sign in sign up
<<
Home , Demagnetizing field, Magnetic moment, Magnetization, Magnetostatics

Chapter 2:

1. The Magnetic 2. Magnetic Fields 3. Maxwell’s Equations 4. Magnetic Calculations 5. Magnetostatic and

Comments and corrections please: [email protected]

Dublin January 2007 1 Further Reading:

• David Jiles Introduction to and Magnetic Materials, Chapman and Hall 1991; 1997 A detailed introduction, written in a question and answer format.

• Stephen Blundell Magnetism in Condensed Matter , Oxford 2001 A new book providing a good treatment of the basics

• Amikam Aharoni Theory of , Oxford 2003 Readable, opinionated phenomenological theory of magnetism

• William Fuller Brown Micromagnetism , 1949 The classic text

Dublin January 2007 2 1. The Moment The m is the elementary quantity in solid state magnetism. Define a local moment - - M(r,t) which fluctuates wildly on a sub-nanometer and a sub-nanosecond scale. Define a mesoscopic average magnetization !m = M!V The continuous medium approximation

M can be the spontaneous magnetization Ms within a ferromagnetic domain A macroscopic average magnetization is the domain average

M = !iMiVi/ !iVi

M (r)

Ms

The mesoscopic average magnetization

Dublin January 2007 3 dl

r m m = IA 1/2 (r"l) O

I m =1/2# r"j(r)d3r

A magnetic moment m is equivalent to a current loop m =1/2# r"j(r)d3r = 1/2# r"Idl = I# dA = m

Inversion Space Time Polar vector -j j

Axial vector M -M

Dublin January 2007 4 1.1 Field due to electric currents and magnetic moments

Biot-Savart Law

B

Unit of B - j Unit of µ T/Am-1 0 ! -7 -1 Right-hand corkscrew µ0=4 $ 10 T/Am

Dublin January 2007 5 1.1 Field due to electric currents and magnetic moments

Field at center of current loop

Dipole field far from current loop - lines of

Dublin January 2007 6 1.1 Field due to electric currents and magnetic moments

2 BA = 4(µ0Idl/4$r )sin% sin%= dl/2r A

% r

& B m Idl

At a general position,

Dublin January 2007 7 2. Magnetic Fields 2.1 The B-field

'.B = 0

dA ’s theorem

Flux: d( = BdA Unit Weber (Wb)

15 Flux quantum (0 = 2.07 10 Wb

Dublin January 2007 8 The B-field

Sources of B " electric currents in conductors " moving charges I " magnetic moments " time-varying electric fields. Not in magnetostatics r

B ' x B = µ j 0 ’s law. Good for very symmetric ex ey ez current paths. )/)x )/)y )/)z

Bx BY BZ B = µ0I/2$r

Dublin January 2007 9 The B-field

Forces: F = q(E + v x B) Lorentz expression. gives dimensions of B and E.

The force between two parallel wires each carrying one ampere is precisely 2 10-7 N m-1. The field at a distance 1 m from a wire t carrying a current of 1 A is 0.2 µ* e e c e ompression C ting Magn rt Spac c tary Spa rain nt Magnetagnetagnet B tar condu M noid er lse plosive Flux uman Hea terplane up u x Human H InterstellarIn 's Field Soleat the SurfacePermaneS HybridP ME Magne 1E-15 1E-12 1E-9 1E-6 1E-3 1 1000 1E6 1E9 1E12 1E15 pT µT T MT

Dublin January 2007 10 Typical values of B Human brain 1 fT

Earth 50 µT

Helmholtz coils 0.01 Am-

Electromagnet 1 T Magnetar 1012 T

Superconducting 10 T

Dublin January 2007 11 2.2 Uniform magnetic fields.

Long B =µ0nI

3/2 Helmholtz coils B =(4/5) µ0NI/a

a

Halbach cylinder B =µ0M ln(r2/r1)I

Dublin January 2007 12 2.3 The H- field.

In free space B = µ0H

' x B = µ0(jc + jm)

'.H = - '.M Coulomb approach to calculate H

3 H = qmr/4$r qm is magnetic charge

Dublin January 2007 13 The H- field.

H = Hc + Hm

Hm is the stray field outside the magnet and the inside it

B = µ0(H + M)

Dublin January 2007 14 2.4 The demagnetizing field The H-field in a magnet depends on the magnetization M(r) and on the shape of the magnet. H d is uniform in the case of a uniformly-magnetized ellipsoid . Tensor relation: H M d = - N A constraint on the values of N when M lies along one of the principal axes, x, y, z, is

Nx + Ny + Nz = 1 • It is common practice to use a demagnetizing factor to obtain approximate internal fields in samples of other shapes (bars, cylinders), which may not be quite uniformly magnetized. N • Examples. Long needle, M parallel to the long axis, a 0 Long needle, M perpendicular to the long axis 1/2

Sphere 1/3

Thin film, M parallel to plane 0 Thin film, M perpendicular to plane 1 Toroid, M perpendicular to r 0

General ellipsoid of revolution Nc = ( 1 - Na)/2

Dublin January 2007 15 Dublin January 2007 16 2.5 External and internal fields

H = H’ + Hd

Inernal field applied field demag field

H ≈ H’ - N M

For a powder sample Np = (1/3) + f(N - 1/3) f is the packing fraction

H’

H’ H’

Ways of measuring magnetization with no need for a demag correction toroid long rod thin film

Dublin January 2007 17

H’ H’

Magnetization of a sphere, and a cube

The state of magnetization of a sample depends on H, ie M = M(H). H is the independent variable.

Dublin January 2007 18 2.6 Susceptibility and permeability

Simple materials are linear, isotropic and homogeneous (LIH) M = "’H’ "’ is external susceptibility M = "H " is internal susceptibility

It follows that from H = H’ + Hd that 1/+ = 1/+’ - N For typical paramagnets and diamagnets + ! 10-5 to 10-3, so the difference between + and +’ can be neglected.

In ferromagnets, + is much greater; it diverges as T , TC but +’ never exceeds 1/N. M M

M

H M s /3 H H' 0 H’ H Magnetization curves for a ferromagnetic sphere, versus the external and internal fields. "’=3

Dublin January 2007 19 • A related quantity is the permeability, defined for a paramagnet, or a soft ferromagnet in small fields as

µ = B/H.

Since B = µ0(H + M), it follows that µ = µ0(1 + +r).

The relative permeability µr= µ/µ0 = (1 + +) µ0 is the permeability of free space.

•In practice it is much easier to measure the mass of a sample than its volume. Measured magnetisation is usually - = M/., the magnetic moment per unit mass (. is the density).

Likewise the mass susceptibility is defined as +m = +/ .

Dublin January 2007 20 2. Maxwell’s Equations

In , there is also an auxiliary field, D. D = %0E + P

(J is defined as the ‘magnetic polarization’ J = µ0M ) Maxwell’s equations in a material medium are expressed in terms of the four fields

In magnetostatics there is no time-dependence of B. D or # Conservation of charge '.j = -)./)t. In a steady state ). /)t = 0 Magnetostatics: '.j = 0; '.B = 0 'xH = j Constituent relations: j = j(E); P = P(E); M = M(H)

Dublin January 2007 21

spontaneous magnetization

virgin curve initial susceptibility

major loop

The hysteresis loop shows the irreversible, nonlinear response of a ferromagnet to an internal magnetic field M = M(H). It reflects the arrangement of the magnetization in ferromagnetic domains.

The B = B(H) loop is deduced from the relation B = µ0(H + M).

Dublin January 2007 22 3 Calculations In magnetostatics, the sources of magnetic field are i) current-carrying conductors and Biot-Savart law ii) magnetic material

------

Dipole sum Amperian approach-currents Coulomb approach-magnetic charge

Dublin January 2007 23 a) Dipole Integrate over the magnetization distribution M(r)

Compensates the at the origin

Dublin January 2007 24 a) Amperian approach Integrate over the equivalent currents j(r)

jm = $ x M and jms = M x en Evaluate from the Biot-Savart law.

Zero for a uniform distribution of M

Dublin January 2007 25 a) Coulomb approach Use the equivalent distribution of magnetic charge

#m = -$.M and #ms = M.en Evaluate from the Biot-Savart law.

Zero for a uniform distribution of M

Dublin January 2007 26 4.1 The magnetic potentials

a) Vector potential for B Maxwell’s =n '.B =0 Now '.('xA) = 0 hence B = ' x A

A is the magnetic vector potential. Units T m.

Latitude in the choice of A: (0, 0, Bz) can be represented by (0, xB,0), (-yB, 0, 0) or (1/2yB, 1/2xB, 0) The of any scalar f(r) can be added to A since 'x'f= 0 B is unchanged by any transformation A ,A’ known as a gauge transformation. Coulomb gauge: choose f( r) so that '.A then A = (1/2)B x r

Dublin January 2007 27 Vector potential for B

Dublin January 2007 28 b) scalar potential for H When the H-field is produced only by , and not by conduction currents, it can be expressed in terms of a potential. The field is conservative, ' x H = 0 Since ' x ' f( r) = 0 for any scalar, we can express H as

H = -'/m

Units of /m are Amps. '.(H + M) = 0 2 Hence ' /m = -.m where .m = - '.M

The potential due to a charge qm is /m = qm /4$r

A dipole m has potential m.r/4$r3

Dublin January 2007 29 4.2 Boundary conditions ! At any interface, it follows from Gauss’s law

B A #S .d = 0

that the perpendicular component of B is continuous.

It follows from from Ampère’s law

H #loop .dl = I0 = 0 (there are no conduction currrents on the surface) that the parallel component of H is continuous.

Since B = ' x A

B A A #S .d = #loop .dl (Stoke’s theorem)

If dollows that the parallel component of A is continuous.

The scalar potential is continuous /m1 = /m2

Dublin January 2007 . 30 Boundary conditions !

In LIH media, B = µ0 µr H

B1en = B2en

H1en = µr2/µr1 H2en

Hence field lies ! perpendicular to the surface of soft iron but parallel to the surface of a superconductor.

Diamagnets produce weakly repulsive images Paramagnets produce weakly attractive images

Images in a ferromagnet (a) and a superconductor (b)

Dublin January 2007 31 4.3 Local magnetic fields !

Hloc = H1 + H2 # $

H1 = -NM + (1/3)M2

H is evaluated as a dipole sum. 2 1 H2 =!

Generally H2 =f M Here f ≈ 1; it depends on the crystal lattice f = 0 for a cubic lattice. Dipole interactions are source of an intrinsic anisotropy contribution.

Dublin January 2007 32 5. Magnetostatic Energy and Forces

Magnetostatic (dipole-dipole) forces are long-ranged, but weak. They determine the magnetic microstructure. ! -1 ! ! 6 -3 M 1 MA m , µ0Hd 1 T, hence µ0HdM 10 J m Atomic volume ! (0.2 nm)3; equivalent temperature ! 1 K.

2 2 Products BH, BM, µ0H , µ0M are all per unit volume.

Magnetic forces do no on moving charges F = q(vxB) or currents F = j x B) No associated with the magnetic force.

"

0 = m x B In a non-uniform field, F = -'Um

U = #mBsin &’d&’ B F = m.'B

U = -m.B ! m m

Dublin January 2007 33 Interaction of two :

m2 m1 U = -m B = -m B B12 p 1 21 2 12 B21 Up =-(1/2)(m1B21 + m2B12)

Reciprocity theorem :

H2 H1

M1 M2

Dublin January 2007 34 5.1 Self-energy Energy of a body in the field Hd it creates itself.

Dublin January 2007 35 Dublin January 2007 36 5.2 Energy associated with a magnetic field

Dublin January 2007 37 3 Energy product - #i µ0B.Hd d r

Dublin January 2007 38 5.2 Energy in an external field

For hysteretic material, B " µH. The energy needed to prepare a state depends on the path followed. The work done to produce a small flux change is

1W = -%I1t = I1(. By Ampere’s law, I = #loopHdl. 3 1W = #loop1( Hdl. 1W = #1BHd r

It would be better to have an expression for the energy of M( r) in the external, applied field H’, because we don’t know what H( r) is like throughout the body. The real H-field is the one in Maxwell’s equations

H = H’ + Hd The constitutive relation is M = M(H) nor M = M(H’)

Dublin January 2007 39 Energy in an external field

The applied field H’ is created by some current distribution j’ '.H’ = 0 ' x H’ = j’ The field created by the body satisfies

'.Hd = - '.M ' x Hd = 0

B = µ0(H + M) = µ0(H’ + Hd + M)

3 3 The magnetic work 1W’ = #1B(H’ + Hd) d r Subtract the term µ0 #1H’H’d r for space 3 Energy change due to the body is 1W’ = #(1B H’ - µ0 1H’H) d r

= 0

Dublin January 2007 40 Dublin January 2007 41 Energy in an external field B M

H H! a) b) #HdB #µ0H’dM

Dublin January 2007 42 5.4 of magnetic materials

M dU = HxdX + dQ

dQ = TdS 2!FG Four thermodynamic potentials U(X,S) -"2!FG

E(HX,S) F(X,T) = U - TS dF = HdX - SdT H’

G(HX,T) = F- HXX dG = -XdH - SdT S = -()G/)T)H’ µ0 M = -()G/)H’)T’ Magnetic work is H1B or µ0H’1M Maxwell relations dF = µ0H’dM - SdT dG = -µ MdH’ - SdT 0 ()S/)H’)T’ = - µ0()M/)T) H’ etc.

Dublin January 2007 43