Practically useful form of Kim model from hysteresis loop of a superconductor
Considering a theoretical hysteresis loop within the Kim model for the Kim constant $H_0$ = 0 it has been found that the expression of $H_0$ obtained in Lal [R. Lal, Physica C 470 (2010) 281] provides a nonzero value 0.3424$H_p$ for this constant. ($H_p$ is full penetration field.) This different value of the Kim constant for the same hysteresis loop has been made a base for a different version of the Kim model, the hysteretic Kim model, such that the value zero of $H_0$ corresponds to the original Kim model and the value 0.3424$H_p$, denoted by a different notation $H_{0,hys}$, corresponds to the hysteretic Kim model. The two versions of the Kim model are interrelated so that $H_{0,hys}$ is a function of $H_0$. An empirical relation, $H_{0,hys}$=0.3424$H_p$exp(1.4$^{0.5}$($H_0$/$H_p$)), has been worked out on the basis of the theoretical $H_0$>0 hysteresis loops. The hysteretic Kim model has been cast in a practically useful form by obtaining an expression of $H_p$ in terms of the hysteretic magnetization. The importance of the hysteretic Kim model has been illustrated by applying it to the YBa$-2$Cu$_3$O$_7$, Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ and Ba$_0.72$K$_0.28$Fe$_2$As$_2$ superconductors by taking hysteresis loops of these systems from literature. It has been found that in these superconductors the dependence of $H_p$ on the sample width is mainly like $^{0.5}$(2$a$), and not like 2$a$ (Bean model). (2$a$ is the sample width.) The empirical relation of $H_{0,hys}$ and $H_0$ has been found to provide a reasonably good understanding of the intergranular matrix of the YBa$_2$Cu$_3$O$_{7-\delta}$ superconductor, where Kim model has not been found successful earlier.