Geology Reference
In-Depth Information
m
¼
m
0
exp
DE
þ
F dx
dx
ð
Þ
kT
where m
0
is the attempt frequency, F the applied force (F
¼
dE
=
dx in the absence
of the barrier), DE
the height of the barrier, dx
;
dx the displacements from
equilibrium position 1 to peak 2 and to the next equilibrium position 3, respec-
tively, k the Boltzmann constant, and T the absolute temperature. The net forward
rate of jumping is then
exp
exp
F dx
dx
m
¼
m
þ
m
¼
m
0
exp
DE
kT
Fdx
kT
ð
Þ
kT
ð
3
:
12a
Þ
The expression (
3.12a
) leads in particular cases to three well-known forms, as
follows:
(1) When dx
¼
2
dx
;
that is, the equilibrium positions 1 and 3 are symmetrically
positioned about the barrier 2 (Fig.
3.3
) we always have
2kT
exp
DE
m
¼
2m
0
sinh
Fdx
ð
3
:
12b
Þ
kT
(2) When
2
Fdx
2
Fdx
kT
;
that is, at relatively low stress and high tem-
perature, we have the approximation
1
exp
DE
kT
m
m
0
Fdx
kT
ð
3
:
12c
Þ
1
(3) When
2
Fdx
kT
;
that is, at relatively high stress and low temperature, we
have the approximation
m
m
0
exp
DE
Fdx
kT
ð
3
:
12d
Þ
The main application of this formalism is in treating the thermally activated
