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Magnetic levitation of YBa2Cu3Oysingle crystals W.C. Chart a,., D.S. Jwo a, Y.F. Lin a, y . Huang b a Department of Physics, Tamkang University, Taiwan 25137 b Materials Science Center, National Tsing Hua University, Hsinchu, Taiwan 30043

Received 5 April 1994; revised manuscript received 3 June 1994

Abstract By using an ordinary electric balance, we are able to measure the magnetic levitation forces acting on a superconducting YBa2Cu30 r (YBCO) single crystal. It is found that when the external magnetic field is parallel to the c-axis of the single crystal, the hysteretic levitation curve is similar to that of a melt-processed YBCO superconducting sample. However, when the external magnetic field is perpendicular to the c-axis of the YBCO single crystal, the levitation forces are too small to be measured by our equipment. Also, we have introduced a simple model with the Bean's approximations to explain the levitation forces. The critical current density derived from this model by fitting with experimental data is quite close to the value obtained from magnetization measurements.

1. Introduction Ever since the discoveries of high-temperature superconductors [ 1 ], magnetic levitation forces of YBCO superconductors have been measured by many research groups with different experimental setups [ 2-6 ]. Theoretical investigations of these levitation forces have given some agreements with the experimental data at some particular magnetic fields [ 59 ]. However, since the magnetic field o f the permanent magnet ( P M ) is usually not uniform, a detailed calculation o f these levitation forces is very difficult. In this paper, we will first introduce an experimental method to measure the levitation forces by the use of an electric balance [4] for two superconducting YBCO single crystals with volumes much smaller than the volume of the PM. Then, we will use a simplified model with the Bean's approximations [ 10,11 ] to find the critical current density Jc of the two super* Corresponding author.

conducting samples (SS) from our levitation force measurements. Also, we will compare these values o f critical current density with the values obtained from magnetization measurements. Hopefully, this will offer us another simple method, instead of performing magnetization measurements, to find the critical current density of a SS at low magnetic field and liquid nitrogen temperature.

2. Experimental The superconducting YBCO single crystals were grown from a mixture of stoichiometric YBCO powder and Ag20 powder at the weight ratio of 3 to 1 by a special process as described in Ref. [ 12 ]. Two single crystals of different sizes were chosen for our measurements. The dimensions of the large sample are approximately 1 . 2 × 0 . 8 × 0 . 6 m m 3 with 3.6 mg mass, and the dimensions of the small sample are approximately 1.0 X 0.8 × 0.2 m m 3 with 1.0 mg mass.

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Magnetization measurement at 77 K and 1 kG gives a critical current density of ~ 1.3X 10 4 A/cm 2 for magnetic field parallel to the c-axis and a critical current density of ~ 0.25 X 104 A / c m 2 for magnetic field perpendicular to the c-axis. A cylindrical N d - F e - B permanent magnet 0.63 cm long and 0.63 cm in diameter with its dipole axis in the vertical direction was used. The axial components of the magnetic field H~(0, h) of the PM as a function of vertical distance h were measured by a gaussmeter and the results are shown in Fig. 1. Magnetic levitation forces as a function of the vertical distance between the PM and the SS were measured by an ordinary electric balance as shown in Fig. 2. Special care must be taken in order to reduce the amount of small drops of moisture that condense on the surface of the PM. A glass tube was used to insulate the PM from the inner container of the liquid nitrogen. The readings of the electric balance as a 3000 x

o -._/ N ~-I-

2500 1 ~ ~ ~

2000 1SO0 l

a

t

e

Experimental data

d

3. Calculations Let us consider a large PM placed at a height h above a small SS (assumed to be cylindrical) as shown in Fig. 4. Then, the vertical force dFz acting on the ring at position (r, z) due to the magnetic flux density B (r, z + h ) of the PM and the current j , dr dz flowing inside the ring can be expressed as dFz = (2nr) (j'~ d r d z ) B r ( r , z + h ) ,

values

( 1)

where Br is the radial component of B. If the dimensions of the SS are very small in comparison with the PM, then

IJ_ L~ 1000 O) E::: ET~ 500 0

function of the vertical distance without the SS were taken first before taking the data for the magnetic levitation forces. It was found that these readings also increased as h decreased but the maximum reading was less than three dynes. Fig. 3 shows the two hysteretic levitation curves for the two YBCO single crystals with their c-axes in the vertical direction. One can see that these two hysteretic levitation curves are quite similar to the results obtained for the melt-processed YBCO samples [4,9]. When the c-axes of the single crystals are in the horizontal direction, the levitation forces are too small to be measured by our equipment.

i 0.5

- I"" .~A "- J- z 1.5 2

i 1

V er t ic a l D i s t a n c e

\ Or lr=o,:=O

\ Oz 1~=o,:=o

2.5

h ( cm )

Fig. 1. Axial components of the magnetic field H, of the PM as a function of the vertical distance h. Symbols represent experimental data and solid line represents values calculated from Eq. (6). ( ~ Electric Balance (~) Cylindrical Magnet (~) Liquid Nitrogen (~) Sample

150 ~'X c

~. N IJ_

+

,

+

Large sample

100

50

u kl_ 0

g 4--'

(~) Supporting Frame (~) 3 - D Micrometer (~

Glass Tube

Fig. 2. Experimental setup for measuring the levitation forces of a small superconducting sample.

2 > ~ _..1

-50

o'.(

o'.~

0:8

Ve rti ca l D i s t a n c e

~

,'.~

1.4

h ( cm )

Fig. 3. Hysteretic levitation curves for two YBCO single crystals. Symbols represent experimental data and solid lines represent values calculated from our model.

W.C. Chan et al. / Physica C 230 (1994) 349-353

351

side the SS. For detailed calculations, the range of the magnetic field H, at the cylindrical surface of the SS is divided into three different regions, as shown in Fig. 5. For the case of decreasing field, we have drawn here only the pictures with the maximum magnetic field H, lying in the region 2H*> H,- H,, > H*, which corresponds to the values of the two parameters H,, and J, we will use in our calculations. As for the other values of H,, the calculations can be performed in similar ways. 3.1. Increasing magnetic field

(1) ForHz

Fig. 4. A simplified calculations.

@

’ dr model for magnetic levitation

force

aB

--!I

0

g-~HzIKl

(6)

and, from Eq. (4)) one has F, = 2vH,B,

Supposing that the magnetic field of the PM is symmetric about its own axis, then one also has:

(az> =o

1

j = &I~ y

(A)

,

(7) (B) Dscreaaing field with

Increasing field

H

l

< Hzm-Hc 1<2H

l

1. Hz < Hcl

vz.

(3)

r=O

By using Maxwell’s equation V-B=O, the total levitation force F, becomes F==-2~(%r2j~dr)[t(~)~=,==~].

(4) 2. Hcl C Hz < Hcl+H*

Here, we have neglected the z-dependence of j, because B, is very small at r close to zero in comparison with B,. As shown in Fig. 4, the SS is divided by the penetration depth rZinto two parts. When the magnetic field H,(g, h+z) (rH,(O, h+z) =H,) at the cylindrical surface of the SS does not exceed H,,, the first critical field of the SS, the induced current is only due to the Meissner effect. However, when HZis larger than H,r, one also has to consider the induced current caused by the critical state of the sample. By using the Bean’s model, one has: J$s=fL,

3. Hz > Hcl+H

l

(5)

where j, is the critical current density of the SS. The positive or negative sign depends upon whether the total magnetic field HZ is increasing or decreasing in-

Fig. 5. Distributions of total magnetic field IT&r, z+ h) (solid lines) inside the SS. Here H’ stands for (a,).

144C.Chan et aL / Physica C 230 (1994) 349-353

352

where v= r~g2t, stands for the volume of the SS and it is assumed for all our calculations that 2 << g. (2) For H~l < H - < H * + H ~ L ,

~H~l/2

g-2<~r<~g,

Jo

g+ <~r<~g-2,r<~g+,

J~

(8)

and

•

[Hd/2 Jc

g--2<~r<~g. r<~g-2.

c

[-I ~< ~2

(18)

(9)

(10)

and F: =2vB=,(Hc~ + H * / 3 ) .

(17)

In order to simplit}' our calculations, the magnetic field tt=(0, h ) of the PM is approximately calculated with z-direction distributed current loops (I), of the same radius b and height c as the PM, by the following formula:

d =]~1(1 -k2-2)

where g+ = g - ( H : - H~ ) /Jc. (3) For H*+H~

J~-~

F_=2t,B=,[ttc,+(lt*/3)(2(g_/g)3-1)].

dl

~ = 2vB.,[H~, +(H=-H,.,) × (1 + (g+/g) + (g+/g)2)/3],

and

(11

3.2. Decreasing magnetic field

The solid curve shown in Fig. 1 represents our calculated results for the magnetic field H=(0, h) with the two fitting parameters ],1 = 15 and ],2 = 7/c ~. One can see thai they are quite close to the measured values. A complete calculation on the levitation forces 1:_ as a function of the vertical distance h between the PM and the SS is done on a personal computer. The calculated results are also shown in Fig. 3. The fitting parameters we used are: tt~.~= 30 G and j~ = 0.8 × 104 A / c m 2.

(1) F o r / / ~ < Hcl,

(Ftcl/~ g-2H_~/H~t <~r<~g, • J 0 g-2<~r<~g-2H.,/H~ , J~ ~- ~ -.J~ go <~r <~g - 2, k J~ r<~go,

4. Discussion and conclusion (12

and

F: =2vBz,[H:+ ( H * / 3 ) ( 2 ( g o ~ g ) 3 - 1)] ,

(13

where g o = g - (H_-m-H~, )/2L. (2) For Hc~

~

H~/2 L ~ - ] -Jc I k Jc

g-2<<,r<~g, g - A - g _ <~r<~g-2, r<~g-2-g_,

(14)

and

F:=2vBz,[Hcl + ( H * / 3 ) ( 2 ( g _ / g ) 3 - 1 ) ] ,

(15)

where g_ = g - ( H ~ - H : ) / 2 j c . (3) For HcI + H*

-jj

g-2<~r<<.g, g-2-g_<~r<~g-2, r<~g-2-g_ ,

In this investigation, we have introduced a method to measure the hysteretic levitation curves for two small YBCO single crystals. We have also introduced a theoretical model with Bean's approximations to find the levitation forces for small SS. From Fig. 3, one can see that our calculated results agree quite well with the experimental data. Also, the critical current density of the two single crystals found in our calculations ( j c = 0 . 8 × 10 4 A / c m 2) is not too far off from the magnetization measurements (jc= 1.3× 104 A / cm z). That means, by measuring the hysteretic levitation curve only, one can also obtain the critical current density of a small superconducting sample at low magnetic field.

Acknowledgement (16)

This project was supported in part by the National Science Council of Taiwan with grant No. NSC-830212-M-032-010.

W.C. Chan et al. / Physica C 230 (1994) 349-353 References

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