Modeling of normal-state like contribution on the pinning induced magnetostriction F. _ Inanır a , M. Okutan b, * , F. Yakuphanog ˘lu c a Department of Physics, Faculty of Science and Arts, Rize University, Rize 53100, Turkey b Department of Physics, Gebze Institute of Technology, Gebze 41400 Kocaeli, Turkey c Department of Physics, Firat University, 23169 Elazig, Turkey Received 7 June 2007; received in revised form 24 September 2007; accepted 9 October 2007 Available online 26 October 2007 Abstract To describe the change of the sign of the magnetostriction at the field close to the upper critical field, B c2 , which were observed in some magnetostriction measurements carried out on the type-II superconductors, theoretical magnetostriction curves for the superconducting state were obtained considering the normal-state like contribution. The calculations were performed employing Bean, Kim, Linear, and the exponential critical-state models. We have reproduced the magnetostriction curves measured by Nabialek et al. [A. Nabialek, B. Kundys, Yu Bukhantsev, A. Wisniewski, S.M. Kazakov, J. Karpinski, H. Szymczak, Supercond. Sci. Technol. 16 (2003) 707] on MgB 2 superconductors and Ikuta et al. [H. Ikuta, Y. Nakayama, N. Hirota, K. Kishio, K. Kitazawa, Appl. Supercond. 1 (1993) 635] on (La 1x Sr x ) 2 CuO 4 superconductors, in which was observed such sign alternation, employing the exponential model. It is evaluated that inhomogeneity of the magnetic induction across the sample volume owing to the existence of critical current causes the normal-state like contribution to the flux-pinning induced magnetostriction. Ó 2007 Elsevier B.V. All rights reserved. 1. Introduction When an external magnetic field between the lower and the upper critical field, thus being in the mixed state, is applied to a superconductor, the magnetic forces resulting from the interaction of the applied field with current in the superconductor may produce its magnetostriction. The magnetostriction observed in type-II superconductors owing to the strong pinning of the flux lines in supercon- ductor defects is highly irreversible as verified by a number of papers [1–14]. The critical state model ensures en excel- lent tool to describe this phenomenon in both low-T c superconductors (LTSC) and high-T c superconductors (HTSC). The flux-pinning induced effect in a single crystal of Bi 2 Sr 2 CaCu 2 O 8 was first observed by Ikuta et al. [3]. The hysteretic dilatation discovered in this sample versus the external field was successfully modeled in one-dimensional by them in the frame of critical-state approach taking into account the infinite slab of thickness 2d [3,15]. The model- ing calculation of the internal stress in a cylindrical super- conductor for the field-cool process in the state of descended field was made by Ren et al. [16]. The pinning induced magnetostriction expressions for more realistic geometrical cases were developed by Johansen and his co-workers [17–20]. The demagnetization effect above all calculations was ignored. Two major problems considering this effect are the following: the magnetostriction expres- sions for a thin flat sample of thickness d and width L =2a under the action of a varied field perpendicular to the face of the sample were derived analytically by Nabi- alek et al. [21] and Eremenko et al. [22] and in both works the modified current and flux distribution equations were used. The similar problem for a thin superconducting circu- lar disk of radius R and thickness d, where d  R, under the field applied perpendicular to the disk plane was solved by Johansen et al. [23] by solving a couple of integral 0921-4534/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.10.010 * Corresponding author. Tel.: +90 262 605 13 22; fax: +90 262 6538497. E-mail address: mokutan@gyte.edu.tr (M. Okutan). www.elsevier.com/locate/physc Available online at www.sciencedirect.com Physica C 468 (2008) 39–46