Department of condensed matter physics
Přírodovědecká fakulta, Masarykova univerzita, Brno
Physical laboratory 4
Magnetization curve
Task goals
Measurement options:
• Calibration of the magnetometer with a known sample (nickel plate) – mandatory
part.
• Changes in magnetization of an iron sample after annealing for diﬀerent cooling rates.
• Comparison of hysteresis loop with magnetic ﬁeld shielding permeability measure-
ments.
• Paramagnetic and diamagnetic susceptibility, anisotropy of diamagnetic susceptibility
in graphite.
• Magnetocrystal anisotropy – diﬀerences in magnetization according to diﬀerent directions
in a crystal.
Theory
The relationship between magnetic intensity H and magnetic induction B is given by
B = µ0 (H + M) , (1)
where M is the magnetization vector that gives the bulk magnetic moment density. The magnetic
moment is then the volume of the sample multiplied by the magnetization
m = V M. (2)
With the known density of the sample and its mass, we can then easily calculate the total magnetic
moment per atom. It is convenient to give the magnetic moment per atom in Bohr magneton units
µB = 9,274 · 10−24 Am2
. Paramagnetic and diamagnetic materials at normal temperatures show
a linear dependence of magnetization on the external ﬁeld
M = χH , (3)
where χ is the magnetic susceptibility, positive for paramagnetics and negative for diamagnetics.
Usually, the susceptibility is very small, in the order of 10−4 to 10−8. The relative permeability
is equal to mur = 1 +c hi. Ferromagnetic materials exhibit a nonlinear hysteresis dependence as
shown in ﬁgure 1. The main parameters of the hysteresis loop are the coercive ﬁeld Hc when
the magnetization is zero, the remanent magnetization MR, and the saturation magnetization.
It is then possible to introduce an external ﬁeld-dependent susceptibility as a derivative of the
magnetization
χ =
dM
dH
; . (4)
Magnetization curve 2
H
H C
S
R
M
M M
Obrázek 1: Hysteresis dependence.
Often the value is given around the origin, i.e., for zero external magnetic ﬁeld. This value is
important, for example, for the eﬃciency of transformers, where we are trying to work only in the
region of small far-saturation ﬁelds.
The change in magnetization of a ferromagnet is due to the movement of the material’s domain
walls and the volume of the individual domains. Defects in the crystal can slow down the movement
of the domain walls and lead to a change in the shape of the hysteresis curve. For example, we
can study the eﬀect of the cooling rate of the material on the shape of the hysteresis loop. During
rapid cooling (quenching), more defects usually remain in the material than during slow cooling,
when the process is closer to thermodynamic equilibrium.
Vibrating sample magnetometer
The vibrating magnetized sample in the vicinity of the coil changes the magnetic ﬁeld ﬂux, which
induces an electromotive voltage in the coil according to Maxwell’s equations
U = −
dΦ
dt
. (5)
m
pick−up coils
1
2
3
4
external field
H
Obrázek 2: Schematic drawing of the relative position of the sample as a magnetic dipole and the
detection coil.
A simple quantitative description of our experiment is possible in the approximation where
replacing the sample with a magnetic dipole. Furthermore, we will approximate the motion of the
sample close to the coil by a harmonic oscillation motion
z(t) = z0 + Az cos(ωt) (6)
Magnetization curve 3
with amplitude Az with mean position z0. The situation is shown in Figure 2. The magnetic ﬁeld
of a magnetic dipole is given by the relation [1, 2]
B(r) =
µ0
4πr3
3(r · m)r
r2
− m , (7)
where r is the position vector relative to the magnetic dipole, m is the magnetic dipole moment,
and µ0 is the vacuum permeability. The real arrangement uses four identical coils, denoted from
1 to 4. Consider ﬁrst the magnetic ﬂux through coil 1
Φ1(z(t)) =
S1
n · B(r)d2
r; , (8)
where the integration is over the surface of the coil.
This magnetic ﬂux depends on the dimensions and number of turns of the coil and the position
of the sample and is directly proportional to the magnetic moment of the sample m. The
quantitative calculation of the magnetic ﬂux is quite challenging; for small sample deﬂections, we
can approximate the dependence on the sample deﬂection by a Taylor expansion to the ﬁrst order.
Φ1(z(t)) ≈ m [−C0 − C1z(t)] . (9)
According the symmetry of the other coils their particular ﬂuxes are
Φ2(z(t)) ≈ m [−C0 + C1z(t)] , Φ3(z(t)) ≈ m [C0 + C1z(t)] , Φ4(z(t)) ≈ m [C0 − C1z(t)] . (10)
The resulting arrangement engages coils 1 and 4 with reversed polarity compared to coils 2 and
3, thus adding up the changes in magnetic ﬂux in each coil. For the magnetic ﬂux through the
coil assembly
Φ(t) = −Φ1(t) + Φ2(t) + Φ3(t) − Φ4(t) = 4mC1Az cos(ωt). (11)
To determine the voltage induced in the coil as the magnet moves, we use Faraday’s law (5). If
we perform the following the time derivative of the magnetic ﬂux (11), we obtain for the voltage
induced in the collection coils
U(t) = −
dΦ
dt
= 4mC1Azω sin(ωt) = mC
√
2 sin(ωt) , (12)
where C is the calibration constant of the magnetometer, determined by measuring a known
sample.
This arrangement also has the advantage of being insensitive to changes in the external magnetic
ﬁeld. The electromotive voltage induced by a change in the external ﬁeld is the same in
each coil. However, because two and two are connected with opposite polarity, the resulting induced
voltage is almost zero, except for variations due to inhomogeneity of the external ﬁeld and
diﬀerences between coils.
Figure 3 shows the schematic of the magnetometer used.
By changing the external magnetic ﬁeld (by changing the current ﬂowing through the electromagnet),
we can then measure the magnetization curve of the sample, i.e., the dependence of the
sample’s magnetic moment on the magnetic ﬁeld.
Lock-in power supply
We use a so-called lock-in ampliﬁer to suppress circuit noise. This device ampliﬁes the AC component
of the measured signal, which has the same frequency as the reference signal. It very
eﬀectively suppresses electromagnetic noise at frequencies other than the measured component.
This way, we can measure AC voltages in fractions of a microvolt. The reference signal source is
a small permanent magnet mounted on a vibrating rod outside the magnet and a reference coil
Magnetization curve 4
motor
pick−up coils
sample
vibration
electromagnet
Obrázek 3: Schematic layout of the magnetometer. The iron core of the electromagnet is plotted
in yellow, electrical windings in green, collecting coils in red, the sample in blue, and the vibrating
rod in black
.
in its vicinity. Thus, the reference signal has the same frequency and phase as the measured signal.
It is advisable to use an experimental frequency diﬀerent from the possible sources of interference;
in particular, it is a good idea to avoid multiples of the mains frequency of 50 Hz, and so on.
The lock-in ampliﬁer measures the centered value
Uout =
1
T
t+T
t
Uin(τ) sin(ωrefτ + φ)dτ; , (13)
where Uin is the input signal, ωref is the frequency of the reference signal, φ is the tunable phase
shift, and T is the selectable time constant of the lock-in ampliﬁer, always much larger than
the period of the measured signal. For a harmonic input signal with the same frequency as the
reference and zero phase shift, the output voltage is equal to the eﬀective value of the harmonic
component of the voltage (amplitude in formula (12) divided by
√
2):
Ulock−in = Cm, (14)
where C is the calibration constant of the magnetometer.
Demagnetization ﬁeld
For a correct interpretation of the measured data, the demagnetization ﬁeld in the sample must
still be considered. The magnetic ﬁeld inside the sample Hi, Bi diﬀers from that outside the sample
He, Be. The magnetic ﬁeld near the sample is aﬀected by it and is inhomogeneous. Obviously holds
Be = µ0He , Bi = µ0(Hi + M) . (15)
Furthermore, the continuities of tangential intensity components and normal components at the
interface are valid. The solutions are related to the ﬁelds inside the sample:
Hi = He − NM; , Bi = µ0(Hi + M) = Be + µ0(1 − N)M , (16)
where N is the demagnetization factor depending on the shape and orientation of the sample with
respect to the magnetic ﬁeld direction. Some values are given in table 1. In our case, we restrict
ourselves to samples from a thin plate with magnetic ﬁeld orientation in the plane of the plate.
The demagnetization factor is negligible, and the correction according to the relation (16) need
not be considered.
Magnetization curve 5
Tabulka 1: Table of demagnetization factor for some special cases.
ﬁeld shape orientation N
sphere arbitrary 1/3
thin plate perpendicular to the plane 1
thin plate in plane 0
long thin cylinder along the axis of the cylinder 0
long thin cylinder perpendicular to the axis 1/2
Magnetic ﬁeld shielding in a cylindrical cavity
Theory
The magnetic permeability of materials µ expresses the relationship between magnetic induction
and magnetic intensity B = µrµ0H and can be studied through magnetic ﬁeld shielding. For
non-ferromagnetic materials, the relative permeability µr takes values very close to one, so their
response in a magnetic ﬁeld is not very diﬀerent from that of a vacuum. The permeability of
ferromagnetics is related to the hysteresis curve; the permeability is proportional to the tangent
directive to the hysteresis curve. For ferromagnetics, it can take on very high values but depends
strongly on the magnitude of the magnetic ﬁeld. Magnetically soft ferromagnetics (small coercive
ﬁelds) have high permeabilities, while magnetically hard materials have low permeabilities. Magnetically
hard materials have permeabilities in the order of tens to hundreds, while magnetically
soft unique materials with high permeabilities can reach values as high as 106. We can assume
a linear dependence of B = µrµ0H with constant permeability for small magnetic ﬁelds and
magnetically soft materials.
Magnetic ﬁeld shielding in a cylindrical cavity
If we place a hollow cylinder of radius R in a homogeneous magnetic ﬁeld of magnitude Bo
perpendicular to the cylinder’s axis, the ﬁeld inside the cylinder is also homogeneous of lower
magnitude Bi. The complete calculation is somewhat tedious [3, 2], here we restrict ourselves to
stating the assumptions:
• Magnetic intensity and induction satisfy Maxwell’s equations in the absence of external
currents
divB = 0, rotH = 0. (17)
• A linear material relation relates the magnetic induction and intensity
B = µrµ0H. (18)
• The tangential component of the magnetic intensity is continuous at the interface of two
environments.
• The normal component of magnetic induction is continuous at the interface of two environ-
ments.
• The magnetic induction at a large distance r from the cylinder axis is
B(r ≫ R) = Bo. (19)
Figure 4 shows the resulting magnetic ﬁeld waveform. The ratio of the induction outside Bo to
that inside the tube Bi is given by the shielding coeﬃcient S, for which the relation [3, 2]
S =
Bo
Bi
=
(µr + 1)2 − b2
a2 (µr − 1)2
4µr
, (20)
Magnetization curve 6
Zdroj
Hall
trubka
A
V
Obrázek 4: Left: The magnetic ﬁeld lines in and around a hollow cylinder made of ferromagnetic
material with permeability µr = 10. In the close vicinity of the cylinder, the homogeneity of
the magnetic ﬁeld is somewhat disturbed. Right: Schematic diagram of the Helmholtz coils. The
position of the shielding tube and the Hall probe for measuring the magnetic ﬁeld are indicated.
where a is the outer radius, and b is the inner radius of the hollow cylinder. For high values of
magnetic permeability µr ≫ 1 and the small wall thickness of the tube d relative to its radius
d ≪ R, we can use the approximate relation
S =
Bo
Bi
≈ 1 +
µrd
2R
. (21)
The above relations hold for small magnetic ﬁelds. Especially inside materials with high permeability,
the maximum value of the magnetic induction (approximately equal to Bmax ≈ µrBo) can
easily exceed the saturation magnetization of the material (for iron about 2.2 T), and the overall
shielding coeﬃcient then comes out eﬀectively lower. This property appears as a dependence of
the shielding coeﬃcient on the external ﬁeld, which decreases with a larger external ﬁeld. The
low-ﬁeld permeability value is obtained from the low-ﬁeld shielding coeﬃcient values, where the
shielding coeﬃcient is independent of the ﬁeld strength.
Homogeneous magnetic ﬁeld in Helmholtz coils
Helmholtz coils are the simplest way to create a homogeneous ﬁeld. These are two coils of the
same number of turns and radius R placed on a common axis at a distance of their radius R from
each other. The magnetic ﬁeld of one coil can be calculated using the Biot-Savart law
H =
I
4πr3
r × dl, (22)
where I is the current ﬂowing through the conductor, r is the distance of the length element dl
from the ﬁeld measurement point. The magnetic ﬁeld on the axis of a narrow coil of radius R with
N turns at a distance z from the center of the coil is easily calculated as
H(z) =
NIR2
4π(R2 + z2)3/2
2π
0
dϕ =
NIR2
2(R2 + z2)3/2
. (23)
The magnitude of the magnetic in the center of the cavity of the Helmholtz coils is obtained as
the sum of the contribution of both coils (the distance of the center of the cavity from the center
of each coil is z = R/2)
H = 2
NIR2
2(R2 + (R/2)2)3/2
=
4
5
3/2
NI
R
. (24)
Magnetization curve 7
−5 0 5
20
22
24
26
28
30
32
34
36
poloha (cm)
H(A/m)
Civka 1 Civka 2
poloha (cm)
poloha(cm)
0.900.99
1.01
Osa civek
Civka 1 Civka 2
−6 −4 −2 0 2 4 6
−6
−4
−2
0
2
4
6
Obrázek 5: Left: Magnetic ﬁeld strength on the axis of Helmholtz coils with radius R = 5 cm.
Right: Distribution in the plane of the axis of Helmholtz coils. Shown are the contours for the
0.90, 0.95, 0.99, and 1.01 values at the center of the cavity. In the star-shaped region’s center
region, the magnetic ﬁeld magnitude deviation is less than 1%.
B
I
F
d w
L
V
Obrázek 6: The principle of the Hall eﬀect.
Hall probe magnetic ﬁeld measurements
Use the Hall eﬀect to measure the magnitude of the magnetic ﬁeld. When charge carriers in a
sample move in a magnetic ﬁeld (electrons or holes in a semiconductor), the Lorentz force acts on
them perpendicular to the direction of their motion
F L = qvd × B, (25)
where q is their charge and vd is their drift velocity. In the steady state, an electric ﬁeld EH is
generated, which eliminates the eﬀect of the Lorentz force
F H = qEH = −F L. (26)
We substitute the drift velocity vd = j
nq = 1
nq
I
wd, where j is the current density, n the concentration
of charge carriers, d the thickness and w the width of the sample. Then, by comparing these
relations, we get the relation for Hall stress
UH = EHw =
RH
d
I B, (27)
where RH = 1
nq is the Hall constant and d is the sample thickness. The sign of the Hall constant
corresponds to the sign of the charge carriers, thus allowing us to determine the conductivity type
and measure the charge carriers’ concentration. Conversely, a Hall probe of known parameters can
measure magnetic induction. We will use a commercial Hall probe with integrated current source
and amplifying electronics of unknown parameters.
Magnetization curve 8
Measurement progress
Make a magnetic ﬁeld measurement without the tube inserted, and measure the Hall voltage Uo
proportional to the external magnetic ﬁeld Bo. With the tube inserted, measure the voltage Ui
proportional to the magnetic induction inside the tube Bi. The shielding coeﬃcient S is equal to
the fraction of the voltage
S =
Uo
Ui
. (28)
To eliminate any possible mids-setting of the zero (zero voltage may not correspond to a state
without a magnetic ﬁeld), we make measurements for both commutations of the current (+ and
−) and then average the resulting voltage concerning the sign
U =
1
2
(U+ − U−) . (29)
Úkoly
1. Plug the Helmholtz coils into the circuit.
2. Measure the shielding coeﬃcient S and the dimensions of the set of cylindrical tubes provided.
Measure the external ﬁeld Bo at the center of the cavity without the shielding tube
inserted and the value of Bi after the shielding tube is inserted. Make measurements for
several current values through the coils (recommended values of 0.5 A, 1.5 A and 2.5 A) and
see if the shielding coeﬃcient is independent of the external ﬁeld strength. Measure for both
directions of the current commutation.
3. Calculate their permeability using the relation (21) or (20).
Recommended procedure and tasks for measurements
The student has three weeks to make the measurements. Recommended procedure:
• First week – familiarization with the magnetometer; calibration of the magnetometer using
a reference sample (nickel plate).
• Second and third week – measuring the magnetization curve for several samples according
to the target.
The output of the practicum will be presented to the instructor in the form of graphs and
measured dependencies, including an estimate of uncertainties where possible and appropriate
Reference
[1] D. Griﬃth, Introduction to electrodynamics, Prentice-Hall (1999).
[2] J.D. Jackson: Classical electrodynamics, Willey (1999).
[3] J. Perry, Proc. Phys. Soc. London 13, 227 (1894).