Mini-thesis study of numerical method

8/18/2019 Mini-thesis study of numerical method

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DIAMAGNETISMAND

PARAMAGNETISM


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Table of contents

1. The Interaction of Solids with Magnetic Fields2. Larmor Diamagnetism

3. Hunds !ules

". #an$#lec% &aramagnetism

'. (uries Law of Free Ions

). (uries Law in Solids

*. +diabatic Demagneti,ation

-. &auli &aramagnetism

. (onduction /lectron Diamagnetism

10. uclear Magnetic !esonance The night Shift

11. /lectron Diamagnetism in Do4ed Semiconductors


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&ro4erties

•

Diamagnetic materials ha5e a wea%6 negati5e susce4tibilit7 tomagnetic fields.

1. Diamagnetic materials are slightl7 re4elled b7 a magnetic field

and the material does not retain the magnetic 4ro4erties when

the e8ternal field is remo5ed.

2. In diamagnetic materials all the electron are 4aired so there is

no 4ermanent net magnetic moment 4er atom.

3. Diamagnetic 4ro4erties arise from the realignment of the

electron 4aths under the influence of an e8ternal magnetic field.

". Most elements in the 4eriodic table6 including co44er6 sil5er6

and gold6 are diamagnetic.


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• Paramagnetic materials ha5e a small6 4ositi5e susce4tibilit7

to magnetic fields.

1. These materials are slightl7 attracted b7 a magnetic field andthe material does not retain the magnetic 4ro4erties when the

e8ternal field is remo5ed.

2. &aramagnetic 4ro4erties are due to the 4resence of some

un4aired electrons6 and from the realignment of the electron 4aths caused b7 the e8ternal magnetic field.

3. &aramagnetic materials include magnesium6 mol7bdenum6

lithium6 and tantalum.


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Diamagnetic

Paramagnetic


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Magnetization density andSusceptibility


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1.Magneti,ation densit7

• +t T90 the magneti,ation densit7 M:H; of a


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!e4lace the magnetic Helmholt, free energ7 F to the energ7

/ we ha5e shorter formula

Here F is defined b7 the fundamental statistical mechanical

rule


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2.The Susceptibility

• The susce4tibilit7 is defined as

In electromagnetism6 the magnetic susceptibility is a

dimensionless 4ro4ortionalit7 constant that indicates the

degree of magneti,ation of a material in res4onse to ana44lied magnetic field.


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Calculate the atomic susceptibilityGene!al "o!mulation


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=e find the shift energ7 is calculated

=hen we a44l7 the magnetic to material6 the energ7 will be

shifted

/ 9 /0n>?/nThe shift energ7 is com4onent which ma%e the characteristic of

material.


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3. Susce4tibilit7 of insulators with

all shells filled Larmor diamagnetism

• The sim4lest a44lication of abo5e result is to a solid com4osedof ions will all electronic shells filled. Such an ion has ,ero

s4in and orbital angular momentum in its ground state


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• (onse


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4. Ground state of ions witha partially lled shellHund’s rules


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4.1 Russell-Saunders Coupling

• To a good a44ro8imation6 the Hamiltonian of the atom or ion

can be ta%en to commute with the total electronic s4in and

orbital angular momenta6 S and L6 as well as with the total

electronic angular momentum @9S>L.

• The states of the ion can be described b7 1;6 S:S>1;6 S, 6 @:@>1;

and @,.

• Since the filled shells ha5e ,ero orbital6 s4in6 and total angular

momentum6 these


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4.2 Hund’s First Rule

• Aut of man7 states one can form b7 4lacing n electrons into

the 2:2l >1; le5els of the 4artiall7 filled shells6 those that lie

lowest in energ76 ha5e the largest total s4in S : that is

consistent with the e8clusion 4rinci4le.

• ote The largest 5alue S can e16 S9n all electrons can ha5e 4arallel s4ins without

multi4le occu4ation of an7 one$electron le5el in the shell.

• If n92l >16 S9l >n two electrons on a shell ha5e o44osite s4ins.

S is reduced form its ma8imum 5alue b7 half a unit for eachelectron after the :2l >1;th

•


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4.2 Hund’s Second Rule

• The total orbital angular momentum L of the lowest$l7ing

states has the largest 5alue that is consistent with Hunds first

rule6 and with the e8clusion 4rinci4le. That 5alue e


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4.3 Hund’s Third Rule

• The states of lowest energ7 can lea5e :2L>1;:2S>1; 4ossible

states6 according to this6 total angular momentum @ can ta%e all

integral 5alues between CL$SC and CL>SC.

• The 5alue @ assumes in the states of lowest energ7 is


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'. Susce4tibilit7 of insulationscontaining ions with a 4artiall7

filled shell

&aramagnetism


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• (onsider the atoms with full the change of energ7 from ground

state. It mean that the shift of energ7 is gi5en b7

=hen the solid contain # such ions 4er unit

5olume6 the susce4tibilit7 is

'.1 If @90


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'.2 @E0

• =e onl7 consider the first term of energ7 because is much

greater than the other two

• The ground state is :2@>1;$fold degenerate in ,ero field

• /5aluating and diagonali,ing :2@>1;$dimentional s


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=ithout the surrounding state 5ectors

If onl7 the :2@>1; states in the ground state multi4let will

contribute a44reciabl7 to the free energ7. The abo5e e


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Magneti,ation of a Set of Identical

Ions of +ngular Momentum @

(uries Law


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• If onl7 the lowest 2@>1 state are thermall7 e8cited with

a44reciable 4robabilit76 the free energ7

The magneti,ation of ions in a 5olume #


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• =here the Grillouin function G H:8; is defined b7

=hen H JJ % GT the small$8 e84ansion


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The susce4tibilit7 is gi5en

or


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• This susce4tibilit7 is larger than the tem4erature$inde4endent

Larmor diamagnetic susce4tibilit7.

• =hen an ion of the shell to the total susce4tibilit7 of the solidcom4letel7 dominates the diamagnetic contribution from the

other :filled; shell.

• Diamagnetic susce4tibilities are of order 10$'

• +t room tem4erature6 4aramagnetic susce4tibilit7 should be oforder 10$2 to 10$3.


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(uries law in Solids

• (uries law fre


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Thermal 4ro4erties of

&aramagnetic insulatorsadiabatic demagneti,ation


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• The Helmholt, free energ7 F=U-TS

Where U is internal energy

The magnetic entropy S(H,T) is given by:

KF onl7 de4ends on the 4roduct KH


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• The entro47 has form

In the case of S unchanged6 HT cannot change6 therefore

This can be se! as a practical metho! "or achieving lo#temperatres only in a temperatre range #here the speci"ic

heat o" the spin system is the !ominant contribtion to the

speci"ic heat o" the entire soli!$


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Susce4tibilit7 of metals

&auli &aramagnetism


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/ach electron will contribute :ta%ing g92;

$G# if its s4in is parallel to the field H

G# if antiparallel If n% is the nmber o" electrons per nit volme6 the magneti,ation

densit7 will be

If the electrons interact with the field onl7 through their magnetic

moments6 the shift energ7 of each electronic le5el b7 GH


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• The susce4tibilit7 is

It is %nown as the &ali paramagnetic ssceptibility and in case

of conduction electron is essentiall7 inde4endent of tem4erature.

In the free electron case6 the densit7 of le5el has form

• The magneti,ation


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(onduction electron diamagnetism

• There are also diamagnetic effects arising from the cou4ling of

the field to the orbital motion of the electrons.

• The de4endence of M on H doesnt a5erage out to ,ero. There

is a net non5anishing magneti,ation anti4arallel to H6 %nown

as the 'an!a !iamagnetism

• For free electrons


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/lectron diamagnetism in do4ed

semiconductors• Do4ed semiconductors is one %ind of conducting material which

has conduction electron diamagnetism can be larger than

4aramagnetism.

• (onsider the case in which the carries go into bands with s4herical

s7mmetr76 so that ()=*+ + m• This is 4ro4ortional to m for free electrons6 the &auli susce4tibilit7

of carriers will be reduced b7 mm : 0.1 or smaller;

• +s a result


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Mini-thesis study of numerical method