#### Magnetism, Physics tutorial

Magnetization:

Material substance obtains magnetic polarization when placed in the magnetic field just as a dielectric medium obtains electric polarization in the electric field. The orbital motion of electrons in atoms and molecules give currents that give rise to magnetic dipoles. In several materials small electric currents related with orbital motion and spin of electrons average to zero. When such atoms are placed in magnetic field, minute electron currents are produced by induction in clouds of electrons, direction of which is such that magnetic field related with them opposes inducing field B. These are called as diamagnetic substances.

Every atom or molecule may be considered as small magnetic dipole with magnetic moment

Qml = e^nIdS

Where Qm is magnetic pole strength, l is pole separation, I is current and dS is area of the loop. Effect of atomic magnets is explained by quantity known as magnetization M which is expressed as magnetic dipole moment per unit volume. In several materials it is found that magnetization M is linearly proportional to magnetic field intensity H, that is

M‾ = χmH‾

Where χm is dimensionless constant known as magnetic susceptibility of material. Susceptibility is function of temperature.

Diamagnetism:

Substances with the negative magnetic susceptibility are known as diamagnetic. Diamagnetism, to greater or lesser degree, is the property of all materials and will always make weak contribution to material's response to the magnetic field. Though, for materials which show some other form of magnetism (like ferromagnetism or paramagnetism), diamagnetic contribution becomes negligible. Substances which generally display diamagnetic behaviour are called diamagnetic materials, or diamagnets. Magnetic susceptibility of different molecular fragments is known as Pascal's Constants.

Diamagnetic materials have relative magnetic permeability that is less than or equal to 1, and thus a magnetic susceptibility that is less than 0. This signifies that diamagnetic materials are repelled by magnetic fields. Though, as diamagnetism is such a weak property its effects are not visible in everyday life. Magnetic susceptibility per unit volume is stated as

χm = M/H

Where M is magnetic moment per unit volume, or magnetization, and H is magnetic field intensity. Diamagnetism describes tendency of electrical charges to partly shield interior of body from applied magnetic field. For the atom in magnetic field motion of electrons is same as possible motion in absence of H except for superposition of the common precession of angular frequency ωL = -eH/2mc

As magnetic moment μ of current loop is provided by product of current by area of loop, we have μ/H = -(Ze2/4mc2)(Ρ2)‾

For Z electrons, where (Ρ2)‾ = (x2)‾ + (y2)‾ is average of square of perpendicular distance of electron from field axis. In terms of mean square distance (r2) ‾ = (x2)‾ + (y2)‾ + (z2)‾ from nucleus, we have (r2)‾ = 3/2(Ρ2)‾ for a distribution of charge which on average is spherically symmetrical, so that (x2)‾ = (y2)‾ = (z2)‾. Then diamagnetic susceptibility per unit volume is, if N is number of atoms per unit volume,

χ = -(Ze2N/6mc2)(r2)‾

This is Langevin expression that be utilized to compute diamagnetic susceptibility.

Paramagnetism:

Substances with the positive susceptibility are known as paramagnetic. Positive susceptibility can be found in:

• All atoms and molecules which has odd number of electrons.
• Examples of such atoms and molecules comprise: sodium atoms; nitrogen II oxide (NO)
• All free atoms and ions with a partly filled inner shell: transition elements; ions isoelectronic with transition elements; rare earth and actinide elements. Examples comprise Mn2+, Gd3+, U4+
• metals

Paramagnetism is the form of magnetism whereby paramagnetic material is only attracted when in presence of the externally applied magnetic field. In contrast with this behaviour, diamagnetic materials are repelled by magnetic fields. Paramagnetic materials have relative magnetic permeability greater or equal to unity (that is., positive magnetic susceptibility) and therefore, are attracted to magnetic fields. Magnetic moment induced by applied field is linear in field strength and rather weak. Paramagnetic materials have small, positive susceptibility to magnetic fields. Such materials are slightly attracted by magnetic field and material doesn't retain magnetic properties when external field is removed. Paramagnetic properties are because of the presence of some unpaired electrons, and from realignment of electron paths caused by external magnetic field. Paramagnetic materials comprise magnesium, molybdenum, lithium, and tantalum.

Consider the paramagnetic material having N atoms per unit volume, each having magnetic moment μ. If the magnetic field H is applied to material magnetization occurs from orientation of magnetic moments and thermal disorder resists tendency of applied field to orient moments. Energy of interaction with applied magnetic field is

V = -μ.H

For thermal equilibrium, magnetization can be deduced as

M = NμL(a)

Where a = μH/kT,

In limit μH / kT <<1, magnetic susceptibility is

χ = M/H = Nμ2/3kT = C/T

Where C = Nμ2/3k is called as Curie constant. The 1/T temperature dependence is called as Curie law. Given equation is called as Langevin equation.

Ferromagnetism:

Ferromagnetism is basic mechanism by which certain materials (like iron) form permanent magnets, or are attracted to magnets. In physics, various different kinds of magnetism are distinguished. Ferromagnetism (comprising ferrimagnetism) is the strongest type; it is the only kind which creates forces strong enough to be felt, and is liable for common phenomena of magnetism faced in everyday life. Other substances respond weakly to magnetic fields with two other kinds of magnetism, paramagnetism and diamagnetism, but forces are so weak that they can only be detected by sensitive instruments in the laboratory.

Ferromagnetism is very significant in industry and modern technology, and is basis for several electrical and electromechanical devices like electromagnets, electric motors, generators, transformers, and magnetic storage like tape recorders, and hard disks.

Curie-Weiss Law:

Ferromagnetic materials are substance which has a magnetic moment even in absence of applied magnetic field. Saturation magnetization Ms is stated as spontaneous magnetic moment per unit volume. Curie point Tc is temperature above which magnetic moment vanishes.

If ionic and atomic magnetic moments of the paramagnetic substance can be set to line up same way by addition of an interaction then ferromagnetic substance will be formed. This interaction is known as Weiss field or molecular field or exchange field. Motion of thermal agitation of elementary particles opposes orienting effect of Weiss field.

Assume that Weiss field BE is proportional to magnetization:

BE = λM

Where λ is Weiss field constant, independent of temperature. Susceptibility above Curie point can be derived from Curie law. That is:

M/(BE + λM) = C/T

χ = M/H = C/(T - Cλ)

This provides non-zero magnetization for zero applied field at Curies point stated by

Tc = Cλ Thus,

χ = C/(T - Tc)

This equation is called as Curie-Weiss law. This law explains observed susceptibility variation in paramagnetic region above Curie point.

Spontaneous Magnetization:

A ferromagnetic substance is said to contain spontaneous magnetic moment if it has a magnetic moment even in absence of applied magnetic field. To compute spontaneous magnetization as a function of temperature

Ms = NSgμBBs(x)

In absence of applied magnetic field

x = SgμBλMs/kT

Domain Theory of Ferromagnetism:

Every piece of ferromagnetic material doesn't have strong magnetic field, despite the fact that all spins are aligned. Iron and other ferromagnets are frequently found in unmagnetised state. At temperatures below Curie point electronic magnetic moments of the ferromagnetic specimen are fundamentally all lined up.

Ferromagnetic materials spontaneously separate into magnetic domains because exchange interaction keeps nearby spins aligned with each other. This is short-range force, so rotations can take place over hundreds of times distance between atoms. To keep magnetostatic energy low, this rotation is concentrated in boundary between domains, or domain wall. Direction that magnetization rotates within the domain wall differs. The domains don't go back to original minimum energy configuration when field is turned off as domain walls tend to become pinned or snagged on defects in crystal lattice, preserving parallel orientation. This is shown by Barkhausen effect: as magnetizing field is changed, magnetization changes in thousands of small discontinuous jumps as domain walls suddenly snap past defects. This magnetization as the function of external field is explained by hysteresis curve. Though this state of aligned domains is not a minimal-energy configuration, it is very stable and has been seen to persist for millions of years in seafloor magnetite aligned by Earth's magnetic field.

Bloch Wall:

The term Bloch wall denotes transition layer that separates adjacent domains magnetized in different directions. Necessary idea of Bloch wall is that entire change in spin direction between domains magnetized in different directions takes place in gradual way over several atomic planes.

This gradual change in direction is because of the fact that for given total change of spin direction exchange energy is lower when change is distributed over several spins than when change takes place suddenly.

Exchange energy between two spins making small angle φ with each other is given as:

wex = JS2Φ2

Here J is exchange integral and S is spin quantum number. Total exchange energy of line of N + 1 atoms is therefore

Eex = JS2Φ20/N

As exchange energy of wall is inversely proportional to thickness, wall might spread out until it filled sizable proportion of crystal, were it not for restraining effect of anisotropy energy that acts to limit width of transition layer.

Total wall energy per unit area is

σw = 2π(JKS2/a)1/2

Domain Dimensions: Wall energy per unit area of crystal surface is about

wwall = σwL/D

Volume contained within domains of closure is oriented in direction of hard magnetization and involves energy K per unit volume, where K is anisotropy constant. Per unit area of crystal surface on one side, volume in domains of closure on both sides is D/2, and so anisotropy energy per unit area is

wanis = KD/2

Therefore, total energy per unit area is

w = σwL/D + KD/2

Energy per unit volume is

fdomain = (2σwK/L)1/2

Antiferromagnetism:

Physical origin of Weiss field is in quantum-mechanical exchange integral. It can be illustrated that energy of interaction of atoms i, j bearing spins Si, Sj has a term

Eex = -2JSi‾.Sj

Where J is exchange integral and is related to overlap of charge distributions i, j. When exchange integral J is positive, we have ferromagnetism; when J is negative, we have antiferromagnetism. In materials which show antiferromagnetism, magnetic moments of atoms or molecules, generally related to spins of electrons, align in the regular pattern with neighboring spins pointing in opposite directions

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