Analytical Study of AC Magnetic Susceptibility of (Bi, Pb) Sr-Ca-Cu-O Superconducting Systems Using Bean Critical State Model

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1 Kasetsart J. (Nat. Sci.) 43 : (9) Analytical Study of AC Magnetic Susceptibility of (Bi, Pb) Sr-Ca-Cu-O Superconducting Systes Using Bean Critical State Model Supreya Trivijitkase *, Adisak Boonchun, Chirawat Chantharangsi and Chutia Paksunchai ABSTRACT AC agnetic susceptibility, χ ac, as a function of teperature, T, and applied field aplitude, H, of (Bi.6 Pb.4 )Sr Ca Cu 3 O β and (Bi.7 Pb.3 )Sr Ca Cu 3 O γ superconducting systes was analyzed using Bean critical state odel. The critical teperature and grain volue fraction of speciens were utilized fro the real part of AC susceptibility easureents, while the pinning force density was utilized fro the iaginary part of the susceptibility easureents. Calculations using Matheatica progra indicated that (Bi.6 Pb.4 )Sr Ca Cu 3 O β superconductor of noinal coposition (34), prepared by solid reaction with an interediate ground and pressed, provided the biggest hysteresis area, and the calculated χ ac (H,T) results were coparable with the experiental χ ac (H,T) data. Key words: BPSCCO, χ ac, critical state, Bean odel INTRODUCTION The critical states of sintered highteperature superconductors, containing weakly coupled superconductive grains have been studied by AC agnetic susceptibility, χ = χ + iχ, easureents (Chen et al., 99; Ishida and Goldfarb, 99; Celebi et al., 998). Measureent of the superconducting transition by eans of coplex AC susceptibility provides a sharp decrease in χ due to diaagnetic shielding and a peak in χ representing losses. Both χ and χ are sensitive to both the teperature,t, and the aplitude of the AC agnetic field,h, (Trivijitkase and Sratongluan, 999). Ceraic superconductors are coposed of an array of superconducting grains, which are interconnected by weak links. Consequently, the intergranular and granular pinning depinning properties are used to define the irreversibility line in the H-T plane (Gonzalez et al., 995). Below this line, a finite critical current density exists, while above this line, it turns to zero. Soe critical state odels are eployed to calculate χ ac in ceraic superconductor, such as the Rollins- Silocox (R-S), Kiishia, Bean, and Anderson- Ki odels. Ji et al. (989) proposed a acroscopic critical state odel to predict both odd and even haronic susceptibility in high critical teperature (T C ) superconductors. Lee and Departent of Physics, Faculty of Science, Kasetsart University, Bangkok 9, Thailand. Division of Physics, Faculty of Science and Technology, Rajaangala University of Technology Krungthep, Bangkok, Thailand. * Corresponding author, e-ail: fscisu@ku.ac.th

2 354 Kasetsart J. (Nat. Sci.) 43(5) Kao (995) used the Anderson Ki critical state odel to analyze χ ac in YBCO ceraic superconductors, which showed satisfactory agreeent with the experiental results. In the present work, Bean critical state odel was used to analyze the fundaental susceptibility, χ ac, as a function of teperature, T, and the AC agnetic field aplitude, H, of high T c BSCCO superconductors. The calculated χ ac (H,T) results were copared with the easured data on thin slab saples iersed in a pure AC field. THEORY When a superconductor is placed in an AC agnetic field, H=H ac cos(ωt), the Fourier expansion of tie dependent agnetization M(t) is given by Equation (): () where, n =,, 3,..., χ n- and χ n- are the n th order coponents of the real and iaginary part of coplex AC susceptibility, respectively. The real and iaginary fundaental susceptibility, χ and χ (n equals ), can be calculated fro the following Equation (a) and (b) respectively: and π χ = (t)cos(ωt)d(ωt) π (a) π χ = (t)sin(ωt) d(ωt) π (b) where, reduced agnetization (t) = M(t)/H ac. The critical state odel proposed by Bean assued that the critical current density, J c, is constant for a sall total field, H. Then the pinning force density (α = J C B), which is Lorentz force per volue, holds for the equilibriu state of the local flux density, B, in the saple. In order to calculate the hysteresis loop of reduced agnetization, (t), and the reduced AC field, h(t) = H/H ac = cos(ωt), the virgin agnetization, vir = ( B /µ )-h, is introduced, where B is the average local field for the inner space of the saple. For the slab saple, the virgin agnetization, vir, can be expressed by Equation (3): α ( h α) vir = + α, for h α, = α, for α h. (3) For α, the tie dependent reduced agnetization (t) is given by Equation (3a): (t) = α {cos( ωt ) cos θ}, for ωt θ, 4α = α, for θ ωt π, α = + {cos( ωt) + cos θ}, 4α for π ωt π+θ, = α, for π+θ ωt π. (3a) where, θ = cos - (-α). For α, the reduced agnetization (t) is expressed by Equation (3b):: α 4α, for ωt π, (t) = cos θ cos(ωt) {cos (ωt) } + α 4α, for π ωt π. (3b) = cosθ cos(ωt) {cos (ωt) } In order to calculate χ (H, T), the following expressions, Equation (4), are introduced: χ = ( f )χ + f χ, g g g χ = ( f )χ + f χ (4) g g g

3 Kasetsart J. (Nat. Sci.) 43(5) 355 where, f g is the effective volue fraction of superconduction grain, χ g and χ are the g coponents of grain susceptibility. The theoretical atrix susceptibilities, χ, χ and χ, χ, are derived by perforing g g the following analytical integrations, only the expressions for intergranular susceptibilities are shown in Equation (5). 4 = ( α )sin α πα α ( α ) α ( α ), fo r and 3 χ = α (3 α ) 3π α,, for χ = α χ α,, for α = 3πα,,, for α (5) The pinning force density, α, as a function of teperature consists of the intergranular atrix pinning force density, α (T), and the granular pinning force density, α g (T). For (B,P)SCCO superconduction syste, the pinning force densities are assued as Equation (6) and (7): α (T) α () T C.9 T = (6) Equation (8) and (9): α () =.75 (T /T ) (8) p C.9 = (9) pg Cg and α ().75 (T /T ) g where, T p is the teperature at χ (axiu). MATERIALS AND METHODS (Bi.6 Pb.4 )Sr Ca Cu 3 O β and (Bi.7 Pb.3 ) Sr Ca Cu 3 O γ superconducting syste were prepared by the conventional solid state reaction technique with an interediate ground and either pressed (P) or not pressed (N). The preparation ethod was described in Trivijitkase and Sratongluan (). The AC agnetic susceptibility easureent was perfored using an AC agnetoeter (Lake Shore odel 73). The saples were cooled down to 55 K by the zero field cooled (ZFC) ethod. Various AC agnetic fields of 5 Hz fundaental frequency were gradually applied to the saple, following by increasing the teperature to K at a heating rate of.86 C/in. Using Bean critical state odel and Matheatica progra, the hysteresis loops of reduced agnetization, (t), and the reduced AC agnetic field, h(t), were calculated fro Equation (3). The theoretical real and iaginary AC agnetic susceptibility as a function of teperature and agnetic field were also calculated fro Equation (4) and (5). RESULTS AND DISCUSSION α (T) α () g T = g (7) TCg where, α g () is the granular pinning force density at K, and T C, T Gg are the Josephson intergranular and granular critical teperature, respectively. According to Bean odel, α(t) at χ ax equals.75, so α() can be deterined fro Figure shows the easured χ and χ as a function of teperature, T, fro 55-5 K and applied field aplitude, H =.,.,,,, 3, 4 and 5 A/ of (Bi.6 Pb.4 ) Sr Ca Cu 3 O β and (Bi.7 Pb.3 )Sr Ca Cu 3 O γ superconductors. Critical teperature T c and granular volue fraction f g of the saples are listed in Table and were deterined fro the real part of the AC agnetic susceptibility χ

4 356 Kasetsart J. (Nat. Sci.) 43(5) easureent at H =. A/. The intergranular atrix pinning force density, α (), and the granular pinning force density, α g (), at absolute teperature were calculated fro Equation (8) and (9) respectively, where T p is the peak teperature deterined fro the iaginary part of the AC agnetic susceptibility χ easureent. The hysteresis loop of reduced agnetization, (t), and the reduced agnetic field, h(t), were calculated fro Equation (3) using the χ experiental data at K and. A/. The results are presented in Figure, which shows that the biggest hysteretic area belonged to the (Bi.6 Pb.4 )Sr Ca Cu 3 O b (P) superconductor prepared by an interediate ground and pressed. In contrast, the saple prepared without an interediate ground and pressed had a saller hysteretic area. The real and iaginary agnetic susceptibility (χ and χ ) as a function of teperature, T, fro 5 5 K and the AC field aplitude, H =.,.,,,, 3, 4 and 5 A/, of the (Bi.6 Pb.4 )Sr Ca Cu 3 O β and (Bi.7 Pb.3 )Sr Ca Cu 3 O γ superconductors were calculated fro Equation (4) and (5) using the data fro Table. Figure The observed AC susceptibility, χ, at 5 Hz and field aplitude, fro right to left, H =.,,,,, 3, 4 and 5 A/, respectively. Table Critical teperature, T C, granular volue fraction, f g, pinning force density of grain, α g () and intergrain, α () at K of (Bi, Pb)Sr-Ca-Cu-O superconductors. Saple T C (K) f g α g () α () (Bi.6 Pb.4 )Sr Ca Cu 3 O β (N) (Bi.6 Pb.4 )Sr Ca Cu 3 O β (P) (Bi.7 Pb.3 )Sr Ca Cu 3 O γ (N) (Bi.7 Pb.3 )Sr Ca Cu 3 O γ (P)

5 Kasetsart J. (Nat. Sci.) 43(5) 357 The calculated values of χ (H, T) and χ (H, T) are shown in Figure 3, which indicates that the calculated χ and χ curves are siilar to the experiental χ and χ curves in Figure. A coparison of the observed χ (H, T) behavior with the calculated results, at H =. A/, for the superconducting syste are presented in Figure 4, which shows that the superconducting syste prepared by an interediate ground and pressed provided a ore consistent calculated and observed AC susceptibility χ ac (H, T). This confirs that Bean critical state odel can be used Figure Calculated hysteresis loops of the thin slab superconductors at K and field aplitude. A/. Figure 3 The calculated AC susceptibility χ at 5 Hz and field aplitude, fro right to left, H =.,,,,, 3, 4 and 5 A/, respectively.

6 358 Kasetsart J. (Nat. Sci.) 43(5) Figure 4 A coparison of the observed and calculated χ (H, T) at 5 Hz and H =. A/ for P=pressed and N=not pressed. for reproducing the observed χ ac data, especially for a lower H-field of the (Bi.6 Pb.4 ) and (Bi.7 Pb.3 ) superconductors prepared by an interediate ground and pressed. CONCLUSION Bean critical state odel was used to analyze the fundaental AC susceptibility as a function of teperature, T, and AC agnetic field aplitude, H, of high T C (Bi.6 Pb.4 )Sr Ca Cu 3 O β and (Bi.7 Pb.3 )Sr Ca Cu 3 O γ superconducting syste. The pinning force density was deterined fro the susceptibility easureent data using the zero field-cooled ethod. Calculations fro Matheatica progra indicated that the biggest hysteresis area belonged to the (Bi.6 Pb.4 ) superconductor of noinal coposition (34) prepared by solid reaction with an interediate ground and pressed. Bean critical state odel was used to reproduce the observed χ ac data. The results showed a consistent calculated and observed AC susceptibility χ ac, especially at. A/ field aplitude, in the (Bi, Pb) superconducting syste prepared by an interediate ground and pressed. LITERATURE CITED Celebi, S., I. Karaca, E. Aksu and A. Gencer Frequency dependence of the intergranular AC loss peak in a high-t C Bi-(Pb)-Sr-Ca-Cu- O bulk superconductor. Physica. C 39: Chen, D. X., Y. Mei and H. L. Luo. 99. Criticalcurrent density in sintered high T C (Bi, Pb)- Sr-Ca-Cu oxide. Physica. C 67: Gonzalez, J. L., P. Mune, L. E. Flores and E. Altshuler AC susceptibility study of the intergranular irreversibility line in BSCCO ceraic superconductor. Physica. C 55: 79-8.

7 Kasetsart J. (Nat. Sci.) 43(5) 359 Ishida, T. and R. B. Goldfarb. 99. Fundaental and haronic susceptibilities of YBa Cu 3 O 7-δ. Phys. Rev. B 4(3): Ji, L., R. H. Sohn, G. C. Spalding, C. J. Lobb and M. Tinkha Critical-state odel for haronic generation in high teperature superconductors. Phys. Rev. B 4(6): Lee, C. Y. and Y. H. Kao Low-field agnetic susceptibility studies of high- T C superconductors. Physica. C 4: Trivijitkase, S. and W. Sratongluan Microstructured evolution in foring Bi-3 superconductor fro Bi-Pb-Sr-Ca-Cu-O syste. Kasetsart J. (Nat. Sci.) 33: Trivijitkase, S. and W. Sratongluan.. Superconducting properties of (Bi, Pb)-Sr-Ca- Cu-O ceraics. Kasetsart J. (Nat. Sci.) 34: 59-7.