Diamagnetism and Paramagnetism

Diamagnetism and Paramagnetism

Physics 355

Free atoms…

The property of magnetism can have three origins:

1. Intrinsic angular momentum (Spin)2. Orbital angular momentum about the nucleus3. Change in the dipole moment due to an applied field

In most atoms, electrons occur in pairs. Electrons in a pair spin in opposite directions. So, when electrons are paired together, their opposite spins cause their magnetic fields to cancel each other. Therefore, no net magnetic field exists. Alternately, materials with some unpaired electrons will have a net magnetic field and will react more to an external field.

Diamagnetism: Classical ApproachDiamagnetism: Classical Approach

nucleus

electron

Consider a singleclosed-shell atomin a magnetic field.Spins are all paired andelectrons are distributedspherically around theatom. There is nototal angularmomentum. r

B

E

v

nucleus

electron

r

B

v, 0

20 0F m r

20 02

e r Lorentz ForceF = -e(v x B)F = eBr0 2

eB

m

Diamagnetism: Larmor PrecessionDiamagnetism: Larmor Precession

Diamagnetism: Quantum ApproachDiamagnetism: Quantum Approach

2 2

22 4

e e r

m m L S B

startingpoint

Quantum mechanics makes some useful corrections. The components of L and S are replaced by their corresponding values for the electron state and r2 is replaced by the average square of the projection of the electron position vector on the plane perpendicular to B, which yields

where R is the new radius of the sphere.

2 223R r

Diamagnetism: Quantum ApproachDiamagnetism: Quantum Approach

22 2

2 2 2 2

22 4

where

2

z z z

z z

z

e eL S x y B

m m

L L d

S

x y x y d

If B is inthe z direction

Diamagnetism: Quantum ApproachDiamagnetism: Quantum Approach

Consider a singleclosed-shell atomin a magnetic field.Spins are all paired andelectrons are distributedspherically around theatom. There is nototal angularmomentum.

• The atomic orbitals are used to estimate <x2 + y2>.

• If the probability density * for a state is spherically symmetric <x2> = <y2>= <z2> and <x2 + y2>=2/3<r2>.

• If an atom contains Z electrons in its closed shells, then

22

6zZe

r Bm

• The B is the local field at the atom’s location. We need an expression that connects the local field to the applied field. It can be shown that it is

10local applied 3 B B M

DiamagnetismDiamagnetism

22

220 0

6

6

znZeM n r Bm

M nZer

B m

• Diamagnetic susceptibilities are nearly independent of temperature. The only variation arises from changes in atomic concentration that accompany thermal expansion.

Core ElectronContribution

Diamagnetism: ExampleDiamagnetism: ExampleEstimate the susceptibility of solid argon. Argon has atomic number 18; and at 4 K, its concentration is 2.66 x 1028 atoms/m3. Take the root mean square distance of an electron from the nearest nucleus to be 0.62 Å. Also, calculate the magnetization of solid argon in a 2.0 T induction field. ccp structure

220

2 27 28 3 19 11

31

5

6

4 10 T m/A 2 66 10 m 18 1 60 10 C 6 2 10 m

6 9 11 10 kg

1 08 10

nZer

m

. ( ) . .

.

.

Diamagnetism: ExampleDiamagnetism: Example

ccp structure

5

70

1 08 10 2 0 T17 2 A/m

4 10 T m/A

BM

. ..

Estimate the susceptibility of solid argon. Argon has atomic number 18; and at 4 K, its concentration is 2.66 x 1028 atoms/m3. Take the root mean square distance of an electron from the nearest nucleus to be 0.62 Å. Also, calculate the magnetization of solid argon in a 2.0 T induction field.

ParamagnetismParamagnetism

Core ParamagnetismCore Paramagnetism

If <Lz> and <Sz> do not both vanish for an atom, the atom has a permanent magnetic dipole moment and is paramagnetic.

Some examples are rare earth and transition metal salts, such as GdCl3 and FeF2. The magnetic ions are far enough apart that orbitals associated with partially filled shells do not overlap appreciably. Therefore, each magnetic ion has a localized magnetic moment.

Suppose an ion has total angular momentum L, total spin angular momentum S, and total angular momentum J = L + S.

22g J L S J/

Core ParamagnetismCore Paramagnetism

Landé g factor

B

B24

B

where is the Bohr magneton

2 9 27 10 J/T

g

e m

Jμ

/ .

Hund’s RulesHund’s Rules• For rare earth and transition metal ions, except Eu and Sm,

excited states are separated from the ground state by large energy differences – and are thus, generally vacant.

• So, we are mostly interested in the ground state.• Hund’s Rules provide a way to determine J, L, and S.

• Rule #1: Each electron, up to one-half of the states in the shell, contributes +½ to S. Electrons beyond this contribute ½ to S. The spin will be the maximum value consistent with the Pauli exclusion principle. Frederick Hund

1896-1997

Hund’s RulesHund’s Rules• Each d shell electron can contribute either 2, 1, 0, +1, or

+2 to L.• Each f shell electron can contribute either 3, 2, 1, 0, +1,

+2, or +3 to L.• Two electrons with the same spin cannot make the same

contribution.

• Rule #2: L will have the largest possible value consistent with rule #1.

Hund’s RulesHund’s Rules

• Rule #3:

if shell half full

if shell half full if shell half full

J L S

J L SJ S

Hund’s Rules: ExampleHund’s Rules: ExampleFind the Landé g factor for the ground state of a praseodymium (Pr) ion with two f electrons and for the ground state of an erbium (Er) ion with 11 f electrons.

Pr• the electrons are both spin +1/2, per rule #1, so S = 1• per rule #2, the largest value of L occurs if one electron is +3 and the other +2, so L = 5• now, from rule #3, since the shell is less than half full,

5 1 4 J L S

2 2 2

21

21 1 1 4 5 1 2 5 6

1 1 0 552 1 2 4 5

J S Lg

JJ J S S L L

J J

( ) ( ) ( ) ( ) ( ) ( )

.( ) ( )( )

Hund’s Rules: ExampleHund’s Rules: ExampleFind the Landé g factor for the ground state of a praseodymium (Pr) ion with two f electrons and for the ground state of an erbium (Er) ion with 11 f electrons.

Er• per rule #1, we have 7(+1/2) and 4(1/2), so S = +3/2• per rule #2, we have 2(+3), 2(+2), 2(+1), 2(0), 1(1), 1(2),

and 1(3), so L = 6• now, from rule #3, since the shell is more than half full,

J = L + S = 15/2

2 2 2

21

215 2 17 2 3 2 5 2 6 71 1 1

1 1 1 22 1 2 15 2 17 2

J S Lg

J

J J S S L L

J J

/ ( / ) / ( / ) ( )( ) ( ) ( )

.( ) / ( / )

Consider a solid in which all of the magnetic ions are identical, having the same value of J (appropriate for the ground state).

• Every value of Jz is equally likely, so the average value of the ionic dipole moment is zero.

• When a field is applied in the positive z direction, states of differing values of Jz will have differing energies and differing probabilities of occupation.

• The z component of the moment is given by:

• and its energy is

B Bz

z JJ

g g M

ParamagnetismParamagnetism

B z JE B g M B

As a result of these probabilities, the average dipole moment is given by

B J B

J

B J B

J

B J

z B J B B

J

2 12 1 1where

2 2 2 2

JM B k T

M JJ

g M B k T

M J

g M e

g J g JB k Te

J xJ xx

J J J J

/

/( / )

( ) coth coth

z B J B B M n ng J g JB k T( / )

Brillouin FunctionBrillouin Function

J (x)

x

B BIf then nearly all of the ions will be

in the lowest state. All dipoles will be aligned with the

applied field and the magnetization is said to be

saturated. The Brillouin function 1 a

g JB k T ,

B

nd the

magnetization M ng J .

ParamagnetismParamagnetism

B B

2 21B3

If the ion has nearly the same probability

of being in any of the states and the magnetization will be

small. In the limit of small the Brillouin function 1 3

and 1

g JB k T

x (J )x / J

M ng J J

,

,

( BB k T) / .

where p is the effective number of Bohr magnetons per ion.

ParamagnetismParamagnetism

0

2 210 B B3

The magnetic suspectibility

is

where

C 1

M C

B T

ng J J k( ) /

The Curie constant can be rewritten as 2 210 B B3C np k /