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PHYSICS 208 – 2nd TERM
Outline Notes (continued) Section 5. Magnetic Properties (This
part of the notes will be relatively brief, because the material is
based rather closely on parts of Callister chapter 20). The
properties to be considered are “weak” magnetic materials
(paramagnets, diamagnets); “strong” magnetic materials
(ferromagnets, antiferromagnets, ferrimagnets); applications of
strong magnets (e.g., in magnetic recording, computer memory
storage,
fast switching, etc.); relation to superconductivity.
5.1 Basic definitions and concepts (see also Callister, sections
20.1 and 20.2) In electricity we have electric charges (electric
monopoles) q, and an electric dipole can be formed from +q and −q
close together. In magnetism we have magnetic dipoles, but no
magnetic monopoles under normal conditions. A magnetic dipole is
equivalent to a small current loop (it produces an equivalent
distribution of magnetic field lines) − see Fig. 20.1 in
Callister.
(from Callister) We can define two different magnetic field
vectors H and B as follows:-
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H is usually defined first in terms of an electric current
flowing along a wire. For example, if we take a long solenoid of
length , N turns, and current I, the field inside the solenoid is
(see Fig. 20.3)
(from Callister)
NIH =
This is just a special case of Ampere’s law, which says that The
line integral (or sum) of H round a closed path = The total current
enclosed
Therefore H = NI, provided we assume no field outside the
solenoid and ignore end effects. The units of H are Amp/m. It is
called the magnetic field strength (or intensity). It is the same
for whatever medium is filling the solenoid. We define another
field quantity B (called the magnetic induction) to represent the
resultant internal field within a material that is subjected to a H
field. In general we write B = µ H where µ is a property of the
material called the permeability. The units of B are Tesla (T),
basically kg/s-C, and so the units of µ can be deduced. In the case
of a vacuum we write B0 = µ0 H where µ0 = permeability of vacuum =
4π × 10-7 kg-m/C2 It is convenient to define a few other quantities
from the above definitions:- The difference between B0 (in vacuum)
and B (when a material is present) is used to define the
magnetization M in the material:
B = B0 + µ0 M = µ0 (H + M) M is the magnetic moment per unit
volume in the material. It has the same dimensions (and units) as
H. Some other relations are to write: M = χm H where χm is called
the magnetic susceptibility
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and B = µ H = µ0 µr H where µr is called the relative
permeabilityBoth χm and µr are dimensionless numbers. Substituting
into B = µ0 (H + M) gives µ0 µr H = µ0 (H + χm H) , which
simplifies to µr = (1 + χm) What is the origin of the magnetic
moments in materials? In other words, what makes B different from
the vacuum B0? Basically, it comes from more current loops (but
this time on an atomic scale or smaller):-
1. Electrons may have an orbital motion around the nucleus
(orbital angular momentum → like a small current loop → magnetic
moment)
2. Electrons also have an intrinsic spin (spin angular momentum)
→ also gives a magnetic moment (see Callister, Fig. 20.4)
Add these together to get the total contribution from each
electron, and then add again for all the electrons in the atom.
Usually this is nonzero (except in a few cases where there is
cancellation).
(from Callister) The basic unit on the atomic scale is taken as
that due to the electron spin: it is called the Bohr magneton and
denoted by µB. µB = 9.27 × 10-24 A-m2 (from QM, it is e /2m, where
m = electron mass). In “weak” magnetic materials, χm > 1
typically. 5.2 Weak magnetic materials (see also Callister, section
20.3) There are two cases:
χm is small and negative (µr is slightly less than 1) →
diamagnetism χm is small and positive (µr is slightly greater than
1) → paramagnetism
Diamagnetism This is observed in materials when The magnetic
moment of each atom = 0
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This occurs if the contributions from all the different
electrons add up (vectorially) to zero, e.g., because of complete
shell (as in He, Ne, etc) or just because of cancellation (as in
Si, Hg, etc). When a field H ≠ 0 is applied, the electron orbits
distort slightly because electrons experience a Lorentz force. This
changes the current loops and induces a small magnetic moment (in
the opposite direction to H) − see Fig 20.5.
(from Callister)
∴ If H > 0, it implies induced M < 0, and since B = µ0 (H
+ M) it implies B < µ0 H. Hence χm < 0 (typically around
−10−5)and µr < 1 (but only slightly). (See Fig. 20.6 and Table
20.2).
(from Callister)
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(from Callister) In principle, the diamagnetic effect occurs in
all materials, but it is so small that it is only observed when
other magnetic effects are zero (i.e., when the atomic magnetic
moments are zero). Paramagnetism This occurs when The magnetic
dipole moment of each atom is nonzero, and There are negligible
magnetic interactions between neighbouring atoms.
When H = 0 the dipole moments are pointing randomly in all
directions and cancel out, so M = 0. When H ≠ 0 the dipole moments
preferentially line up in the direction of H (in order to reduce
their potential energy, which is given by −M⋅H).
∴ M > 0 (with magnitude depending on the strength of H).
Hence χm > 0 (typically 10−5 to 10−2)and µr > 1 (slightly).
See Callister, Figs. 20.5b and 20.6. 5.3 Strong magnetic materials
(see also Callister, sections 20.4 and 20.5) These occur when
• The magnetic dipole moment of each atom is nonzero, and •
There are strong magnetic interactions between neighbouring atoms
(by contrast with
paramagnetism). The interactions are QM in origin (and are
called exchange interactions)
• They depend on the overlap of wave functions on neighbouring
atoms • They give a potential energy contribution that makes
neighbouring magnetic moments
align either parallel (like ↑ ↑) or antiparallel (like↑ ↓) There
are three important types of strong magnetic behaviour:-
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Ferromagnetism: this is when all atoms have the same magnitude
of magnetic moment and the interactions favour parallel alignment
(even when H = 0) → Large M ≠ 0 Antiferromagnetism: same as above
but the interactions favour antiparallel alignment (even when H =
0) → M = 0 (by cancellation) Ferrimagnetism: this occurs in
compounds where there are two (or more) types of atoms having
different magnetic moments, and the interactions favour
antiparallel alignment of some of these (even when H = 0) → M ≠ 0
(only partial cancellation), but not as large as in
ferromagnetism.
Ferromagnetism M is large, so that typically M >> H, i.e.
χm >> 1 (can be up to about 106), and so B = µ0 (H + M) ≈ µ0
M This can occur even when H = 0 (unlike both diamagnetism and
paramagnetism). (See Fig. 20.7 in Callister).
(from Callister) The max value of M in any material occurs when
all the atomic dipoles in the solid line up parallel − this is
called the saturation magnetization Ms The corresponding value of B
is denoted by Bs (≈ µ0 Ms). Example: Calculate Bs and Ms for Ni
given that the magnetic moment of each atom is 0.6µB
and the density of Ni is 8.9 g/cm3. Answer: Ms = max magnetic
moment/unit volume = 0.6µBN
where N = no. of atoms/unit volume (m3)
But N = ρNi NA /ANi , where ρNi = density, NA = Avogadro’s
number, and ANi = atomic weight
∴ N = 6 3 23(8.9 10 g/m )(6 10 atoms/mol)
58.7 g/mol× × = 9.1 × 1028 atoms/m3
so Ms = 0.6 × 9.27 × 10-24 × 9.1 × 1028 A/m = 5.1 × 105 A/m Then
Bs = µ0Ms = 4π × 10-7 × 5.1 × 105 Tesla (T) = 0.64 T Other common
ferromagnets are Fe and Co (and metallic alloys).
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Antiferromagnetism These materials have a high degree of
magnetic order (antiparallel arrangement of nearest neighbours) but
M = 0 due to cancellation. Occurs in many insulating compounds and
ceramics. An example is MnO where the Mn2+ ions have canceling
magnetic moments and the O2− ions have zero magnetic moment (see
Callister, Fig. 20.8).
(from Callister) Ferrimagnetism Again there are antiparallel
alignment of some magnetic moments, but cancellation is only
partial and so M is nonzero. Some of the most common ferrimagnets
are: Cubic ferrites M Fe2O4 , where M is a metal e.g. Fe3O4 (when M
= Fe), which is magnetite Iron garnets M3Fe5O12 , where M is a rare
earth ion such as Sa, Eu, Gd, Y, etc e.g. Y3Fe5O12 , which is
yttrium iron garnet (or YIG). See Callister, Fig. 20.9 to
illustrate the behaviour in Fe3O4.
(from Callister)
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5.4 Temperature dependence and domains (see also Callister,
sections 20.6 − 20.10) Temperature dependence
In ferromagnets and ferrimagnets, M decreases as T is increased,
because thermal effects reduce the magnetic order caused by the
exchange interactions. The M value starts at Ms at low T (with T ≈
0) and steadily decreases and becomes zero (if H = 0) at a
transition temperature TC (called the Curie temperature). For T
> TC the material becomes paramagnetic. [See Callister, Fig.
20.10].
(From Callister) In antiferromagnets, M is zero because there is
the good antiparallel ordering at low T. This ordering is reduced
by thermal fluctuations as T is increased. Again it disappears at a
transition temperature TN (called the Neel temperature). For T >
TN the material becomes paramagnetic. Domains
These are small regions of uniform magnetic alignment (in
ferromagnets and antiferromagnets). Their size depends very much on
the type of material (i.e., whether it is in a field H of given
strength) and on the temperature − it might often be of ~ µm or mm
− domains are separated by small boundary regions (domain walls)
where there is a
transition from one direction of M to another (see Callister,
Figs. 20.11 and 20.12).
(From Callister)
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(From Callister) Magnetization process and hysteresis
Suppose we start from an initially unmagnetized piece of
ferromagnet (i.e., one where all the domains cancel out, so B =0
when H = 0). If H is gradually increased, the domains pointing
roughly in the direction of H tend to grow in size. Those in the
“wrong” direction shrink. So B increases as H increases, until
there is saturation (B = Bs) when H is large enough to make all
domains parallel. (See Callister, Fig. 20.13).
(From Callister) If H is now gradually reduced, B becomes
smaller but does not follow the original magnetization curve
(because the domains do not form again exactly as before) − this
effect is known as hysteresis. By cycling H from large positive to
large negative values, the B versus H plot traces out a closed loop
− the hysteresis loop. See also Callister, Figs. 20.14and
20.15.
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(From Callister)
(From Callister) Note the definitions of remanence Br (the value
of B when H is reduced to zero) and coercivity Hc (the magnitude of
H required to make B = 0). Hard and soft magnetic materials
“Hard” − large area hysteresis loop, large Br and Hc. “Soft” −
small area hysteresis loop, small Hc. Hard materials are suitable
for applications as permanent magnets
(e.g., tungsten steel − see also Callister, Table 20.6).
(From Callister)
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Soft materials are suitable for applications where the magnetic
state is constantly being changed, as in high-frequency magnetic
devices, switches, transformer cores, etc.
(e.g. soft iron, Si-iron, Fe/Ni alloys, etc − see also Table
20.5).
(From Callister) It can be shown that the work needed to take a
material through a complete hysteresis cycle is proportional to the
area of the hysteresis loop. ∴ A small area loop is an advantage in
soft materials, as is small Hc. Magnetic information storage The
domain structure can be used for information storage (e.g. in
computer memories or magnetic recording). Read section 20.10 of
Callister for a brief account of magnetic storage in soft
materials. The basis idea is for some magnetic materials
(particularly when produced as a thin film) there are just two
stable directions for the domain magnetizations (say ← and →).
These can be used to represent “0” and “1” in the binary code. An
external circuit (a “recording head” can be used to read or write
the information). See Callister, Fig. 20.18)
(From Callister) In more recent developments it has become
advantageous to use films magnetized in the perpendicular direction
(rather than the parallel orientation shown above). Currently
arrays of artificially produced magnetic nanometer-sized elements
(such as nanowires) are being studied as a way to achieve
higher-density magnetic memories.
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5.5 Superconductivity (see also Callister, section 20.11)
Superconductivity (SC) is concerned with the low-T behaviour of the
electrical conductivity σ or the resistivity ρ (= 1/σ). In 1911 it
was discovered that for T < 4 K a sample of Hg has infinite σ
(or zero ρ). Soon the SC effect was discovered in other materials,
mainly metals, below some critical temperature TC. (See also Fig.
20.21).
(From Callister) In metals, TC ranges from a fraction of 1 K up
to ~10 K. In metallic alloys, TC values up to ~25 K can be
obtained. This how it remained for many years (and in fact it was
believed that ~30 K was an upper limit on TC). Then in 1986 a new
class of SC was discovered in some particular ceramic materials.
Remarkably, TC values up to ~100 K or even 200 K were obtained. In
the conventional SCs (metals, alloys) the ρ = 0 property is very
well tested (e.g., by observing currents in SC loops remaining
undiminished over large periods of time). The TC values can be
reduced by either applying a large field H or making a large
current density J pass through the material. For large enough H or
J (exceeding some critical values) the SC property can be
destroyed. (See Fig. 20.22).
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(From Callister) The occurrence of SC is closely related to some
magnetic behaviour called the Meissner effect − the field lines of
B are excluded from the interior of the SC material (see Fig.
20.23).
(From Callister) But B = µ H = µ0 µr H = µ0 (1 + χm) H , so B =
0 implies χm = −1, i.e., the SC is like a “perfect” diamagnet. The
above property holds for H < HC. In some materials there is just
one HC value
− these are called Type I SC. In other materials there are two
values HC1 and HC2, such that for H < HC1 the Meissner effect is
complete (B = 0), but for HC1 < H < HC2 there is a partial
Meissner effect in which the B field is partially excluded from the
SC (and so −1 < χm < 0)
− these are called Type II SC. Why does SC occur? A successful
theory was developed in the 1950s by Bardeen, Cooper and Schrieffer
(called the BCS theory). It depended on identifying an attractive
component to the interaction forces between two electrons inside
the SC (in addition to the Coulomb force of repulsion). This could
come about from the distortion effect on the crystal lattice. In
some circumstances the electrons could move as pairs through the
material, giving rise to the SC effect.
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The new ceramic-based SC compounds, discovered in 1986, have a
much higher TC (~ 100 to 200 K). It is believed that an
electron-pairing effect still takes place, but it must occur in
quite a different way from the original BCS theory. An important
factor for applications involving the high-TC SC materials is that
the TC values are above liquid-N2 temperatures (~ 77 K). See Table
20.7 for listings of SC materials.
(From Callister)
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Outline Notes (continued)Section 5. Magnetic Properties