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Thermodynamics of the hydrogen-nickel system
~cra metall. Vol. 32. No. 12 pp. 2233-2239. 1984 Printed in Great Britain. All rights resewed
THERMODYNAMICS
Copyright 0
OF THE HYDROGEN-NICKEL
OOOI-6160/84 $3.00 +O.OO 1984 Pergamon Press Ltd
SYSTEM
REX B. McLELLAN and P. L. SUTTER Department of Mechanical Engineering and Materials Science, William Marsh Rice University, Houston, TX 77251, U.S.A. (Received 4 Ocrober 1983) Abstract-Careful measurements of the hydrogen solubility at atmospheric pressure in nickel in equilibrium with Hz-gas have confirmed previous observations that the partial thermodynamic functions of H in Ni exhibit a strong temperature dependence. This effect, rendered visible by the large range of reciprocal temperature spanned by the data, is not due to interactions between dissolved H-atoms and lattice imperfections, but simply to the thermal excitation of the quantum energy levels of the dissolved H-atoms which may be approximated to harmonic oscillators. R&sum&-Des mesures so&&s de la solubiliti de l’hydrogkne B la pression atmosphCrique dans Ie nickel en &quilibre avec de l’hydrogtne gazeux ont confirm& des observations anttrieures selon lesquelles les fonctions thermodynamiques partielles de I’H dans le Ni varient fortement avec la tempCrature. Get &et, Hindu visible par la large gamme de I’inverse de la tempirature balayie par les expiriences, n’est pas dO aux interactions entre Ies atomes d’H dissous et les imperfections r&ticulaires, mais simplement d Texcitation thermique des niveaux d’imergie quantique des atomes d’H dissous que l’on peut assimiler % des oscillateurs harmoniques. Bassung-Genaue Messungen der WasserstollZslichkeit in Nickel, welches unter Atmosphifrendruck im Gleichgewicht mit Hz-Gas steht, haben friihere Eeobachtungen best&& nach denen die partiellen thermodynamischen Funktionen dea H im Niikel sehr teniperaturabhiingig sind. Dieser Effekt, der in den Ergebnissen wegen des sehr @en Bereiches in der rexiproken Temperatur deutlich wird, geht nicht auf die Wechselwirkungen zwischen gel&ten H-Atomen und den Gitterfehlem zurkk, sondem einfach auf die thermische Anregung der quantenmechanischen hergicnivcaus des H-Atoms, welches niiherungsweise als ein hamonischer Oszillator angesehen werden kann.
1. INTRODUCI’ION
The nickel-hydrogen system is a most interesting binary system from a thermodynamic point of view. The high diffusivity of H in Ni and the absence of polymorphism have enabled both thermodynamic and kinetic data to be obtained in a large range of reciprocal temperature [m (5-35) x IO-’ K-l]. The situation for Ni-H may be contrasted to that for the Fe-H system. In the latter case both the H-solubility and Hdiffusivity in the bee range, which also span about the same range of (l/T), show deviations from “normal” behavior which is in part due to spurious surface-related experimental effects and partly due to the nature of the H-Fe interaction and to H-trap interactions. The complicated situation has been reviewed recently [l] and “selection rules” outlined for choosing experimental data which are free of spurious surface effects. The diffusivity measurements are, as would be expected, far more sensitive to surface effects than are solubility determinations. The situation is much simpler for the “chemically cleaner” metal Ni and much of the available data are reliable. The situation has been discussed previously for solutions of H in Ni, Pt, and Co [2]. A careful choice of the Ni-H solubility data consistent with the avoidance of surface effects is given in Table 1. The last three studies listed in Table
1 were carried out using single crystals of Ni and the last two (11,121 do not give data points, but representative equations. The data of Table 1 have been plotted in the usual representation in Fig. 1 using the symbol scheme given in Table 1. The concentration unit 0, is the atom ratio of H (i.e. H/Ni). The values of 6,, in all cases, were found by direct equilibration with Hz gas. Where the gas pressure was not atmospheric, in a few cases, the solubilities have been calculated so as to refer to this pressure using Sieve&s Iaw. It can be clearly seen that the plot of In 0, vs l/T is curved and, in keeping with previous findings for Fe-Ni-H f.c.c. solutions [ 131,there is no discernable discontinuity at the Curie temperature. The lower dashed line in Fig. 1 is the least-squares fit to the data above T = lo3 K. Table 1. Hydrogen volubility in nickel Symbol T-range (K) Reference 573-1473 3 0 485-1673 4 573-l 173 5
2233
f :
973-1573 293-1293 476-1083 989-1516
67 98
Ee -_-_-_sc -I-l-l-SC
573-1423 723-1423 -
10 IO II I2
McLELLAN and SUTTER:
2234
IO”
1101111
5
I
9
11
13
15
THERMODYNAMICS OF H-Ni SYSTEM
It
19
21
23
23
21
29
31
33
33
IO'/T Fig. 1. Arrhenius plot of the solubility 0, of H in Ni at atmospheric pressure against reciprocal temperature. The upper dashed line (---) represents the equation of Jones and Pehlke [I I] and the lower dashed line (-I-I-) the sin&crystal data of Eichenauer et al. [12]. Data from this work are indicated by diamond symbols (0 and Q). Now the difftivity (D) data collection of Viilkl and Alefeld [14], which span virtually the same (l/r)-range as Fig. 1, shows that the Arrhenius plot of ln D vs V, is remarkably linear in the entire range. The purpose of the present work is two-fold. On the other hand, we wish to confirm the lowtemperature solubility data in the range below 800 K and to take special care that surface effects are obviated. Secondly we wish to explain the origin of the curvature in Fig. 1. 2. EXPERIMENTAL The basic method employed to measure the Hsolubility in Ni uses the equilibratequench-analyze technique and has been described previously in its basic features [lo, 131. In the present case some modifications were made to the technique. The Ni samples of MARZ-grade material of thickness 5 x 10m5m were annealed in vacuum (5 x IO-’ Pa) at 1373 K for 3 h and slowly cooled. Selected samples were also electrochemically polished prior to equilibration. The samples were quenched in < 1 s into refrigerated water, cleaned in alcohol and acetone and analyzed by outgassing them through a N,-cold trap into a vessel of known volume. The pressure increase during the outgassing was measured by an MKS Barytron pressure gauge connected to a chart recorder. The entire apparatus was calibrated for each measurement on a Ni-sample in two ways: Pd samples were equilibrated and analyzed along with the Ni, and NBS standards were also analyzed for hydrogen.
The experimental solubilities are given in Table 2. The surface area and equilibration times (after saturation) were varied to show that these factors did not affect the. results. Each solubility value given in Table 2 is the mean of three separate determinations made at the indicated temperature. The difference between the mean solubility and an extremum value never exceeded 6%. The new data points are plotted in Fig. 1 using diamond symbols (0). The measurements resulting from electropolished samples are denoted with diamonds with “tails” (Q). It is clear that the previous low-temperature observations are confirmed by the present measurements.
3. DISCUSSION The origin of the curvature displayed in Fig. 1 has been the object of previous speculation. Firstly it should be realized that there is no reason to expect Fig. 1 to show a linear relation between In 0, and l/T since the chemical potentials of H both in the gas phase and the solid are temperature dependent. For
Table 2. Hydrogen solubilitiu in nickel, this work Temperature (K)
Symbol
-357 381 423 526
Q
464 498 554
0
8,x 105 4.51 5.45 6.41 Il.2 7.62 9.78 12.57
McLELLAN
and SU’ITER:
THERMODYNAMICS
the majority of thermodynamic data sets the temperature range spanned is short enough so that temperature variations in the relative partial molar quantities are not revealed in Arrhenius solubility plots. The temperature dependency in the gas phase may be removed by replotting the data in the representation In (U,T’j4) vs l/T based upon the solubility equation (13). ln(B,T’14) = ln(P’/*I)
--k_r
+ $
(I)
where P is the HI-gas pressure, I, is a known (13) constant, - E,D is the dissociation energy per atom of the Hz-molecule at O”K, and Ri and’ 3; are the partial enthalpy and excess entropy of H-atoms in the solution. Equation (I) is model-independent except for the assumption that the partial configurational entropy is given by its ideal value of In Bi. A plot of In(fIiT7”) vs (l/T) is given in Fig. 2 using the symbol convention of Table 1. The nonlinearity exhibited in Fig. 2 is related to the temperature-dependence of Hi and 3;. Two broad approaches are possible. In one case we can consider the interactions between solute atoms and either themselves, lattice defects, or both. On the other hand, the curvature may be due to the intrinsic interaction between isolated H-atoms and the Ni-lattice, and thus independent of the interstitial concentration.
I.
fi
11
4
6
6
10
OF
H-Ni SYSTEM
2235
3. I. Interaction models The interaction between H-atoms in Ni and trapping sites related to mechanical discontinuities has been discussed [IO]. It was however concluded that, although such a mechanism is consistent with the solubility behavior portrayed in Fig. 2, the depth of trapping sites and/or their concentration would be unreasonably large. However, the fact that no discontinuities corresponding to trapping site interactions are observed in the diffusivity of H in Ni provides overwhelming evidence for the rejection of such ideas. The notion of H-H pairs or higher order clusters causing the observed behavior in the tcmperature variation of the solubility [IS] can also be rejected on the same grounds. It should be noted in passing that in the case of strong interactions between i-atoms and trapping sites (or other i-atoms) or at temperatures where kT <(trapping depths), the distribution of the iatoms in the available energy level in the solid phase is no longer a Boltzmann distribution even when Oi<<1. However, the factor T7j4 [cf. equation (l)] still occurs in the solubility equation because of the degrees of freedom in the gas. 3.2. Intrinsic models The ferromagnetic transformation in metals does affect the chemical potential of interstitially dissolved
a
8
0
*
a’*
12
14
16
16
20
22
24
*
6
.a
26
26
30
.I 32
34
Fig. 2. Arrhenius plot of 0,T”4 vs l/T for H in Ni. The symbol code is as in Fig. 1. The dashed lines are. calculations referred to in the text.
2236
McLELLAN
and SLITTER:
THERMODYNAMICS
solute atoms. Such behavior can be observed in respect to its manifestation regarding the diffusivity of the heavy interstitials in iron around the Curie temperature [16] and the self-diffusion @es9 tracer diffusion) behavior of iron [17].However there is little evidence for the manifestation of nonlinear temperature variations of the elastic parameters of the solvent metal in regard to H-metal solutions. Therm~yna~c me~uremen~ of H in the Fe-Ni-H system, where the Curie temperature varies over a wide range, failed to reveal any effects [13]. Belyakov and Ivanov [IS] have reported a small jog in D for H in Ni at the Curie temperature. However, this is not visible in the data compilation of Vlilkl and Alefeld 1141t. Thus it would seem reasonable to reject mechanisms involving changes in the elastic properties of the Ni matrix due to the ferromagnetic transformation or due to any other nonlinear variations with temperature in the eiastic mod&i. ’ Another possibility, for which there is ~bi~ty in the case of the Fe-H system [I], is that there exists in Ni-H solutions a temperature-dependent distribution of solute atoms between the available octahedral and tetrahedral sites. However this mechanism would lead to a nonlinear Arrhenius plot of In D vs l/T, which is not observed. (Such effects cureseen in the diffusivity of C in b.c.c. Fe.) The only positional information regarding the Ni-H system stems from the neutron scattering work of Woolan et al [19]who found octahedral occupancy for the B-phase. Thus we will assume that in the dilute solution the H-atoms are all in the octahedral interstices independent of the temperature. This is, in fact, the only assumption made in deriving equation (1). Let us write & = R, - TS; in equation (1) in the form & = R;r - TSrn + n; - Tsp’ = ~9”’ + I?; - Tsy’.
(3)
The symbol & denotes that remainder to the chemical potential of the i-atoms after subtraction of the partial molar configurational entropy -k In 0,. The superscript rn denotes the changes in J?, and S: resulting from the effect on the solvent matrix of introducing an isolated i-atom into an interstitial site, and the superscript i denotes that part of Criand 3; ascribable to the dissolved i-atom itself. We may then write equation (1) in the following form, showing the slope of the plot in Fig. 2
OF H-Ni SYSTEM
where If, as a first approximation, @ is ascribed to the elastic deformation of the matrix, it is fairly easy to show [2OJ,from the definition of the strain energy function in terms of the stress tensor [21], that for a reversible isotropic change in a body, the strain energy may be identified with the Helmholtz free energy per unit volume, and this is a linear function of p, the shear modulus, for an elastically isotropic Hookean solid. If ~1decreases linearly with temperature in the form p = p” -pT and p>“’= Ccc, where C is a constant, then
and all the curvature in Fig. 2 can be ascribed to the term (6. The elastic constants of Ni show a nonlinear temperature variation at low temperatures (T < 300 k) due to the departure from classical behavior at low tern~rat~~. This has been discussed recently in a lucid manner by Weiner [22]. The simplest way to calculate f$ is to assume that the H-atom behaves as a harmonic oscillator of frequency v so that [22] Ts>‘= $ hv f 3kT In( I- e-h’*T). (7)
&m&
In this approximation the solubility equation becomes
P”“A
*I=-
(1
_
e-hv/kT)-f
x exp
Et - /lF” - 3hv kT
.
(8)
The next question to be resolved is the appropriateness of the approximation (7) and the most suitable value of v. The “best” value of v was obtained by using a least-squares regression to fit all the data of Fig. 2 to equation (8) where the numerator of the exponential term is a constant determined in the limit as T-r 0 (lower solid line in Fig 2). The best fit occurs for (h/k) = 1204K and is indicated by the uppermost dashed line in Fig. 2. This dashed line is a good representation of the actual data. The simplest approximation to the frequency v obtainable from Q, the activation energy for diffusion is given by [23] / A \I/2 1 (9)
where d is the jump distance and m is the mass of H. Taking d as the octahedral-octahedral distance and using the value of Q given by Viilkl and Alefeld [14] (Q = 0.41 e.V = 39.42 kJ/mol), formula (9) yields the *The degree of mutual scatter in the available data (141 value v = 2.52 x 10” or (lfv/k)= 1210K. This is below 300 K is larger than that at higher temperatures. close to the vatue derived from the solubility data andWork is currently underway in the author’s laboratory equation (8). in which an electrolytic technique, using u high-stubi~ity Now Katz ef uf. (KGB) [24] have measured the electrolyte, is being used to measure D from the Curie difIusivity of H, D, and T in Ni single crystals in the point down to 250 K.
McLELLAN
and SUTfER:
THERMODYNAMICS I
OF H-Ni
SYSTEM
2237
this value of E. and the definition of the canonical partition function Q( P’,T) = f, e-&*‘&’ n-0
leads to the resultt Q(
v,
T)
=
e-u/*
eeur/* x
(1 [I
-
-e-“)-’
2xu e-“(I -e-“)-*I
(12)
where u = hv,/kT and x = hc/(2n)’
ox II 4
”
___-
0
-‘-1-1
F
ELW
KGB
14
-T
0 ELW --mm_ 12
D
w
KG6
i
104/T
Fig. 3. The diffusivity (upper part) and diffusivity ratios (lower part) of the hydrogen isotopes in nickel as a function of reciprocal temperature.
range 673-l 198 K. The diffusivity results of KGB (solid lines and symbols) are compared with those of Eichenauer et al (ELW) [12] (dashed lines, no symbols) in the upper part of Fig. 3. The lower part of Fig. 3 shows the temperature dependence of the isotopic diffusivity ratio r = D,/D,r taken from these authors. It can be clearly seen that the values of r deviate from the classical mass dependence KGB have discussed expression r = (MH/M,,)‘fl. this finding, together with the observed isotope dependence of the solubility ratios, in terms of the AK-effect {AK = ratio of the kinetic energy carried forward by the diffusing atom in the transition mode to the total kinetic energy associated with this mode) and quantum mechanical corrections to v arising both from the coarseness of the energy levels for the jumping atom with respect to kT and tunneling effects. KGB drew the conclusions that these corrections could not explain the observed thermodynamic and kinetic data for H-Ni and proposed a model involving an anharmonic potential for the bound H-atom. Taking the anharmonic potential U in terms of the displacement Ax in the form.
temperature
16 Kmv,. How KGB have shown [24] that the partition function (12) leads to good agreement between absolute rate theory predictions and the isotopic dependence of the diffusivity Patio r and the solubility ratio a = er/eyr. The latter calculations are shown in Fig. 4, in which In a is plotted against l/T. The tritium data (@) are taken from Hawkins [25] and those for deuterium from ELW (V) [ 121,Sieverts and Danz (0) [26] and Ebisuzaki et al. (0) [27]. The anharmonic potential calculations, using the fitted values Iwo/k = 920 K and x = 0.174, are shown by the upper dashed line (T) and lower dashed line (D). The degree of scatter in the data in Fig. 4 is rather large, especially for T, but the calculated a-values using the anharmonic potential are clearly consistent with the experimental data. Now in order to properly assess the plausibility of the anharmonic model, it should be used to recalculate the solubility equation for H in solid Ni since in this case the large volume of data is selfconsistent and spans a large range of (l/T). This calculation will now be performed. Using the partition function [12], the chemical potential pi is given by pi= -kTIn[Q(V,T)]’ =?(1
+x)-3kTIn{(l
-e-‘)-I
x [ 1 - Zxu e-“( 1 - e-“)-2]) =$(l
+x)+@.
following KGB, gives the energy level scheme E,=&+hv,(n
+f> 01
+ (C/16K2)(hv,)2[(n + 1,’ + f]
(11)
where K and C are constants and vi = K/4x2m is the frequency
of the unperturbed
tA clear derivation
linear oscillator.
is given in the appendix
Using
to Ref. [24].
0.6
1
0.6
1
1.0
I
1.2
L
I
I
L
f.4
16
1.8
2.0
IO'/ T -
2.2
K-’ Fig. 4. Variation of the solubility ratio u = @‘/tJp, 0: with temperature for the hydrogen isotopesin nickel. The symbols are experimental data points and the solid and dashed lines are calculated a-values.
McLELLAN
2238
and SUITER:
THERMODYNAMICS OF H-Ni SYSTEM
In equation (13) the unperturbed frequency v,, has been written simply as v for convenience and the temperature-dependent part of pj is written in the form @= -3kTIn{(l
-e+)-’ x [l -Zxue-‘(1
-e-“)-‘I}.
(14)
The complete chemical potential in the solid phase, &, is formed by adding r: to the “matrix” component ~7”’ and +kT In 6,, the partial configurational entropy contribution, This yields &=~~“+kT’lnfl,+3hv/2(1
+x)+0.
(15)
Equating & to /.~f, the chemical potential in the gas phase (IO) pf=E[+kTln
LP"2
- 2-714 ( )
It can be seen that the a-values calculated from the simple model are uniformly greater than those obtained by including the anharmonic correction, but are still not incompatible with the somewhat imprecise data. It must not be forgotten that the harmonic mode1 can not explain the diffusivity isotope effect. It is interesting to note that for the f.c.c. metals Ni, Pd, and Cu, although the classical mass dependence of the diffusivity is not observed, the frequency factors D,, are mass dependent in the classical manner [14]. This fact is a rough indication that the harmonic approach is a good approximation when the displacements are small, but more sophisticated calculations are needed to determine the properties of the saddle point configuration and the mass dependence of the activation energy for diffusion.
gives the solubility equation
r?"- [Et- 3hv/2( 1 + x)] kT xexp
-- k; . (16) ( ) This solubility equation may be evaluated by determining the constant &“’ in the limit T-0 and calculating 8, using the values of v and x given by KGB (h/k = 970 K and x = 0.174) from their data fitting work. The result of this calculation is shown by the center dashed line in Fig 2. It can be seen that 6, calculated thus is too small, and, as T increases over 1000 K has the opposite curvature with respect to the experimental data. It must be concluded that the anharmonic approach, despite its consistency with the experimental isotope effects, is inconsistent with the measured H-solubility in Ni. The obvious question which must now be posed relates to the agreement between the simple harmonic model expressed by equation (8) and the isotope effect data illustrated in Fig 4. It is a simple matter to calculate a = fly/e: by writing an analogeous equation to (8) for D and taking the ratio in the form lna=lng+ D
with obvious notation. The ratio (lH/,lD) is given by 2(i)J/2/(f)‘fl = 5.657 [I 31, H-H dissociation energy E$" and then D-D dissociation energy are taken from the compilation of Gaydon [28], (hvH/k) is 1204 as deduced from the simple harmonic model, and vD is taken simply as its classical value v”/J2. The calculated variation of a with reciprocal temperature is shown by the lower solid line in Fig. 4. An analogeous calculation using the same source for EC7 [28] yields the a-values given by the upper solid line in Fig. 4 for T.
4. CONCLUSIONS
The nonlinearity of the Arrheuius plots of the solubility 0, or 6, T7"vsreciprocal temperature for the hydrogen-nickel system is not due to the trapping of H-atoms at low-energy sites, but simply a consequence of the temperature-variation of the occupied energy states of the dissolved H-atoms acting as harmonic oscillators. This effect is undoubtedly a “normal” phenomenon for interstitial solid solutions, but may only be clearly observed when the solubility data span a large continuous range of reciprocal temperatures. The high-temperature range in Fig. 2 covering 1073-1773 K (see arrows) exhibits data whose Arrhenius plot appears linear when taken alone. This same remark is also valid in the low temperature range from 600 K downwards. The simple harmonic model predicts that, at low temperatures, the plot of In (6,T”‘) vs l/Tbecomes increasingly linear as T decreases. This is reflected in Fig. 2. The value of v, the Einstein frequency for a dissolved H-atom in nickel, is very close to that deduced classically from the known activation energy for the diffusion of hydrogen in nickel. The simple harmonic model is in accord with the somewhat imprecise experimental data for the isotopic mass effect in the solubility ratio, but can not explain the dependence of diffitsivity upon isotopic mass. Further refinements to the theory are required. It is hoped that the low-temperature diffusivity experiments currently in progress in the author’s laboratory will shed further light on this problem. Acknowledgements-The authors would like to express their gratitude for the support provided by the U.S. Department of Energy and to the Robert A. Welch Foundation. REFERENCES I. K. Kiuchi and R. B. McLellan, Acre mercrll. 31, 961 (1983).
McLELLAN
and SUTTER:
THERMODYNAMICS
2. K. Kiuchi and R. B. McLellan, J. less-common. Mew/s. In press.
_3 L. Luckemeyer-Hasse and H. Schenck, Archs Ei.senw. 6, 290 (1932). 4. A. Sieverts. Z. Mefullk. 21, 37 (1929). 5. M. H. Armbruster. J. Am. Chetn. Sot. 65, 1043 (1943). 6. J. Smittenberg, Rec. Truu. Chim. Pays-Bus. 53, 1065 (1934). 7. J. S. Blackemore, W. A. Oates and E. 0. Hall, Trans. Am. Ins/. Min. Engrs 242, 332 (1968). 8. P. Combette and P. Azou. Mem scien!. Revue Mew/l. 67, I7 (1970). 9. R. B. McLellan W. A. Oates, Acta mefull. 21. 181 (1973). IO. S. Stafford and R. B. McLellan, Acfa mefull. 22, 1463 (1974). II. F. G. Jones and R. D. Pehlke, Mefull. Truns. 2, 2655 (1971).
12. W. Eichenauer, W. Loser and H. Whitte, Z. Metal/k. 56, 287 (1965).
13. S. Stafford and R. B. McLellan, Actu metull. 24, 553 (1976).
14. J. Viilkl and G. Alefeld, D@sion in Solidr (edited by A. S. Nowick and J. J. Burton), Chap. 5, p. 231. Academic Press, New York (1975).
OF H-Ni SYSTEM
2239
IS. K. 0110 and L. A. Rosa&. Truns. Am. Insr. Min. Engr.v 242, 244 (I 968). 16. M. Wuttig. Scripta nw/tr//. 5. 33 (1971). 17. G. Hettich, H. Mchrer and K. Maier, Scrip/u me/~/l. I I, 795 (1977). 18. Yu. I. Belakov and N. 1. lonov. Sooicf Ph,r.s.. Tech. fhvs. 6. I46 ( I961 ). 19. E.-O. Woodlan. J. W. Cable and W. Koehlcr. J. P/rJ~s. C/tern. Solicf.~24. I 14I ( 1963). 20. J. D. Eshclby, So/it/ S/. /‘/I& 3. 79 (1956). 21. H. B. Callen. T/~c,n~toc!~,t~u,~rir.r. Chap. 13. Wiley, New York (1966). 22. J. H. Weiner. Slu/isficu/ Mechunics uf Ekusriciry, Chap. 10.3. Wiley. New York (1983). 23. R. B. McLellan. Truns. Am. 6x1. Min. Engrs 230, 1468 (1964). 24. L. Katz. M. Guinan and R. J. Borg. Phys. Rev. 84, 330 (1971). 25. N. J. Hawkins, Knolls Atomic Power Lab. Rept. KAPL-868, Schenectady. New York (1953). 26. Z. Sieverts and W. Danz, Z. unorg. u//g. Chem. 274, I38 (1914). 27. Y. Ebisuzaki, W. J. Kass and M. O’Keefe, Phil. Mug. 15, 1071 (1967).
28. A. G. Gaydon, Diffusion Energies and Spectra of Diatomic Molecules. Chapman & Hall, London (1968).
THERMODYNAMICS
Copyright 0
OF THE HYDROGEN-NICKEL
OOOI-6160/84 $3.00 +O.OO 1984 Pergamon Press Ltd
SYSTEM
REX B. McLELLAN and P. L. SUTTER Department of Mechanical Engineering and Materials Science, William Marsh Rice University, Houston, TX 77251, U.S.A. (Received 4 Ocrober 1983) Abstract-Careful measurements of the hydrogen solubility at atmospheric pressure in nickel in equilibrium with Hz-gas have confirmed previous observations that the partial thermodynamic functions of H in Ni exhibit a strong temperature dependence. This effect, rendered visible by the large range of reciprocal temperature spanned by the data, is not due to interactions between dissolved H-atoms and lattice imperfections, but simply to the thermal excitation of the quantum energy levels of the dissolved H-atoms which may be approximated to harmonic oscillators. R&sum&-Des mesures so&&s de la solubiliti de l’hydrogkne B la pression atmosphCrique dans Ie nickel en &quilibre avec de l’hydrogtne gazeux ont confirm& des observations anttrieures selon lesquelles les fonctions thermodynamiques partielles de I’H dans le Ni varient fortement avec la tempCrature. Get &et, Hindu visible par la large gamme de I’inverse de la tempirature balayie par les expiriences, n’est pas dO aux interactions entre Ies atomes d’H dissous et les imperfections r&ticulaires, mais simplement d Texcitation thermique des niveaux d’imergie quantique des atomes d’H dissous que l’on peut assimiler % des oscillateurs harmoniques. Bassung-Genaue Messungen der WasserstollZslichkeit in Nickel, welches unter Atmosphifrendruck im Gleichgewicht mit Hz-Gas steht, haben friihere Eeobachtungen best&& nach denen die partiellen thermodynamischen Funktionen dea H im Niikel sehr teniperaturabhiingig sind. Dieser Effekt, der in den Ergebnissen wegen des sehr @en Bereiches in der rexiproken Temperatur deutlich wird, geht nicht auf die Wechselwirkungen zwischen gel&ten H-Atomen und den Gitterfehlem zurkk, sondem einfach auf die thermische Anregung der quantenmechanischen hergicnivcaus des H-Atoms, welches niiherungsweise als ein hamonischer Oszillator angesehen werden kann.
1. INTRODUCI’ION
The nickel-hydrogen system is a most interesting binary system from a thermodynamic point of view. The high diffusivity of H in Ni and the absence of polymorphism have enabled both thermodynamic and kinetic data to be obtained in a large range of reciprocal temperature [m (5-35) x IO-’ K-l]. The situation for Ni-H may be contrasted to that for the Fe-H system. In the latter case both the H-solubility and Hdiffusivity in the bee range, which also span about the same range of (l/T), show deviations from “normal” behavior which is in part due to spurious surface-related experimental effects and partly due to the nature of the H-Fe interaction and to H-trap interactions. The complicated situation has been reviewed recently [l] and “selection rules” outlined for choosing experimental data which are free of spurious surface effects. The diffusivity measurements are, as would be expected, far more sensitive to surface effects than are solubility determinations. The situation is much simpler for the “chemically cleaner” metal Ni and much of the available data are reliable. The situation has been discussed previously for solutions of H in Ni, Pt, and Co [2]. A careful choice of the Ni-H solubility data consistent with the avoidance of surface effects is given in Table 1. The last three studies listed in Table
1 were carried out using single crystals of Ni and the last two (11,121 do not give data points, but representative equations. The data of Table 1 have been plotted in the usual representation in Fig. 1 using the symbol scheme given in Table 1. The concentration unit 0, is the atom ratio of H (i.e. H/Ni). The values of 6,, in all cases, were found by direct equilibration with Hz gas. Where the gas pressure was not atmospheric, in a few cases, the solubilities have been calculated so as to refer to this pressure using Sieve&s Iaw. It can be clearly seen that the plot of In 0, vs l/T is curved and, in keeping with previous findings for Fe-Ni-H f.c.c. solutions [ 131,there is no discernable discontinuity at the Curie temperature. The lower dashed line in Fig. 1 is the least-squares fit to the data above T = lo3 K. Table 1. Hydrogen volubility in nickel Symbol T-range (K) Reference 573-1473 3 0 485-1673 4 573-l 173 5
2233
f :
973-1573 293-1293 476-1083 989-1516
67 98
Ee -_-_-_sc -I-l-l-SC
573-1423 723-1423 -
10 IO II I2
McLELLAN and SUTTER:
2234
IO”
1101111
5
I
9
11
13
15
THERMODYNAMICS OF H-Ni SYSTEM
It
19
21
23
23
21
29
31
33
33
IO'/T Fig. 1. Arrhenius plot of the solubility 0, of H in Ni at atmospheric pressure against reciprocal temperature. The upper dashed line (---) represents the equation of Jones and Pehlke [I I] and the lower dashed line (-I-I-) the sin&crystal data of Eichenauer et al. [12]. Data from this work are indicated by diamond symbols (0 and Q). Now the difftivity (D) data collection of Viilkl and Alefeld [14], which span virtually the same (l/r)-range as Fig. 1, shows that the Arrhenius plot of ln D vs V, is remarkably linear in the entire range. The purpose of the present work is two-fold. On the other hand, we wish to confirm the lowtemperature solubility data in the range below 800 K and to take special care that surface effects are obviated. Secondly we wish to explain the origin of the curvature in Fig. 1. 2. EXPERIMENTAL The basic method employed to measure the Hsolubility in Ni uses the equilibratequench-analyze technique and has been described previously in its basic features [lo, 131. In the present case some modifications were made to the technique. The Ni samples of MARZ-grade material of thickness 5 x 10m5m were annealed in vacuum (5 x IO-’ Pa) at 1373 K for 3 h and slowly cooled. Selected samples were also electrochemically polished prior to equilibration. The samples were quenched in < 1 s into refrigerated water, cleaned in alcohol and acetone and analyzed by outgassing them through a N,-cold trap into a vessel of known volume. The pressure increase during the outgassing was measured by an MKS Barytron pressure gauge connected to a chart recorder. The entire apparatus was calibrated for each measurement on a Ni-sample in two ways: Pd samples were equilibrated and analyzed along with the Ni, and NBS standards were also analyzed for hydrogen.
The experimental solubilities are given in Table 2. The surface area and equilibration times (after saturation) were varied to show that these factors did not affect the. results. Each solubility value given in Table 2 is the mean of three separate determinations made at the indicated temperature. The difference between the mean solubility and an extremum value never exceeded 6%. The new data points are plotted in Fig. 1 using diamond symbols (0). The measurements resulting from electropolished samples are denoted with diamonds with “tails” (Q). It is clear that the previous low-temperature observations are confirmed by the present measurements.
3. DISCUSSION The origin of the curvature displayed in Fig. 1 has been the object of previous speculation. Firstly it should be realized that there is no reason to expect Fig. 1 to show a linear relation between In 0, and l/T since the chemical potentials of H both in the gas phase and the solid are temperature dependent. For
Table 2. Hydrogen solubilitiu in nickel, this work Temperature (K)
Symbol
-357 381 423 526
Q
464 498 554
0
8,x 105 4.51 5.45 6.41 Il.2 7.62 9.78 12.57
McLELLAN
and SU’ITER:
THERMODYNAMICS
the majority of thermodynamic data sets the temperature range spanned is short enough so that temperature variations in the relative partial molar quantities are not revealed in Arrhenius solubility plots. The temperature dependency in the gas phase may be removed by replotting the data in the representation In (U,T’j4) vs l/T based upon the solubility equation (13). ln(B,T’14) = ln(P’/*I)
--k_r
+ $
(I)
where P is the HI-gas pressure, I, is a known (13) constant, - E,D is the dissociation energy per atom of the Hz-molecule at O”K, and Ri and’ 3; are the partial enthalpy and excess entropy of H-atoms in the solution. Equation (I) is model-independent except for the assumption that the partial configurational entropy is given by its ideal value of In Bi. A plot of In(fIiT7”) vs (l/T) is given in Fig. 2 using the symbol convention of Table 1. The nonlinearity exhibited in Fig. 2 is related to the temperature-dependence of Hi and 3;. Two broad approaches are possible. In one case we can consider the interactions between solute atoms and either themselves, lattice defects, or both. On the other hand, the curvature may be due to the intrinsic interaction between isolated H-atoms and the Ni-lattice, and thus independent of the interstitial concentration.
I.
fi
11
4
6
6
10
OF
H-Ni SYSTEM
2235
3. I. Interaction models The interaction between H-atoms in Ni and trapping sites related to mechanical discontinuities has been discussed [IO]. It was however concluded that, although such a mechanism is consistent with the solubility behavior portrayed in Fig. 2, the depth of trapping sites and/or their concentration would be unreasonably large. However, the fact that no discontinuities corresponding to trapping site interactions are observed in the diffusivity of H in Ni provides overwhelming evidence for the rejection of such ideas. The notion of H-H pairs or higher order clusters causing the observed behavior in the tcmperature variation of the solubility [IS] can also be rejected on the same grounds. It should be noted in passing that in the case of strong interactions between i-atoms and trapping sites (or other i-atoms) or at temperatures where kT <(trapping depths), the distribution of the iatoms in the available energy level in the solid phase is no longer a Boltzmann distribution even when Oi<<1. However, the factor T7j4 [cf. equation (l)] still occurs in the solubility equation because of the degrees of freedom in the gas. 3.2. Intrinsic models The ferromagnetic transformation in metals does affect the chemical potential of interstitially dissolved
a
8
0
*
a’*
12
14
16
16
20
22
24
*
6
.a
26
26
30
.I 32
34
Fig. 2. Arrhenius plot of 0,T”4 vs l/T for H in Ni. The symbol code is as in Fig. 1. The dashed lines are. calculations referred to in the text.
2236
McLELLAN
and SLITTER:
THERMODYNAMICS
solute atoms. Such behavior can be observed in respect to its manifestation regarding the diffusivity of the heavy interstitials in iron around the Curie temperature [16] and the self-diffusion @es9 tracer diffusion) behavior of iron [17].However there is little evidence for the manifestation of nonlinear temperature variations of the elastic parameters of the solvent metal in regard to H-metal solutions. Therm~yna~c me~uremen~ of H in the Fe-Ni-H system, where the Curie temperature varies over a wide range, failed to reveal any effects [13]. Belyakov and Ivanov [IS] have reported a small jog in D for H in Ni at the Curie temperature. However, this is not visible in the data compilation of Vlilkl and Alefeld 1141t. Thus it would seem reasonable to reject mechanisms involving changes in the elastic properties of the Ni matrix due to the ferromagnetic transformation or due to any other nonlinear variations with temperature in the eiastic mod&i. ’ Another possibility, for which there is ~bi~ty in the case of the Fe-H system [I], is that there exists in Ni-H solutions a temperature-dependent distribution of solute atoms between the available octahedral and tetrahedral sites. However this mechanism would lead to a nonlinear Arrhenius plot of In D vs l/T, which is not observed. (Such effects cureseen in the diffusivity of C in b.c.c. Fe.) The only positional information regarding the Ni-H system stems from the neutron scattering work of Woolan et al [19]who found octahedral occupancy for the B-phase. Thus we will assume that in the dilute solution the H-atoms are all in the octahedral interstices independent of the temperature. This is, in fact, the only assumption made in deriving equation (1). Let us write & = R, - TS; in equation (1) in the form & = R;r - TSrn + n; - Tsp’ = ~9”’ + I?; - Tsy’.
(3)
The symbol & denotes that remainder to the chemical potential of the i-atoms after subtraction of the partial molar configurational entropy -k In 0,. The superscript rn denotes the changes in J?, and S: resulting from the effect on the solvent matrix of introducing an isolated i-atom into an interstitial site, and the superscript i denotes that part of Criand 3; ascribable to the dissolved i-atom itself. We may then write equation (1) in the following form, showing the slope of the plot in Fig. 2
OF H-Ni SYSTEM
where If, as a first approximation, @ is ascribed to the elastic deformation of the matrix, it is fairly easy to show [2OJ,from the definition of the strain energy function in terms of the stress tensor [21], that for a reversible isotropic change in a body, the strain energy may be identified with the Helmholtz free energy per unit volume, and this is a linear function of p, the shear modulus, for an elastically isotropic Hookean solid. If ~1decreases linearly with temperature in the form p = p” -pT and p>“’= Ccc, where C is a constant, then
and all the curvature in Fig. 2 can be ascribed to the term (6. The elastic constants of Ni show a nonlinear temperature variation at low temperatures (T < 300 k) due to the departure from classical behavior at low tern~rat~~. This has been discussed recently in a lucid manner by Weiner [22]. The simplest way to calculate f$ is to assume that the H-atom behaves as a harmonic oscillator of frequency v so that [22] Ts>‘= $ hv f 3kT In( I- e-h’*T). (7)
&m&
In this approximation the solubility equation becomes
P”“A
*I=-
(1
_
e-hv/kT)-f
x exp
Et - /lF” - 3hv kT
.
(8)
The next question to be resolved is the appropriateness of the approximation (7) and the most suitable value of v. The “best” value of v was obtained by using a least-squares regression to fit all the data of Fig. 2 to equation (8) where the numerator of the exponential term is a constant determined in the limit as T-r 0 (lower solid line in Fig 2). The best fit occurs for (h/k) = 1204K and is indicated by the uppermost dashed line in Fig. 2. This dashed line is a good representation of the actual data. The simplest approximation to the frequency v obtainable from Q, the activation energy for diffusion is given by [23] / A \I/2 1 (9)
where d is the jump distance and m is the mass of H. Taking d as the octahedral-octahedral distance and using the value of Q given by Viilkl and Alefeld [14] (Q = 0.41 e.V = 39.42 kJ/mol), formula (9) yields the *The degree of mutual scatter in the available data (141 value v = 2.52 x 10” or (lfv/k)= 1210K. This is below 300 K is larger than that at higher temperatures. close to the vatue derived from the solubility data andWork is currently underway in the author’s laboratory equation (8). in which an electrolytic technique, using u high-stubi~ity Now Katz ef uf. (KGB) [24] have measured the electrolyte, is being used to measure D from the Curie difIusivity of H, D, and T in Ni single crystals in the point down to 250 K.
McLELLAN
and SUTfER:
THERMODYNAMICS I
OF H-Ni
SYSTEM
2237
this value of E. and the definition of the canonical partition function Q( P’,T) = f, e-&*‘&’ n-0
leads to the resultt Q(
v,
T)
=
e-u/*
eeur/* x
(1 [I
-
-e-“)-’
2xu e-“(I -e-“)-*I
(12)
where u = hv,/kT and x = hc/(2n)’
ox II 4
”
___-
0
-‘-1-1
F
ELW
KGB
14
-T
0 ELW --mm_ 12
D
w
KG6
i
104/T
Fig. 3. The diffusivity (upper part) and diffusivity ratios (lower part) of the hydrogen isotopes in nickel as a function of reciprocal temperature.
range 673-l 198 K. The diffusivity results of KGB (solid lines and symbols) are compared with those of Eichenauer et al (ELW) [12] (dashed lines, no symbols) in the upper part of Fig. 3. The lower part of Fig. 3 shows the temperature dependence of the isotopic diffusivity ratio r = D,/D,r taken from these authors. It can be clearly seen that the values of r deviate from the classical mass dependence KGB have discussed expression r = (MH/M,,)‘fl. this finding, together with the observed isotope dependence of the solubility ratios, in terms of the AK-effect {AK = ratio of the kinetic energy carried forward by the diffusing atom in the transition mode to the total kinetic energy associated with this mode) and quantum mechanical corrections to v arising both from the coarseness of the energy levels for the jumping atom with respect to kT and tunneling effects. KGB drew the conclusions that these corrections could not explain the observed thermodynamic and kinetic data for H-Ni and proposed a model involving an anharmonic potential for the bound H-atom. Taking the anharmonic potential U in terms of the displacement Ax in the form.
temperature
16 Kmv,. How KGB have shown [24] that the partition function (12) leads to good agreement between absolute rate theory predictions and the isotopic dependence of the diffusivity Patio r and the solubility ratio a = er/eyr. The latter calculations are shown in Fig. 4, in which In a is plotted against l/T. The tritium data (@) are taken from Hawkins [25] and those for deuterium from ELW (V) [ 121,Sieverts and Danz (0) [26] and Ebisuzaki et al. (0) [27]. The anharmonic potential calculations, using the fitted values Iwo/k = 920 K and x = 0.174, are shown by the upper dashed line (T) and lower dashed line (D). The degree of scatter in the data in Fig. 4 is rather large, especially for T, but the calculated a-values using the anharmonic potential are clearly consistent with the experimental data. Now in order to properly assess the plausibility of the anharmonic model, it should be used to recalculate the solubility equation for H in solid Ni since in this case the large volume of data is selfconsistent and spans a large range of (l/T). This calculation will now be performed. Using the partition function [12], the chemical potential pi is given by pi= -kTIn[Q(V,T)]’ =?(1
+x)-3kTIn{(l
-e-‘)-I
x [ 1 - Zxu e-“( 1 - e-“)-2]) =$(l
+x)+@.
following KGB, gives the energy level scheme E,=&+hv,(n
+f> 01
+ (C/16K2)(hv,)2[(n + 1,’ + f]
(11)
where K and C are constants and vi = K/4x2m is the frequency
of the unperturbed
tA clear derivation
linear oscillator.
is given in the appendix
Using
to Ref. [24].
0.6
1
0.6
1
1.0
I
1.2
L
I
I
L
f.4
16
1.8
2.0
IO'/ T -
2.2
K-’ Fig. 4. Variation of the solubility ratio u = @‘/tJp, 0: with temperature for the hydrogen isotopesin nickel. The symbols are experimental data points and the solid and dashed lines are calculated a-values.
McLELLAN
2238
and SUITER:
THERMODYNAMICS OF H-Ni SYSTEM
In equation (13) the unperturbed frequency v,, has been written simply as v for convenience and the temperature-dependent part of pj is written in the form @= -3kTIn{(l
-e+)-’ x [l -Zxue-‘(1
-e-“)-‘I}.
(14)
The complete chemical potential in the solid phase, &, is formed by adding r: to the “matrix” component ~7”’ and +kT In 6,, the partial configurational entropy contribution, This yields &=~~“+kT’lnfl,+3hv/2(1
+x)+0.
(15)
Equating & to /.~f, the chemical potential in the gas phase (IO) pf=E[+kTln
LP"2
- 2-714 ( )
It can be seen that the a-values calculated from the simple model are uniformly greater than those obtained by including the anharmonic correction, but are still not incompatible with the somewhat imprecise data. It must not be forgotten that the harmonic mode1 can not explain the diffusivity isotope effect. It is interesting to note that for the f.c.c. metals Ni, Pd, and Cu, although the classical mass dependence of the diffusivity is not observed, the frequency factors D,, are mass dependent in the classical manner [14]. This fact is a rough indication that the harmonic approach is a good approximation when the displacements are small, but more sophisticated calculations are needed to determine the properties of the saddle point configuration and the mass dependence of the activation energy for diffusion.
gives the solubility equation
r?"- [Et- 3hv/2( 1 + x)] kT xexp
-- k; . (16) ( ) This solubility equation may be evaluated by determining the constant &“’ in the limit T-0 and calculating 8, using the values of v and x given by KGB (h/k = 970 K and x = 0.174) from their data fitting work. The result of this calculation is shown by the center dashed line in Fig 2. It can be seen that 6, calculated thus is too small, and, as T increases over 1000 K has the opposite curvature with respect to the experimental data. It must be concluded that the anharmonic approach, despite its consistency with the experimental isotope effects, is inconsistent with the measured H-solubility in Ni. The obvious question which must now be posed relates to the agreement between the simple harmonic model expressed by equation (8) and the isotope effect data illustrated in Fig 4. It is a simple matter to calculate a = fly/e: by writing an analogeous equation to (8) for D and taking the ratio in the form lna=lng+ D
with obvious notation. The ratio (lH/,lD) is given by 2(i)J/2/(f)‘fl = 5.657 [I 31, H-H dissociation energy E$" and then D-D dissociation energy are taken from the compilation of Gaydon [28], (hvH/k) is 1204 as deduced from the simple harmonic model, and vD is taken simply as its classical value v”/J2. The calculated variation of a with reciprocal temperature is shown by the lower solid line in Fig. 4. An analogeous calculation using the same source for EC7 [28] yields the a-values given by the upper solid line in Fig. 4 for T.
4. CONCLUSIONS
The nonlinearity of the Arrheuius plots of the solubility 0, or 6, T7"vsreciprocal temperature for the hydrogen-nickel system is not due to the trapping of H-atoms at low-energy sites, but simply a consequence of the temperature-variation of the occupied energy states of the dissolved H-atoms acting as harmonic oscillators. This effect is undoubtedly a “normal” phenomenon for interstitial solid solutions, but may only be clearly observed when the solubility data span a large continuous range of reciprocal temperatures. The high-temperature range in Fig. 2 covering 1073-1773 K (see arrows) exhibits data whose Arrhenius plot appears linear when taken alone. This same remark is also valid in the low temperature range from 600 K downwards. The simple harmonic model predicts that, at low temperatures, the plot of In (6,T”‘) vs l/Tbecomes increasingly linear as T decreases. This is reflected in Fig. 2. The value of v, the Einstein frequency for a dissolved H-atom in nickel, is very close to that deduced classically from the known activation energy for the diffusion of hydrogen in nickel. The simple harmonic model is in accord with the somewhat imprecise experimental data for the isotopic mass effect in the solubility ratio, but can not explain the dependence of diffitsivity upon isotopic mass. Further refinements to the theory are required. It is hoped that the low-temperature diffusivity experiments currently in progress in the author’s laboratory will shed further light on this problem. Acknowledgements-The authors would like to express their gratitude for the support provided by the U.S. Department of Energy and to the Robert A. Welch Foundation. REFERENCES I. K. Kiuchi and R. B. McLellan, Acre mercrll. 31, 961 (1983).
McLELLAN
and SUTTER:
THERMODYNAMICS
2. K. Kiuchi and R. B. McLellan, J. less-common. Mew/s. In press.
_3 L. Luckemeyer-Hasse and H. Schenck, Archs Ei.senw. 6, 290 (1932). 4. A. Sieverts. Z. Mefullk. 21, 37 (1929). 5. M. H. Armbruster. J. Am. Chetn. Sot. 65, 1043 (1943). 6. J. Smittenberg, Rec. Truu. Chim. Pays-Bus. 53, 1065 (1934). 7. J. S. Blackemore, W. A. Oates and E. 0. Hall, Trans. Am. Ins/. Min. Engrs 242, 332 (1968). 8. P. Combette and P. Azou. Mem scien!. Revue Mew/l. 67, I7 (1970). 9. R. B. McLellan W. A. Oates, Acta mefull. 21. 181 (1973). IO. S. Stafford and R. B. McLellan, Acfa mefull. 22, 1463 (1974). II. F. G. Jones and R. D. Pehlke, Mefull. Truns. 2, 2655 (1971).
12. W. Eichenauer, W. Loser and H. Whitte, Z. Metal/k. 56, 287 (1965).
13. S. Stafford and R. B. McLellan, Actu metull. 24, 553 (1976).
14. J. Viilkl and G. Alefeld, D@sion in Solidr (edited by A. S. Nowick and J. J. Burton), Chap. 5, p. 231. Academic Press, New York (1975).
OF H-Ni SYSTEM
2239
IS. K. 0110 and L. A. Rosa&. Truns. Am. Insr. Min. Engr.v 242, 244 (I 968). 16. M. Wuttig. Scripta nw/tr//. 5. 33 (1971). 17. G. Hettich, H. Mchrer and K. Maier, Scrip/u me/~/l. I I, 795 (1977). 18. Yu. I. Belakov and N. 1. lonov. Sooicf Ph,r.s.. Tech. fhvs. 6. I46 ( I961 ). 19. E.-O. Woodlan. J. W. Cable and W. Koehlcr. J. P/rJ~s. C/tern. Solicf.~24. I 14I ( 1963). 20. J. D. Eshclby, So/it/ S/. /‘/I& 3. 79 (1956). 21. H. B. Callen. T/~c,n~toc!~,t~u,~rir.r. Chap. 13. Wiley, New York (1966). 22. J. H. Weiner. Slu/isficu/ Mechunics uf Ekusriciry, Chap. 10.3. Wiley. New York (1983). 23. R. B. McLellan. Truns. Am. 6x1. Min. Engrs 230, 1468 (1964). 24. L. Katz. M. Guinan and R. J. Borg. Phys. Rev. 84, 330 (1971). 25. N. J. Hawkins, Knolls Atomic Power Lab. Rept. KAPL-868, Schenectady. New York (1953). 26. Z. Sieverts and W. Danz, Z. unorg. u//g. Chem. 274, I38 (1914). 27. Y. Ebisuzaki, W. J. Kass and M. O’Keefe, Phil. Mug. 15, 1071 (1967).
28. A. G. Gaydon, Diffusion Energies and Spectra of Diatomic Molecules. Chapman & Hall, London (1968).