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1 Numerical study of critical state to investigate magnetization and pinning properties of the Yb211-doped bulk Sm123 superconductor Dr. Kemal ÖZTÜRK K. Ozturk 1,*, A Patel 2, B. A. Glowacki 2, E. Yanmaz 2 1 Department of Physics, Faculty of Sciences, Karadeniz Technical University, 61080 Trabzon, Turkey 2 Department of Materials Science and Metallurgy, University of Cambridge, Pembroke Street, Cambridge CB2 3QZ, UK * Email of corresponding author: kozturktu@hotmail.comkozturktu@hotmail.com

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2 The magnetic force between a superconductor and a magnet is proportional magnetic moment in superconductor. The value of magnetic moment dependent on the critical current density and radius of a shielding current loop of the superconductor can be increased considerable by improving of the structural properties such as grain orientation, density the of crack, and quantity of the pinning center. In this study the magnetization and magnetic force properties between magnetized Yb211-doped bulk Sm123 superconductor and magnetic source were investigated. Critical state modelling was carried out using the FEM (Finite Element Method) software to simulate field magnetization and to investigate magnetic force properties of doped-superconductor. A critical comparison was made between the modelling and experimental results and it was seen that the modelling is successful on clarify magnetization and pinning properties of Yb211-doped Sm123 bulk superconductor. Abstract

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3 Contents 1. Experimental results 2. Modelling of bulk superconductor magnetization 2.1 The H formulation for modelling the critical state using FEM in COMSOL 3. The Comparison of the experimental and modelling results 4. Results

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Fig. 1. The variation of the transition temperature T c offset and c-lattice constant of nominal (Sm123) 1 x (Yb211) x bulk samples depending on Yb211 doping ratio x. 4 1. Experimental results

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5 Fig. 2. The Electron Dispersive X-ray (EDX) spectra taken from Sm123 matrix region of (Sm123) 1 x (Yb211) x sample for doping ratio x= 0.10.

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6 Fig. 3. Magnetic field dependence of the critical current densities of (Sm123) 1 x (Yb211) x sample for doping ratio of x= 0.00, 0.05, 0.10, 0.15, 0.20, 0.25 and 0.35 at (a) 50 K and (b) 77 K measurement temperatures.

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7 Fig. 4. Variation of volume pinning force F p as a function of magnetic field of (Sm123) 1 x (Yb211) x sample for different Yb211 doping ratios at 5 K and 77 K (inset plot). Fig. 4 shows plot of volume pinning force F p B (in which F p = J c x B and B= 0 H) curves at 5 K and 77 K (inset plot). It is seen from Fig. 4 the value of volume pinning force increase with increasing of doping ratio x at 5 K except for x=0.35 (which this case can be clarified as superconducting properties are destroyed because of excess doping ratio). The maximum volume pinning force at 5 K is ~ 1.5x10 9 Nm 3 for x=0.25 doping ratio which is approximately two times larger than undoped sample. In additionally the inset plots in Fig. 4, the pinning force values increase with increasing doping ratio in the low field region for 0 H 1 T and pinning force values reach a maximum for x=0.05 in the high field region imply that dominant pinning properties are different depending on the field degree and doping ratio. As a result, it can be said that the non- superconducting phases as Yb211 particles and crystal defects contribute to pinning in low field regions with increasing Yb211 doping ratio. On the other hand the RE-rich low T c clusters such as Yb123, (Sm,Yb)123 and nano-sized compositional fluctuation regions contribute to pinning for optimum Yb211 doping ratio in high field region at 77 K.

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Fig. 5. Plots of f p = F p /F pmax versus b = B/B max for (Sm123) 1 x (Yb211) x samples with different Yb211 doping ratio at 5 K (a) and (b) doping ratio x=0.25 sample at 5 K and 77 K measurement temperatures. In figure, the solid line represents pinning, the dashed line normal point pining and the dotted line surface pinning In order to obtain an explicit insight into the origin of the pinning properties, an extended analysis between the normalized volume pinning force f p and normalized magnetic field b have been examined. The normalized pinning force f p = F p /F pmax is often scaled with b = B/B max, where B max is the field at which the F p reaches its maximum value. The scaling of f p b for HTS superconductor samples is often analyzed by using following Eqs. (1) (3) [1, 2]. In Fig. 5, f p = F p /F pmax versus b = B/B max are plotted for (a) (Sm123) 1-x (Yb211) x samples with different Yb211 doping ratio x=0.00 0.35 at 5 K and (b) doping ratio x=0.25 sample at 5 K and 77 K measurement temperatures.

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9 Fig. 6. The vertical levitation force density vs. vertical distance between the MPMG-processed (Sm123) 1 x (Yb211) x samples and the Permanent magnet for reheat-temperatures at 1170 and 1100 C (inset plot [3]). The levitation force is then proportional with the magnetic moment of the superconductor and magnetic field gradient of the magnet, where m is the magnetic moment of a superconductor, is the field gradient produced by the permanent magnet. The magnetization of the superconductor is, where the magnetization per unit volume (V) is defined as, A is a constant depending on the sample geometry, J c is a critical current density of a superconductor and r is the radius of a shielding current loop [3]. This formulation indicates that it is necessary to have r, J c and as large as possible to acquire a high levitation force.

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2. Modelling of bulk superconductor magnetization 2.1 The H formulation for modelling the critical state using FEM in COMSOL 10 The space over which the magnetic field is modelled consisted of superconductor, air and normal conductor subdomains. The field lies in the plane whilst induced superconducting currents and forced normal conductor currents are perpendicular to the plane flowing azimuthally. The critical state in superconductors can be modelled using FEM. The non-zero resistance below J c takes account of flux creep. The E-J law used in the model was Where E 0 = 1 ×10 −4 V m −1 and n = 21 are typical values for a type-II superconductor. (6) (7) The critical state in superconductors can be modelled using FEM technique in commercial COMSOL Multiphysics package [4]. Many different methods have been implemented, mostly in 2D, with the main two formulations solving for the magnetic vector potential (A-V formulations) and the magnetic field components. The latter formulation can be called the H-formulation and solves for the variables H x and H y which indirectly give current distribution. The critical state modelling reported in this section uses the H-formulation in COMSOL as first reported by Z Hong et al. [5], who applied the method to the 2D case of a bulk magnetized in uniform and non-uniform fields and the case of current forced through a superconducting wire. For the superconductor, the electric field is given by an E-J power law in which the E field is proportional to a high power of the current density. This form is a good practical approximation for the resistance of a bulk superconductor [6, 7]. (6) Fig. 7

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The E-J law used for the air and normal conductor subdomains was simply Ohm’s law.. If the critical current density J c is field dependent, then J c (B) relation expressed as Kim model in 11 (9) (10) Combining Equations (8)-(10) in (7) to eliminate J and E , the final two PDEs expressed in terms of the field components is given by : (11)

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H r = 0 H z = H 0 sin t 12 The r = 0 boundary condition was simply H r = 0 as no radial applied field components are allowed by symmetry. The internal SC-air and NC-air boundary conditions were continuity of the electric and magnetic field components parallel to the boundary surface and these are described as: A Dirichlet boundary condition was used for the outer boundary and r = 0 boundary. When applying a uniform external field to magnetize the superconducting region, the outer boundary condition was, (12) For modelling the coefficients for both subdomains shown below give Equation (11) when substituted into (12). (13)

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13 The mesh used to model a 7x6mm superconducting domain in axial symmetry exposed to a uniform external magnetic field (the total mesh consisted of 1204 elements). Fig. 8

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14 Kim term Lorentzian (Peak) term with b=B/B0 ; J0 and B0 are the original parameters of the Kim model. The secondary peak is represented by the Lorentzian term characterized by its center position b1, its width b2, and the relative amplitude a [8]. Subdomain x=0.00 at 50K 12 Jc(B) Jc0*(1+D*sqrt(abs(B))+B0/(B0+B)+a/((B/B0-b1)^2+b2^2)) E _scrho_air*J _scE0*(Jz_sc/Jc)^n Subdomain x=0.15 at 77K 12 Jc(B) Jc0*(B0/(B0+B)+a/((B/B0-b1)^2+b2^2)) E _scrho_air*J _scE0*(Jz_sc/Jc)^n

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15 3. The Comparison of the experimental and modelling results (a) Magnetization loops from the FEM modelling of a 14 mm diameter with 6mm thick (Sm123) 1 x (Yb211) x bulk exposed to a uniform external field of H(t) = (B 0 /μ 0 )sin(10πt) in axially symmetric geometry (B 0 =2.74T) for a full cycle of the applied field in 5 Hz. (b) The experimental magnetization loops of (Sm123) 1 x (Yb211) x bulk at 50K measurement temperature. (a) (b) Fig. 9 The magnetization of the sample in the z-direction can be calculated by

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16 (a) Magnetization loops from the FEM modelling of a 14 mm diameter with 6mm thick (Sm123) 1 x (Yb211) x bulk exposed to a uniform external field of H(t) = (B 0 /μ 0 )sin(10πt) in axially symmetric geometry (B 0 =2.96T) for a full cycle of the applied field in 5 Hz. (b) The experimental magnetization loops of (Sm123) 1 x (Yb211) x bulk at 77K measurement temperature. (a) (b) Fig. 10

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17 Fig. 11 The magnetization loop from the FEM modelling of a 14 mm diameter with 6mm thick (Sm123) 1 x (Yb211) x bulk for x=0.00 exposed to a uniform external field of H(t) = (B 0 /μ 0 )sin(10πt) in axially symmetric geometry (B 0 =2.74T) for a full cycle of the applied field in 5 Hz. The points on the curve represent the time steps shown in Fig. 12. Full penetration occurs between (c) and (d). Fig. 12 Magnetization modelling of a 14mm diameter with 6mm thick (Sm123) 1 x (Yb211) x bulk for x=0.00 exposed to a uniform external field in axially symmetric geometry. Surface plot shows induced superconducting currents and arrows show the total magnetic field. The extended Kim model was used for J c (B) with Yb211 doping ratio dependent parameters. It can be clearly seen from the plots how the superconducting tries to oppose the penetration of the flux lines as the external field increases. When the external field decreases the superconducting also resists the motion of the flux lines and at the end of the cycle (ωt = 2π that is t=0.2s) the superconducting has trapped some of the field and remains magnetized.

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18 Fig. 13 The magnetization loop from the FEM modelling of a 14 mm diameter with 6mm thick (Sm123) 1 x (Yb211) x bulk for x=0.00 exposed to a uniform external field of H(t) = (B 0 /μ 0 )sin(10πt) in axially symmetric geometry (B 0 =2.74T) for a full cycle of the applied field in 5 Hz. The points on the curve represent the time steps shown in Fig. 14. Full penetration occurs in (d) t=0.06s as shown in Fig 14. Fig. 14 t=0.1s half cycle

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19 Fig. 15 The magnetization hysteresis loops modelled at 50K for different sample size of (Sm123)1-x (Yb211)x for x=0.00 doping ratio

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20 Fig. 16 (a) Magnetization loops from the FEM modelling of a 14 mm diameter with 6mm thick ( Sm123) 1 x (Yb211) x for x=0.15 bulk exposed to a uniform external field of H(t) = (B 0 /μ 0 )sin(10πt) in axially symmetric geometry for a full cycle of the applied field in 5 Hz. The three curves illustrate the results obtained when the power constant n value of the used in the modelling is changed. (E–J power law n = 17, 21 and 23 field-dependent Jc.) A simple extension of the model presented would be to calculate the power dissipated in the bulk as the applied field does work against the Lorentz force during flux penetration. This power, which is from the dot product of E and J integrated over volume is given by: This power can then be integrated over time to give the total energy dissipated, which will indicate how significant heating effects would be. Where the power n used was 21 which is typical value for type II superconductor. The E-J law used in the model was

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21 Trapped field curves for full cycle of applied field in (Sm123)1- x(Yb211)x sample for x=0.15 (at 50K measurement temperature). Trapped field data taken from above of the sample surface with 1mm measurement steps.

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4. Results In this study the effect of Yb211 doping on the critical transient temperature (T c ), c-lattice parameter, critical current density (J c ), volume pinning force (F p ), pinning mechanism and the vertical levitation force density properties of the MPMG-processed (Sm123) 1 x (Yb211) x samples were investigated. The (Sm123) 1 x (Yb211) x sample with x=0.25 shows the best J c (0) value which is almost five times larger than that of the undoped one at zero field and the measurement temperature at 77 K, though in the peak effect region the best J c performance is the sample with x=0.05 Yb211 doping ratio. It was determined as experimentally that the different structural properties of doped Yb211 from the main superconductor Sm123 compound change the J c (H), F p (H) and levitation force density performance of (Sm123) 1 x (Yb211) x samples depending on Yb211 doping ratio and applied magnetic field quantity. In addition, a finite element method code based on the critical state model is carried out to investigate the current distribution, the magnetization and pinning properties in the Yb211 doped Sm123 bulk superconductors depending on doping ratio. In study the numerical method is based on solving the partial differential equations time dependently and is adapted to the commercial finite element software Comsol Multiphysics 3.5a. It was seen that the numerical results shows a good agreement with the experimental data and the critical state modelling using FEM in COMSOL Multiphysics can be provides a useful tool for predicting the order of magnitude of magnetization, critical current density and levitation force for new bearing design of HTS superconductors.

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23 References [1] S.Y Chen, A. Gloter, C. Colliex, I.G. Chen, M.K. Wu, IEEE Trans. Appl. Supercond. 17 (2007) 2957 2960. [2] T. Goto, K. Inagaki, K. Watanabe, Physica C 330 (2000) 51 57. [3] K. Ozturk, S. Celik, A. Cansiz, Phys. Status Solidi A 206 (2009) 2569 2575. [4] A Patel, R Palka and B A Glowacki, Supercond. Sci. Technol. 24 (2011) 015009 (8pp) [5] Z. Hong, A. M. Campbell, and T. A. Coombs, Supercond. Sci. & Technol. 19 (2006) 1246 1252. [6] J. Rhyner, Physica C 212 (1993) 292-300. [7] Z. Hong, Q. Jiang, R. Pei, A. M. Campbell and T. A. Coombs, Supercond. Sci. Technol.20 (2007) 331–337. [8] T. H. Johansen, M. R. Koblischka, H. Bratsberg, and P. O. Hetland, Phys. Rev. B 56 (1997) 11273 11278.

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