A
S2 † For S ˆ 0 it reduces to D …0; T† ˆ …0† ‡

  B
T: n0

Thus …0† is the residual resistivity and

…2†   B n0

is the

temperature coecient of resistivity in the disordered state. In a recent article by Lang et al. [2] it was shown, for CuPt, that by making use of the Rossiter formula the resistometry turns out to be an ultra-precise method for determination of a long-range order parameter.

* Corresponding author. E-mail address: [email protected]€.cuni.cz (B. SprusÏ il).

We have in vain searched the literature for the values   of the Rossiter parameters A and nB0 in the case of the CuAu alloy. Therefore, we decided to measure them by making use of dynamic resistometry. Here we report the results.

2. Experimental part We used a meander-shaped specimen spark-cut from a 0.3 mm foil of (49.40.5) at% Cu, (50.50.6) at% Au. The measurements were done under argon atmosphere in the apparatus for dynamic resistometry described in [3]. The specimen was heated up to 206 C in a complicated manner (with a heating rate vH ˆ 10 K/min from room temperature to 120 C, then with 0.2 K/min to 200 C and ®nally with 0.02 K/min to 206 C) and held there for 14 h. The heating proceeded with vH ˆ 0:02 K/min to 436 C and continued with vH ˆ 2 K/min up to 480 C. At this temperature it was held for 40 min; this isotherm is believed to fully destroy the long-range order in the alloy. Then it was cooled with the rate vC =2 K/min to 422 C. Afterwards the cooling rate was reduced to 0.02 K/min and the cooling proceeded down to the room temperature. The measured temperature dependence R…T† of specimens resistance is given in Fig. 1. In the disordered region (above approx. 425 C upon heating and down to approx. 405 C upon cooling) detailed inspection reveals, that R…T† is well

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B. SprusÏil, B. Chalupa / Intermetallics 8 (2000) 831±833

was measured with a specimen quenched from 650 C [4]. Its value Ð established from a ®gure Ð is approximately equal to 14:1 m cm. We have assumed that the resistance in the disordered state at 20 C can be calculated from Eq. (3), i.e. far below the temperature interval for which experimental data exist. Furthermore, it was assumed that the state of disorder reached experimentally by quenching from 650 C is the same as that formally extrapolated by the RD formula. Under these assumptions one gets FD ˆ 14500 cmÿ1. Dividing RD [Eq. (3) ]by FD one gets D…0;T † [Eq. (2)] and ®nds …0† ˆ …11:8  1:2† m cm and

B n0

ˆ …7:71  0:02†
10ÿ3 m cm/K.

The parameters A and S can be evaluated without the knowledge of the value of F provided that Fig. 1. The temperature dependence of specimens resistance observed upon heating (full curve) and cooling (dashed curve).

FD ˆ FOR

linear, with parameters slightly dependent on the sign and magnitude of the rate of temperature changes. We believe, that the function

holds, i.e. neglecting the dependence of F on the temperature and the degree of long-range order. Then from Eqs. (1) and (2) it follows that

RD ‰m Š ˆ aD ‡ bD
T

bOR 1 ˆ bD …1 ÿ A
S2 †

…7†

ÿ
aOR 1
1 ÿ S2 ˆ 2 aD …1 ÿ A
S †

…8†

ˆ …171  18† ‡ …111:8  0:25†
10ÿ3
T‰KŠ

…3†

which represents the data observed upon cooling with vC ˆ 2 K/min, is the best possible approximation ( standard error) to the temperature dependence of the resistance in disordered state RD up to 480 C. Also the ®nal part of the cooling curve, extending from approx. 180 C to room temperature, is linear. We believe that it is representative of the ordered state with a ®xed value of S, close to 1. For the resistance ROR in this ordered state we get by least square ®t

…6†

i.e.     aOR aD = S ˆ1ÿ bOR bD

…9†

   bD =S2 ˆ A: 1ÿ bOR

…10†

2

ROR ‰m Š ˆ aOR ‡ bOR
T ˆ …17:05  0:044† ‡ …166  0:1†
10ÿ3
T‰KŠ

…4†

3. Discussion Starting from Eqs. (3) and (4) one can calculate the   constants …0†, A and nB0 in the Rossiters Eq. (1) and the value of S reached at the room temperature. In the ®rst step it is necessary to ascertain the value of the factor F Fj ˆ

Rj ; j

j ˆ D; OR:

…5†

The form of our specimen does not allow to calculate F in the usual simple way. Therefore, we have made use of the fact that the value of resistivity D (20 C) is known. It

In this way one gets S ˆ 0:97 and A ˆ 0:35. Provided for strict ful®lment of Eq. (6) it is possible to calculate the standard errors of A and S from the known standard errors of aD, aOR ; bD and bOR . One arrives at 0.05 for S and at 0.04 for A. With A ˆ 0:35 one would get from Eq. (1) for a fully ordered alloy …S ˆ 1† …S ˆ 1; 20 C† ˆ 3:48 m cm:

…11†

In [4]  ˆ 3:6  cm was measured for a specimen quenched from 650 C and annealed at 200 C, evidently representing a well ordered state of CuAu. Thus the value of A does not contradict experimental data. The two-step decrease observed on R…T† upon cooling indicates that ordering proceeded ®rst by the formation of the CuAu II phase, then of the CuAu I phase. This is in accordance with expectations based on in situ TEM observations for cooling rates lower than 2 K/min [5]. Possible remains of CuAu II initially present at

B. SprusÏil, B. Chalupa / Intermetallics 8 (2000) 831±833

temperatures within the equilibrium CuAu I ®eld of the phase diagram are expected to transform to CuAu I before reaching room temperature. It has been observed, that the transformation into the CuAu I form takes ``several hours'' at 300 C [6] and 24 h at 355 C [7]. In our experiment it took 50 min to change the temperature by 1 K below 435 C so that there was enough time for the transformation into CuAu I to take place. Thus our parameters belong to the CuAu I ordered state. 4. Conclusions Dynamic resistometric measurements have shown that the temperature dependence of resistance in stoichiometric CuAu is well linear in the disordered state as well in the CuAu I ordered state at low temperatures (for which a constant degree of long range order can be assumed). Therefore, they can be formally described by the formula derived by Rossiter [1]. For constants of this formula we found   the following values: …0† ˆ

…11:8  1:2† m cm,

B n0

ˆ …7:71  0:02†
10ÿ3 m cm/K

and A ˆ …0:35  0:05†. With these values the Rossiter formula predicts resistivity values in a good accordance with the available experimental data.

833

Acknowledgements The authors wish to express their thanks to Professor P. KratochvõÂl (of the Charles' University, Prague) for valuable comments. References [1] Rossiter PL. Long-range order and the electrical resistivity. J Phys F 1980;10:1459±65. [2] Lang H, Mohri T, Pfeiler W. L11 Long-range order in CuPt; a comparison between X-ray and residual resistivity measurements. Intermetallics 1999;7:1373±81. [3] SprusÏ il B, SÏõÂma V, Chalupa B, Smola B. Phase transformations in CuAu and Cu3Au: a comparison between calorimetric and resistometric measurements. Z Metallkde 1993;84:118±23. [4] Johansson CH, Linde JO. RoÈntgenographische und elektrische Untersuchungen des CuAu-Systems. Ann Physik, 5 Folge, Bd 1936;25:33±48. [5] Bonneaux J, Guymont M. Study of order±disorder transition series in AuCu by in-situ temperature electron microscopy. Intermetallics 1999;7:797±805. [6] Dehlinger U, Graf L. UÈber Umwandlungen von festen Metallphasen. I. Die tetragonale Gold-Kupferlegierung AuCu. Z Physik 1930;64:359±77. [7] KoÈster W. ElastizitaÈtsmodul und DaÈmpfung der geordneten Phasen CuZn AuCu3. AuCu, PdCu3 und PtCu3. Z Metallkde 1940;32:145±50.