Environmental Engineering Reference
In-Depth Information
Table 10. Continued
E b
(kJ/g-atom)
a
(nm)
γ sv0
(J/m 2 )
γ′ sv0
(J/m 2 )
γ′′ sv0
(J/m 2 )
( h k l )
Sb
(sc * )
265
0.336
(100)
(110)
0.66
0.77
0.61
0.66
0.60,
0.54
Bi
(sc * )
210
0.326
(100)
(110)
0.55
0.64
0.54
0.54
0.49,
0.49
Po
(sc * )
144
0.334
(100)
(110)
0.38
0.44
0.44
0.37
Si
(A4)
446
0.771
(110)
1.06
1.14
Ge
(A4)
372
0.810
(110)
0.80
0.88
Table 8 shows some necessary parameters in the Eqs. (4.3) and (4.5). Tables 9~11 give
the predicted γ sv0 values for fcc, bcc, hcp, diamond and sc structure crystals in terms of Eqs.
(4.3) and (4.5) where two sets of experimental results γ′′ sv0 [114-115] and the first principle
calculations γ′ sv0 [116] are also shown. Note that the experimental results are not orientation-
specific but are averaged values of isotropic crystals. Thus, they should be close to those of
the most close-packed surface.
For both noble and transition metals, our predictions agree nicely with the experimental
results and FCD calculations as shown in tables 9~11 although our predictions for transition
metals have slightly larger deviations than those for the noble metals due to the fact that their
d-bands are not fully filled and they present peaks at the Fermi level, which can slightly
change from one surface orientation to the other and consequently the energy needed to break
a bond changes also a little.
As shown in these tables, γ sv0 values of transition metals increase along an isoelectronic
row where a heavier element has a larger γ sv0 value. This is because the d -level of a heavier
element is higher in energy and the corresponding d -wave function with a stronger bonding is
more extended. This is also true for elements in the same row in the Periodic table where a
heavier element has more d -electrons [118]. An exception is in VA series where γ sv0 value of
Nb is smaller than that of V possibly due to the rehybridization of Nb where Nb, whose d
shell is less than the half-full, rehybridizes in the opposite direction, i.e., depletes their d z 2
orbitals based on a charge density difference analysis [146-148].
γ sv0 values for sp metals except for Be are smaller than those for d -metals due to their
bond nature of s - and p -electrons, which are more mobile than the localized d -electrons and
therefore less energy is needed to break these bonds.
For fcc metals except Ca, Sr and Al, the mean-square root error χ between the predicted
and the experimental results for the most close-packed (111) is about 7.5%. For aluminum,
the degree of covalent Al-Al bonding increases or the nature of the bonding changes with
reduced CN [149], which leads to model prediction deviating from the experimental results
since our formula neglects the variation of bonding type. However, the reason of deviations
for Ca and Sr is not clear.
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