Physica C 216 (1993) 89-93 North-Holland PHMCA Ii

Behavior of superconducting samples levitated in a static magnetic field Study in a non-vertical configuration

Victor Sosa and J.L. Pefia Departamento de Fisica Aplicada. Centro de Investigacidn y de Estudios Avanzados de1 IPN, Unidad Mkrida, AP 73 Cordemex, 97310 Mkrida. Yucath. Mexico

Received 3 1 March 1993 Revised manuscript received 19 May 1993

The mechanical response of superconducting samples in a static magnetic field was investigated. Levitation of superconducting YBalCusOx samples was performed in an unusual configuration. Undamped oscillations were observed; most of the samples oscillated for times as long as 2 min, but a few of them showed a highly damped behavior. Volume magnetization of samples was estimated from the levitation; the magnetic stiffness was also measured, and we found a dependence on the density. The samples levitated rigidly after oscillating; a discussion in terms of flux pinning is given.

1. Introduction

It is well known that high-temperature supercon- ductors (HTSC) are type-II superconductors, i.e., they undergo flux penetration under magnetic fields higher than their lower critical field (H,, ). For the YBazCusO, (YBCO) compound at 77 K, H,, is around 100 Oe. This fact, together with strong flux pinning, allows the stable levitation of a magnet to occur above a HTSC sample, or vice versa, in a ver- tical configuration [ 1- 111. Moreover, magnets and samples can oscillate and/or rotate with respect to each other, with or without damping; this feature can be applied in superconducting bearings [ 12,13 1. However, there are only a few reports on levitation of HTSC samples above a magnet [ 1,3,10]; this is because of the problem of keeping the sample in the superconducting state. When pinning is strong enough, suspension of samples below [ 141 or even beside [ 151 a magnet is possible. Also, it has been shown that suspension can be achieved for samples with low pinning [ 161. Pinning improvement is very important for application of HTSC, especially for obtaining very high fields and thus for the devel- opment of technology (for instance, trains which could be magnetically levitated) [ 171. Various

models have been proposed for vertical levitation; most common are flux exclusion (Meissner effect ) and complete flux penetration [ 91.

The study of the mechanical response of super- conducting samples in a magnetic field gives infor- mation on their magnetic properties; usually, an al- ternating field is applied to the sample. Some techniques used are vibrating reed [ 18 1, torsional pendulum [ 19 3, mechanical pendulum [ 20,2 11, swinging beam [ 22,231, and mechanical oscillator [24,25]. However, most of the authors who have made quantitative experiments have studied only the vertical configuration; we only know of one excep- tion [26]. The origin of the restoring lateral force that causes oscillations is not very clear yet; it has been attributed to flux pinning [27,28], or to in- homogeneities in the distribution of the current den- sity [29,30]. It is also known that flux pinning can be eliminated by thermal effects, or by a long dis- placement of the sample relative to the lines of the magnetic field. In this case, lines behave as pluck springs, and this is manifested macroscopically by a resistance to motion, i.e., there is a rigid levitation (dry friction) [31].

In this work, we report measurements of the vol- ume magnetization and magnetic stiffness from lev-

0921-4534/93/$06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.

90 V. Sosa, J.L. Peiia /Samples levitated in a static magneticj?eld

itation and oscillation of YBCO samples placed be- tween two ring-shaped permanent magnets, having their axes in a horizontal direction. As far as we know, this is the first time that such a configuration is re- ported. Observation of dry-friction behavior is also possible.

2. Experimental procedure

The geometric contiguration is shown in fig. 1 (a). Ring-shaped ferrite magnets were placed into a liq- uid-nitrogen bath, and a ring made of aluminum (not shown in the figure) was put between the magnets to deep them separated. Their dimensions were: inner radii 3.5 and 5.2 cm, outer radii 8.9 and 10.4 cm, and thicknesses 1.7 and 1.4 cm, respectively. In equi- librium, the samples levitated at a position slightly above the edge of the smaller magnet. Samples stud- ied in this work have an average mass of 0.07 g, 1 mm thickness, and 3-5 mm length. They were pre- pared by a solid state reaction and cut from different pellets, with densities in the range of 2.5 to 6.1 g/ cm’; their critical temperature (Z’,) was 92 K. Dif- ferent separators between the magnets were used; for a distance larger than 3 mm, levitation was not achieved. The magnetic field was measured with a Hall probe, and a typical strength was 600 Oe at the magnet surface. This value is lower inside the sample due to demagnetization; however, this correction will be ignored. We used small iron filings to observe the magnetic lines in the levitation zone, and it can be

Permanent + magnets

L- --. __ Superconducting sample levitating _

_.I - - - Liyid Nitrogen

I I

a

concluded that the dominant component is the one perpendicular to the magnet surface; other compo- nents, if any, are negligible.

A magnetized material with a vector magnetic mo- ment p in an inhomogeneous magnetic field B undergoes a force given by [ 321

F=V(pB) ) (1)

and levitation is observed when the vertical com- ponent of this force equals the weight of the sample. In this way, it is possible to estimate the value of p from a measurement of the field gradient and the weight of the sample. The field gradient was deter- mined by measuring B at different points, separated by 1 mm.

Samples were initially cooled by putting them into the liquid-nitrogen bath. Then they were put be- tween the magnets by means of non-ferromagnetic pincers. After being released, YBCO samples always exhibited a stable levitation at a position located a few mm above the edge of the smaller magnet, as in- dicated in both figs. 1 (a) and (b). They oscillated freely around their equilibrium position when gently pushed with the pincers. Oscillations were observed to occur along the edge of the smallest magnet (fig. 1 (b) ), and a typical amplitude was 2mm . For this reason, we will consider that the motion is one-di- mensional, in the x direction. A light damping at- tributed to air friction was also detected. An alter- native way to move the samples was to push the magnets, giving a sudden rotational displacement around the symmetry axis.

b

Fig. 1. (a) Cross-sectional view of the experimental set-up. The magnets have their opposite poles facing, so a metallic ring is required (but not shown in the fgure), to keep them apart. (b) Frontal view of magnets and oscillating sample, as filmed by the camera. The amplitude of the oscillations was between 1 to 3 mm. This condition was maintained for 2 min.

V. Sosa, J.L. Peiia /Samples levitated in a static magnetic field 91

Since a sample exchanges heat with the atmo- sphere continuously, it will no longer levitate once its temperature reaches the T, value. The heating rate was monitored by using a type-K thermocouple lo- cated in the zone where the sample levitated, a rate of 4 K/min was measured. A maximum levitation time of 2 min was observed. Oscillation frequencies cf) were measured in two ways; 1) using a strobo- scope, and 2) filming the motion with an 8 mm cam- era and reproducing the tape at velocities 5 to 30 times lower than normal.

The measured frequencies were in the range be- tween 3.44 and 4.24 Hz. In both methods, an un- certainty of less than 5Oh was achieved. The hori- zontal magnetic stiffness (k) was then calculated from the relation

f= (k/m)“2(2n)-’ ) (2)

where m is the sample mass. The density was esti- mated dividing m by the apparent volume. This method provides the additional advantage of not having to consider other contributions to the stiff- ness, as in the pendulum and oscillator techniques [ 19-21,24,25].

3. Results and discussion

3.1. Levitation condition

We can assume that B=B(x, y)z (see fig. 1 (b) for the coordinate system), given the pattern that small iron filings form when they are placed between the magnets. Then, it can be assumed that the magnet- ization has the form p = - ,uz, and we take p as a pos- itive constant. Assuming homogeneity, we can write p =MI’, where M is the volume magnetization (ab- solute value) and I/is the volume of the sample, and then we can conclude that the magnetic force is given

by

F= - VMVB(x, y) )

and the vertical component is

F~=- vikfaB/ay.

Thus, the condition for levitation is

M= -Pgi (aBiaY) . (3)

The field gradient measured in the levitation zone was aB/ay= - 1 kG/cm. For a sample having p=4 g/cm’, a magnetization value of M=4 emu/cm’ is obtained. This is comparable to values measured by traditional methods (typically less than 10 emu/cm3 for YBCO at 77 K). It is worth mentioning that Marinelly et al. [ 261 did some careful measurements in order to calibrate a magnetic levitometer pro- posed by them, with a geometry and method similar to the ones presented here; however, they applied the study only to micron-sized particles. Since M de- pends on temperature and this is always increasing, at a certain time M will be too low to maintain the levitation condition, and the sample will fall down (and consequently, the oscillation will end). We ob- served that this happened at around 86 K.

3.2. Oscillatory motion

In fig. 2, we show the magnetic stiffness k as a function of the density of the sample. Values ob- tained here are similar to other measurements [ 2 1. It can be seen that a limit value is reached as the den- sity approaches 6.4 g/cm’, i.e., the ideal crystallo- graphic value.

The horizontal force from eq. ( 1) is given by

F, = - MVaB/ax .

Because of the symmetry, we can say that in the re- gion of levitation the field is symmetric in x and reaches its minimum value at the equilibrium point, which is also the origin (0, 0). Then, the field gra- dient is positive and we can assume as a first ap proximation that B depends quadratically on x:

B(x, y=O)=B(O, O)+tj?x2,

where fi= a2Blax2 IXzXO is a positive constant. Therefore, the horizontal magnetic force has the form

F,=--MV/3x,

and this restitutive force causes the oscillations [ 3 11. Harter et al. [ 15 ] conclude that forces for a lateral suspension must arise from complex interaction be- tween induced currents and fields in the sample. Davis [ 29,301 gives an explanation of lateral forces in a vertical levitation based upon the asymmetry of the supercurrents generated when the lateral dis- placement starts. Calculations of the stiffness using

92 V. Sosa, J.L. Pefia /Samples levitated in a static magneticjeld

60

20

10

0 1 2 3 4 5 6 7

density (g/cm3)

Fig. 2. Magnetic stiffness vs. density of the sample. The measurements were realized while the sample was levitating.

these ideas are quite complicated for this two-mag- net configuration and are not intended in this work. An additional contribution to the stiffness coming from flux pinning can be important: occasionally, some samples had a highly damped oscillation, with a damping constant of around 1.2 s - ‘. This also can be due to flux depinning, originating in large dis- placements and by a very low pinning potential in these samples.

3.3. Friction regime

As mentioned above, the samples eventually ob- eyed the free-levitation condition no more because of a partial loss of magnetization, and fell slowly onto the aluminum separator. However, we observed that the contact between sample and metal was rather weak, as shown in fig. 3 (a). Hence, the separator only provided the extra force required to hold up the sam- ple, i.e., the weight of the sample is balanced now by the sum of the residual magnetic force and the nor- mal force. This starts to happen when the temper- ature is 86 K, then, samples still remain in the su- perconducting state. However, they show now a different and well known behavior, with an oppo- sition to motion as if they were stuck in sand (fric- tion regime). This transition is a result of the de- pinning of flux lines originating when the samples fell, since the line distortion was too large. A thermal

METALLIC SEPARATOR

Fig. 3. (a) Position of the sample after levitation. The metallic separator supports in part the sample, and compensates the de- crease of the magnetic force originating in the decrease of the magnetization. Magnets are not shown. (b) Different positions taken by the sample when pushed by pincers.

effect also contributes to the flux depinning. After this, friction between sample and magnetic

lines remains and is opposed to any motion of the sample, but allows for a continuous range of stable positions. As suggested in fig. 3 (b), samples can be displaced along the separator and rotated with the pincers, and they stay in the position that they are put in, just as reported before [ 1,311. We observed this condition for more than 4 mitt, until supercon- ductivity was tinally lost and samples fell completely onto the separator. This took such a long time be- cause the separator was submerged in liquid nitrogen and in contact with the sample. It is interesting to

V. Sosa, J.L. Peiia /Samples levitated in a static magnetic field 93

note that our configuration allowed us to observe all these different magnetic behaviors.

4. Conclusions

We presented a configuration for the observation of the mechanical response of superconducting ce- ramics in a static magnetic field. With this geometry, we could study both levitation and oscillation of su- perconducting samples, although not strictly in a thermodynamic equilibrium state. Estimations of the volume magnetization were done, and we found val- ues of the order of those reported by other authors. The magnetic stiffness was measured directly, with- out any other contribution; this was done by the stro- boscopic method and by filming the motion, with an uncertainty of less than 5%. The magnetic stiffness presented a maximum value at a density of around 45O!6 of the ideal one, and the trend was to reach a constant value as the density was increased. This is because of a competition between the compactness of the samples and the density of fluxons inside it. After levitation, samples remained superconducting and presented a typical rigid motion due to flux depinning.

Ackowledgements

We thank Gaspar Montaiio, Leidy Vazquez, Rob- erto Sanchez and Monica Vales for their help in the realization of the experiments.

This work was supported in part by CONACyT- Mexico and SEP-Mexico.

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