Absolute vacancy concentrations in noble metals and some of their alloys

1. Phys. Chem. Solids Vol. 55, No. 10, pp. 907-915, 1994 Copyright 8 1994 Eisevier Science Ltd Printed in Great Britain. All rights rescm!d GQ22-3697/94 $7.00 + 0.00

Pergamon

ABSOLUTE VACANCY CONCENTRATIONS IN NOBLE METALS AND SOME OF THEIR ALLOYS TH. HEHENKAMP Institut fiir Metallphysik, University of Giittingen, Giittingen, Germany Abstract-Absolute vacancy concentrations are being determined by Differential Dilatometry (DD) measuring simultaneously the relative change in length by a laser/optical slit technique and in lattice parameter by a Debye-Scherrer technique in one and the same specimen. Accuracy has been improved to some parts in 106.Precautions are taken to prevent selective evaporation of constituents making such measurements in alloys feasible up to 1150°C presently. The temperature range, in which vacancy formation can be reliably observed, can be extended further to lower temperatures by adding results of positron lifetime measurements, which give absolute vacancy concentrations only, if the trapping rate of vacancies for positrons is well known. Since the DD measurements and the ones employing positron annihilation show some regions of overlap, the trapping rate may be determined combining both techniques. The results are discussed in terms of mono~divacancy formation for the pure metals Cu, Ag and Al and of the “complex model” for some Cu and Ag alloys, from which reliable data for the enthalpy and entropy of binding may be deduced. The Gibbs free energy of binding is being compared to literature data and models published so far. In this context, the following question may be discussed again: which quantities govern the binding effect most? Excess charge, misfit or others? Keyword: A. alloys, A. metals, C. positron annihilation spectroscopy, D. defects, D. thermal expansion.

After the first introduction of lattice defects in thermal equilibrium by Frenkel [l], as pairs of vacancies and interstitial vacancies without simultaneous formation of interstitials were postulated by Wagner and Schottky [2]. In 1936 Beyer and Wagner [3] presented the idea of distinguishing between these two types of equilibrium defects experimentally by comparing the macroscopic volume (pyknometer) to the microscopic one, measured by the lattice parameter. For Frenkel defects the difference between these two volumes should be zero, for preferential vacancy formation larger than zero. In 1942, Gott [4] tried to measure this in GGttingen for pure aluminum by a mechanical dilatometer and X-ray diffraction. Unfortunately the accuracy obtainable in those days was just not sufficient to detect the small effect known now to be less than 10e3. The first successful measurements were reported by van Dujn and van Galen [5] and Feder and Nowick [6] in lead and aluminum. Simmons and Balluffi 17-101 started a series of measurements in aluminum, silver, gold and copper which are the best known and served as standard values for absolute vacancy concentrations in the higher melting metals for many decades. Some other measurements followed. d’Heurle et al., [l l] and Feder and Nowick in lead [12] and dilute lead alloys, Feder and Charbnau [ 131in sodium. Feder [ 141in lithium and von Guerard et al. in aluminum [IS].

There are numerous other techniques available to determine vacancy concentrations such as resistivity, specific heat, positron annihilation or solvent diffusion enhancement. All these techniques, however, are indirect and not absolute in nature and require knowledge of at least one additional parameter which is either not measurable or uncertain. The primary drive to take up the question of absolute measurements again by my own group was the complete lack of absolute data for (noble metal) alloys and the feasibility of more accurate experiments than before due to substantial developments in the basic techniques needed to perform such measurements.

BASIC TECHNIQUES For cubic crystals, no interstitial formation and small vacancy concentrations their molar fraction c, may be obtained from ~--

A@-) a0 > (1)

The second bracket lowers the vacancy concentration at the melting point by 2% at most according to

TH. HE~EN~MP

908

Seeger [16] and is therefore, a negligible correction at present accuracies. Further corrections (1 - K(T)) may be needed for a possible temperature dependence of the atomic or formation volume of the lattice or the vacancy, respectively, as has been pointed out by Heumann [ 171.This effect is small again but not easy to calculate because of temperature dependent anharmonicity of the lattice. Both corrections have been neglected so far, but become increasingly important as a result of the improved accuracy of the experiments. Further work on these questions is in progress. In order to measure small absolute vacancy concentrations precisely enough, an apparatus has been developed by Hehenkamp and co-workers over many years [18-211. The relative change in length Al/J, and the relative change of the lattice parameter Aa/% have to be measured simultaneously with as much accuracy as is obtainable. For AIJI~, a laser slit technique has been developed. Figure 1 shows the arrangement. The cylindrical specimen of about 18 mm diameter and 22 mm length is encapsulated by a cage made of graphite, both sitting on a T-shaped fixture being centered to the X-ray goniometer. Between the top of the sample and the cage, an optical slit is being formed which measures the difference in

B

I I/

F G P PP S GP Z

-

HeNe laser

Window Measuring cell Specimen Penta prism Optical slit Optical grey filter Photodiode array

Fig. 2. Optical setup mounted to X-ray goniometer.

length change between cage and sample. The slit is ill~inat~ by a He-Ne laser at 632 nm. The laser beam is passing through a pinhole, is widened to 25 mm diameter and is made exactly parallel. The interference pattern can be observed with a linear camera consisting of a chain of photodiodes connected to a multichannel analyzer. Many interference minima are detectable from which the slit width b can be calculated by an on-line computer according to

AB CDI3 F 0 HI K -

EAeasuriog cell ewer

Slit cheek Specimen cwer Specimen Measuring cell Sample holder Heater Differential thermocouple Specimen thermocouple Window

Fig. 1.

Sample, cage and optical slit.

where k is the order of the minimum, 1 the wavelength of the laser, E the distance between slit and detector and xk the distance between the minimum of kth order and the maximum of xeroth order. E can be determined precisely by varying the distance of the linear camera in an exactly measurable way keeping b constant. The precision of b measurements is presently about 2 x 10T6 averaging over 30 data points. Since the zero point of the primary beam is always determined from each measurement, a shift of the position of the zero* order as result of thermal expansion of the sample or its holder is automatically eliminated. Figure 2 demonstrates the final optical setup. An interference filter at 632 nm prevents saturating the photodiodes by the sample glow at high temperatures. An additional grey filter permits

909

Absolute vacancy concentrations adjustment of the light intensity for these diodes. However, since the arrangement measures the difference in thermal expansion between sample and cage only, the expansion of the latter has to be determined additionally with the same accuracy. This is accomplished by replacing the sample by an identical one made from molybdenum which forms no thermal vacancies in the temperature range of the apparatus up to 1150°C. Therefore, Af/l,, and Au/u, have to be identical for MO. A further check is possible for any subsequent run with actual samples because f?r the temperature range, in which no thermal vacancy formation takes place, Al/l,, and Aala, have to be identical again. Figure 3 shows schematically the complicated design of the actual sample. It is a cylindrical rod basically, from which a tetrahedral sector has been cut out at one side to permit the X-rays to enter and to be reflected back. In the center, powder of the same composition (particle size < 25 pm) has been pressed into it (2 mm x 10 mm) with a precision press and sintered to the basic cylinder. This permits reflections at any orientation of the sample provided that the Bragg conditions are met. Two slits house a window of pyrolytic carbon which is amorphous up to 2000°C and may be thinned sufficiently to prevent substantial X-ray absorption and selective evaporation of alloy constitutents, which would ruin any alloy measurement otherwise completely. By this arrangement the thermodynamic activity of alloy constituents in the gas phase above the sintered material always equals the one in the solid material like in a Knudsen cell at any temperature. Figure 4 demonstrates the setup for the X-ray lattice parameter measurements. To gain sufficient accuracy both measurements, Al/l, and Aala,,, have to be performed simultaneously in one and the of

rotation

I i

,

:

Detector

Fig. 4. Bragg-Brentano configuration for lattice parameter measurements. same specimen since the temperature difference has to be kept lower than 0.1 K. Furthermore, only the back-reflection range is permissible in which two reflections are being followed in order to detect any effect of misalignment during a run. X-Rays are obtained from broad-focus Cu, Ni or Co X-ray tubes. Power required is around 2500 watts. Resolution is limited by the natural line width of the X-rays (3 x 10m4Cu K,) and the accuracy to determine the center of gravity of a broader X-ray line reducing the uncertainty to values a little lower than 10m5per data 10-3

0 di.di. 0.75 eV 0 pos. 0.67 eV

IO-4

10-s

10-6

.3 . Center

circle

Measuring

0.0012

e z

10-3

z::

10-4

0.0014

0.0016

0.0018

0 di.di. 1.19 eV 0 pos. 1.19eV

Pyrolytic graphite window Powder specimen Differential thermocouple

0.0008

Thermocouple bore

0.0010

T-shaped fixture

I

0 di.di.

0.0010

5mm

1.07 eV

0.0012

0.0014

Reciprocal temperature (K-l) Fig. 5. Arrhenius

Fig. 3. Sample.

0.0012

plots for vacancy Ag.

formation

in Al, Cu and

910

TH. HEHENKAMP

point presently. The apparatus puter-controlled and performs steps of the temperature up and (1 run N 30 h). Further details

is completely comruns at adjustable down automatically are given elsewhere

[lgl.

RESULTS

indicating that H, the formation enthalpy and S, the formation entropy are partial thermodynamic quantities. k In c, is the configurational partial entropy of mixing vacancies to the lattice. Rewriting eqn (3) leads to the usual expression for the molar fraction of vacancies c, c, = exp(S, /k)exp( - H, /kT).

FOR PURE MJZTALS

In Fig. 5, the vacancy concentrations are plotted logarithmically versus the reciprocal absolute temperature for pure aluminum, copper and silver (open circles). Vacancy formation can be followed to much lower temperatures as is possible with differential dilatometry by evaluating lifetime spectra (closed circles). Kluin [22] demonstrated that the trapping rate of vacancies for positrons can be determined. This can be done since both methods show some region of overlap. in this context one can refer also to Fig. 12. The trapping rate turns out to be almost temperature independent within experimental accuracy. The formation enthalpy of vacancies shows a clearly detectable temperature dependence for aluminium. Table 1 collects several absolute DD and positron e+ data of the literature for this metal, comparing them with our latest results 123,281. The high temperature (WT) measurements always indicate a larger effective formation enthalpy H, than the low temperature (LT) positron data. Since positron annihilation has the tendency to give slightly higher values due to systematic errors inherent in the evaluation of the data, this result for aluminum clearly indicates a curved Arrhenius plot which may be interpreted as the simultaneous fo~ation of single and divacancies. Single vacancy formation may be understood easily from minimizing the Gibbs free energy of the system with respect to the vacancies, which in thermal equilibrium is given by

p,d+O Y = H, - T(S, + k in c,),

(3)

Plotting In c, versus l/T should give a straight Arrhenius plot, the slope of which permits unambiguous determination of H,. If, however, divacancies play a significant role the Arrhenius plot should be curved and the slope would give only an effective HF”‘, since there would be an underlying second eq~librium 2c*,Ffc*, * Total vacancy concentration is given by cy = C,”+ 2C&

= clv + 12c:, expe)exp@),

with AS,, = S,, - 25,“; AH,, = 2H,v - H,, .

(3

In terms of a mono/divacancy interpretation of the results for AI the following set of data may be deduced from our own measurements [23,28]: H,, = 0.67 eV

S,, = 1.1 k

AH,, = 0.20 eV

AS,, = 0.7 k

The divacancy is, therefore, bound attractively. At the melting point, 18% of all vacancies would be divacancies. However, this is not the only possible interpretation of the data. It has been pointed out by many authors 129-351 that not so much the isobaric

Table 1. Method

Temp. range

DD DD DD DD DD e+ e+ e+ e+ e+

HT HT HT HT HT LT LT LT LT LT

S(k) 2.3(S) 2.4

1.76 3.12

(4)

K (ev) 0.15 (4) 0.77 0.76 0.71 (4) 0.81 0.69 (3) 0.71 0.66 (3) 0.66 (2) 0.67

c,(T,)

x 10-j

0.92 (8) 0.88 0.85 0.96

Ref. 23 12 7 24 15 22 25 26 27 28

911

Absolute vacancy concentrations formation but rather the isochoric one is more tem~rature-inde~udent. They arrive at an equation relating the (isobaric) vacancy concentration to the compression modulus B (cab-model) c, = exp (

C%(T)R(T) kT >’

(6)

where Q is the average atomic volume and c’, a dimensionless constant independent of temperature

AgSb

100

50

700

800

900

1000

1100

Temperature (K) Fig. 7. Normalized vacancy concentration versus temperature for several Ag-Sb alloys.

I

o = Wa,,!(T-T,,)

I 400

I

I

I

600

800

1000

T K)

Fig. 6. Al&(T - Z’,) and Aa/n,,fT - r,) as A(T); a = Ag, 1.36, d = 2.46, e = 3.10 andf= 4.00 at% Sb.

b = 0.69, c =

and pressure. Fitting this model to data available for Al gives a slope of 0.75 eV at 8.50K and of 0.71 eV at 550 K, a difference which is certainly a little smaller than the observed one of 0.75 eV against 0.47 eV but would be still acceptable within the margin of error of the measurements. However, one weak point in the present data is the formation enthalpy as observed by positron annihilation supposed to be close to the one of the monovacancy. By comparing the other results for copper and silver it turns out, that most positron measurements give 1.28 eV for Cu [36,37] and 1.1l-l. 15 eV for silver 136431 which are 0.05-0.09 eV higher than the absolute data. This could neither be understood from the mono~divacancy nor from the cEQ point of view. In Fig. 5, the Arrhenius plot for the vacancy concentration in Cu as obtained from positron annihilation is shown here with a different slope of 1.19eV and by accident, the same as for the absolute data. This is a result of the correction to the raw positron data by Kluin et al. 119,221. This correction seems to point to deviations from the two state trapping model as their origin, possibly produced by trapping prior to thermalization, a process criticized by Jensen and Walker [44]. Positrons according to their opinion do not stay long enough in apparently existing states of strong trapping at nonthermal energies. Vacancy formation enthalpies obtained from positron lifetimes seem to exhibit a systematic trend to be too high as discussed earlier 137,431, even for equivalent evaluation procedures of the same set of data. One is tempted, therefore, to look for an artifact of the evaluation by which the formation data are extracted from an observed lifetime spectrum. This question is taken up by Franz et ai. [45] using computer simulations. Although in this first investigation artifacts could be

912

TH. HEHENKAMP

found they did not explain the observed deviations for H, sufficiently. However, if one takes further realistic boundary conditions into account such as combined limited time resolution and noise, this situation may possibly change [28]. Therefore, at present it is not possible to,make realistic estimates of a divacancy contribution for Cu and Ag. The only conclusion one can safely draw from the results in Fig. 5 for all three metals is, that a divacancy contribution is certainly much smaller than most data in the literature (461. Furthermore the total number of vacancies at the melting point is quite close to the Simmons and Balluffi data for aluminum [7] but considerably higher for silver [21] and copper [19] (7.6 x 10m4 compared to 2 x 10m4 and 5.2 x IO-’ compared to 1.5 x 10v4). Since the formation enthalpies are close, this points to an increased formation entropy (3 k for Cu, 2.5 k for Ag).

RESULTS FOR ALLOYS

As shown in Fig. 6 for DD measurements in silver-antimony alloys as a function of Sb concentration, the number of vacancies at their respective melting points is increasing and the onset of vacancy formation is shifted to lower temperatures, the larger the Sb content. Vacancy formation in alloys is much more complicated than in pure base metals. In the noble metals and some transition metals, the range of interaction between a vacancy and an added impurity atom is not completely, but essentially limited to the first neighbor shell of the vacancy. The alloy

OS9

0.7

T=SOOK

._ ,a 0.5 ii < 0.3

0.1 0

0.01

0.02

0.03

0.04

0.05

%b

Fig. 8. Degree of association ai = cd/c, of i impurity atoms to a vacancy as a function of temperature for Ag-I .36 at% Sb.

0.9 ._

A@b

0.7

350

5.50

1.36 at %

750

950

1150

T W

Fig. 9.Degree of association as a function of S&-content at 800K.

vacancies can, therefore, he subdivided into complexes of vacancies without impurity atoms in that shell or with one, two, three or even more, since, due

to its short range, binding is not limited to one impurity atom alone. The number of impurity atoms bound to a vacancy is indicated by the number of the complex i (i = 0. . . 12 for fee). By minimizing the Gibbs free energy again Berces and Kovacs [47] arrive at an expression for the molar fraction of vacancies in an alloy which can be normalized to the one in the pure base metal, ct:

x exp[(i(&, - TS,,)+

err?‘>

A#?‘].

(7)

Here NA and Ns are the molar fractions of matrix A and impurity B; and i, the complex number (number of impurities at a given vacancy). H,,, the enthalpy of impurity-vacancy binding, S,, its entropy the average number of impurity-impurity neighborhoods as calculated as shown in Ref. 47 and A, the impu~ty-impu~ty interaction energy, which is usually repulsive. z is the coordination number of the lattice. Since c,,,, is a function of Hb,, S,,, and A only, these quantities may be determined from the temperature dependence of c,,,~~ as shown for Ag-Sb as an example in Fig. 7. The normalized vacancy concentration varies strongly with temperature in the usual way for association equilibria being large for low temperatures. This indicates that at low temperature higher numbers of impurity atoms will be associated with a given vacancy than at higher ones as shown in Fig. 8, plotting the degree of association versus temperature for complexes from 0 to 4 at 1.36% Sb and in Fig. 9 as a function of composition at 800 K. By fitting eqn (7) to the results, very consistent sets of data for Hb,, S,, and A may be obtained for several samples of different alloy compositions [20,21]. As a result of

Absolute vacancy concentrations

913

function of the impurity excess charge Z. I-&,, values are compared to theoretical predictions by . = por. LeClaire [49] and Dederichs [50] for OK. If,,, is always o - di.di. the lowest for an impurity of the same horizontal in the periodic chart and increases for lower or higher 4 horizontals in contrast to the predictions by LeCIaire, where they should be only dependent on excess charge. Dederichs has performed his calculations for 3 4 0 1 2 the same horizontat only as the host metal. The qualitative trend of I&,, as a function of Z is given by C,, (at. 96) all descriptions correctly. The binding itself, however, Fig. 10. Effective formation enthalpy as a function of has been substantially underestimated by LeClaire, composition. particularly in view of the fact, that the binding at is controlled rather by higher temperatures g,, = Hb, - TS,, which is still larger than If,,, by the temperature variation of c,,,, the formation around 0.05 eV at 1000 K. Therefore, it is clear, that enthalpy obtained from a (curved) Arrhenius plot is excess charge alone cannot describe binding suffino longer a constant as for monovacancies in a pure ciently. Homovalent Cu exhibits a rather large bindmetal eqn (3) but an effective quantity HF depending ing enthalpy in silver. A second contribution to the on composition and temperature, as indicated for binding could originate from misfit. However, if one Ag-Sb alloys at IOOOK as function of Sb content compares for example binding in Ag-As (no misfit) obtained from DD (di di) and positron lifetimes to Ag-Sb (misfit) the other horizontal (As) shows the (pos), indicated in Fig. IO. larger binding energy. Anyhow misfit seems to play Measurements have been performed for many a minor role if any for vacancy-impurity binding. different impurities in Cu and Ag by Kluin [19,22], Empirically the binding enthalpy correlates besides to Wolff [21], Mosig [20,23] and Butscher 1181. excess charge sensitively to a term /Am/M]“?, dm The results for & are presented in Fig. 11 as a being the difference in mass between impurity and host of mass, M. The physical reason for such a behavior is unclear at present because it does not show in entropy but rather in enthalpy. As shown in cu the lowest plot in Fig. I 1 the term, ~n,8~)A, reduces the total binding enthalpy as function of the number, (i) of impurities at a vacancy. ___----_ Finally, the knowledge of absolute vacancy con- - - LeClaire centrations from DD measurements permits, as Dederichs already mentioned, determination of positron trapping rates because of the relation AgSb

P

s

0.8

$

0.6

a0

0.4 0.2 0

Complex i Fig. I 1. Impurity-vacancy binding enthalpies as a function of excess charge in Cu- and Ag-alloys (two plots) and Gibbs free energy of binding versus complex number i (below).

where p is the trapping rate, I2 the intensity of the reciprocal lifetime component 1, = l/r2 of the vacancies and i, = j/r, which may be all determined experimentally. Figure 12 shows some g values for Ag and some Ag-Sb alloys as examples for many similar alloys. Addition of antimony reduces the trapping rate substantially particularly for the first per cent. By applying the complex model and the degrees of association as weight factors one can determine also the change in positron lifetime not only for a given alloy but also for individual complexes [21]. Complexes with small i up to 2 show almost no change in positron lifetime, for higher i, the lifetime is increasing for Ag-Sb.

914

TH. HEHENKAMP Acknowledgemenrs-This work has been continuously supported by Deutsche Forschungsgemeinschaft. I am very much indebted to to many co-workers: Prof. C. Liidecke, Dr J.-E. Kluin, Dr J. Wolff, Dr K. Mosig, G. Butscher and many others, who contributed substantially to the work. Helpful discussions with Prof. F. Faupel are gratefully acknowledged.

25 20 a-

IS

p!

IO

0 z u

_ AgSb 0.69% . .I*.

5

REFERENCES

2 2

.H a 2 g

700

800

900

25 20 I5

10 5 0 205 3 ,a t-

200

195 0

1

3

2 C,,

(at.

4

%I

Fig. 12. Trapping rates for positrons at vacancies for Ag and two Ag-Ab alloys, as a function of temperature upper part, of concentration middle. Lower part: positron lifetime as a function of Sb concentration.

SUMMARY

With a newly designed apparatus it has been possible to measure absolute vacancy concentration in pure metals and particularly in alloys up to 1150°C. Vacancy concentrations range from 5 x 1O-4 (Ag) to 30 x 10e4 (alloys) at their respective melting points. Accuracy is better than IO-* presently, with an older version of 3 x 10-j. Formation enthaipies for the pure metals can be extracted and are compared to values obtained from positron annihilation. Only for aluminum clear indications exist, that the Arrhenius plot is curved by comparing high temperature absolute data to low temperature positron measurements. By preventing selective evaporation of alloy constituents by a glassy carbon window installed in the sample, measurements for alloys have been possible, from which data for vacancy-impurity binding could be extracted. These are compared to theoretical calculations. Neither excess charge nor misfit are correlated completely to the binding energies. By comparison of absolute and positron data trapping rates could be extracted also for alloys for the first time. The complex model was found to serve as a perfect tool to quantify vacancy-impurity binding up to about 5 at.% impurity content.

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vacancy

39. Fukishima H., phD Thesis, University of Tokyo (1979). 40. Schaefer H. E.. Struck W.. Banhart F. and Bauer W., Mater. Sci. Fokm E-18, ‘117 (1987). 41. Dlubek G., Briimmer 0. and Meyendorf N., Appl. Phys. 13. 67 (1977). 42. Campbell J.’ C., Schuhe G. W. and Jackman J. A., J. Phvs. M. 1985 (1977). 43. Campbell J: C., Schultd C. W. and Gingerich R. R., J. Nuclear Mater. 69170, 609 (1978). 44. Jensen K. 0. and Walker A. B., J. Phys. Cond. Matter 2, 9757 (1990).

concentrations

915

45. Franz M, Hehenkamp Th., Kluin J.-E. and McGervey J. D., Phrs. Rev. B48, 3507 (1993). 46. Landolt-&nstein, Atomic Defects in Metals, Vol. 25, Springer Verlag Berlin (1989). 47. Berces G. and Kovacs J., Phil. Mug. A48, 883 (1983). 48. Butscher G., Diploma Thesis Giittingen (1993). 49. LeClaire A..D.,-Phil. Mug. 7, 141 (1962): 50. Dederichs P. H., Hoshino T., Drittler B., Abraham K. and Zeller R., Physica B172, 203 (1991).