week 14: magnetic fields announcements mate 153, dr. gleixner 1

Week 14: Magnetic FieldsWeek 14: Magnetic FieldsAnnouncementsAnnouncements

MatE 153, Dr. Gleixner 1

General Concept Behind MagnetismGeneral Concept Behind Magnetism Circulating current sets up a magnetic moment (Circulating current sets up a magnetic moment (mm) )

perpendicular to the currentperpendicular to the current– This results in a B field (magnetic field) that must terminate This results in a B field (magnetic field) that must terminate

back on itself (all magnets must have a north & south, can’t back on itself (all magnets must have a north & south, can’t be alone like electrical charge)be alone like electrical charge)

Electrons in atoms are the same concept- circulating Electrons in atoms are the same concept- circulating charge that sets up a magnetic moment and a magnetic charge that sets up a magnetic moment and a magnetic fieldfield– The magnetic field is due to both the orbital angular The magnetic field is due to both the orbital angular

momentum and the spin momentum and the spin – Only electrons in unfilled shells contribute to a net Only electrons in unfilled shells contribute to a net

magnetization (as those in full shells will cancel each other magnetization (as those in full shells will cancel each other out)out)


Comapring a Magnetic Moment Comapring a Magnetic Moment with a Bulk Magnetwith a Bulk Magnet

*Used with permission from Kasap

BrO P

µ m NS

µ m

F i g . 8 . 3 : A m a g n e t i c d i p o l e m o m e n t p u t s o u t a m a g n e t i cf i e l d j u s t l i k e b a r m a g n e t . T h e f i e l d B d e p e n d s o n µ m .


Magnetism from an Orbiting Magnetism from an Orbiting ElectronElectron


I

A

–e

orb

L

r

F ig . 8 .4 : A n o rb it t in g electro n is eq u iv a len t to a m a g n eticd ip o le m o m en t o rb .


Remember L??Remember L??


33

z

0

L

Lz

(+1)cos = m

(b)

y

x

z

Orbiting electron

LLz

Bexternal

(a)

Bexternal

0

1

2

–1

–2

m

= 2

z

L = 2(2+1)

(c)

Bexternal

Fig. 3.26:(a) The electron has an orbital angular momentum which has aquantized component, Lz, along an external magnetic field, Bexternal. (b) Theorbital angular momentum vector L rotates about the z-axis. Its componentLz is quantized and therefore the orientation of L, the angle , is alsoquantized. L traces out a cone. (c) According to quantum mechanics, onlycertain orientations () for L are allowed as determined by and m.


Spin Magnetic MomentSpin Magnetic Moment


z µspin

B

z

S Sz

Fig. 8.5: The spin magnetic moment precesses about anexternal magnetic field along z and has an average value ofz along z.


Remember S??Remember S??


35

Spin Down

Spin UpSz (along Bz)

+/2

S

S

0

32

ms = +1/2

ms = –1/2–/2

32

Fig. 3.28: Spin angular momentum exhibits spacequantization. Its magnitude along z is quantized so that theangle of S to the z-axis is also quantized.

l

2

1mS

2

3S

sZ


Net MagnetizationNet Magnetization Net magnetization is due to magnetic Net magnetization is due to magnetic

moments from both forms of angular moments from both forms of angular momentummomentum

However, only electrons in un-filled shells However, only electrons in un-filled shells contribute to an overall magnetic momentcontribute to an overall magnetic moment


Average Magnetic Moment for S Average Magnetic Moment for S shellshell

Consider Consider the examplethe example of an unfilled s shell of an unfilled s shell

In an applied magnetic field, m spin can not In an applied magnetic field, m spin can not align with B because S is space quantized.align with B because S is space quantized.

The torque that results cause the spin The torque that results cause the spin magnetic moment to precess about B.magnetic moment to precess about B.


Bohr MagnetonBohr Magneton Each spin magnetic moment (Each spin magnetic moment (ss) contributes ) contributes

a average magnetic moment on the z axis in a average magnetic moment on the z axis in the presence of a magnetic fieldthe presence of a magnetic field

ee

s

e

zZ m2

emem

meS


Important Macroscopic Magnetism Important Macroscopic Magnetism TermsTerms

BBoo

oo

HH


Important Macroscopic Magnetism Important Macroscopic Magnetism TermsTerms

MM

BB

mm


Solenoid With and Without Solenoid With and Without Magnetizable MaterialMagnetizable Material


I

B

I

M

A

(b)

I

I

Bo

(a)

Fig. 8.6: (a) Consider a long solenoid. With free space asmedium inside, the magnetic field is Bo. (b) A materialmedium inserted into the solenoid develops a magnetizationM.


A Look at Where M Comes FromA Look at Where M Comes From


Surface currents

Surface currents

F rom P rincip le s o f E le ctron ic Ma te ria ls and De vice s , S e cond E d ition , S .O . Ka s a p (© McG ra w-Hill, 2002 )http://Ma te ria ls .Usa sk.Ca

Fig. 8.7: Elementary current loops result in surface currents. There is nointernal current as adjacent currents on neighboring loops are in oppositedirections


Types of MagnetsTypes of Magnets

The M that results from the applied field is The M that results from the applied field is a function of the material in the corea function of the material in the core

The material types can be divided into The material types can be divided into several main categoriesseveral main categories– DiamagnetismDiamagnetism– ParamagnetismParamagnetism– FerromagnetismFerromagnetism– AntiferromagnetismAntiferromagnetism– FerrimagnetismFerrimagnetism


Diamagnetism vs ParamagnetismDiamagnetism vs Paramagnetism

DiamagneticDiamagnetic

ParamagneticParamagnetic


DiamagnetismDiamagnetism


NS FM

Fig. 8.12: A diamagnetic material placed in a non-uniformmagnetic field experiences a force towards smaller fields.This repels the diamagentic material away from a permanentmagnet.


ParamagnetismParamagnetism


µ av = 0 and M = 0

oH

M

µ av ° 0 and M = mH

(b )(a)

F ig . 8 .1 3 : (a ) In a p a ram agnetic m ateria l each ind iv idua la tom p ossesses a p erm anent m agnetic m om ent b u t due totherm al ag ita tion there is no average m om ent p er a tom andM = 0 . (b ) In the p resence of an ap p lied field , ind iv idua lm agnetic m om ents take a lignm ents a long the ap p lied fieldand M is fin ite and a long B .


FerromagneticFerromagnetic Posses magnetization even without the Posses magnetization even without the

presence of an applied fieldpresence of an applied field Exists up to Exists up to TTCC the Curie temperature the Curie temperature

Only certain materials are ferromagneticOnly certain materials are ferromagnetic– criteria 1 is that there is an unfilled shellcriteria 1 is that there is an unfilled shell– criteria 2 is that there is a positive exchange criteria 2 is that there is a positive exchange

energyenergy


FerromagnetismFerromagnetism


M

F i g . 8 . 1 5 : I n a m a g n e t i z e d r e g i o n o f a f e r r o m a g n e t i cm a t e r i a l s u c h a s i r o n a l l t h e m a g n e t i c m o m e n t s a r es p o n t a n e o u s l y a l i g n e d i n t h e s a m e d i r e c t i o n T h e r e i s as t r o n g m a g n e t i z a t i o n v e c t o r M e v e n i n t h e a b s e n c e o f a na p p l i e d f i e l d .


Curie Temperature of Curie Temperature of FerromagnetsFerromagnets


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Iron

Msat(T)

Msat(0)

T / TC

Fig. 8.21: Normalized saturated magnetization vs. reducedtemperature T/TC where TC is the Curie temperature (1043K).


Exchange EnergyExchange Energy

EEexex=-2J=-2JeeSS11SS22

– SS11 and S and S22 are spins of electrons are spins of electrons– JJee is negative for most materials is negative for most materials– So the exchange energy is So the exchange energy is negative (minimum)negative (minimum) if the if the

spins are misalignedspins are misaligned For Fe, Co, NiFor Fe, Co, Ni

– JJee is positive is positive– So the exchange energy ESo the exchange energy Eexex is negative (minimum) if the is negative (minimum) if the

spins are alignedspins are aligned Fe, Co, Ni most common examples of ferromagnetsFe, Co, Ni most common examples of ferromagnets


Exchange Energies of Different Exchange Energies of Different MaterialsMaterials


Je

r / rd0

+

–Mn

FeCo

NiGd

Cr

Fig. 8.20: The exchange intergral as a function of r/rd,where r is the interatomic distance and rd the radius of the d-orbit (or the average d-subshell radius). Cr to Ni aretransition metals. For Gd the x-axis is r/r f where r f is theradius of the f-orbit.


AntiferromagnetismAntiferromagnetism

Individual atoms bond as to give no Individual atoms bond as to give no magnetic moment even in the presence of a magnetic moment even in the presence of a field due to the crystal structurefield due to the crystal structure

Exists only below Neel Temperature: TExists only below Neel Temperature: TNN


AntiferromagnetismAntiferromagnetism


M =0

F ig . 8 .1 6 : In th is an tiferrom agnetic B C C crys ta l ( C r) them agnetic m om ent of the center a tom is cancelled b y them agnetic m om ents of the corner a tom s (a q uarter of thecorner a tom b elongs to the unit cell).


FerrimagnetismFerrimagnetism

Results in a net magnetization even when Results in a net magnetization even when there is no applied field (similar to there is no applied field (similar to ferromagnetic).ferromagnetic).

It comes from opposite magnetizations in It comes from opposite magnetizations in crystal structure of differing magnitudes crystal structure of differing magnitudes resulting in a net permanent magnetization resulting in a net permanent magnetization in one direction .in one direction .


FerrimagnetismFerrimagnetism


M

A B

Fig. 8.17: Illustration of magnetic ordering in aferrimagnetic crystal. All A–atoms have their spins alignedin one direction and all B–atoms have their spins aligned inthe opposite direction. As the magnetic moment of an A–atom is greater than that of a B–atom, there is netmagnetization, M, in the crystal.


week 14: magnetic fields announcements mate 153, dr. gleixner 1

Documents

magnetic field mate

average magnetic moment

spin magnetic moment

overall magnetic moment

magnetic moments

magnetic field electrons

applied magnetic field

b field magnetic field