96
5 Flux Pinning Phenomena
x =
f pt
f pt + f p
; −
a f
2
≤ <
a f
2
(5.64)
independently of . The statistical distribution in the initial condition is shown in
the middle panel of Fig. 5.25. There are unstable regions where flux lines do not
stay. In the initial state, the distribution is symmetric with respect to x = 0, and the
pinning force density is zero. When the flux lines are displaced in the direction of the
positive x-axis, there is no change in the number of flux lines. The left unstable region
becomes wider, while the right one becomes narrower. The critical state is attained
when the flux front reaches x = a f /4. The corresponding distribution is shown in
the lowest figure in Fig. 5.25. For further displacement, new flux lines come in from
the left side, and the same number of flux lines go out on the right side. Thus, the
distribution is in the steady state. The pinning force density in the critical state is [5]
F p = −
N p
a f
a f /4
x 0
−
4f p
a f
x
d
dx
dx = N p f p
f p − f pt
f p + f pt
.
(5.65)
Fig. 5.25 Pinning force
(upper panel), statistical
distribution of flux lines in
the initial state (middle
panel), and that in the critical
state (lower panel) for
k
f < 4f p /a f