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PowerPoint PresentationGoal of this class
Goal of this class
Goal of this class
Goal of this class
Goal of this chapter
We begin with magnetostatics, the classical physics of the magnetic
fields, forces and energies associated with distributions of
magnetic material and steady electric currents. The concepts
presented here underpin the magnetism of solids.
Magnetostatics refers to situations where there is no time
dependence.
2.1 The magnetic dipole moment
The local magnetization M(r ) fluctuates on an atomic scale – dots
represent the atoms. The mesoscopic average, shown by the dashed
line, is uniform.
The elementary quantity in solid-state magnetism is the magnetic
moment m.
the local magnetization M(r,t ) which fluctuates wildly on a
subnanometre scale, and also rapidly in time on a subnanosecond
scale.
But for our purposes, it is more useful to define a mesoscopic
average <M(r,t )> over a distance of order a few Nanometres,
and times of order a few microseconds to arrive at a steady,
homogeneous, local magnetization M(r). The time-averaged magnetic
moment δm in a mesoscopic volume δV is
2.1 The magnetic dipole moment
Continuous medium approximation The representation of the
magnetization of a solid by the quantity M(r) which varies smoothly
on a mesoscopic scale.
The concept of magnetization of a ferromagnet is often extended to
cover the macroscopic average over a sample:
According to Amp`ere, a magnet is equivalent to a circulating
electric current; the elementary magnetic moment m can be
represented by a tiny current loop. If the area of the loop is A
square metres, and the circulating current is I amperes, then
2.1 The magnetic dipole moment
2.1 The magnetic dipole moment
Magnetic moment and magnetization are axial vectors. They are
unchanged under spatial inversion, r →−r, but they do change sign
under time reversal t →−t.
normal polar vectors such as position, force, velocity and current
density, which change sign on spatial inversion but not necessarily
on time reversal.
Strictly speaking, axial vectors are tensors; They can be written
as a vector product of two polar vectors, as in (2.4), but their
three independent components can be manipulated like those of a
vector.
2.1 The magnetic dipole moment
2.1.1 Fields due to electric currents and magnetic moments
the Biot–Savart law
2.1.1 Fields due to electric currents and magnetic moments
2.1 The magnetic dipole moment
calculate the field created by a magnetic moment associated with a
small current loop, firstly at the centre, and secondly at a
distance r much greater than the size of the loop.
2.1 The magnetic dipole moment
To simplify the second calculation, we choose a square loop of side
δl r and first evaluate the field at two special positions, point
‘A’ on the axis of the loop and point ‘B’ in the broadside
position.
2.1 The magnetic dipole moment
2.1 The magnetic dipole moment
The field falls off rapidly, as the cube of the distance from the
magnet. It has axial symmetry about m.
2.1 The magnetic dipole moment
Faraday represented magnetic fields using lines of force. (The
basic idea dates back to Descartes.) The lines provide a picture of
the field by indicating its direction at any point; its magnitude
is inversely proportional to the spacing of the lines. The
direction of the field of a point dipole relative to the normal to
r is known as ‘dip’.
the differential equation for the line of force
c is a different constant for each line
2.1 The magnetic dipole moment
The field of the magnetic moment has the same form as that of an
electric dipole p = qδl formed of positive and negative charges±q
which are separated by a small distance δl. The vector p is
directed from −q to +q.
Hence the magnetic moment m may somehow be regarded as a magnetic
dipole; its associated magnetic field is called the magnetic dipole
field.
One uses Cartesian coordinates, with m in the z-direction
another resolves the field into components parallel to r and
m:
2.2 Magnetic fields
The magnetic field that appears in the Biot–Savart law and in
Maxwell’s equations in vacuum is B, but the hysteresis loop of Fig.
1.3 traced M as a function of H. It is time to explain why we need
these two magnetic fields, with different units and
dimensions.
Fields with this property are said to be solenoidal; the lines of
force all form continuous loops.
Gauss’s theorem:
magnetic flux flowing out of the surface S through area dA
an alternative name for the B-field is magnetic flux density
2.2.1 The B-field
Flux has a named (but little used) unit of its own, the weber,
abbreviated to Wb. A unit equivalent to T is Wb /m2. The other
synonym for B is magnetic induction.
Sources of the B-field are: (i) electric currents flowing in
conductors; (ii) moving charges (which constitute an electric
current); and (iii) magnetic moments (which are equivalent to
current loops).
Time-varying electric fields are also a source of magnetic fields,
and vice versa, but we are restricting our attention to
magnetostatics
2.2.1 The B-field
Sources of the B-field are: (i) electric currents flowing in
conductors;
(ii) moving charges (which constitute an electric current);
(iii) magnetic moments (which are equivalent to current
loops).
2.2.1 The B-field
All the sources of B are moving charges, but B itself interacts
with charges only when they move. The fundamental relation between
the fields and the force f exerted on a charged particle is the
Lorentz expression
The electric and magnetic fields can therefore be expressed in
terms of the basic quantities of mass, length, time and current.
The units of these four quantities in the Syst`eme International
(SI) are the kilogram, metre, second and ampere. A coulomb is an
ampere second.
Hence the units of E and B are, respectively, newtons per coulomb
(N C−1), and N C−1 m−1 s. The latter reduces to kg s−2A−1, or
tesla. Units and dimensions are discussed in Appendix A.
2.2.1 The B-field
The ampere is then defined as the current flowing in conductors in
vacuum which produces a force of precisely 2 × 10−7 N m−1 when the
two conductors are 1 metre apart
=
2.2.2 Uniform magnetic field
Structures that produce a uniform magnetic field in their bore: (a)
a long solenoid, (b) Helmholtz coils and (c) a Halbach
cylinder.
An infinitely long solenoid creates a uniform field that is
parallel to its axis in the bore, and zero everywhere
outside.
Helmholtz coils are a pair of matched coaxial coils whose
separation is equal to their radius a (Fig. 2.6(b)). The field is
uniform, with zero second derivative at the centre. It is given
by
2.2.3 The H-field
Now we come to the H-field, also known as the magnetic field
strength or magnetizing force. It is an indispensable auxiliary
field whenever we have to deal with magnetic or superconducting
material. The magnetization of a solid reflects the local value of
H. The distinction between B and H is trivial in free space. They
are simply related by the magnetic constant μ0:
where jc is the conduction current in electrical circuits and jm is
the Amp`erian magnetization current associated with the magnetized
medium.
In a material medium,
Amp`ere’s law
where Ic is the total conduction current threading the path of the
integral. The new field is no longer divergenceless, but has
sources and sinks associated with nonuniformity of the
magnetization.
2.2.3 The H-field
where Ic is the total conduction current threading the path of the
integral. The new field is no longer divergenceless, but has
sources and sinks associated with nonuniformity of the
magnetization.
We can imagine that H, like the electric field E, arises from a
distribution of positive and negative magnetic charge qm. The field
emanating from a single charge would be
Units of qm are A m.
2.2.3 The H-field
magnetic charges do offer a mathematically convenient way of
representing the H-field, and some force and field calculations
become much simpler if we make use of them. Charge avoidance is a
useful principle in magnetostatics.
Any magnet will produce an H-field both in the space around it and
within its own volume. We can write the field as the sum of two
contributions
where Hc is created by conduction currents and Hm is created by the
magnetization distributions of other magnets and of the magnet
itself.
The second contribution is known as the stray field outside a
magnet or as the demagnetizing field within it. It is represented
by the symbol Hd
2.2.3 The H-field
Units of H, like those of M, areAm−1. One tesla is equivalent to
795 775 A m−1 (or approximately 800 kA m−1).
Inside the magnet the B-field and the H-field are quite different,
and oppositely directed. H is also oppositely directed to M inside
the magnet, hence the name ‘demagnetizing field’.
The field lines of H appear to originate on the horizontal surfaces
of the magnet, where a magnetic charge of density σm = M · en
resides; en is a unit vector normal to the surface. The H-field is
said to be conservative (∇ × H = 0), whereas the B-field, whose
lines form continuous closed loops, is solenoidal (∇ · B =
0).
When considering magnetization processes, H is chosen as the
independent variable, M is plotted versus H, and B is deduced from
(2.33). The choice is justified because it is possible to specify H
at points inside the material in terms of the demagnetizing field,
acting together with the fields produced by external magnets and
conduction currents.
2.2.4 The demagnetizing field
It turns out that in any uniformly magnetized sample having the
form of an ellipsoid the demagnetizing field Hd is also uniform.
The relation between Hd and M is
Along the principal axes of the ellipsoid, Hd and M are collinear
and the principal components of N in diagonal form (Nx ,Ny ,Nz) are
known as demagnetizing factors. Only two of the three are
independent because the demagnetizing tensor has unit trace:
2.2.4 The demagnetizing field
2.2.5 Internal and external fields
The external field H', acting on a sample that is produced by
steady electric currents or the stray field of magnets outside the
sample volume, is often called the applied field. The sample itself
makes no contribution to H'.
2.2.5 Internal and external fields
Ways of measuring magnetization with no need for a demagnetizing
correction: (a) a toroid, (b) a long rod and (c) a thin
plate.
a sphere for which the magnetization is uniform
2.2.6 Susceptibility and permeability The simplest materials are
linear, isotropic and homogeneous (LIH). For magnetism, this means
that the susceptibility or applied field is small and a small
uniform magnetization is induced in the same direction as the
external field:
where χ’ is a dimensionless scalar known as the external
susceptibility. Where χ is the internal susceptibility.
For single crystals, the susceptibility may be different in
different crystallographic directions, and M = χH becomes a tensor
relation with χij a symmetric second-rank tensor, which has up to
three independent components in the principal-axis coordinate
system.
2.2.6 Susceptibility and permeability
Permeability is related to susceptibility. It is defined in the
internal field. In LIH media the permeability μ is given by
The relative permeability
2.2.6 Susceptibility and permeability
Consider the example of an isotropic, soft ferromagnetic sphere
with high permeability and no hysteresis.
M is zero in the multidomain state that exists before the field is
applied.
The ideal soft material has
a hard ferromagnetic sphere
2.2.6 Susceptibility and permeability
2.2.6 Susceptibility and permeability
Generally, magnetic media are not linear, isotropic and homogeneous
but nonlinear and hysteretic and often anisotropic and
inhomogeneous as well!
Then B, like M, is an irreversible and nonsingle-valued function of
H, represented by the B(H) hysteresis loop deduced from the M(H)
loop
2.3 Maxwell’s equations
Maxwell’s equations in a material medium are expressed in terms of
all four fields:
In order to solve problems in solid-state physics we need to know
the response of the solid to the fields. The response is
represented by the constitutive relations
portrayed by the magnetic and electric hysteresis loops and the
current– voltage characteristic. The solutions are simplified in
LIH media, where
2.3 Maxwell’s equations
2.3 Maxwell’s equations
2.4 Magnetic field calculations
magnetic moments currents magnetic charge
They yield identical results for the field in free space outside
the magnetized material but not within it.
Alternative approaches for calculation of the magnetic field
outside a uniformly magnetized cylinder
AmperianDirect Coulombian
C.f. Internal magnetic field of a dipole
The two models for a dipole (current loop and magnetic poles) give
the same predictions for the magnetic field far from the source.
However, inside the source region they give different predictions.
The magnetic field between poles (see figure for Magnetic pole
definition) is in the opposite direction to the magnetic moment
(which points from the negative charge to the positive charge),
while inside a current loop it is in the same direction (see the
figure to the right). Clearly, the limits of these fields must also
be different as the sources shrink to zero size. This distinction
only matters if the dipole limit is used to calculate fields inside
a magnetic material.[4]
If a magnetic dipole is formed by making a current loop smaller and
smaller, but keeping the product of current and area constant, the
limiting field is
Unlike the expressions in the previous section, this limit is
correct for the internal field of the dipole.[4][9]
https://en.wikipedia.org/wiki/Magnetic_moment#cite_note-Brown62-4
2.4 Magnetic field calculations The second approach considers the
equivalent current distributions in the bulk and at the surface of
the magnetized material:
Using the Biot–Savart law (2.5) and adding the effects of the bulk
and surface contributions to the current density,
2.4 Magnetic field calculations
The third approach uses the equivalent distributions of magnetic
charge in the bulk and at the surface of the magnetized
material:
2.4.1 The magnetic potentials
Vector potential
The flux density invariably satisfies
where A is a magnetic vector potential. Units of A are Tm. There is
substantial latitude in the choice of vector potential for a given
field.
B = (0,0,B)
A = (0,xB,0)
A = (-yB,0,0)
c.f. some useful vector calculus
The definition of A is not unique, because it is permissible to add
on the gradient of any arbitrary scalar function
2.4.1 The magnetic potentials
Gauge transformation
A useful gauge is the Coulomb gauge, where f is chosen so
that
A convenient expression for A in the Coulomb gauge is
It does not matter that the definition of A is not unique, because
the observed effects depend on the magnetic field, not on the
potential from which it is mathematically derived.
2.4.1 The magnetic potentials
The expression for the field due to a distribution of currents
obtained by integrating the Biot–Savart law (2.5)
2.4.1 The magnetic potentials
At large distances the vector potential for a magnetic moment m
equivalent to a current loop is
2.4.1 The magnetic potentials
Amp`ere’s law
we see that in the Coulomb gauge, the vector potential satisfies
Poisson’s equation
2.4.1 The magnetic potentials
Scalar potential When the H-field is produced only by magnets, and
not conduction currents, it too can be expressed in terms of a
potential. The field is then conservative, and Amp`ere’s law (2.28)
becomes
so the scalar potential satisfies Poisson’s equation:
By solving Poisson’s equation:
2.4.1 The magnetic potentials
Magnetostatic calculations are easier with the scalar pmotential,
but it should be understood that it is only permissible to use it
for problems where no conduction currents are present.
2.4.2 Boundary conditions The perpendicular component of B is
continuous.
The parallel component of H is continuous.
2.4.2 Boundary conditions
If medium 1 is air and medium 2 has a high permeability, this shows
that the lines of magnetic field strength inside a highly permeable
medium tend to lie parallel to the interface, whereas in air they
tend to lie perpendicular to the interface. This is the reason why
soft iron acts as a magnetic mirror
The situation is reversed if the iron is replaced by a sheet of
superconductor. Ideally, the superconductor is a perfect diamagnet
into which flux does not penetrate. It follows that B is parallel
to the surface, and the image is repulsive.
2.4.2 Boundary conditions
2.4.3 Local magnetic fields
The question then arises: ‘What is the value of the local magnetic
field Hloc at a point in a solid?’ The point may be an atomic
site.
The calculation of B at any point r may be carried out in principle
by replacing the integral (2.51) by a sum over atomic point dipoles
mi .
Region 1 : a continuum Region 2 : the Lorentz cavity, where the
atomic-scale structure is taken into account
a is the interatomic separation.
The Lorentz cavity is chosen to be spherical. The field due to
region 1 can be evaluated from the distribution of surface charges
σm = M · en on the inner and outer surfaces.
The field H2 produced by the atoms contained within the cavity is
evaluated as a dipole sum
2.4.3 Local magnetic fields
2.5 Magnetostatic energy and forces
There are two main contributions to the energy of a ferromagnetic
body:
1. Atomicscale electrostatic effects like exchange or single-ion
anisotropy, and magnetostatic effects.
2. The magnetostatic effects, which involve the self-energy of
interaction of the body with the field it creates by itself, as
well as the interaction of the body with steady or slowly varying
external magnetic fields, are considered here.
Exchange and other electrostatic effects are the subject of Chapter
5.
The magnetostatic interactions are rather weak compared to the
short- range exchange forces responsible for ferromagnetism, but
they are important in ferromagnets nonetheless because the domain
structure and magnetization process depends on them. It is the
long-range nature of the dipole–dipole interaction, varying as
r^−3, that allows these weak interactions to determine the magnetic
microstructure.
2.5 Magnetostatic energy and forces
Magnetic fields do no work on electric currents or moving charges
because the magnetic part of the Lorentz force ( j × B) per unit
volume or (v × B) per unit charge is always perpendicular to the
motion. We cannot associate a potential energy function with the
magnetic force.
2.5 Magnetostatic energy and forces
Let us first consider a small rigid magnetic dipole m in a
pre-existing steady field B.
Taking the reference state at θ = 0, where θ is the angle between m
and B, and integrating the torque gives the ‘potential
energy’
We assume that turning the magnetic dipole has no effect either on
its moment or on the sources of B.
Zeeman energy
2.5 Magnetostatic energy and forces
there is no net force on a magnetic moment in a uniform field; the
‘potential energy’ does not depend on position. However, if B is
nonuniform, the energy of the dipole does depend on its
position.
The energy εm is minimized for a ferromagnet or paramagnet by this
force tending to pull the material into a region where the field is
greatest, but a diamagnet is pushed out to a region where the field
is smallest.
2.5 Magnetostatic energy and forces
the mutual interaction of two parallel dipoles
Nose-to-tail broadside
Reciprocity
The two dipoles are an example of the reciprocity theorem, a useful
result in magnetostatics, which states that the energy of
interaction of two separate distributions of magnetization M1 and
M2 producing fields H1 and H2 is
The reciprocity theorem is used to simplify magnetic energy
calculations such as the interaction of a magnetic medium with a
read head, for example.
2.5.1 Self-energy
We consider the energy of a body with magnetization M(r) in a
magnetic field. The result is different according to whether the
field in question is an external field H or the demagnetizing field
Hd created by the body itself.
Here we discuss the second case, considering first a small moment
δm at a point inside the macroscopic magnetized body.
The factor of ½ which always appears in expressions for the
self-energy is needed to avoid double counting because each element
δm contributes as a field source and as a moment.
2.5.1 Self-energy
2.5.1 Self-energy
We have used the handy result that for a magnet in its own field,
when no currents are present,
where the integral is again over all space.
Far from the magnet, , so the integral over a surface of infinite
radius is zero.
2.5.1 Self-energy
This shows that the energy associated with a permanent magnet can
be either associated with the integral of H2d over all space
(2.79), or with the integral of −Hd · M over the magnet (2.78), but
not both. These are alternative ways of regarding the same energy
term.
The expression (2.78) assumes the magnetization is known in
advance, so we can evaluate the magnetostatic energy in the field
produced by the magnetization configuration. In practice, the
magnetization tends to adopt a configuration which minimizes its
self-energy. For an ellipsoid, there may be uniform magnetization
along the axis where N is smallest.
2.5.2 Energy associated with a magnetic field
In free space,
An expression for the energy associated with a static magnetic
field may be obtained by considering an inductor L consisting of a
current loop which creates a flux
The electromotive force (emf) developed in a circuit
2.5.2 Energy associated with a magnetic field
The power needed to maintain a current I in the inductor is
Integrating from 0 to I gives an expression for the energy
associated with the inductor:
The same energy can be associated with the field in space created
by the current in the inductor.
2.5.2 Energy associated with a magnetic field
In free space,
This is actually a general statement irrespective of whether the
field is created by electric currents or magnetic material
(2.79).
2.5.2 Energy associated with a magnetic field
When designing magnetic circuits that include permanent magnets,
the aim is usually to maximize the energy associated with the field
created by the magnet in the space around it.
where the indices o and i indicate integrals over space outside and
inside the magnet.
The integral on the left is the one to be maximized.
The energy product:
It is twice the energy stored in the stray field of the
magnet.
2.5.2 Energy associated with a magnetic field
The energy stored in the field outside the magnet
2.5.3 Energy in an external field
2.5.3 Energy in an external field
They exhibit hysteresis, so that the energy needed to prepare a
state described by B and H depends on the path followed.
When the magnetization is uniform, this expression becomes
The increment of work δw done to produce a small change in flux
δ
2.5.3 Energy in an external field
More generally, we would love to have an expression for the energy
of the magnetization distribution M(r) in the external, applied
field H' which is supposedly undeformed by the presence of the
magnetic material.
The basic constitutive relation for the material is
The applied field H' is supposed to be created by some external
current distribution j' .
2.5.3 Energy in an external field
The work associated with the energy changes of the magnetized body
alone
a term associated with the H-field in empty space
2.5.3 Energy in an external field
The first integral is zero
The second integral is the contribution to the magnetostatic
self-energy
by reciprocity
2.5.3 Energy in an external field
This expression relates the magnetic energy to the self-energy and
the constitutive relation M = M(H).
The energy increment per unit volume
2.5.3 Energy in an external field
The energy expended to magnetize a sample is related to its
anisotropy energy, including shape anisotropy, since the
magnetization process in the external field depends on the
orientation of the sample.
the energy associated with the
applied field
the hysteresis energy loss per cycle
2.5.3 Energy in an external field
An expression for the energy needed to magnetize an LIH
paramagnetic material in an external field may be deduced. Here the
moment is induced by the field, according to (2.40). Hence, from
(2.93)
2.5.4 Thermodynamics of magnetic materials
HX represents some external action on the system, and X is a state
variable
The internal energy per unit volume
The system is usually defined by fixing one variable in each of the
(T,S) and (HX,X) pairs. Four thermodynamic potentials can be
defined by fixing two variables experimentally and leaving the
other two variables free.
When T is fixed, as is often the case,
When the magnetization is uniform
At thermodynamic equilibrium
2.5.4 Thermodynamics of magnetic materials
2.5.4 Thermodynamics of magnetic materials
The changes in F and G at constant temperature are associated with
the areas under the reversible H(M) or −M(H) curves.
In the case of an LIH medium, the changes of Helmholtz free energy
and Gibbs free energy on magnetizing the medium are
2.5.4 Thermodynamics of magnetic materials
The spontaneous magnetization of a ferromagnet falls with
increasing temperature. The fall becomes precipitous just below the
Curie point, where the entropy of the spin system increases rapidly
as the spin moments become disordered.
In this temperature range, large entropy changes can be produced by
modest applied fields. The entropy and magnetization of the
ferromagnet are obtained as partial derivatives of the Gibbs free
energy
2.5.4 Thermodynamics of magnetic materials
2.5.4 Thermodynamics of magnetic materials
the specific heat of magnetic origin
Moreover, from the second derivatives of the four thermodynamic
potentials, four Maxwell relations are obtained.
2.5.5 Magnetic forces
Forces in thermodynamics are related to the gradient of the free
energy, which represents the ability of the system to do
work.
When M is uniform and independent of H‘
The Kelvin force
2.5.5 Magnetic forces
A general expression for the force density when M is not
independent of H is
Note that H in this expression is the internal field, not the
applied field H’ , and υ = 1/d, where d is the density.
When this is independent of density, as it is for dilute solutions
or suspensions of magnetic particles, the first term is zero and
the force Fm is given by the Kelvin expression for a paramagnet
with H’ = H. The demagnetizing field is negligible in dilute
paramagnetic solutions, but in more concentrated samples such as
ferrofluids, the first term takes care of the dipole–dipole
interactions.
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