Effet of magnetization proess on levitation fore between a superonduting disk and a permanent magnet L. Liu, Y. Hou, C.Y. He, Z.X. Gao Department of Physis, State Key Laboratory for Artifiial Mirostruture and Mesosopi Physis, Peking University, Beijing 1871, P. R. China L. Xiao, H.T. Ren, Y.L. iao, M.H. Zheng General Researh Institute for Non-ferrous Metals, Beijing 188, China Abstrat The levitation fores between a permanent magnet and a oaxial superonduting disk after different magnetization proesses are measured. Signifiant effet of the magnetization proess on the levitation fore is observed. Theoretial alulations of levitation fore based on the ritial state model with temperature and field-dependent ritial urrent density, and the heat dissipation due to the flux motion are in perfet agreement with the experimental data. Keywords: superonduting disk, levitation fore, field ooling, zero-field-ooling PACS number(s): 74.6.2w, 74.25.Ha, 74.25.Ld Introdution It is well known that the very high trapped magneti field in bulk high temperature superonduting disk (SD) and the levitation fore between SD and a permanent magnet (PM) have possible industrial appliations [1-7]. Several groups reported their theoretial and experimental results in reent years. Qin et al. reported the theoretial alulation of the hystereti levitation fore between
a PM and a non-magnetized SD after zero-field-ooling (ZFC) from first priniples [8]. Wang et al. onsidered the magnetization of the SD, and alulated the levitation fore between a PM and a SD after field-ooling (FC) [9]. In the above alulations, however, the ritial urrent density as a funtion of the magneti field and temperature, as well as the heat dissipation in SD was not taken into aount. Ohsaki et al. onsidered both fators and alulated the distributions of the urrent density and temperature in SD after a magneti field pulse, but their alulations did not distinguish the normal state region from the superonduting state region in the pulsed field magnetization proess [1]. Hou et al. proposed a systemati theoreti framework to disuss the basi equation of the urrent motion in SD by taking into aount the heat dissipation, the temperature and field-dependent ritial urrent density, and the different urrent dependent eletri fields E( j ) in the normal state region and the superonduting state region [11]. Aording to this framework they simulated the trapped magneti field in SD, whih is in good agreement with experimental results of several groups [12-13]. However, no experiment results on levitation fore are reported so far. In this paper, the levitation fores between a PM and a oaxial SD were measured after different magnetization proesses. The rate β at whih the applied magneti field is swept down in FC proess was hanged and signifiant effet on the levitation fore was observed. The alulated levitation fores aording to the systemati theoreti framework [11] oinide with the experimental data very well. Experimental The SD used in experiments is a melt-textured-grown (MTG) YBCO sample of 3 mm in diameter, 2 mm in thikness [14]. The permanent magnet (Nd-Fe-B) is the same size and the magneti field on its surfae is.53t. The SD sample is set in a ontainer first. In ZFC proess the SD is ooled by liquid nitrogen to 77 K below its ritial temperature (92K) without applied magneti field. In FC ase, a
hosen magneti field (.5T,.375T,.25T generated by the eletromagnets) is applied on the SD along the axial diretion of the disk at room temperature, then the SD is ooled below its ritial temperature in liquid nitrogen and the applied magneti field is swept down at a given rate β (.1T/s or.25t/s) to zero. After above proess, we wait for 1 minute and set the SD on the sample holder in the levitation fore measurement system, whih is shown in Fig. 1. With a jak the SD an approah the PM the levitation fores in this proess are measured by a fore sensor, and the distane between the SD and the PM is measured preisely by a displaement sensor. These eletrial signals are transmitted to the omputer and proessed by LABVIEW program. Theoretial Basi equations for urrent motion Aording to the systemati theoreti framework [11] we onsider the SD with radius a and thikness b. The desired equation of motion for the urrent density (, t) r in the superonduting disk as following [9]: ( r, ) 1 a b = µ 1 ρ' ' (, ') ( ) ( ', ') r r yl + φ ρ t d dzq E A z (1) where A ϕ is the vetor potential of applied magneti field, yl (, ') = ( ρ, ρ', ') Q rr f z z (2) and 1 Q is the reiproal kernel, whih is defined by a b ' ' 1 (, ') ( ', '') = δ ( '') dρ dzq rr Q r r r r (3) with ( ρ, ρ', ') f z z = π dφ ρ'osφ 2π 2 2 2 ( z z' ) + ρ + ρ' 2 ρρ'osφ 12
1 ρ ' 1 2 2 2 = 1 k K 2 ( k ) E( k ) πk ρ and k 2 = 4 ρρ ' ( ρ + ρ' ) + ( z z' ) 2 2 Eq. (1) an be easily time integrated by starting with ( ) zt ρ,, = ( is the initial urrent density distribution in the SD disk) and then by putting (,, = + ) = (,, ) + (,, ) soon as the indued urrent density (, zt, ) ρ zt t dt ρ zt ρ zt dt. As ρ is obtained, the vetor potential generated by the indued urrent density A an be derived as, (, ) a b ρ µ ρ' ' (, ') ( ') A z d dz Q = rr r. (4) And the radial and axial trapped magneti field an be written in the form of, B ρ A =, z B z ( ρ A ) 1 = ρ ρ (5) respetively. These equations should be supplemented by relationships between and the magneti field B and the eletri field E, whih depends on the material. In the superonduting state region, where <, is the ritial urrent density, the power-n model is used to desribe the nonlinear harateristis of the superondutor [9]: = E E n (6a) n = σ + 1 and σ is the flux reep exponent, and in the normal state region, where, we use E( ) = E (6b) Generally the ritial urrent density depends on both the loal field B and the temperature T. The Kim model is used to desribe the flux density dependene of [1]: = B B + B (7) where is when B = and B is a parameter. We inlude the temperature dependene of as the following equation [1]:
2 2 T α = 1 T (8) where T is the ritial temperature at B = T and α is onstant. Heat dissipation When a superondutor is subjeted to a non-stationary external magneti field, the heat generation rate per unit volume is W = E (9) The temperature hange due to the heat generation is desribed by the heat diffusion equation. T κ = t 2 C T W (1) here κ is the thermal ondutivity, and C is the heat apaity per unit volume. The levitation fore Aording to the similar disussion as before [9], the top surfae enter of the SD approahes the bottom surfae enter of PM as, s = z vt (11) where veloity v represents the speed at whih the PM approahes the SD, z is the initial distane between the top surfae of the SD and the bottom surfae of the PM. As the urrent density (, zt, ) ρ and the radial magneti field PM B ρ inside the SD have been derived, the vertial levitation fore along the z-axis an be readily obtained as [9], Seletion of Parameters z a b PM = 2 π ρ ρ ( ρ, ) ( ρ, ) (12) F d dz z B z b ρ The parameters used in our alulation are taken as follows: T = 92 K [9]; µ 4π 1 7 = ; 6 3 C =.88 1 /m K [8]; κ 1 1 = 6 W m K [8]; 4 E = 1 1 V m [1]; 9 2 α = 1.9 1 A / m ; v = 2 mm s ; B =.5Τ and the flux reep exponent σ = 4. All these parameters remain unhanged
in the following disussions. Results and Disussions The experimental results of the levitation fore measured in both the FC and ZFC ases are shown in Fig.2. It an be observed that the levitation fore in the FC ase ( β =.25 T/s) has a muh larger value than that in the ZFC ase. Taking the distane of 1 mm for example, the levitation fore is 45 N and 1 N for the FC and ZFC ases respetively. In the FC magnetization proess, after the applied magneti field is swept down to zero, large indued urrent remains in the SD, whih dominates the levitation fore, while the indued urrent in the SD by PM s approahing is muh smaller than that after FC [9]. But no indued urrent exists in the SD after the ZFC. There is only a small indued urrent in the SD when the SD approahes the PM. So it is reasonable that the high levitation fore is obtained after the FC. The solid urves in Fig.2 represent our alulated levitation fores using Equation (12). The alulation results are in perfet agreement with the experimental data. How the rate β affets the levitation fore is also shown in Fig.2. At lower rate the gradient of magneti field in the edge is smaller, driving fore is weaker, and less magneti flux esape from the SD. Thus the trapped magneti field at the enter of the SD an be higher after the applied magneti field is swept down to zero and the indued urrent density in the SD is larger, whih is orresponding to higher levitation fore. When the rate β keeps onstant.25 T/s, the magnitude B of the applied magneti field beomes a ruial fator. Fig.3 shows the levitation fore as a funtion of the distane s with different B. We measure larger levitation fores with a larger applied magneti field, whih is reasonable beause a larger urrent is indued in the SD with a larger applied magneti field and thus generates a larger levitation fore when the PM approahes at a onstant veloity v.
Conlusion We have set up a levitation fore measurement system and measured the levitation fores between a permanent magnet and a oaxial superonduting disk after different magnetization proesses. The effets of the magnitude of applied magneti field and the rate on the levitation fore are studied in both experiment and theoretial alulation. The alulated results oinide with the experimental data very well, whih strongly supports the systemati theoreti framework proposed by Hou et al.. Aknowledgments This work was supported by the National Siene Foundation of China (NSFC 11744), the Ministry of Siene and Tehnology of China (Projet No. NKBRSF-G19996462) and the TaiZhao Foundation of Peking University. Referenes [1] B.R. Weinberger, L. Lynds,.R. Hull, and U. Balahandran, Appl. Phys. Lett. 59, 1132 (1991). [2] M. Matsunaga, et al. Superond. Si. Tehnol. 15, 842 (22). [3] H. Minami, and. Yuyama, pn. Appl. Phys. 34, 346 (1995). [4] L.K. Kovalev, et al. Superond. Si. Tehnol. 15, 817 (22). [5] T. Oka, et al. Physia C 335, 11 (2). [6] S.I. Yoo, T. Higuhi, N. Sakai, H. Fujimoto and M. Murakami, Mater. Si. Eng. B 52, 23 (1998). [7]. Wang, et al. IEEE Trans. Appl. Superond. 9, 94 (1999). [8] M.. Qin, et al. Phys. Rev. B. 66, 24516(22). [9].. Wang, et al, Superond. Si. Tehnol. 16, 527 (23). [1] H. Ohsaki, T. Shimosaki and N. Nozawa, Superond. Si. Tehnol. 15, 754 (22). [11] Y. Hou, et al. http://arxiv.org/abs/ond-mat/311499.
[12] G. Fuhs, et al. Appl. Phys. Lett. 76, 217 (2). [13] S. Gruss, et al. Appl. Phys. Lett. 79, 3131 (21). [14] H.T. Ren, et al. Physia C,282-287,485 (1997) Figure aptions: Fig. 1: Experimental setup for the levitation fore measurement. Fig. 2: The measured levitation fores F as a funtion of distane s with different β. The solid urves z are alulated results. Fig. 3: The measured levitation fores F as a funtion of distane s with different magnitudes of the z applied magneti field. Fig. 1
Fig. 2
Fig. 3