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component to electronegative component within the matrix. However, the
correct value of electronic charge associated with any solute is difficult to
establish in most cases. Therefore, it is difficult to account properly for this
effect. All the same, Eq. (35) remains valid for each single-alloy system.
Equation (35) can be extended to solute diffusion as well.
[57]
2.2.7 Summary of Empirical Correlations
To summarize Section 2.2, the correlations between the diffusion and
melting parameters give the best results compared to any other relation-
ship involving diffusion parameters and bulk properties. In this way, these
correlations are unique for three reasons:
1. Simplicity
2. Different possible ways in which the two phenomena corre-
late with each other
3. Their universal character in the sense that their validity is not
restricted by the crystal type and nature of cohesive forces.
92 DIFFUSION PROCESSES IN ADVANCED TECHNOLOGICAL MATERIALS
Figure 2.15 ∆H
*
a
2
versus (e/A )
53
plots for silver-base systems: Ag-Ge, Ag-Cd
and Ag-In, Ag-Sb, and Ag-Tl alloys.
Notes: 1. The diffusion data for Ag-Ge alloys were obtained from R. E. Hoffman, D. Turnbull, and E. W.
Hart, Acta Metall., 3:417 (1955).
2. The diffusion data for Ag-Cd and Ag-In alloys were obtained from A. Schoen, Ph. D. thesis,
University of Illinois (1958).
3. The diffusion data for Ag-Sb alloys were obtained from E. Sonder, Phys. Rev
., 100:1662 (1955).
4. The diffusion data for Ag-Tl alloys were obtained from R. E. Hoffman, Acta Metall., 6:95 (1958).
Ch_02.qxd 11/30/04 8:45 AM Page 92
SOLID STATE DIFFUSION AND BULK PROPERTIES, TIWARI ET AL.93
The correlations with the elastic constants and the bulk properties
have somewhat restricted applicability, mainly because the nature of elas-
tic deformation associated with the saddle-point configuration is not
known with any degree of certainty. The success of Zener’s hypothesis in
metallic systems is due to the fact that the temperature dependence of
Young’s modulus and other elastic constants is nearly the same. However,
the applications of Zener’s hypothesis have so far been restricted mainly
to the metals. Lack of appropriate data in other types of solids may be
responsible for this situation.
The correlations with Debye temperature, valence bond parameter
and electron-to-atom ratio should be viewed as first-order effects such that
no violations of the proposed relationships have been observed. Areason-
able degree of scatter in data points in these cases is not unexpected when
we consider matrices with different crystal structure and the subtle varia-
tions in the nature of cohesion. The equations given in this section to esti-
mate the values of ∆H
*
and ∆S
*
can be used to obtain the magnitudes of
the diffusion parameters from the knowledge of bulk properties. Their
substitution in Eq. (1) will give the diffusion coefficient.
Finally, it should be emphasized that because the correlations dis-
cussed in this section are generally empirical or semi-empirical in nature,
their success should be judged more by the systematics rather than by the
numbers only. From fundamental considerations, it is more important to
consider the diverse nature of solids for which any particular relationship
is valid. Further, any case of nonconformability within a group should be
closely examined to ascertain the reasons for its deviation from the pat-
tern followed by others within the same group.
In this context, two examples are relevant. When Wert and Zener
[2, 29]
first proposed their correlations between D
o
and ∆H
*
T
m
for interstitial
diffusion in BCC metals, some of the results deviated from the predicted
behavior. Subsequently, these data were found to be spurious, and the
experiments on the well-characterized material yielded the expected
agreement.
[1]
Much later, Barr and Dawson
[58]
showed that the deviations
in the activation energy for cationic diffusion in alkaline earth oxides from
the value expected from the melting point correlations were due to the
extrinsic factors arising from the presence of impurities in the matrix.
2.3 Equilibrium Thermodynamic Calculation of
Diffusion Parameters
In this section, we derive an equation relating the activation volume
∆V
*
with the activation entropy ∆S
*
. The derivation of this equation is
based on equilibrium thermodynamics.
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2.3.1 The Activation Volume
Vaidya showed
[59]
that the logarithm of the diffusion coefficient
exhibits a regular variation with the specific volume of the matrix. Vaidya
noted that, despite the discontinuous change in the volume upon melting,
the extrapolated value of the diffusion coefficient agrees satisfactorily
with the experimental value for the self-diffusion in liquid state at the
melting point. Earlier, a Taylor series expansion of ∆G
*
as a function of
temperature was used by Vineyard and Dienes
[60]
to analyze the effect of
thermal expansion on the formation energies of vacancies and interstitials.
These authors assumed that the crystal vibrational frequencies as well as
its potential energies depend on crystal volume alone and on no other ther-
modynamic variable. This view is supported by Borelius,
[61]
who showed
that the vibrational entropies of liquids are the same as those of solids, and
the difference in the entropies of solid and liquid arises mainly from the
changes in the volume. It has also been shown that the elastic con-
stants
[62, 63]
and the compressibility
[64]
vary in a regular way with specific
volume. Mott and Gurney
[65]
used a similar approach to estimate the tem-
perature variation of activation energy for self-diffusion in ionic crystals.
Following the same approach, we expand here the free energy of acti-
vation for self-diffusion as a function of specific volume in a Taylor
series.
[66]
A prime requirement of Taylor series expansion is the fixing of
origin. The value of the variable parameter at the origin constitutes the
first term of the expansion. We choose T 0 K as the origin, and Eq. (2.2)
gives the value of ∆G
*
at origin as ∆H
*
. Hence, ∆H
*
becomes the first term
in the expansion. Therefore, the expansion of ∆G
*
in a Taylor series as a
function of specific volume (V) yields:
∆G
*
∆H
*
[∂(∆G
*
)∂V](∆V) [∂
2
(∆G
*
)∂V
2
](∆V)
2
2! ......... (36)
Equation (36) is a general equation for ∆G
*
as a function of V. It is
implicitly assumed in writing this equation that ∆G
*
is completely defined by
change in the magnitude of the specific volume, which can be brought about
by changing temperature and pressure, either individually or collectively. It
is also implied that when we try to evaluate the derivative of ∆G
*
with
respect to specific volume, it is of little consequence which one of the two
parameters, T or P, is held constant. At any particular value of V, ∆G
*
has a
fixed value. Accordingly, the slope of the plot of ∆G
*
with respect to Vat any
given point is controlled only by the magnitude of V (see Appendix 2A).
In evaluating the first derivative of ∆G
*
in Eq. (36), it would therefore
be proper to use the basic equation, which defines the activation volume as:
∆V
*
[∂(∆G
*
)∂P]
T
[∂(∆G
*
)∂V]
T
bV. (37)
94 DIFFUSION PROCESSES IN ADVANCED TECHNOLOGICAL MATERIALS
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SOLID STATE DIFFUSION AND BULK PROPERTIES, TIWARI ET AL.95
As before, b represents compressibility. Ignoring second and higher order
terms in Eq. (36) and combining it with Eq. (37), we get:
∆G
*
∆H
*
(∆V
*
b)(∆VV). (38)
The specific volume of the matrix at any temperature is related to that at
0 K, V
0
, by
V V
0
(1 aT), (39)
where a is the volumetric coefficient of thermal expansion. Furthermore,
∆V V V
0
V
o
aT. (40)
From Eqs. (39) and (40), we have:
∆VV (aT)(1 aT). (41)
Substituting for ∆VV from Eq. (41) in Eq. (38), we obtain:
∆G
*
∆H
*
(∆V
*
b)(aT)(1 aT). (42)
Comparison of Eq. (42) with Eq. (2.2) gives the desired relationship:
∆V
*
∆S
*
(ab)
1
(1 aT). (43)
The thermodynamic principle used in the derivation of Eq. (43) is
contained in Eq. (37). Equation (43) represents a formal relationship
between the diffusion parameters and bulk properties.
The calculation of activation volume with the help of Eq. (43)
involves the parameter ab, which is given by:
(∂S∂V)
T
ab. (44)
Here, S denotes the entropy of the matrix. Thus the magnitude of ab
controls the variation of entropy with the specific volume of the matrix.
This, in turn, is dependent on the nature of chemical bonding between the
atoms. Using ab values of the matrix implies that the variation of
entropy with specific volume follows identical behavior for the matrix as
well as for the activated complex. This is a reasonable assumption because
in both situations, the chemical species involved are identical and there-
fore the nature of bonding remains unaltered.
Ch_02.qxd 11/30/04 8:45 AM Page 95
Table 2.3 compares activation volumes estimated from Eq. (43) with
the experimental values. Data for the activation volume for self-diffusion
as a function of temperature are available only for zinc,
[67]
cadmium,
[68]
and sodium.
[69]
The differences between the experimental and the calcu-
lated values range from 7.5 to 39.1%. Two opposite trends are seen in
Table 2.3. For zinc and cadmium, the agreement improves with the tem-
perature, while the opposite is true for sodium. With sodium, the meas-
ured compressibility values are affected by the extreme plasticity of the
matrix at high temperature. Bridgman
[70]
pointed out that the plasticity of
metals like sodium at T 0.5 T
m
introduces error in the compressibility
measurements. Therefore, the agreement is poorer at higher temperatures.
With zinc and cadmium, the available compressibility data have been
obtained on polycrystalline specimens, whereas the diffusion experiments
were performed on single crystals. The presence of grain boundaries will
lower the compressibility. The difference between the compressibility of
the matrix and grain boundary will decrease with the increase of temper-
ature. In accordance with this suggestion, the agreement between calcu-
lated and experimental values improves with the increase in temperature.
It is concluded that discrepancies in the estimated value of ∆V
*
arise
mainly due to the uncertainties in the values of parameters used in its
calculation.
The relationship given in Eq. (43) is otherwise perfectly valid as it
does not contain any empirical or semi-empirical constant. Therefore, as
96 DIFFUSION PROCESSES IN ADVANCED TECHNOLOGICAL MATERIALS
Table 2.3. Comparison of Experimental and Calculated Values of the
Activation Volume for Self-Diffusion in Zinc, Cadmium and Sodium
a
Activation Volume (10
6
m
3
mol
1
)
Serial No. Matrix Temp (K) Experimental Calculated Differences (%)
1Zn573 4.28 3.20 25.2
2 623 3.97 3.22 18.9
3 673 3.72 3.44 7.5
4Cd524 6.90 4.84 29.8
5 549 7.14 5.08 28.8
6 574 7.65 5.49 28.2
7 592 7.49 5.87 21.6
8Na287.8 11.46 10.23 10.7
9 364.3 12.40 17.25 39.1
a. G. P. Tiwari, Scripta Mater., 39:991 (1998)
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SOLID STATE DIFFUSION AND BULK PROPERTIES, TIWARI ET AL.97
shown in the following section, it can be used to obtain estimates of other
diffusion parameters.
2.3.2 Activation Entropy and Diffusion Frequency
Temperature variation of diffusion coefficients usually follows an
Arrhenius relationship and is written as:
D D
0
exp(∆H
*
RT). (45)
If D
0
and ∆H
*
are both constant in Eq. (45), a logarithmic plot of the dif-
fusion coefficient as a function of reciprocal of absolute temperature will
be a straight line. The slope of such a plot yields ∆H
*
, whereas D
0
is given
by the intercept on the y-axis. Thus, D
0
and ∆H
*
are both experimentally
determinable quantities.
A comparison between Eqs. (1) and (45) shows that the frequency
factor,
D
0
, is:
D
0
fa
2
u exp(∆S
*
R). (46)
Equation (46) gives the value of ∆S
*
with the help of the experimentally
determined value of D
0
. The value of ∆S
*
estimated from Eq. (46) depends,
however, on the assumed value of the diffusion frequency (usually the
Debye frequency, ν
D
). Equation (43) permits us to obtain an independent
value of ∆S
*
without the input of the assumed value of diffusion fre-
quency. Substituting for (1 aT) from Eq. (39) in Eq. (43) and rewriting,
we obtain:
∆S
*
(ab)(∆V
*
V)V
0
. (47)
The only inputs needed to calculate ∆S
*
from Eq. (47) are the experi-
mental values of ab, ∆V
*
V, and V
0
. Substituting the above value of ∆S
*
in Eq. (46), the diffusion frequency u may be independently estimated
because f and a are known and D
0
is an experimentally determined quan-
tity. The estimated values of ∆S
*
and u as well as the data used in their eval-
uation are recorded in Table 2.4 for a number of metals. Values of ∆S
*
thus
estimated are compared with those computed traditionally from Eq. (46) by
taking Debye frequency as the diffusion frequency derived from the Debye
temperatures. The differences between the two values are also recorded in
the table. Similarly, independent estimates of diffusion frequencies com-
puted with the help of independently estimated ∆S
*
are compared with
Debye frequencies. Differences between them are also listed in Table 2.4.
Ch_02.qxd 11/30/04 8:45 AM Page 97
98
Table 2.4. Calculated Values of Activation Entropy and Diffusive Frequency for Self-Diffusion in a Number of Metals
Basic Data Activation Entropy Diffusive Frequency
∆S
*
∆S
*
nn
D
ab V
0
aD
0
Debye T, (calculated) (from n
D
) Difference (calculated) ( kθh) Difference
Metal (10
5
K
1
)(m
2
/dyne) (10
6
m
3
/mol) ∆V
*
Vf(10
10
m) (10
4
m
2
/s) θ (K) (J/mol/K) (J/mol/K) (%) (sec
1
) (sec
1
) (%)
Al 6.96 1.28 10
16
9.993 0.87 0.78146 4.0491 0.047 428 47.24 11.76 75.1 1.25 10
11
8.92 10
12
98.6
Cu 5.04 7.51 10
17
7.092 0.90 0.78146 3.6153 0.16 343 42.80 25.67 40.0 9.10 10
11
7.15 10
12
87.3
Au 4.23 6.00 10
17
10.207 0.85 0.78146 4.078 0.04 165 61.12 18.22 70.2 1.97 10
10
3.44 10
12
99.4
Ni 3.81 5.69 10
17
6.597 0.80 0.78146 3.5238 0.92 450 35.31 38.38 8.7 1.36 10
13
9.37 10
12
44.6
Pd 3.48 5.28 10
17
8.852 1.01 0.78146 3.8902 0.21 274 58.88 28.58 51.5 1.49 10
11
5.71 10
12
97.4
Ag 5.76 1.00 10
16
10.284 0.86 0.78146 4.086 0.04 225 50.91 15.61 69.3 6.72 10
10
4.69 10
12
98.6
Pt 2.67 3.60 10
17
9.095 1.09 0.78146 3.931 0.33 240 73.47 33.26 54.7 3.97 10
10
5.00 10
12
99.2
Pb 8.67 2.37 10
16
18.272 0.80 0.78146 4.9489 1.37 105 53.44 48.14 9.9 1.16 10
12
2.19 10
12
47.1
g-Fe 3.53 5.94 10
17
7.097 0.77 0.78146 2.8664 0.49 470 32.43 36.21 11.7 1.54 10
13
9.79 10
12
57.6
a-Fe 3.53 5.94 10
17
7.097 0.77 0.72722 2.8644 2 470 32.43 48.51 49.6 6.77 10
13
9.79 10
12
95.1
d-Fe 3.53 5.94 10
17
7.097 0.77 0.72722 2.8644 1.9 470 32.43 48.08 48.2 6.43 10
13
9.79 10
12
84.8
Na 20.9 1.56 10
15
23.672 0.41 0.72722 4.289 0.004 158 12.98 0.80 106.1 6.27 10
11
3.29 10
12
94.8
K 24.9 3.57 10
15
45.465 0.55 0.72722 5.334 0.16 91 17.43 30.83 76.9 9.50 10
12
1.90 10
12
80.0
Rb 27.0 5.20 10
15
55.869 0.40 0.72722 5.63 0.23 56 11.60 36.99 219.0 2.47 10
13
1.17 10
12
95.3
Li 14.1 8.70 10
16
12.996 0.26 0.72722 3.5089 0.038 344 5.47 14.79 170.2 2.20 10
13
7.17 10
12
67.4
Zn-
‘c11.9 1.67 10
16
9.170 0.65 0.78146 2.6649 0.13 327 42.48 29.41 30.8 1.41 10
12
6.81 10
12
79.2
Zn⊥c11.9 1.67 10
16
9.170 0.65 0.78121 2.6649 0.18 327 42.48 32.12 24.4 1.96 10
12
6.81 10
12
71.2
Mg
‘c 8.13 2.82 10
16
13.977 0.77 0.78146 3.2088 1 400 31.01 41.61 34.2 2.98 10
13
8.33 10
12
72.0
Mg⊥c 8.13 2.82 10
16
13.977 0.77 0.78121 3.2088 1.5 400 31.01 44.98 45.1 4.48 10
13
8.33 10
12
81.4
Cd
‘c 8.94 2.14 10
16
12.955 0.63 0.78146 2.9787 0.12 209 34.18 30.62 10.4 2.84 10
12
4.35 10
12
34.8
Cd⊥c 8.94 2.14 10
16
12.955 0.63 0.78121 2.9787 0.18 209 34.18 33.99 0.6 4.26 10
12
4.35 10
12
2.2
Ref. a,b a, c b c d b e f
a. R. E. Hanneman and H.C. Gatos, J. Appl. Phys., 36:1794 (1965)
b. H. E. Boyer and T.L. Gall (eds.), Metals Handbook, Desk Edition, ASM, OH (1985)
c. A. M. Brown and M. F. Ashby, Acta Metall., 28:1094 (1980), table 3
d. A. D. LeClaire, in Physical Chemistry: An Advanced Treatise (H. Eyring et al., eds.), vol. 10, Academic Press, New York (1970), chap. 15
e. N. L. Peterson, J. Nucl. Mater., 69–70:3 (1978)
f. C. Kittel, Introduction to Solid State Physics, Wiley Eastern, New Delhi (1977), chap. 5, table 1, p. 126
Ch_02.qxd 11/30/04 8:45 AM Page 98
SOLID STATE DIFFUSION AND BULK PROPERTIES, TIWARI ET AL.99
2.3.3 Specific Heat of the Activated Complex
This section further explores Eq. (43) to arrive at the value of specific
heat of the activated complex formed during the diffusive jump. The spe-
cific heat of the activated complex signifies the difference in the specific
heats between two states of the diffusing atom: activated and normal.
The Grüneisen relationship is:
ab gC
V
V, (48)
where g and C
V
are the Grüneisen constant and the specific heat at con-
stant volume, respectively. Applying this relationship to the activated
state, we obtain:
∆C
V
*
(∆V
*
g)(ab). (49)
Here, ∆C
V
*
is the specific heat of the activated complex at constant vol-
ume. Substituting for ab from Eq. (43) in Eq. (49), we obtain:
∆C
*
V
∆S
*
(1 aT)g . (50)
The importance of Eq. (50) lies in the fact that it yields a relationship
between two diffusion parameters and can be used to calculate the specific
heat of the activated complex in diffusion, since all the parameters in the
right-hand side of this equation can be obtained experimentally. The cor-
responding value of specific heat at constant pressure, ∆C
P
*
, is obtained
from the thermodynamic relationship:
∆C
*
P
∆C
*
V
{(a
2
T∆V
*
)b}. (51)
Table 2.5 lists the values of ∆C
*
V
and ∆C
*
P
estimated from Eqs. (50) and
(51) for a number of metals, along with the data used in their calculation.
The variation of ∆C
P
*
with temperature for some metals such as Cu, Pt, Na,
K, and Fe is given in Table 2.6.
2.3.4 Magnitude of Estimated Values
The Entropy of Activation ∆S
*
. Table 2.4 shows that the two estimated
values of activation entropy, one from Eq. (47) and the other from
Eq. (46), using Debye frequency in the latter, are of the same order: the
differences in these values range from 0.6% for Cd to 75.1% for Al. In the
case of alkali metals, however, the disparity between the two values is
Ch_02.qxd 11/30/04 8:45 AM Page 99
100
Table 2.5. Calculated Values of ∆C
*
V
and ∆C
*
P
of the Activated Complex for a Number of Metals
∆C
*
V
∆C
*
P
Temp ∆S
*
a V
0
b (calculated) (calculated) ∆C
*
P
Difference
Metal (K) (J/mol/K) ∆V
*
V (10
5
K
1
) g (10
6
m
3
/mol) (m
2
/dyne) (J/mol/K) (J/mol/K) (J/mol/K) ∆C
*
P
(%)
Al 700 47.24 0.87 6.96 2.19 9.993 1.28 10
16
22.620 25.034
Cu 700 42.80 0.90 5.04 2.00 7.092 7.51 10
17
22.156 23.719 14.644 38.3
Au 700 61.12 0.85 4.23 3.04 10.207 6.00 10
17
20.702 22.565
Ni 900 35.31 0.80 3.81 1.88 6.597 5.69 10
17
19.427 20.679
Pd 900 58.88 1.01 3.48 2.28 8.852 5.28 10
17
26.635 28.537
Ag 700 50.91 0.86 5.76 2.36 10.284 1.00 10
16
22.440 24.576 38.074 35.5
Pt 1000 73.47 1.09 2.67 2.56 9.095 3.60 10
17
29.467 31.481
Pb 400 53.44 0.80 8.67 2.62 18.272 2.37 10
16
21.103 23.021
g-Fe 1400 32.43 0.77 3.53 1.66 7.097 5.94 10
17
20.502 22.183
a-Fe 700 32.43 0.77 3.53 1.66 7.097 5.94 10
17
20.020 20.840
d-Fe 1700 32.43 0.77 3.53 1.66 7.097 5.94 10
17
20.709 22.771
Na 350 12.98 0.41 20.9 1.31 23.672 1.56 10
15
10.633 11.651 65.898 82.3
K 320 17.43 0.55 24.9 1.37 45.465 3.57 10
15
13.735 15.235 44.183 65.5
Rb 312.64 11.60 0.40 27.0 1.67 55.869 5.20 10
15
7.529 8.591
Li 400 5.47 0.26 14.1 1.18 12.996 8.70 10
16
4.889 5.225
Zn 400 42.48 0.65 11.9 2.01 9.170 1.67 10
16
22.140 24.260 16.736 31.0
Mg 600 31.01 0.77 8.13 1.48 13.977 2.82 10
16
21.971 23.558
Cd 600 34.18 0.63 8.94 2.23 12.955 2.14 10
16
15.874 17.140
Ref. a – f c a, b g b a, h i
a. R. E. Hanneman and H.C. Gatos, J. Appl. Phys., 36:1794 (1965)
b. H. E. Boyer and T.L. Gall (eds.), Metals Handbook, Desk Edition, ASM, OH (1985)
c. A. M. Brown and M. F. Ashby, Acta Metall., 28:1094 (1980), table 3
d. A. D. LeClaire, in Physical Chemistry: An Advanced Treatise (H. Eyring et al., eds.), vol. 10, Academic Press, New York (1970), chap. 15
e. N. L. Peterson, J. Nucl. Mater., 69–70:3 (1978)
f. C. Kittel, Introduction to Solid State Physics, Wiley Eastern, New Delhi (1977), chap. 5, table 1, p. 126
g. K. A. Gschneider, Jr., in Solid State Physics, vol. 16 (F. Seitz and D. Turnbull, eds.), Academic Press (1964), table XXIV, p. 412
h. C. Kittel, Introduction to Solid State Physics, Wiley Eastern, New Delhi (1977), chap. 3, table 4, p.85
i. H. M. Gilder and D. Lazarus, Phys. Rev., B11:4916 (1975)
Ch_02.qxd 11/30/04 8:45 AM Page 100
SOLID STATE DIFFUSION AND BULK PROPERTIES, TIWARI ET AL. 101
higher and may be attributed to the uncertainty in the magnitude of com-
pressibility values for these metals.
[70]
The Diffusion Frequency u. Table 2.4 also shows the comparison
between the calculated values of diffusion frequency and the Debye fre-
quency. In a large majority of cases, the differences between them are
below 100%. The calculated values are smaller than the Debye frequency
for Cd, Pb, Zn, Cu, Pd, Ag, Al, Pt, Au, etc., whereas these values are larger
for Ni, Fe, Mg, Li, K, and Rb. Surprisingly, the value is smaller for Na
than for the other alkali metals. The implication is that the calculated diffu-
sion frequency values are somewhat different from the Debye frequencies.
Table 2.6. ∆C
*
P
as a Function of Temperature for a Few Common Metals
∆C
*
P
Difference
Temp ∆C
*
P
(ref. a) in ∆C
*
P
Temp ∆C
*
P
Metal (K) (J/mol/K) (J/mol/K) (%) Metal (K) (J/mol/K)
Cu 600 23.381 12.552 46.6 g-Fe 1184 21.765
700 23.716 14.644 38.3 1200 21.795
800 24.059 16.736 30.4 1300 21.989
900 24.401 18.828 22.8 1400 22.183
1000 24.745 20.920 15.5 1500 22.378
1100 25.092 23.012 8.3 1600 22.574
1200 25.440 25.104 1.3 1665 22.702
1300 25.791 27.196 5.5 d-Fe 1665 22.702
Na 298.15 11.384 56.137 79.7 1700 22.771
350 11.651 65.898 82.3 1800 22.968
371 11.760 69.852 83.2 Pt 600 30.356
K 298.15 15.056 41.166 63.4 700 30.636
320 15.235 44.183 65.5 800 30.917
336.35 15.369 46.442 66.9 900 31.198
a-Fe 600 20.652 1000 31.481
700 20.840 1100 31.765
800 21.030 1200 32.050
850 21.125 1300 32.336
900 21.220 1400 32.623
950 21.315 1500 32.911
1000 21.411 1600 33.200
1100 21.603 1700 33.490
1184 21.765 1800 33.781
1900 34.073
2000 34.366
a. H. M. Gilder and D. Lazarus, Phys. Rev., B11:4916 (1975)
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