Download Magnetisation and Susceptibilty

Susceptibility and Magnetisation-Measurements
Evan Sheridan with Niall Robertson
11367741
February 2013
Abstract
Using a Hall Probe, electromagnet and a set of weighing scales the magnetic phenomena of paramagnetism and ferromagnetism are investigated. The current was calibrated with the magnetic field so that further measurements could be made, where
their relationship is found to be non-linear. Using a sample of paramagnetic Mohr’s salt
(χ ≈ 0.33) the field calibration constant C was calculated to be C = 31.88 ± 0.03m−1 .
Using this, the susceptibilty of Gadolinium Galium Garnet (Gd3 Ga5 O12 ) was found
to be χ = 0.44 ± 10. Using the Curie Law on Gd+
3 it’s susceptibilty was found to be
χ ≈ 0.8 providing itself as a benchmark. Using a sample of ferromagnetic hematiteαF e2 O3 hysteresis was exhibited where it showed a low covercity suggesting that it
may be a soft magnetic material. Hysteresis was investigated for Gd3 Ga5 O12 and the
results showed hints at ferromagnetism but the results were not concrete. However,
using the results the susceptibility of χGd3 Ga5 O12 ≈ 0.49 ± 0.05 which is quite close,
and within error, to the χ found using the force method.
1
Introduction and Theory
A central idea when discussing magnetism is the notion of a magnetic moment. There are
two equivalent definitions:
Magnetic Pole: Much in the same way as an electric dipole exists for electrostatic charges
a similar situation occurs for magnetic dipoles. Essentially, the magnetic moment is defined:
m
~ = p~l
Even though no magnetic monopoles exist, the magnetic field lines look similar to the electric
field lines generated by an electric dipole and hence the definition.
Current Loop: We consider a current loop of radius R of circulating current I. The moving
charge generated a magnetic field and the magnetic moment is defined as:
~
µ
~ = IS
The two defintions are equivalent and the units show that 1A.m2 = 1J.T −1 . Magnetisation
is directly related to magnetic moments, therefore demonstrating their importance.
The Types of Magnetism
Ferromagnetism: Occurs below the Curie Temperature TC and has a spontaneous magnetisation M due to the alignment of spins in a material into magnetic domains. A domain is
a region in the material where there is a uniform magnetisation. Without the application of
a magnetic field the domains will generally cancel out and there is no magnetisation in bulk
media. With the application of an external magnetic field all the domains1 will align and
the magnet is said to be magnetized. There are 4 ferromagnets at room temperature. The
response due to an external magnetic field is detailed by the Hysteresis Loop. Even when
the applied magnetic field is removed some of domains will remain pointed in the direction
of the field, giving rise to a net magnetisation in the absence of a field.
Figure 1: Domain Transition
Having computationally modelled2 a ferromagnet using the 2-D Ising Model here were the
results that illustrate the phase transition of a ferromagnet:
1
My friend, Ruarı́ Brett, has a nice gif of a magnetic phase transition, illustrating the formation of
domains at the Curie Temperature, here: http://www.maths.tcd.ie/~brettr/images/ising500.gif
2
The Ising Model code and results can be found here: http://www.maths.tcd.ie/~sheridev/Ising.
html
1
Figure 2: Ferromagnetic Phase Transition
In this experiment, since Gd3 Ga5 O12 is known to have a phase transition at room temperature, then the most relevant transition would be for the exchange constant J = 70.
Antiferromagnetism: Occurs below the Neel Temperature and neighbouring spins are
mutually opposite and give rise to a zero magnetisation. It is a manifestation of ordered
magnetism. Above the Neel Temperature the material is usually paramagnetic.
Paramagnetism: Occurs above the Curie Temperature TC for ferromagnets and if an external magnetic field is applied then there is an induced magnetisation. Suceptibilty is the
measure of the response of the material to the application of an external field. Paramagnets
have a positive susceptibilty. They will align to the applied field. The phenomenon of paramagnetism is mainly due to the presence of some unpaired electrons.
Diamagnetism: The materials will create a magnetic field that is in the opposite direction
to the applied field, and have a negative susceptibilty. Generally thought of as non-magnetic
behaviour because it is a property of all materials and is negligble if the material exhibits
any other form of magnetism.
Magnetic Hysteresis
This phenonema occurs when a ferromagnet is placed in a varying magnetic field. The first
part of the curve is labelled the “virgin curve”. As the field strength is increased the domains
in the ferromagnet tend to align with the field and at a certain strength of field the curve
asymptotically approaches a value and this is called magnetic saturation. The field is then
decreased to 0. However, the net magnetisation of the material is no longer zero. The value
of the magnetisaion at H = 0 is called the remanence. The field is then reversed and
increased until the magnetisation reaches zero again. The value of the H-field at this point
2
is called the covercity of the ferromagnet. The process is then repeated and the Magnetic
hysteresis loop is what ends up being traced out; illustrating how the current state of the
magnet is dependent on it’s past.
Figure 3: The Magnetic Hysteresis Loop
Such hysteresis is a method of creating permanent magnets. If the loop is narrow with small
covercity then it is a “soft magnetic material”. It it is wide with a high covercity it is know
as a “hard magnetic material”.
Curie’s Law
Curie’s law is given by:
M=C·
H
T
where C is the Curie constant such that:
µ0 N g 2 µ2B J(J + 1)
3k
giving the expression for the susceptibility (using B = µ0 H):
C=
χ=
N g 2 µ2B J(J + 1)
3kT
3
where in the case of Gd3 Ga5 O12 g = 2, J = 27 , T = 293K and:
µB =
eh̄
= 9.274 × 10−24 Am2
2me
N = 3N0 = 3 × (6.022 × 1023 )
k = 1.38 × 10−23 m2 kgs−2 K−1
the reson N = 3N0 is because in Gd3 Ga5 O12 the only magnetic ion is Gd+
3 and there are 3
moles of this.
Using these constants we come to a value of:
χ ≈ 0.8
for the susceptibility of Gd3 Ga5 O12 .
Experimental Theory
The force F on a magnetic dipole in a nonuniform magnetic field B is:
~
F~ = (m
~ · B)
writing in component form we get:
∂By
∂Bz
∂Bx
+ my
+ mz
Fz = mx
∂z
∂z
∂z
Therefore in a uniform magnetic field there is no force on a magnetic dipole.
In the experiment this expression reduces to:
Fz = mx
∂Bx
∂z
The gravitational force must also be taken into account in the experiment, when recording
the mass measurments the following formula is used:
FT ot = g(Mef f − Mi )
where Mef f is the “effective mass” of the sample because it’s inertia will change depending
on the magnetic force.
The magnetic susceptibilty (χ) of a paramagnetic or diamagnetic material is defined by the
following relation:
M = χH
it is a proportionality constant relationg M (the magnetisation of a material=density of
magnetic dipole moments) and H (the magnetic field strength). Equivalently it can be
expressed as:
4
M
H
where a paramagnetic material will have a postive susceptibility (it’s moments align with the
field) and a diamagnet will have a negative susceptibilty (the moments align anti-parallel).
χ=
For a ferromagnet the saturated magnetisation is given by:
σs =
Fz
mCBx
5
The Experiment
Part 1
We placed the Hall Probe in the region where the magnetic sample was going to be placed.
The potentiometer, which is proportional to the current, was varied such that the magnetic
field was recorded every 40 turns.
Part 2
A sample of Mohr’s salt of sufficient mass was placed in the sample holder.
The mass was measured and then the scales were zeroed so that when the magnetic field
was incresed we recorded the increasing change in mass.
We used the formula Fz = g(mef f − mi ) to calculate the magnetic force experienced by the
sample.
Part 3
We replaced the magnetic sample with a sufficient mass of Gd3 Ga5 O12 . We then repeated
the exact same procedure.
Using the data we calculated the susceptibility both theoretically and experimentally.
Part 4
We replaced the sample with a sufficient mass of hematite-αF e2 O3 .
We increased the magnetic field incrementally, measuring Fz as we did so, and then decreased
it back to zero.
At zero, we reversed then direction of the magnetic field and increased it to it’s maximum
value and dcreased it back again.
Using the data we plotted a hysteresis loop and investigated the hysteresis of Gd3 Ga5 O12 also.
6
Results and Analysis
Part 1
Plotting Current vs. Magnetic Field:
Figure 4: Non-Linear Calibration Curve
which calibrates the current. We do see that the current varies in a non-linear way with
respect to the magnetic field for larger values.
Part 2
The equation of the line for Fig.5 is given by:
F = (506.451 × 10−6 (±4.9 × 10−6 ))B 2
and given that:
F = CmχB 2
=⇒ C =
=⇒ C =
506.451 × 10−6
mχ
506.451 × 10−6
(4.82 × 10−5 )(0.330)
=⇒ C = 31.88 ± 0.03m−1
with units got by dimensional analysis and the error given by:
s
2 2
1
∆M
∆m
∆C =
+
0.33
M
m
7
Figure 5: Mohr Salt Linearity Graph
where ∆M is the uncertainty in the slope.
Part 3
Figure 6: Force vs Magnetic Field Squared
The equation of the line for Fig.6 is given by:
F = (14.717 × 10−4 (±6.661 × 10−6 ))B 2
8
and again using that:
F = CmχB 2
=⇒ χ =
=⇒ χ =
14.717 × 10−4
Cm
14.717 × 10−4
(31.88)(10.46 × 10−5 )
=⇒ χ = 0.44 ± .10
The uncertainty in the susceptibilty is given by:
v
u 2 2 2 !
u
∆M
M t
∆C
∆m
∆χ =
+
+
Cm
M
C
m
Such a value of the susceptibility is reasonable. In comparision to the theoretical value
which is given by χ ≈ 0.8 we are out by roughly a factor of two. We must take into account
that Gd3 Ga5 O12 doesn’t not mirror the ideal properites that we assumed in the Curie Law
calculation, and so we shouldn’t necessarily expect direct agreement.
Part 4
The hysteresis loop for hematite-αF e2 O3 is given by:
Figure 7: Hysteresis Loop for hematite-αF e2 O3
The uncertainty in the magnetisation is given by:
v
u 2 2 2 2 !
F u
∆F
∆m
∆C
∆B
t
∆σ =
+
+
+
mcB
F
m
C
B
9
The remanance of hematite-αF e2 O3 by inspection is ≈ 0.98. The loop is narrow with a
small covercity suggesting that hematite-αF e2 O3 is a soft magnetic material. The rather
large uncertainty close to the origin arises in the mathematics of the error equation. We are
essenitally dividing by a very small number so the uncertainty shoots up.
Out of curiosity an attempt at plotting a “hysteresis loop” for Gd3 Ga5 O12 is given:
Figure 8: Hysteresis Loop for Gd3 Ga5 O12
The motivation for this is that Gd3 Ga5 O12 has a magnetic phase transition at 293.4 K. Referring back to Fig.2 we see that for a ferromagnet there will be a period where material
“transitions” and the temperature at which it will become paramagnetic is called the Curie
Temperature. So, given that we are roughly at room temperature and we do not know how
long it will take for Gd3 Ga5 O12 to make a full transition from ferromagnet to paramagnet
it is reasonable to investigate are there any remnants of hysteresis. Therefore, from Fig.8
there are inidications to hysteresis, though most of it is within the error of the experiment,
especially close to the origin. If considered as a “hysteresis” loop then it is extermely narrow.
Another, and perhaps more revealing, use of this plot is to try and predict the susceptibility
. Using table 4 in the appendix and averaging over all of the
of χ using the fact that χ = M
H
susceptibilites we find that χ ≈ 0.49 which is similar and within error of our calculation of
χGd3 Ga5 O12 with the Force method.
10
Discussion
The ferromagnetic behaviour of hematite-αF e2 O3 was sucessfully illustrated with its hysteresis. Where it’s remanence is was found to be quite large. The “virgin curve”, however,
was not observed. The reason this did not happen may be because the hematite-αF e2 O3
retained some of it’s reamanence from the last time it was magnetised. If it were a nonmagnetised sample of hematite-αF e2 O3 then we would expect to observe the “virgin curve”.
Regarding the hysteresis loop of Gd3 Ga5 O12 it is open to conjecture whether it shows any
hysteresis at all, since measuring a reasonable susceptibilty, and therefore confirming it’s
paramagnetic behaviour, should exclude this notion in the first place anyhow. It would have
been nice to try and place the sample of Gd3 Ga5 O12 in a cooling chamber and repeat the
hysteresis experiment through a loop of temperatures. If for each temperature the hysteresis
is plotted then I’d expect as the sample was cooled that the hysteresis loop would broaden;
illustrating the magnetic phase transition nicely.
Another conclusion that can be made of the “hysteresis” loop of Gd3 Ga5 O12 is in reference to
Fig.2. Since the argument for ferromagnetism is weak and we are roughly at room temperature we could conclude that the temperature range transtion for Gd3 Ga5 O12 is not extremely
narrow, again to be understood with respect to Fig.2. Since in Fig.2 we could theoretically
assign the behaviour of Gd3 Ga5 O12 to the J = 70 curve then such reasoning is justified.
Though Fig.8 and it’s slight hints at hysteresis could indicate that it has not yet achieved
it’s Curie Temperature but is quite close to the full ferromagnetic-paramagnetic transition.
Regarding the susceptibilty of Gd3 Ga5 O12 and the two independent methods of achieving it,
we can make a strong statement that it is roughly about χ ≈ 0.47. In comparison to Mohr’s
Salt it is about one and a half times greater.
11
Appendix
Part 1
B (T)
0.015 ± 0.0001
0.098 ± 0.0001
0.823 ± 0.0001
0.205 ± 0.0001
0.323 ± 0.0001
0.445 ± 0.0001
0.560 ± 0.0001
0.652 ± 0.0001
0.728 ± 0.0001
0.789 ± 0.0001
0.823 ± 0.0001
0.879 ± 0.0001
−0.015 ± 0.0001
−0.098 ± 0.0001
−0.205 ± 0.0001
−0.323 ± 0.0001
−0.445 ± 0.0001
−0.560 ± 0.0001
−0.652 ± 0.0001
−0.728 ± 0.0001
−0.789 ± 0.0001
−0.823 ± 0.0001
−0.879 ± 0.0001
I (No. turns (A))
0±1
40 ± 1
360 ± 1
80 ± 1
120 ± 1
160 ± 1
200 ± 1
240 ± 1
280 ± 1
320 ± 1
360 ± 1
400 ± 1
0±1
40 ± 1
80 ± 1
120 ± 1
160 ± 1
200 ± 1
240 ± 1
280 ± 1
320 ± 1
360 ± 1
400 ± 1
Part 2
Mef f (kg)
(482 ± 2) × 10−7
(486 ± 2) × 10−7
(504 ± 2) × 10−7
(540 ± 2) × 10−7
(598 ± 2) × 10−7
(641 ± 2) × 10−7
(707 ± 2) × 10−7
(762 ± 2) × 10−7
(807 ± 2) × 10−7
(847 ± 2) × 10−7
(880 ± 2) × 10−7
F (N)
(472.36 ± 19) × 10−6
(476.28 ± 19) × 10−6
(493.92 ± 19) × 10−6
(529.20 ± 19) × 10−6
(586.04 ± 19) × 10−6
(637.98 ± 19) × 10−6
(692.86 ± 19) × 10−6
(746.76 ± 19) × 10−6
(790.86 ± 19) × 10−6
(830.06 ± 19) × 10−6
(862.40 ± 19) × 10−6
12
B (T)
(225 ± 3) × 10−6
(9604 ± 19) × 10−6
(42025 ± 41) × 10−6
(104329 ± 646) × 10−6
(198025 ± 89) × 10−6
(313600 ± 112) × 10−6
(425104 ± 1304) × 10−6
(52984 ± 1456) × 10−6
(622521 ± 1578) × 10−6
(702244 ± 1676) × 10−6
(772641 ± 1758) × 10−6
Part 3
Mef f (kg)
(1046 ± 2) × 10−7
(1064 ± 2) × 10−7
(1114 ± 2) × 10−7
(1207 ± 2) × 10−7
(1339 ± 2) × 10−7
(1517 ± 2) × 10−7
(1682 ± 2) × 10−7
(1835 ± 2) × 10−7
(1975 ± 2) × 10−7
(2101 ± 2) × 10−7
(2220 ± 2) × 10−7
F (N)
(1025 ± 19) × 10−6
(1042 ± 19) × 10−6
(1091 ± 19) × 10−6
(1181 ± 19) × 10−6
(1312 ± 19) × 10−6
(1486 ± 19) × 10−6
(1648 ± 19) × 10−6
(1798 ± 19) × 10−6
(1935 ± 19) × 10−6
(2058 ± 19) × 10−6
(2175 ± 19) × 10−6
13
B (T)
(225 ± 3) × 10−6
(9604 ± 19) × 10−6
(42025 ± 41) × 10−6
(104329 ± 646) × 10−6
(198025 ± 89) × 10−6
(313600 ± 112) × 10−6
(425104 ± 1304) × 10−6
(52984 ± 1456) × 10−6
(622521 ± 1578) × 10−6
(702244 ± 1676) × 10−6
(772641 ± 1758) × 10−6
Part 4
B(Am−1 )
(0.205 ± 0.001)
(0.323 ± 0.001)
(0.445 ± 0.001)
(0.560 ± 0.001)
(0.652 ± 0.001)
(0.728 ± 0.001)
(0.789 ± 0.001)
(0.838 ± 0.001)
(0.879 ± 0.001)
(0.838 ± 0.001)
(0.789 ± 0.001)
(0.728 ± 0.001)
(0.652 ± 0.001)
(0.560 ± 0.001)
(0.445 ± 0.001)
(0.323 ± 0.001)
(0.205 ± 0.001)
(0.098 ± 0.001)
(−0.205 ± 0.001)
(−0.323 ± 0.001)
(−0.445 ± 0.001)
(−0.560 ± 0.001)
(−0.652 ± 0.001)
(−0.728 ± 0.001)
(−0.789 ± 0.001)
(−0.728 ± 0.001)
(−0.838 ± 0.001)
(−0.879 ± 0.001)
(−0.838 ± 0.001)
(−0.789 ± 0.001)
(−0.728 ± 0.001)
(−0.652 ± 0.001)
(−0.560 ± 0.001)
(−0.445 ± 0.001)
(−0.323 ± 0.001)
(−0.205 ± 0.001)
(−0.098 ± 0.001)
σ(Am−1 )
(0.1048502959 ± 0.4766927845)
(0.1605550749 ± 0.1254194231)
(0.2261751969 ± 0.046922101)
(0.2759898732 ± 0.024291034)
(0.3244345264 ± 0.0152515371)
(0.365549501 ± 0.0108633916)
(0.400421101 ± 0.0084478456)
(0.427899392 ± 0.0070114898)
(0.4545179006 ± 0.0060027019)
(0.4327850178 ± 0.0006556419)
(0.4073398242 ± 0.0007411659)
(0.3805464036 ± 0.0008744936)
(0.3474589122 ± 0.0011063146)
(0.30767083 ± 0.0015521812)
(0.2675767583 ± 0.0025981077)
(0.1975249935 ± 0.0059485655)
(0.1464575562 ± 0.0185973302)
(0.0835541717 ± 0.1345930785)
(−0.0998574247 ± −0.0260787591)
(−0.1573862248 ± −0.0070909593)
(−0.2116079808 ± −0.0030160755)
(−0.2814731157 ± −0.0016246373)
(−0.3223414004 ± −0.0011426139)
(−0.365549501 ± −0.0008873458)
(−0.3991238404 ± −0.0007459505)
(−0.4258637146 ± −0.0006586111)
(−0.4529653202 ± −0.0005969927)
(−0.4327850178 ± −0.0006556419)
(−0.406907404 ± −0.0007414113)
(−0.38148371 ± −0.0008737344)
(−0.3464123492 ± −0.0011076926)
(−0.3464123492 ± −0.0015521812)
(−0.30767083 ± −0.0026024435)
(−0.2668100627 ± −0.0060197122)
(−0.1943561433 ± −0.0189610923)
(−0.1431289753 ± −0.1872943329)
(−0.0591842049 ± −0.1872943329)
14
χ
(0.461821527 ± 0.05)
(0.511464858 ± 0.05)
(0.4970745354 ± 0.05)
(0.5082588693 ± 0.05)
(0.4928390593 ± 0.05)
(0.4975989669 ± 0.05)
(0.5021284354 ± 0.05)
(0.507504564 ± 0.05)
(0.5106197995 ± 0.05)
(0.5170852111 ± 0.05)
(0.5164499019 ± 0.05)
(0.5162735414 ± 0.05)
(0.5227285763 ± 0.05)
(0.532912442 ± 0.05)
(0.5494121964 ± 0.05)
(0.6012960861 ± 0.05)
(0.6115324876 ± 0.05)
(0.7144271032 ± 0.05)
(0.3907720613 ± 0.05)
(0.4871093886 ± 0.05)
(0.4872638538 ± 0.05)
(0.4755235523 ± 0.05)
(0.5026305638 ± 0.05)
(0.494388651 ± 0.05)
(0.5021284354 ± 0.05)
(0.5058603808 ± 0.05)
(0.5081905902 ± 0.05)
(0.5153189081 ± 0.05)
(0.5164499019 ± 0.05)
(0.5157254803 ± 0.05)
(0.5240160852 ± 0.05)
(0.531307284 ± 0.05)
(0.5494121964 ± 0.05)
(0.5995731747 ± 0.05)
(0.601721806 ± 0.05)
(0.6981901236 ± 0.05)
(0.6039204584 ± 0.05)