Environmental Engineering Reference
In-Depth Information
For hcp metals, χ ≈ 10% except Mg, Zn, Cd and Tl. For Cd and Tl, both of our
predictions and FCD calculations deviate evidently from the experimental results. In the case
of Zn and Cd,
c
/
a
ratios (1.86 and 1.89) are larger than the ideal value of 1.633. Thus, the
nearest
CN
values will differ from the ideal condition, which should contribute the deviation
of our prediction.
For sc metals, χ = 2.6% where Sb and Bi with the rhombohedral structure are assumed to
have slightly a distorted sc structure [116].
For bcc metals, χ = 10%. The smallest value of χ in all considered structures appears for
diamond structure crystals with χ = 1.4%, which implies that the pure coherent bond does not
change after a
CN
deduction.
Note that the temperature dependence of surface energy in our model is ignored although
the experimental results listed in tables 9~11 are calculated at 0 K while the most lattice
constants cited are measured at room temperature. This temperature effect decreases the
prediction accuracy of our model and can be partly responsible for the disagreement with
other experimental and theoretical results.
In the FCD calculations, there are often exceptions that the most close-packed surface
does not have the lowest γ
sv0
values or there exists a weak orientation-dependence [116].
These physically unacceptable results are fully avoided in our calculation. In addition, the
anisotropy in our formula is perfectly considered.
(
100
)
(
111
)
and
(
110
)
(
111
)
γ
/
γ
≈
1
16
γ
/
γ
≈
1
27
sv
sv
sv
sv
_
sv
γγ
for hcp metals, which are in agreement with the
latest theoretical values of 1.15 and 1.22 [118,137]. In addition,
for fcc metals as well as
(
0001
)
(
10
1
0
)
/
≈
1
22
sv
sv
γγ
for sc and
bcc metals, which is especially comparable with 1.14 for monovalent
sp
metals based on the
jellium model [150].
If the experimental results are taken as reference, 60% of our predicted γ
sv0
values of the
most close-packed surfaces of 52 elements shown in tables 9~11 have better agreements with
the experimental ones than those of the FCD calculations do while 20% of the FCD
calculations are in reverse. Note that LDA is used in our formula while GGA is used in FCD.
Recently, it has been shown that both methods need to be corrected due to the neglect of
surface electron self-interactions where GGA is worse than LDA [146-148]. This is surprising
because GGA is generally considered to be the superior method for energetic calculations
[116].
For the transition metals and noble metals, the formula works better than for others as the
greatest contribution to bonding is from the
s-d
interaction and the orbitals of the latter
localize, which is more like a pair interaction. According to tables 9~11, the predicted γ
sv0
values of divalent sp metals have bad correspondence with the experimental results since the
many body (e.g. trimer) terms are here critical to understanding the cohesive energy. Thus,
the used pair potentials physically may be not fully correct. It is possible that the background
of our formula, namely the broken-bond model, is not universally applicable although the
lattice constants used in Eqs. (4.3) and (4.5) have measuring error of about 2% [114,130].
According to the first principles calculations, the effect of relaxation on the calculated γ
sv0
value of a particular crystalline facet may vary from 2 to 5% depending on the roughness
[147,151]. The semi-empirical results indicate further that the surface relaxation typically
affects the anisotropy by less than 2% [123]. Surface relaxations for vicinal surfaces have
been studied mainly using semi-empirical methods due to the complexity arisen by the
(
100
)
(
110
)
/
≈
1
16
sv
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