magnetic properties of materials

7/31/2019 Magnetic Properties of Materials

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PHYS 44064

SOLID STATE PHYSICS

BRIEF REPORT OF

MAGNETIC PROPERTIES OF SOLIDS

Student name: M.A.A.N Gunawardana

Student no: PS/2007/086


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MAGNETIC PROPERTIES OF SOLIDS

Index

Introduction..2

Diamagnetism, Paramagnetism and ferromagnetism ..3

Langevins theory of diamagnetism.4

Langevins theory of Paramagnetism.6

Failure of Langevins theory .8

Weiss Modification8

Ferromagnetism.8

Antiferromagnetism and Ferrimagnetism..9

Weiss theory of Ferromagnetism ..9

Applications12

Nuclear Magnetic Resonance.12

NMR Spectrometer.13


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Introduction

According to their magnetic properties ,magnetic materials are divided into 3 groups

they are

Diamagnetic Paramagnetic Ferromagnetic

The vector quantity M defined as the intensity of magnetization or more often

magnetization ,which is used to characterization of substance, it is the vector sum of the

magnetic moments of atoms (or molecules) contained in unit volume

Where N=the number of particles in volume V of magnetic material

=magnetic moment of I th atom (or molecule)For large number of magnetic materials it is found out that the intensity of

magnetization is directly proportional to the magnetic field intensity i.e. Where

is known as the magnetic susceptibility of the substance


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Diamagnetism

In 1846 Michael Faraday discovered that a specimen of bismuth brought near to the

pole of strong magnet is repelled. He called such substance diamagnetic. Antimony,

bismuth, mercury, gold and copper are some example of diamagnetic substance. For

diamagnetic substance and is independent of temperature. Unlike paramagneticmaterials, whose atoms or molecules have a net magnetic moment, the atoms ormolecules of a diamagnetic material have zero magnetic moment in absence of an

external magnetic field. With external magnetic field a net dipole moment, opposing the

field is induced in the atoms or molecules. In paramagnetic materials there is also

diamagnetic effect but it is much weaker than that of the paramagnetism

Paramagnetism

Paramagnetic substances are attracted towards the region of stronger magnetic field.

Aluminum oxygen and platinum are some examples of paramagnetic materials. Their

magnetic susceptibility and depends on temperature according to the Curie Lawi.e. Where C is Curie constant

Paramagnetism occurs on substances where the individual atoms, ions, or molecules

possess a permanent magnetic dipole moment. In the absence of external magnetic field,

the atomic dipoles point in random directions. So there is no resultant magnetization ofsubstance as a whole in any direction. This random orientation is the result of thermal

agitation within the substance. When external field is applied, the atomic dipoles tend to

orient themselves parallel to the field, since this is a state of lower energy than the anti

parallel position. This gives net magnetization parallel to field and a positive

contribution to susceptibility. At lower temperatures, the thermal agitation, which tends

to give a random orientation to the atomic dipoles, is less. Hence bigger proportions of

dipoles are able to align themselves parallel to field, and the magnetization is greater for

a given field. For ordinary field temperatures . For large fields at lowtemperatures the magnetization produced is no longer proportional to the applied field,and tends to a constant value. This saturation effect is produced when all atomic dipoles

are aligned parallel to the field, so that magnetization reaches to limiting maximum

value.

Ferromagnetism

Ferromagnetic materials are very strongly magnetic. For a given field, magnetization in

ferromagnetic substance is abouttimes stronger than dia or Para magneticsubstances. Irons, Nickel, cobalt are some examples of ferromagnetic substances. As

temperature increases, the value of decreases. Above a certain temperature, Knownas the Curie temperature, ferromagnets becomes paramagnets


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Langevins theory of Diamagnetism

Consider an electron (mass =m, charge = e) rotating about the nucleus (charge=Ze) in a

circular orbital of radius r. letbe angular velocity of electron. Then

Or .(1)The magnetic moment of electron is

=currentarea=

(2)Let a magnetic field of induction be now applied. is normal to and into the page

Electron

Electron

Figure 1

An additional force called Lorentz force acts on the electron ( )

The condition of stable motion is now given by

.(3) Or Solving the quadratic equation in

,

r

Ze

r

Ze


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Thus the angular frequency is now different from . The result of establishing a fieldof flux density is to set up a precessional motion of the electronic orbits with angularvelocity , this is called the Larmor theorem. ThenChange in frequency of Revolution of electron= The corresponding change in the magnetic momentum of the electron is

{ }

On summing over all electrons in the atom, the induced moment per atom becomes

Let N be the number of atoms per unit volume. Then the magnetization M is given by

All the electron orbits are not oriented normal to the magnetic field. Hence in Eq. (6)should be replaced by the average of square of the projection of orbit radii for various

electrons in a plane perpendicular to . Hence we should replace in Eq.(6) by therefore

Volume susceptibility of material


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Eq.(7) shows that is independent of the field strength and temperature. This is inaccordance with Curies experimental results.

Langevins theory of Paramagnetism

Langevin assumes that each atom has a permanent magnetic momentum m. The onlyforce acting on atom is that due to the external field .Let be the angle of inclinationof the axis of the atomic dipole with the direction of the applied field. Then magneticpotential energy of the atomic dipole is

Now, on classical statistics, the number of atoms making an angle between

is

Where K is Boltzmanns constant and T is the absolute temperature

Put . Then

Hence the total number of atomic magnets in unit volume of paramagnetic material

Put then

The component of each dipole moment parallel to B in . The total magneticmoment of all the n atoms contained in unit volume of the gas is the magnetization M. itis given by

Put therefore, we get


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Evaluating integral and substituting the value of C from (3), we get

[ ] Where * +is called the langevins function

The variation of M with is shown in figure 2.

M

mn

Figure 2

Case(i):At low temperatures or large applied field,

Hence, the magnetization M in this case will be

So saturation is reached when all the atomic dipoles are parallel to B

Case (ii): Under normal conditions is very small. Then,

Initial slope


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Where is called the Curie constantFailure of langevins theory

(i) Langevins theory was unable to explain a more complicated dependence ofsusceptibility upon temperature exhibited by several paramagnetics such as highly

compressed and cooled gasses, very concentrated solutions of salts etc...

(ii) Langevins theory could not account for the intimate relation between Para -andferromagnetism.

Weiss Modification

Langevins theory applies only to gasses, where the molecules are sufficiently far apart for their

mutual interactions to be negligible. In liquids and solids such interactions may be large, and many

substances obey the modification Curie-Weiss law

is called the curie temperature and is characteristic of the substance. Eq.(10) holds only attemperatures where ||Eq.(10) is of the same form as Eq.(9), except that the origin oftemperature is shifted from 0 to Ferromagnetism

Ferromagnetic substances are very strongly magnetic, the best known examples of

ferromagnets are the transition metals Fe, Co, and Ni. A ferromagnet has a spontaneous

magnetic moment a magnetic moment even in zero applied fields. The atoms (or

molecules)of ferromagnetic materials have a net intrinsic magnetic dipole moment which is

primarily due to the spin of electrons. The interaction between the neighbouring atomic

magnetic dipoles is very strong. It is called spin exchange interaction and is present even in

the absence of an external magnetic field. It turns out that the energy of two neighbouring

atomic magnets due to interaction is the least when their magnetic moments are parallel.

The neighbouring magnetic moments are therefore strongly constrained to take parallel

orientation (see figure 3,a). This effect of exchange interaction to align the neighbouring

magnetic moments parallel to one another spreads over a small finite volume of the bulk

a. b. c. Figure 3


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This small (1-0.1 mm across) volume of the bulk is called a domain. All magnetic moments

within a domain will point in the same direction, resulting in a large magnetic moment. Thus

the bulk material consists of many domains. The domains are oriented in different

directions. The total magnetic moment of a sample of substance is the vector sum of the

magnetic moments of the component domain

In unmagnetised piece of ferromagnetic material, the magnetic moments of the domains

themselves are not aligned. When an external field is applied, those domains that are

aligned with the field increasing size at the expense of the others. In a very strong field all

the domains are lined up in the direction of the field and provide the high observed

magnetization.

Antiferromagnetism and Ferrimagnetism

The only type of magnetic order which has been considered thus far is ferromagnetism, inwhich , in fully magnetized state, all the dipoles are aligned in exactly the same direction

(figure 3,a). There are, however, substances which show different types of magnetic order.

In antiferromagnetic materials such as Cr and MnO, the dipoles have equal moments, but

adjacent dipoles joint in opposite direction (see figure 3, b). Thus the moments balance each

other, resulting in a zero net magnetization. In ferrimagnetic materials (also called ferrites)

such as, the magnetic moments of adjacent ions are anti parallel and of unequalstrength (see figure 3, c) so there is a finite net magnetization. By suitable choice of rare

earth ions in the ferrite lattices it is possible to design ferrimagnetic substances with specific

magnetizations for use in electronic components.

Weiss theory of Ferromagnetism

According to Weiss, the atomic magnets of a ferromagnetic substance are grouped into

certain regions or domains. When the substance is in the unmagnetised condition, the

domains form closed chains with no free poles. When the substance is magnetized, the

chains break up and the domains gradually set themselves with their magnetic axes all

pointing in he field direction. Thus ferromagnetism is a crystal phenomenon

Weiss assumed that a molecular magnetic field exists at the position of every atom ormolecule this field arises due to the interaction of all neighbouring molecules. The

molecular field is proportional to the magnetization vector

Here =molecular field coefficientThe effective field strength may be regarded as the vector sum of external applied fieldstrength B and the internal molecular field strength

Hence


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Consider 1 gram mole of the substance. Let

Then

(

)

(

)

Then

The domains will obey the general theory of paramagnetism by langevin

Here,

When external field is zero,

Now,

Here N=Avogadros number


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Or

Eqs.(3) and (4) simultaneously determine the condition of spontaneous magnetization

figure 4 shows a graph drawn between and corresponding to Es. (3) and (4)curve 1 is langevin curve corresponding to Eq.(3) while straight line 2 is corresponding to

Eq.(4) . the two curves meet at 0 and A. the solution is not true. Hence is the

true solution

A B

C

(1) (2)0.5

1 2 3 4

Figure 4

It is obvious from the graph that A represents a stable state of spontaneous magnetization.

If the molecules in a domain assume sate C, then the local magnetization equilibrium sate A,

now the magnetization and the value of will in consequence increase until the state AD isreached . on the other hand , if the molecules in domain assume state B, then the localmagnetization is more than the equilibrium value. Now the magnetization and the value ofwill tend to decrease until the state AD is reachedWe know that the slope of the tangent at the origin of the langevin curve is

since , when is small. Hence the condition for stable spontaneousmagnetization is given by


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But , the Curie point. Hence T


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NMR spectrometer

Figure 4 shows a schematic arrangement for observing the nuclear magnetic resonance, and

is based on the energy absorption method

(Static field) s

Figure-5

The specimen S, about 1 c. c. of the material being investigated, is placed between the poles

of an electromagnet. This magnet produces the field . An r-f coil surrounding thespecimen is carefully positioned to produce a second field perpendicular to . An r-fgenerator not only serves to drive the coil but also supplies a signal to auxiliary circuitry

which measures the r-f power absorbed by the specimen. To trace out the absorption line,

an auxiliary low frequency oscillator supplies power to secondary coils wound on the main

magnet core. These coils permit the main field. Such a signal is sketched in figure 6

Power absorbed

Figure 6

Rf-

generator

Electromagnet

Rf-coil

Amplifier mixer and

other circuitry

Oscilloscope


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NMR finds a number of practical applications. It can be used for the accurate determination

of nuclear moments. In a sensitive form of magnetometer NMR can be used for measuring

magnetic fields. In medicine NMR tomography has been developed. In this technique

images of tissues are produced by the magnetic resonance. Whole plants, small animals or

parts of bigger animals are placed between the pole pieces of suitably designed magnet.Using surface coils of wire radio frequency radiation is directed towards the part under

study. Spectra obtained in this way are used for medical diagnosis. This method is supposed

to be superior diagnosis by X rays because the potential damage to living tissue by X rays is

not present in the case of radio frequency radiation

magnetic properties of materials

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