5.6 Coherent Potential Approximation Theory
105
η p =
1 − k f a f /4f p
1 + k f a f /4f p
.
(5.90)
The value of η p is almost constant, and the linear summation holds over a relatively
wide range of f p .
In the case of weak pinning, in which 4f p /k f a f is very small, the right-hand side of
(5.86) is small. This suggests that there is a solution in the vicinity of t = 1 around
which the left-hand side is small. We replace t by one except in the numerator on
the left-hand side and neglect t in the denominator on the right-hand side. Thus, we
have
t ∼ = 1 −
8f p
βk f a f
,
(5.91)
and the pinning force density is given by
F p ∼ =
4N p f
2
p
βk f a f
.
(5.92)
The obtained pinning force density obeys the statistical summation given by (5.5)
[17]. As shown here, the coherent potential approximation theory explains the
experimental results over a wide range of the elementary pinning force in Fig. 5.4.
The most important result obtained in this section is that the threshold value of the
elementary pinning force exists formally, but it is always smaller than the elementary
pinning force. That is, no substantial threshold value exists. Hence, the pinning loss in
an AC magnetic field of sufficiently large amplitude is hysteresis loss, which depends
only on the pinning force density, but not on the flow resistivity, as described by the
critical state model.
The theoretical result obtained in this section can be extended to the flux flow
state above the critical current density. This is supported by the fact that the time
average and statistical average agree as
1
T
T
0
f (x)dt =
1
a f
a f
0
f (x)d,
(5.93)
in the range of small v, where (5.68) and the relationship T = a f /v are used. In fact,
the agreement has been shown theoretically for various pinning models [12, 18].
This supports the dynamic critical state model of (5.41).
105
η p =
1 − k f a f /4f p
1 + k f a f /4f p
.
(5.90)
The value of η p is almost constant, and the linear summation holds over a relatively
wide range of f p .
In the case of weak pinning, in which 4f p /k f a f is very small, the right-hand side of
(5.86) is small. This suggests that there is a solution in the vicinity of t = 1 around
which the left-hand side is small. We replace t by one except in the numerator on
the left-hand side and neglect t in the denominator on the right-hand side. Thus, we
have
t ∼ = 1 −
8f p
βk f a f
,
(5.91)
and the pinning force density is given by
F p ∼ =
4N p f
2
p
βk f a f
.
(5.92)
The obtained pinning force density obeys the statistical summation given by (5.5)
[17]. As shown here, the coherent potential approximation theory explains the
experimental results over a wide range of the elementary pinning force in Fig. 5.4.
The most important result obtained in this section is that the threshold value of the
elementary pinning force exists formally, but it is always smaller than the elementary
pinning force. That is, no substantial threshold value exists. Hence, the pinning loss in
an AC magnetic field of sufficiently large amplitude is hysteresis loss, which depends
only on the pinning force density, but not on the flow resistivity, as described by the
critical state model.
The theoretical result obtained in this section can be extended to the flux flow
state above the critical current density. This is supported by the fact that the time
average and statistical average agree as
1
T
T
0
f (x)dt =
1
a f
a f
0
f (x)d,
(5.93)
in the range of small v, where (5.68) and the relationship T = a f /v are used. In fact,
the agreement has been shown theoretically for various pinning models [12, 18].
This supports the dynamic critical state model of (5.41).