t affiliated with the authors of this page or responsible for its content.
ac Susceptibility Measurements in High-T
ac Susceptibility Measurements in
High-T
c
Superconductors
Experiment ACS
University of Florida Department of Physics
PHY4803L Advanced Physics Laboratory
Objective
You will learn how to use an alternating cur-
rent susceptometer to study the magnetiza-
tion induced in various magnetic materials in
response to the alternating magnetic eld in-
side a driven solenoid. In particular, a high-
T
c
superconductor is studied as a function of
temperature near the superconducting transi-
tion. A two-phase lock-in amplier is used to
measure the amplitude of the sample magne-
tization and its phase relative to that of the
driving magnetic eld.
References
M. Nikolo, Superconductivity:
A guide
to alternating current susceptibility mea-
surements and alternating current suscep-
tometer design, Am. J. Phys. 63, 65
(1995).
W. Lin, L.E. Wenger, J.T. Chen, E.M. Logo-
thetis, and R.E. Soltice, Nonlinear mag-
netic response of the complex AC suscep-
tibility in the YBa
2
Cu
3
O
7
superconduc-
tors, Physica C 172, 233-241 (1990).
C. Kittel, Introduction to Solid State Physics,
6th ed. (Wiley, 1986). See Chapter 12
and Appendix K.
N. W. Ashcroft and N. D. Mermin, Solid
State Physics, (Holt, Rinehart and Win-
ston, 1976).
Chapter 34 gives intro-
ductory descriptions of superconductiv-
ity and lists references for more advanced
reading.
W. E. Pickett et al., Fermi Surfaces, Fermi
Liquids, and High-Temperature Super-
conductors, Science 255, 46 (1992).
M. Tinkham, Introduction to Superconductiv-
ity (McGraw-Hill, 1996).
Introduction
The equipment list for this experiment in-
cludes a vacuum pump, liquid nitrogen (LN
2
)
cryostat, ac susceptometer, function genera-
tor, two-phase lock-in amplier, thermocou-
ple temperature sensor and 5 1/2 digit multi-
meter, computer data acquisition system, and
YBa
2
Cu
3
O
7
(YBCO) superconductor. The-
oretical considerations include superconduc-
tivity, magnetization, susceptibility, Faradays
law of induction and ac circuit analysis. Many
experimental and theoretical aspects will only
be treated at a level necessary to understand
their application to this experiment. The in-
terested student is encouraged to explore the
ACS 1 ACS 2
Advanced Physics Laboratory
references, textbooks, and other material to
ll in the gaps.
High-Tc superconductor
Superconductivity was discovered in 1911 by
H. Kammerlingh Onnes in Leiden.
After
decades of detailed experiments, a thorough
understanding of conventional superconduc-
tors became available in 1956 with the ad-
vent of the Bardeen-Cooper-Schrieer (BCS)
theory. There are now hundreds of known
compounds which exhibit this remarkable phe-
nomenon. The eld had a major upheaval in
1986, when Bednorz and M¨uller at the IBM
Research Laboratory in Zurich discovered su-
perconductivity in La
2x
Sr
x
CuO
4
. The dis-
covery has led to a new class of oxide super-
conductors with signicantly higher transition
temperatures T
c
. These high-T
c
oxide super-
conductors are now a subject of intense ex-
perimental and theoretical study, as there are
experimental indications that they dier from
conventional superconductors. They also pro-
vide a convenient demonstration of supercon-
ductivity in the laboratory, requiring only LN
2
as the cryogen.
Superconductors possess a unique collection
of physical behaviors: zero electrical resis-
tance, the Meissner eect (perfect dc diamag-
netism), an energy gap in their electronic
excitation spectrum, the quantization of mag-
netic ux, and the Josephson eect. This ex-
periment studies their magnetic response, in
particular their ac diamagnetism at low fre-
quency (¯h
).
Any conductor will partially shield its in-
terior from changing magnetic elds through
the generation of eddy currents. Because of
resistive losses, the shielding is incomplete. In
a perfect conductor (zero resistance material)
the shielding becomes perfect; the magnetic
eld cannot change in a perfect conductor.
Figure 1: The Meissner eect in a superconduct-
ing sphere cooled in a dc magnetic eld. Upon
passing below the transition temperature, the
eld is expelled from the material (right).
Superconductors are perfect conductors, but
they shield their interiors from dc magnetic
elds as well. Not only dB/dt but also B is
zero inside a superconductor. The expulsion
of static magnetic elds from the interior of
superconductors is called the Meissner eect
and will not be considered in this experiment.
However, it should be pointed out that the
Meissner eect is not a consequence of zero
resistance. Figure 1 (left) shows a supercon-
ducting sphere in a dc magnetic eld above
the transition temperature. The material is in
the normal state and the eld completely pen-
etrates it. The sample is then cooled into the
superconducting state and spontaneously gen-
erates eddy currents completely expelling the
magnetic eld. If the superconducting transi-
tion is simply a transition to a perfectly con-
ducting state (where dB/dt = 0), the mag-
netic eld could not change on cooling through
the transition, and would be trapped in the
material. Superconductors are more than per-
fect conductors!
September 24, 2008 ac Susceptibility Measurements in
High-T
c
Superconductors
ACS 3
Susceptibility
A magnetic material is described by its magne-
tization M(r), or magnetic dipole moment per
unit volume at points r throughout its volume.
Recall that the SI unit of magnetic dipole mo-
ment is A·m
2
and thus the SI unit of magne-
tization is A/m.
Determining the magnetic eld associated
with an arbitrary magnetization distribution,
M(r), can be dicult.
However, if the
material is in the shape of a long cylinder
(solenoidal) and the magnetization is constant
and points along the cylinder axis, the situa-
tion simplies.
Before considering this special case, recall
that for a long solenoid carrying a current I,
the magnetic eld outside the solenoid is zero
and inside B = µ
0
nI where n is the solenoids
number of turns per unit length. The eld
depends only on the product nI which can be
called the current density and has SI units of
A/m, the same as M.
The microscopic model for bulk magnetiza-
tion is that it arises from the contributions
of a great many atomic or molecular current
loops. In the interior of the material, the cur-
rent from adjacent loops are in opposite direc-
tions and these currents produce no magnetic
eld. (The cancelation is strictly correct only
if the magnetization is constant.) On the ma-
terials boundary, however, the loop currents
are unopposed, and lead to a surface current
density (current per unit length) M×n, where
n is the surface normal.
For a constant axial magnetization M
throughout a long, cylindrically-shaped sam-
ple, the surface current density is constant and
equal to M. This current circulates around
the cylinder like the current in a solenoid, and
the magnetic eld is the same as that of a
solenoid with nI = M, i.e., inside the ma-
terial B
in
= µ
0
M (a constant), and outside
B
out
= 0.
If such a sample is placed in a uniform mag-
netic eld B
a
oriented parallel to the sample
axis, the net magnetic eld is the superposi-
tion of the applied eld and the eld due to the
magnetization. If the external eld is written
B
a
= µ
0
H
a
, (which denes the magnetic in-
tensity H
a
) the magnetic eld outside is just
B
out
= µ
0
H
a
and the eld inside becomes
B = µ
0
(H
a
+ M)
(1)
For some materials the magnetization is
zero in the absence of an applied eld and only
becomes non-zero as a response to the applied
eld. For linear response materials, the mag-
netization will be aligned with (or against) the
eld and proportional to it. That is, we can
write
M = H
a
(2)
which denes the materials susceptibility .
Equation 1 can then be expressed
B = µ
0
(1 + )H
a
(3)
For paramagnetic materials, > 0 and the
magnetic eld is enhanced in the presence of
the material. For diamagnetic materials, <
0 and the eld inside the material is reduced.
Non-linear materials in dc magnetic elds
can show saturation eects, hysteresis, and
magnetization containing terms proportional
to higher powers in the applied eld.
If the applied magnetic eld is not constant
but is instead oscillating sinusoidally along the
sample axis at a frequency , i.e.,
B
a
= µ
0
H
0
cos t
(4)
the situation (for a linear magnetic material)
is only slightly more complicated than that for
a dc eld. The general form for a materials
magnetization in an ac magnetic eld is also
sinusoidal at the frequency of the applied eld,
September 24, 2008 ACS 4
Advanced Physics Laboratory
but it can be shifted in phase relative to the
applied eld. Relative to the applied eld as
given by Eq. 4, the general form for the mag-
netization can be expressed
M = H
0
( cos t + sin t)
(5)
Phasor representations are very useful for
working with ac quantities. Recall phasors as
rotating vectors in the complex plane. Eulers
identity in the form
e
it
= cos t + i sin t
(6)
illustrates the rotational behavior of a pha-
sor of unit length with oscillating real (cos t)
and imaginary (sin t) components. Also re-
call that the real physical quantity associated
with a phasor is the projection of the phasor
on the real axis and is obtained by taking the
real part (abbreviate