Environmental Engineering Reference
In-Depth Information
Eq. (4.3) implies that γ
sv0
values still depend on the bond-broken rule although they are
scaled by both of
Z
S
and the square root of
Z
S
.
In Eq. (4.3),
Z
S
can be determined according to the crystalline structure through
determining
Z
(h k l)
by a geometric consideration [140-141]. For an fcc or hcp structure,
Z
B
=
12; For a bcc lattice, although
Z
B
= 8 is taken according to the nearest-neighbor definition by
some authors (probably the majority), others prefer to regard
Z
B
= 14 since the difference
between the nearest neighbor bond length and the next-nearest neighbor bond length is small
[142]. Here, the latter is accepted. By assuming that the total energy of a surface atom is the
sum of contributions from both of the nearest neighbor and the next-nearest neighbor atoms,
Eq. (4.3) should be rewritten for bcc metals after normalization [139],
γ
sv0
= [(2-
k
1
-
k
1
1/2
)+ϕ(2-
k
1
′
-
k
1
′1/2
)]
E
b
/[(2+2ϕ)
N
a
A
S
]
(4.4)
where the superscript comma denotes the next-nearest
CN
on a surface and ϕ shows the total
bond strength ratio between the next-nearest neighbor and the nearest neighbor [143]. To
roughly estimate the size of ϕ, the Lennard-Jones (LJ) potential is utilized [143]. The
potential is expressed as
u
(
r
) = -4η[(τ
/r
)
6
-(τ
/r
)
12
] with
η being the bond energy and τ insuring
d
u
(
r
)/d
r
(
r=h
)
= 0, i.e. τ
= 2
-1/6
h
where
r
is the atomic distance. For bcc crystal,
h
= 3
1/2
a
/2 and
h
′
=
a
, respectively. Let
r
=
a
, η
′
= 2η/3. Thus, ϕ = [(2/3)×6]/8 = 1/2. Adding this value into
Eq. (4.4) [139],
γ
sv0
= [3-
k
1
-
k
1
1/2
-
k
1
′
/
2-(
k
1
′
/
4)
1/2
]
E
b
/(3
N
a
A
S
).
(4.5-a)
Note that the bonding of the LJ potential, which is utilized to justify ϕ value in Eq. (4.4),
is different from the metallic bond in its nature. For instance in a LJ bonded system, the
surface relaxation is outwards whilst in the transition metals it is inwards. However, this
difference leads to only a second order error in our case and has been neglected.
The effect of next-nearest
CN
also occurs for simple cubic (sc) and diamond structure
crystals because there are twice and thrice as many the second neighbors as the first
neighbors, respectively. Similar to the above analysis, for sc crystals: η
′
≈ η/4 and ϕ =
[(1/4)×12]/6 = 1/2, which is the same for bcc and thus Eq. (4.5-a) can also hold for sc crystals.
For diamond structure crystals, η
′
≈ η/10 and ϕ = [(1/10)×12]/4 = 3/10. With this ϕ value, Eq.
(4.4) is rewritten as [139],
γ
sv0
= [26-10
k
1
-10
k
1
1/2
-3
k
1
′
-(9
k
1
′
)
1/2
]
E
b
/(26
N
a
A
S
).
(4.5-b)
Z
(h k l)
can be determined by some known geometrical rules. For any surface of a fcc
structure with
h
≥
k
≥
l
[140-141],
Z
(h k l)
= 2
h
+
k
for
h
,
k
,
l
being odd
(4.6-a)
Z
(h k l)
= 4
h
+2
k
for the rest
(4.6-b)
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