522 MAGNETIC PROPERTIES OF SUPERCONDUCTORS
Fig. 16.5 Field penetration in a type II hard (flux-pinned) superconductor slab of thickness d, according to the Bean model. (a) Applied field increasing in steps H1, H2, H3. (b) Field decreasing in steps H3, H4, H5, H6. (c) Resulting magnetization behavior.
In a superconducting solenoid used to create a high field, the magnetic field at the midplane of the coil is directed along the axis of the solenoid, but in opposite directions at the inside and outside diameters. At the end-plane of the solenoid, however, the field direction is radial (Fig. 16.7). In both cases, the field is perpendicular to the direction of current flow, so superconductors for use in high-field solenoids must be tested in an appropriate field geometry. A short length of the conductor is placed across the inside diameter of a Bitter-type resistive magnet or a superconducting solenoid, connected to an appropriate current source, and anchored securely to resist the large lateral force resulting from the
Fig. 16.6 Geometry of the Kim experiments.
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16.4 SUSCEPTIBILITY MEASUREMENTS |
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Fig. 16.7 Field distribution around a superconducting solenoid.
flow of a large current perpendicular to a large magnetic field. Then at a series of fixed fields, the current is increased until a voltage drop across the short test length is first detected.
Note that if the superconducting conductor is a wire, there is no difference in the geometry of a conductor at the central plane or at the end-plane of the solenoid. In the case of superconducting tapes, the two geometries are not the same. At the midplane, the field is parallel to the tape surface, while at the end of the solenoid, the field is perpendicular to the tape surface. In this case, two tests may be necessary. In practice, the highest fields occur at the inside diameter of the coil, and at the midplane, so only this configuration may need to be tested.
16.4SUSCEPTIBILITY MEASUREMENTS
Low-field measurements are often useful in the study of superconductors, since they provide information on critical temperature approaching the condition H ¼ 0, J ¼ 0 (J denotes current density). A low-field magnetization measurement is effectively a measurement of susceptibility. Usually it is easiest to measure ac susceptibility, since it can provide a continuous signal (usually as a function of temperature) without requiring any relative motion of the sample and measuring coil. A common arrangement uses two identical pickup coils symmetrically located in a larger field coil. The two pickup coils are connected in series opposition, so their output is zero when no sample is in place. The sample is placed in one of the two coils, and the resulting signal is a direct measure of the sample moment. For greater accuracy, the sample can be moved between the two coils and the difference in signal recorded.
Since measurements over a range of temperature are usually required, a decision must be made between locating the pickup coils in the variable-temperature region, which gives good coupling to the sample and high sensitivity but means the pickup coil resistance will change with temperature, or locating the pickup coils outside the controlled
524 MAGNETIC PROPERTIES OF SUPERCONDUCTORS
temperature region, which keeps them at constant temperature and resistance but substantially lowers the coupling and the sensitivity. The voltage sensor is normally a lock-in amplifier locked to the frequency of the drive field or one of its harmonics.
Samples of the high-temperature cuprate materials often turn out to be granular superconductors; they consist of small superconducting regions or grains separated by boundary layers of different properties that act to couple the grains together. The properties of the grains are termed intrinsic properties, and those of the boundary layers are called coupling properties. Susceptibility measurements as a function of temperature are wellsuited to determine the critical temperature (or temperatures) Tc of a sample. The inphase susceptibility shows a sharp drop at Tc of the intrinsic material, and a further, more gradual drop beginning at a somewhat lower temperature as the grains are further shielded by superconductivity in the coupling regions. A resistivity measurement on the same sample will show zero resistance only when there is a continuous superconducting path through the entire length of the sample.
The out-of-phase susceptibility ideally shows a small but sharp peak at Tc of the intrinsic phase, corresponding to a power dissipation, and a larger, broader maximum at a slightly lower temperature, corresponding to the development of superconductivity in the coupling regions.
If a granular sample is mechanically pulverized, so that the grains are physically separated, the coupling effects disappear and the susceptibility shows only the intrinsic properties of the grains.
The susceptibility behavior of granular superconductors depends on the magnitude of the field applied during the measurement, and also to some extent on the frequency of the measurement. Some exploratory measurements are often needed to establish satisfactory measurement conditions.
Superconducting materials have a characteristic quantity l called the magnetic penetration depth, which is the equivalent thickness of the surface supercurrent-carrying layer. If the grains of a granular superconductor are small enough so that the volume of this surface layer is not negligible compared to the volume of a grain, then the physical
Fig. 16.8 Demagnetizing fields acting on a thin film sample with the applied field Ha perpendicular to the sample plane. Hdz is the demagnetizing field perpendicular to the sample plane. (a) Ferromagnet (x .. 1). The perpendicular demagnetizing field Hdz is large because of the large perpendicular demagnetizing factor, and this field acts to decrease the magnetization of the sample.
(b) Superconductor (x ¼ 21). The large perpendicular demagnetizing field in this case acts to increase the perpendicular component of magnetization.
16.5 DEMAGNETIZING EFFECTS |
525 |
size of the grain is larger than the size of the superconducting part, and results intended to show the total volume of superconductor will need to be corrected.
16.5DEMAGNETIZING EFFECTS
Demagnetizing fields are important in many magnetic measurements, and are not only important but non-intuitive in the case of superconducting samples. When dealing with magnetic materials, it is often reasonable to regard the magnetization M as a constant quantity, whose direction but not magnitude is influenced by the demagnetizing
Fig. 16.9 Calculated demagnetizing factors for superconducting cylinders (x ¼ 21). (a) cylinder magnetized along the axis. Lower curve is fluxmetric demagnetizing factor Nf; upper curve is magnetometric factor Nm. (b) Magnetometric factor Nm for cylinder magnetized along a diameter. Note that in (a) the x-axis is length/diameter and in (b) it is diameter/thickness.
526 MAGNETIC PROPERTIES OF SUPERCONDUCTORS
field. Superconducting materials do not have a saturation magnetization; in the fully superconducting state, the magnetization is always a function of the total field, which is the applied field plus the demagnetizing field. A magnetic thin film is difficult to magnetize perpendicular to the film surface, because the large demagnetizing field opposes the applied field. In a superconducting thin film, the large demagnetizing field aids the applied field, and magnetization is easier in the perpendicular direction than in the parallel direction. These two cases are illustrated in Fig. 16.8.
As noted in Chapter 2, demagnetizing factors depend on the susceptibility x of the sample material, as well as on the shape of the sample. Demagnetizing factors for cylinders with x ¼ 21, corresponding to the fully-shielded superconducting state, have been calculated by D.-X. Chen, E. Pardo, and A. Sanchez [J. Magn. Mag. Mater., 306 (2006) p. 135]. Figure 16.9 gives the results. As in Chapter 2, Nf is the fluxmetric demagnetizing factor, appropriate when the sample magnetization is measured with a tightly fitting coil around the midplane, and Nm is the magnetometric factor, which applies when the total sample moment is measured. The paper also lists values of Nf and Nm for axially magnetized cylinders with x between 21 and 0, but these are of limited practical utility since in the intermediate or the mixed state, the effective susceptibility depends strongly on the applied field.
APPENDIX 1
DIPOLE FIELDS AND ENERGIES
We consider a magnet, or dipole, consisting of two point poles of strength p, interpolar distance l, and magnetic moment m ¼ pl.
The field H1 of the magnet at a point P distant r from the magnet center and in line with
the magnet (Fig. A1.1a) is given by, from Equation 1.3, |
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H1 ¼ |
p |
p |
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2prl |
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¼ |
[r2 (l2=4)]2 |
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[r (l=2)]2 |
[r þ (l=2)]2 |
If r is large compared to l, this expression becomes |
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H1 ¼ |
2pl |
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2m |
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(A:1) |
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r3 |
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Similarly, the field H2 at a point P abreast of the magnet center (Fig. A1.1b) is the sum of the two fields H(þ) and H(2), equal in magnitude:
H2 ¼ |
2H(þ) cos a |
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2 |
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p |
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l=2 |
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¼ |
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2 |
=4) |
{r2 þ (l2=4)} |
1=2 |
r |
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þ (l |
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¼ |
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pl |
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[r2 þ (l2=4)]3=2 |
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Introduction to Magnetic Materials, Second Edition. By B. D. Cullity and C. D. Graham Copyright # 2009 the Institute of Electrical and Electronics Engineers, Inc.
528 APPENDIX 1
Fig. A1.1 Fields of dipoles.
If r is large compared to l, this expression becomes
H2 ¼ |
pl |
¼ |
m |
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(A:2) |
r3 |
r3 |
In Fig. A1.1c we wish to know the field H at P, where the line from P to the magnet makes an angle u with the magnet axis. The moment of the magnet can be resolved into components parallel and normal to the line to P, so that
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2(m cos u) |
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Hr ¼ |
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Hu ¼ |
m sin u |
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H ¼ (Hr2 þ Hu2)1=2 |
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(3 cos2u þ 1)1=2: |
(A:5) |
r3 |
Fig. A1.2 Interacting dipoles.
The resultant field H is inclined at an angle (u þ f) to the magnet axis, where
tan f ¼ Hu ¼ tan u :
Hr 2
We now want an expression for the mutual potential energy of two magnets. Figure A1.2 shows two magnets of moment m1 and m2 at a distance r apart and making angles u1 and u2 with the line joining them. The field at m2, parallel to m2 and due to m1, is
Hp ¼ Hr cos u2 Hu cos(908 u2):
The potential energy of m2 in the field of m1 is, from Equation 1.5,
Ep ¼ m2(Hr cos u2 Hu sin u2):
Combining this with Equations |
A.3 and A.4 gives |
Ep ¼ |
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m1 m2 |
(2cos |
u1 cos u2 sin u1 sin u2) |
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u2) 3cos u1 cos u2]: |
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Similarly, it can be shown that the potential energy of m1 in the field of m2 is given by the same expression. Thus Ep is the mutual potential energy of the two dipoles. It is also called the dipole–dipole energy or the interaction energy between the two dipoles. It is fundamentally a magnetostatic energy.
APPENDIX 2
DATA ON FERROMAGNETIC ELEMENTS
TABLE |
A2.1 Data on the Ferromagnetic Elements |
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208C |
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0K |
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Element |
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ss (emu/g) |
Ms (emu/cm3) |
4pMs (G) |
s0 (emu/g) |
mH (mb) Tc, 8C ss/s0 |
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Fe |
218.0 |
1,714 |
21,580 |
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221.9 |
2.219 |
770 |
0.982 |
Co |
161 |
1,422 |
17,900 |
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162.5 |
1.715 |
1131 |
0.991 |
Ni |
54.39 |
484.1 |
6,084 |
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57.50 |
0.604 |
358 |
0.946 |
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TABLE A2.2 |
Relative Saturation Magnetization |
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ss/s0 |
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T/Tc |
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Fe |
Co, Ni |
J ¼ 21 (theory) |
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1 |
1 |
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1 |
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0.1 |
0.996 |
0.996a |
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1.000 |
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0.2 |
0.99 |
0.99 |
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1.000 |
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0.3 |
0.975 |
0.98 |
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0.997 |
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0.4 |
0.95 |
0.96 |
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0.983 |
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0.5 |
0.93 |
0.94 |
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0.958 |
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0.6 |
0.90 |
0.90 |
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0.907 |
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0.7 |
0.85 |
0.83 |
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0.829 |
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0.8 |
0.77 |
0.73 |
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0.710 |
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0.85 |
0.70 |
0.66 |
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0.630 |
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0.9 |
0.61 |
0.56 |
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0.525 |
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0.95 |
0.46 |
0.40 |
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0.380 |
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1.0 |
0 |
0 |
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0 |
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From American Institute of Physics Handbook, 2nd ed. (New York: McGraw-Hill, 1963). aValue for Ni only.
Introduction to Magnetic Materials, Second Edition. By B. D. Cullity and C. D. Graham Copyright # 2009 the Institute of Electrical and Electronics Engineers, Inc.
APPENDIX 3
CONVERSION OF UNITS
Note: Table A3.1 gives the conversion factors in the form of the ratio of the cgs unit to the SI unit. So to convert from the SI to the cgs unit, multiply the SI value by the value of the ratio. To convert a cgs unit to SI, multiply the cgs value by the reciprocal of the ratio.
Introduction to Magnetic Materials, Second Edition. By B. D. Cullity and C. D. Graham Copyright # 2009 the Institute of Electrical and Electronics Engineers, Inc.
