5.6 Coherent Potential Approximation Theory
103
that the remaining pinning interactions exert an elastic restoring force proportional
to the displacement − x 0 . The corresponding spring constant is denoted by K. This
is related to the Labusch parameter representing the strength of the pinning field.
It should be noted that the elastic force caused by the surrounding pinning centers,
K( − x 0 ), is an internal force that is conveyed from one to another. Hence, it is
nothing else than the elastic restoring force on the observed flux line given by the
first term in (5.80). This force is balanced by the pinning force on the observed flux
line:
K( − x 0 ) + f (x) = 0.
(5.81)
From (5.80) and (5.81), we obtain the force balance of (5.60) with
k
f
= k
−1
f + K
−1
.
(5.82)
Figure 5.28 schematically illustrates the elastic condition around the pinning
center. The observed flux line interacting with the pinning center is also interacting
elastically with other flux lines within the correlation length. This elastic interaction
is due to the short-range distortion, and the corresponding spring constant is given
by k f . On the other hand, the flux lines in the elementary region are interacting with
the surrounding pinning centers through the elasticity of the flux line lattice, the
spring constant of which is K. The force balance treated here can be expressed as an
elastic interaction connected in series. Based on the present argument, each size of
the three-dimensional elementary region of flux lines is determined in proportion to
the pinning correlation length in the corresponding direction (see Appendix A.5).
If we use k
f given by (5.82), all the results in Sect. 5.4 can be used. Another
important point is shown here. This is a relationship with (5.53). The interaction
distance is generally in proportion to the flux line spacing as
d i =
a f
ζ
,
(5.83)
where ζ is a constant depending on the shape of the pinning centers, and we have
ζ = 2π for point pins. Hence, the pinning force density is given by
Fig. 5.28 Elastic interaction
connected in series between
the pinning center and its
surroundings
flux line
pinning
center
correlated region
103
that the remaining pinning interactions exert an elastic restoring force proportional
to the displacement − x 0 . The corresponding spring constant is denoted by K. This
is related to the Labusch parameter representing the strength of the pinning field.
It should be noted that the elastic force caused by the surrounding pinning centers,
K( − x 0 ), is an internal force that is conveyed from one to another. Hence, it is
nothing else than the elastic restoring force on the observed flux line given by the
first term in (5.80). This force is balanced by the pinning force on the observed flux
line:
K( − x 0 ) + f (x) = 0.
(5.81)
From (5.80) and (5.81), we obtain the force balance of (5.60) with
k
f
= k
−1
f + K
−1
.
(5.82)
Figure 5.28 schematically illustrates the elastic condition around the pinning
center. The observed flux line interacting with the pinning center is also interacting
elastically with other flux lines within the correlation length. This elastic interaction
is due to the short-range distortion, and the corresponding spring constant is given
by k f . On the other hand, the flux lines in the elementary region are interacting with
the surrounding pinning centers through the elasticity of the flux line lattice, the
spring constant of which is K. The force balance treated here can be expressed as an
elastic interaction connected in series. Based on the present argument, each size of
the three-dimensional elementary region of flux lines is determined in proportion to
the pinning correlation length in the corresponding direction (see Appendix A.5).
If we use k
f given by (5.82), all the results in Sect. 5.4 can be used. Another
important point is shown here. This is a relationship with (5.53). The interaction
distance is generally in proportion to the flux line spacing as
d i =
a f
ζ
,
(5.83)
where ζ is a constant depending on the shape of the pinning centers, and we have
ζ = 2π for point pins. Hence, the pinning force density is given by
Fig. 5.28 Elastic interaction
connected in series between
the pinning center and its
surroundings
flux line
pinning
center
correlated region