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UEEP2024 Solid State Physics

Topic 5 Magnetism

Magnetic properties of solids

• The magnetic moment of a free atom has three principal sources :

1. The spin with which electrons are endowed
2. The electron orbital angular momentum about the nucleus
3. The change in the orbital moment induced by an applied magnetic field

Magnetic properties of solids

Alternative names for B and H B name used by magnetic flux density electrical engineers magnetic induction applied mathematicians electrical engineers magnetic field physicists H name used by magnetic field intensity electrical engineers magnetic field strength electrical engineers auxiliary magnetic field physicists magnetizing field physicists

Magnetic properties of solids

Magnetization is defined as magnetic moment per unit volume. For certain magnetic materials, it is found empirically that the magnetization M is proportional to magnetic field strength H

M   H

Magnetic susceptibility  > 0 -- paramagnetism  < 0 -- diamagnetiism

Atomic Magnets

• The orbital magnetic number m l takes values between – l and +l.
• Spin magnetic number ms takes values of 1/2.
• A single electron the component of the spin magnetic moment, μs, in the direction parallel to the magnetic field is given by
• The quantity is known as the Bohr magneton. e e s s m e m em 2       e B m e 2   

Atomic Magnets

• The component of orbital magnetic moment is given by
• When an atom has more than one electron, for a particular subshell, Total spin angular momentum Total orbital angular momentum
• If a subshell is filled, the values of S and L are both zero.
• If a subshell is partially filled, electrons are distributed between different m l and m s states. l B l l m m em      2 

 s S m

 l L m

Total angular momentum J

• The total angular momemtum J is obtained by combining L and S as follows:  If the subshell is less than half filled then J = L-S; If the subshell is more than half filled then J = L + S; If the subshell is exactly half filled then L = 0 and so J = S.
• By applying these rules, the values of S, L and J can be determined.

Landé splitting factor g

• The maximum component of the magnetic moment of the atom in the direction parallel to the magnetic field is where g is the Landé splitting factor
• Most atom exhibit a magnetic moment.

m g J

j B

2 ( 1 ) 3 ( 1 ) ( 1 ) ( 1 )        J J J J S S L L g

Solution

• The 3d subshell corresponds to l = 2, and therefore m l takes integer values between - 2 and +2. i.e. there are five allowed values of m l . so that the subband has a capacity to hold 10 electrons.
• In Cr 3+ , the three 3d electrons can therefore all occupy m s = +1/2 and therefore S = 3/2.
• The maximum value of L is obtained if two electrons occupy the state with m l = +2 and one occupies the m l = +1 state. However, the exclusion principle forbids two electrons with the same value of m s from occupying the same m l state.
• Consequently the largest allowed value of L corresponds to having one electron in each of the states m = +2, +1 and 0, and therefore L = 3.
• Since the subband is less than half filled, the value of J is J = L – S = 3 – 3/2 = 3/2.

Which materials have magnetic moment?

• Filled electron subshell in an atom do not affect the magnetic moment because S = L = 0 so J = 0.
• Inert atoms have a magnetic moment of zero.
• Ionic materials have a magnetic moment of zero because the electrons are transferred from one atom to another so that the resulting ions have only filled subshells.

Which materials have magnetic moment?

• Filled electron subshells in an atom do not affect the magnetic momentum of the atom because they have a net angular momentum of zero. (S=L=0, so J = 0).
• This means that inert atoms have a magnetic moment of zero because they have only filled electron subshells.
• Ionic materials have a magnetic moment of zero because the electrons are transferred from one atom type to another so the resulting ions have only filled subshells.

Which materials have magnetic moment?

• In covalent materials the outer subshell is only partially filled, and so these materials have a finite magnetic moment.
• However, each covalent bond is formed by a pair of electrons with opposite spin and with net orbital angular momentum of zero, covalent solids have a net magnetic moment of zero.
• Although most atoms have a non-zero magnetic moment, it appears that in majority of solids the effects cancel and the resultant magnetization is zero.

Which materials have magnetic moment?

• It appears that most non-metals display only a small magnetic susceptibility.
• In simple metal, such as sodium and aluminium, the valence electrons are assumed to be localized, in other words they are no longer attached to any particular atom. As a result the metal ions contains only filled electron subshells, and so each has a total angular momentum of zero.
• Although the ions do not contribute to the magnetic moment, the delocalised electrons do produce a non- zero magnetic moment. This is known as Pauli paramagnetism.

Which materials have magnetic moment?

• Free electron paramagnetism is a weak effect.
• Paramagnetism is caused by the magnetic dipole moments becoming aligned with the magnetic field, whereas diamagnetism results from an induced magnetic field which opposes the applied field.
• Paramagnetism gives rise to a positive susceptibility, whilst diamagnetism produces a negative susceptibility.