Magnetic Materials

Magnetic materials are classified by how they magnetize in an applied field $H$, i.e. by the magnetization $M = \chi H$ and the arrangement of atomic magnetic moments.

1. Diamagnetism

• Atoms have no net permanent moment (all electrons paired).
• An applied field changes electron orbital motion (Lenz's law) and induces a small moment opposing the field, so $\chi < 0$ and $\mu_r$ slightly less than 1.
• Independent of temperature. Diamagnetic materials are repelled by a magnet. Superconductors are perfect diamagnets ($\chi = -1$).

2. Paramagnetism

• Atoms have permanent moments (unpaired electrons) but they do not interact; thermal motion keeps them random.
• A field partly aligns them, giving a small positive $M$.
• Obeys Curie's law: $\chi = C/T$. Weakly attracted by a magnet.

3. Ferromagnetism

• Strong exchange interaction aligns neighbouring moments parallel, forming domains with spontaneous magnetization.
• Very large $\chi$, hysteresis, saturation, retentivity.
• Above the Curie temperature $T_C$ it becomes paramagnetic: $\chi = \dfrac{C}{T - T_C}$. Fe: $T_C = 770$ °C.

4. Antiferromagnetism

• Exchange interaction aligns neighbouring moments antiparallel and of equal size, so net $M \approx 0$.
• $\chi$ is small and rises with $T$ up to the Néel temperature $T_N$, then falls as paramagnetic: $\chi = \dfrac{C}{T + \theta}$.

5. Ferrimagnetism

• Two sublattices with antiparallel but unequal moments, so a net spontaneous magnetization remains.
• Domains and hysteresis like ferromagnets, but with high electrical resistivity (they are ceramics), giving low eddy-current loss.
• Examples: magnetite Fe₃O₄, MnZn and NiZn ferrites; used in high-frequency transformer cores.

 Para:   ^ / -> v \   random
 Ferro:  ^ ^ ^ ^ ^    parallel
 Anti:   ^ v ^ v ^    antiparallel equal
 Ferri:  ^ v ^ v ^    antiparallel unequal
         (big ^, small v)

The hysteresis loop (B–H loop) shows how flux density $B$ lags behind magnetizing force $H$ when a ferromagnetic material is taken through a full magnetization cycle. Its shape tells the designer which material suits which job.

            B
            ^      Bs
            |    _____
        Br  |   /    /
            |  /    /
   ---------+-/----/------> H
        -Hc |/    / Hc
           /|    /
     _____/ |
            |

Quantities read from the loop and their significance

Use in selection

• Soft magnetic materials (narrow, tall loop): low $H_c$, small loop area, high $\mu$, high $B_s$. Chosen for transformer, motor and generator cores, relays and electromagnets, where the field reverses many times and loss must be low. Examples: silicon steel, permalloy, soft ferrites.
• Hard magnetic materials (wide, fat loop): high $H_c$, high $B_r$, large $(BH)_{max}$. Chosen for permanent magnets in meters, loudspeakers, PMDC motors. Examples: Alnico, NdFeB, SmCo, hard ferrites.
• Square loop materials: two stable states $\pm B_r$; used in magnetic memory and switching cores.

Hysteresis loss per cycle (Steinmetz): $W_h = \eta B_{max}^{1.6} f V$, so for AC machines the material with the smallest loop area is selected.

The domain theory (Weiss, 1907) explains why a piece of ferromagnetic material such as iron can be unmagnetized and yet become strongly magnetized by a small field.

Main ideas

1. Domains: a ferromagnetic crystal is divided into small regions (about $10^{-6}$–$10^{-3}$ m) called domains. Inside each domain all atomic magnetic moments are aligned parallel by the strong exchange interaction, so each domain is magnetized to saturation.
2. Random orientation: in an unmagnetized specimen the directions of magnetization of different domains are random, so their effects cancel and net $M = 0$.
3. Domain walls (Bloch walls): domains are separated by transition layers a few hundred atoms thick in which the moment direction rotates gradually from one domain to the next.
4. Why domains form: a single domain would create a large external field with high magnetostatic energy. Splitting into domains (and closure domains at the ends) lowers this energy. The final size is set by the balance of magnetostatic, exchange, anisotropy, wall and magnetostrictive energies.

  Unmagnetized          Magnetized (H ->)
  +-----+-----+         +-----------+
  |  ^  |  -> |         | -> -> ->  |
  +-----+-----+   H     | -> -> ->  |
  | <-  |  v  |  ===>   | -> -> ->  |
  +-----+-----+         +-----------+

Magnetization process

On applying a field $H$:

1. Reversible wall motion (weak field): domains whose magnetization is nearly along $H$ grow, and others shrink, by small wall movement. Removing $H$ returns the walls.
2. Irreversible wall motion (medium field): walls jump past impurities and defects (Barkhausen jumps). Large rise in $B$; this is the steep part of the B–H curve and the cause of hysteresis.
3. Domain rotation (strong field): the remaining magnetization of domains rotates from the easy axis to the field direction, until the material saturates.

Explanations given by the theory

• Hysteresis and retentivity: walls stuck at defects do not return when $H$ is removed, leaving $B_r$.
• Curie temperature: above $T_C$ thermal energy destroys the exchange alignment, domains vanish, and the material becomes paramagnetic.
• Soft vs hard: pure, defect-free materials let walls move easily (soft); fine-grained, impure alloys pin walls (hard).

A magnetic domain is a small region of a ferromagnetic (or ferrimagnetic) material in which all atomic magnetic moments are aligned in the same direction because of the exchange interaction, so the region is magnetized to saturation. Neighbouring domains are separated by domain walls. In an unmagnetized body the domains point in random directions and cancel.

Behaviour of domains in an external field

As the applied field $H$ increases, the magnetization changes in stages that match the parts of the initial B–H curve.

  B
  ^                 ______ saturation
  |            ___/  (3) rotation
  |          /
  |         / (2) irreversible wall
  |        /      motion (Barkhausen)
  |      _/
  |   _/  (1) reversible wall motion
  +-------------------------------> H

1. Zero field: domains are randomly oriented; net $M = 0$.
2. Weak field: reversible domain wall movement
   • Domains whose magnetization is close to $H$ have lower energy, so they grow by small shifts of their walls; unfavourable domains shrink.
   • If $H$ is removed, walls return to their original places. The curve is gentle and reversible.
3. Medium field: irreversible domain wall movement
   • Walls break free from impurities, voids and grain boundaries and move in sudden jumps (Barkhausen effect).
   • Favourable domains grow rapidly by eating unfavourable ones; $B$ rises steeply.
   • On removing $H$ the walls do not return completely; this gives remanence and hysteresis.
4. Strong field: domain rotation
   • Now most of the crystal is one domain, magnetized along an easy axis close to $H$.
   • Further increase in $H$ rotates the magnetization away from the easy axis into the exact direction of $H$, against anisotropy energy.
   • When all moments are parallel to $H$, the material reaches saturation $M_s$.

 H = 0        weak H ->     strong H ->   very strong
 [^][>]       [^ ][>>>]     [>>>>>>]      [-->-->-->]
 [<][v]       [< ][>>>]     [>>>>>>]      aligned with H
 random       walls move    one domain    rotated (Ms)

If the field is then reduced to zero, the domain structure does not fully go back, leaving residual magnetism $B_r$.

Soft magnetic materials are easily magnetized and demagnetized: high permeability, low coercivity, narrow hysteresis loop, low hysteresis loss and (when laminated or ceramic) low eddy-current loss.

Applications

Why soft materials suit these uses

• In AC devices the flux reverses 50–100 times per second (or kHz–MHz in electronics), so a narrow loop keeps hysteresis loss small.
• High permeability needs little magnetizing current.
• Low coercivity lets relays and electromagnets release quickly when current stops.

These four classes all have atoms with permanent magnetic moments (unpaired electron spins); they differ in how the neighbouring moments interact and line up.

 Paramagnetic      Ferromagnetic
  ^  /  ->  \       ^  ^  ^  ^
  v  <-  /  ^       ^  ^  ^  ^
 (random)          (parallel)

 Antiferromagnetic Ferrimagnetic
  ^  v  ^  v        ^  v  ^  v
  v  ^  v  ^        (big up, small down)
 (equal, opposite) (unequal, opposite)

Paramagnetism

• Moments do not interact, so thermal agitation keeps them random; $M = 0$ without a field.
• A field partly aligns them, giving small positive $\chi$ ($10^{-5}$–$10^{-3}$), $\mu_r$ slightly above 1.
• Curie law: $\chi = C/T$. Examples: Al, Pt, O₂, CuSO₄.

Ferromagnetism

• Strong positive exchange interaction makes neighbouring moments parallel; spontaneous magnetization in domains.
• Very large $\chi$ ($10^2$–$10^5$), hysteresis, saturation.
• Above Curie temperature it becomes paramagnetic: $\chi = C/(T - T_C)$. Examples: Fe, Co, Ni.

Antiferromagnetism

• Negative exchange interaction makes neighbouring moments antiparallel and equal, so net $M = 0$.
• Small positive $\chi$, which increases with $T$ up to the Néel temperature $T_N$, then falls: $\chi = C/(T + \theta)$ above $T_N$.
• Examples: MnO, FeO, NiO, Cr.

Ferrimagnetism

• Two sublattices with antiparallel but unequal moments; the difference gives a net spontaneous magnetization.
• Large $\chi$, domains and hysteresis like ferromagnets, but very high resistivity (ceramic), so low eddy-current loss.
• Examples: Fe₃O₄ (magnetite), MnZn and NiZn ferrites, YIG.

Hard vs soft magnetic materials

     B                      B
     ^  soft                ^   hard
    _|_                 ____|____
   / |/                /    |   /
  /  /------> H      -/-----+--/--> H
 /  /|               /      | /
                    /_______|/

Material for an electromagnetic relay

A soft magnetic material (soft iron, low-carbon steel or Ni-Fe permalloy) is chosen for the relay core, yoke and armature, because:

• High permeability: a small coil current produces enough flux to pull the armature; coil power is low.
• Low coercivity and low remanence: when the coil is switched off, the core loses its magnetism at once, so the armature is released quickly by the spring and does not stick.
• Low hysteresis loss: suits relays that operate frequently or on AC.
• High saturation flux density: gives strong pull from a small core.

A hard material would keep its magnetism after the coil is de-energized, so the contacts would not open reliably.

Domain walls

A domain wall (Bloch wall) is the thin transition layer, a few hundred atoms thick, between two neighbouring magnetic domains, inside which the direction of the atomic moments rotates gradually from the direction of one domain to that of the other. Its thickness is a balance between exchange energy (prefers a wide wall) and anisotropy energy (prefers a narrow wall).

 domain 1   |  wall (spins rotate)  |  domain 2
  ^ ^ ^ ^   |  ^ / -> \ v          |  v v v v

Demagnetization due to domain formation

A ferromagnetic crystal does not stay as a single domain, because a single domain produces poles at its ends and a large external field, which stores a large magnetostatic energy. The crystal lowers this energy by dividing into domains. This splitting is the reason an iron piece can exist in an unmagnetized (demagnetized) state.

 (a) single     (b) two        (c) four        (d) closure
     domain         domains        domains         domains
 +--------+   +--------+     +--+--+--+--+   +--------+
 |  ---->  |   | ---->  |     |^ |v |^ |v |   |\ <--- /|
 | N    S |   | <----  |     |  |  |  |  |   | ^    v |
 +--------+   +--------+     +--+--+--+--+   |/ ---> \|
 large field  field halved   field small     +--------+
                                             no free poles

1. (a) Single domain: all moments parallel; N and S poles at the ends; large external field and maximum magnetostatic energy.
2. (b) Two antiparallel domains: flux partly returns inside the crystal; magnetostatic energy roughly halves, at the cost of one wall.
3. (c) N domains: energy falls to about $1/N$ of (a). Splitting stops when the energy of extra walls equals the energy saved.
4. (d) Closure domains: triangular domains at the ends, magnetized at 90°, close the flux path completely inside the crystal. Magnetostatic energy becomes almost zero and net magnetization is zero.

Role of the external field

When a field is applied and then reduced in a controlled way (an AC field decreasing slowly to zero), domain walls move back and forth through smaller and smaller distances. The domains settle back into this low-energy, randomly oriented, flux-closed arrangement, and the material is demagnetized. Heating above the Curie temperature and cooling in zero field also restores random domains.

Why hard magnetic material for permanent magnets

A permanent magnet must keep strong magnetism for years without a supply. Hard materials (Alnico, NdFeB, SmCo, hard ferrite) have:

• High retentivity $B_r$: strong field left after magnetizing.
• High coercivity $H_c$: stray fields, vibration and temperature changes cannot easily demagnetize them, because domain walls are pinned by fine precipitates and defects.
• Large $(BH)_{max}$: more energy stored per unit volume, so a smaller magnet does the job.

Diamagnetic vs paramagnetic vs ferromagnetic

  Dia: lines bend away      Para/Ferro: lines drawn in
  --- \   /---              ---\     /---
  ---- ( ) ----             ----(===)----
  --- /   \---              ---/     \---

Mechanisms in brief

• Diamagnetism: the field alters electron orbits (Lenz's law) and induces a moment opposite to the field; present in all materials but masked if others exist.
• Paramagnetism: unpaired spins partially align with the field against thermal agitation.
• Ferromagnetism: exchange interaction aligns spins parallel within domains; the field just rotates/grows domains, giving very high magnetization.

Ferromagnetism

The property of some materials to show strong, spontaneous magnetization even without an external field, because the strong exchange interaction aligns atomic magnetic moments parallel within domains. They have very large positive susceptibility ($\chi \approx 10^2$–$10^5$), show hysteresis and saturation, and become paramagnetic above the Curie temperature, $\chi = C/(T - T_C)$. Examples: iron, cobalt, nickel, gadolinium, Alnico.

Paramagnetism

The property of materials whose atoms have permanent magnetic moments (unpaired electrons) that are randomly oriented and do not interact. An applied field partly aligns them, giving a weak magnetization in the field direction. Susceptibility is small and positive ($10^{-5}$–$10^{-3}$) and follows Curie's law $\chi = C/T$. They are weakly attracted by a magnet and lose magnetization when the field is removed. Examples: aluminium, platinum, manganese, oxygen, CuSO₄.

Diamagnetism

The property of materials whose atoms have no permanent magnetic moment (all electrons paired). An applied field changes the orbital motion of electrons and, by Lenz's law, induces a small moment opposing the field. Susceptibility is small and negative ($\approx -10^{-5}$), independent of temperature, and $\mu_r$ is slightly less than 1. They are weakly repelled by a magnet. Examples: copper, bismuth, silver, gold, water, graphite; superconductors are perfect diamagnets ($\chi = -1$).

Closure domains

Closure domains are small triangular (prism-shaped) domains that form at the ends or surfaces of a ferromagnetic crystal, magnetized at 90° to the main domains, so that the magnetic flux forms a closed loop entirely inside the crystal. They remove free poles at the surface, making the external field and magnetostatic energy nearly zero.

Demagnetization by domain formation

A ferromagnetic crystal tends towards the state of lowest total energy. A single, fully magnetized domain has free poles and a large external field, which stores a large magnetostatic energy. The crystal reduces this energy by splitting into domains:

 (a) single     (b) two        (c) many        (d) closure
     domain         domains        domains         domains
 +--------+   +--------+     +--+--+--+--+   +--------+
 |  ---->  |   | ---->  |     |^ |v |^ |v |   |\ <--- /|
 | N    S |   | <----  |     |  |  |  |  |   | ^    v |
 +--------+   +--------+     +--+--+--+--+   |/ ---> \|
 large field  field halved   field small     +--------+
                                             no free poles

1. (a) Single domain: all moments parallel, poles at the ends, large external field and maximum magnetostatic energy $\left(\tfrac{\mu_0}{2}\int H^2 dV\right)$.
2. (b) Two antiparallel domains: part of the flux returns inside the crystal; the magnetostatic energy is roughly halved, at the cost of one domain wall.
3. (c) Many domains: with $N$ domains the energy falls to about $1/N$. Splitting stops when the energy of a new wall equals the magnetostatic energy saved.
4. (d) Closure domains: triangular domains at the ends, magnetized at 90° to the main domains, complete the flux loop inside the crystal. No free poles remain, the external field and magnetostatic energy are almost zero, and the net magnetization is zero: the crystal is demagnetized.

Energy balance

Total energy = exchange + magnetostatic + anisotropy + domain wall + magnetostrictive energy. Domains form only as long as the decrease in magnetostatic energy is larger than the extra wall energy needed. The result is a state in which domains are randomly oriented and flux is closed, which is why a piece of iron is normally unmagnetized. Demagnetizing a magnetized sample (AC field reduced slowly to zero, or heating above $T_C$) lets the material return to this low-energy domain pattern.

Demagnetization is the reduction of the net magnetization of a ferromagnetic material to zero. It happens naturally through the formation of domains, and it can be done deliberately by heating, by a reverse field, or by an alternating field of decreasing amplitude.

Demagnetization through domain formation

A single fully magnetized domain has poles at its ends and a large external field, so its magnetostatic energy is high. The crystal lowers its energy by dividing into domains:

 (a) single     (b) two        (c) many        (d) closure
     domain         domains        domains         domains
 +--------+   +--------+     +--+--+--+--+   +--------+
 |  ---->  |   | ---->  |     |^ |v |^ |v |   |\ <--- /|
 | N    S |   | <----  |     |  |  |  |  |   | ^    v |
 +--------+   +--------+     +--+--+--+--+   |/ ---> \|
 large field  field halved   field small     +--------+
                                             no free poles

1. (a) Single domain: all moments parallel, poles at the ends, large external field and maximum magnetostatic energy $\left(\tfrac{\mu_0}{2}\int H^2 dV\right)$.
2. (b) Two antiparallel domains: part of the flux returns inside the crystal; the magnetostatic energy is roughly halved, at the cost of one domain wall.
3. (c) Many domains: with $N$ domains the energy falls to about $1/N$. Splitting stops when the energy of a new wall equals the magnetostatic energy saved.
4. (d) Closure domains: triangular domains at the ends, magnetized at 90° to the main domains, complete the flux loop inside the crystal. No free poles remain, the external field and magnetostatic energy are almost zero, and the net magnetization is zero: the crystal is demagnetized.

The number and size of domains are fixed by the balance between the saved magnetostatic energy and the energy of the domain walls (plus anisotropy and magnetostrictive energy).

Practical methods of demagnetization

1. Heating above the Curie temperature and cooling in zero field: domains re-form randomly.
2. Applying a reverse field equal to the coercive force $H_c$: brings $B$ to zero (but not stable).
3. AC demagnetization: an alternating field whose amplitude is reduced slowly to zero takes the material round smaller and smaller hysteresis loops until it ends at the origin.

        B
        ^   ___
        |  / _ \
     ---+-/-(o)-\---> H     shrinking loops
        |  \___/            spiral to B = 0

Deperming method

Deperming is a demagnetization process used for large steel structures, mainly ships and submarines, to remove their permanent magnetism (acquired from the earth's field, welding and hammering during building) so that they do not trigger magnetic mines or disturb compasses.

• Large cables are wound round the hull (or the ship is placed in a deperming facility).
• Strong DC pulses of alternating polarity and decreasing magnitude are passed through the cables, e.g. +I, −0.8I, +0.6I, ….
• Each pulse drives the steel round a smaller hysteresis loop, so the permanent magnetization is reduced almost to zero.
• It is repeated periodically. A related method, degaussing, uses permanently installed coils carrying current to cancel the ship's field continuously.

Magnetic dipole moment

A magnetic dipole moment $m$ is a measure of the strength of a magnetic dipole (current loop or pair of poles). For a current $I$ circulating in a loop of area $A$:

$$m = IA \quad (\text{unit: A m}^2)$$

It points normal to the loop by the right-hand rule. In a field $B$ the dipole experiences a torque $\tau = m \times B$.

Atomic magnetic moment

The atomic magnetic moment is the net magnetic moment of an atom, arising from (i) the orbital motion of electrons round the nucleus, (ii) the spin of electrons, and (iii) a very small nuclear spin contribution. Filled shells cancel, so only unpaired electrons contribute. Its natural unit is the Bohr magneton:

$$\mu_B = \frac{e\hbar}{2m_e} = 9.27\times10^{-24}\ \text{A m}^2$$

Example: an iron atom has about $2.2\,\mu_B$ in the metal.

Ferromagnetic vs ferrimagnetic materials

 Ferro:  ^  ^  ^  ^       Ferri:  ^  v  ^  v
                                   (large ^, small v)

Demagnetizing a magnetic material using the B–H curve

After a ferromagnetic material has been magnetized to saturation and the field removed, it keeps a remanent flux density $B_r$ (point b). It can be demagnetized in these ways:

(a) By a reverse field equal to the coercive force: applying $H = -H_c$ takes the material from $B_r$ to $B = 0$ (point c). This is not stable; small disturbances leave some residual magnetism.

(b) By an alternating field of decreasing amplitude (AC demagnetization): the material is taken round a series of hysteresis loops, each smaller than the last. As the amplitude falls to zero, the loops shrink and spiral into the origin, where $B = 0$ and $H = 0$. The domains end up randomly oriented, so this method gives true demagnetization.

               B
               ^  a (sat.)
          b Br |   ___
             __|__/  /
            /  | __ /
           / /_|/ //
   -------c-/--o--/-------> H
         -Hc|  |/ / +Hc
           /___|_/
               |   loops shrink to o

(c) By heating above the Curie temperature and cooling without a field: thermal agitation destroys alignment and domains re-form randomly.

Material for electronic storage of digital data

A hard (semi-hard) magnetic material with a square (rectangular) hysteresis loop is chosen, e.g. γ-Fe₂O₃, CrO₂, barium ferrite, or Co-Cr/Co-Pt alloy thin films in hard disks.

Justification

• Two stable states: a square loop has high remanence $+B_r$ and $-B_r$ at $H = 0$, which represent binary 1 and 0.
• High remanence: gives a strong readable signal.
• Sufficient coercivity: stray fields and neighbouring bits cannot erase stored data, so storage is non-volatile; yet coercivity is not so high that the write head cannot switch it.
• Sharp switching: a field just above $H_c$ flips the bit quickly and cleanly.
• Fine particles/grains: allow high storage density.

The read/write heads themselves use soft materials (permalloy, ferrite) so they follow the signal without retaining it.

 Ferro:      ^  ^  ^  ^  ^
 Antiferro:  ^  v  ^  v  ^

Examples

• Ferromagnetic: iron (Fe), also Co, Ni.
• Antiferromagnetic: manganese oxide (MnO), also FeO, NiO, Cr.

Magnetic domain

A magnetic domain is a small region (about $10^{-6}$–$10^{-3}$ m) inside a ferromagnetic material in which all atomic magnetic moments are aligned in the same direction by the exchange interaction, so the region is magnetized to saturation.

Domain wall

A domain wall (Bloch wall) is the transition layer, a few hundred atoms thick, separating two adjacent domains. Across it the direction of the moments rotates gradually from that of one domain to the other.

 domain 1  | wall          | domain 2
 ^ ^ ^ ^   | ^ / -> \ v    | v v v v

Losses in magnetic materials

When a magnetic core carries alternating flux, part of the input energy is turned into heat. These core (iron) losses are:

1. Hysteresis loss

   • Energy spent in moving domain walls and rotating domains back and forth each cycle; equals the area of the B–H loop per cycle per m³.
   • Steinmetz formula: $P_h = \eta\,B_{max}^{1.6}\,f\,V$
   • Reduced by soft materials with a narrow loop (silicon steel, CRGO).

2. Eddy-current loss

   • Alternating flux induces emfs and circulating currents in the conducting core, causing $I^2R$ heating.
   • $P_e = K_e\,B_{max}^2\,f^2\,t^2\,V$ (t = lamination thickness)
   • Reduced by thin insulated laminations, adding silicon to raise resistivity, or using ferrites/powder cores.

3. Anomalous (excess/residual) loss

   • Extra eddy loss from local, jerky domain wall motion (Barkhausen jumps); becomes important at high frequency.

Total core loss $P_c = P_h + P_e (+ P_{anom})$; it appears as heat and lowers efficiency of transformers and machines.

These three types all have atoms with permanent magnetic moments that interact strongly through the exchange interaction. They differ in how the magnetic moment vectors of neighbouring atoms line up.

 Ferromagnetic       ^  ^  ^  ^  ^   all parallel
 Antiferromagnetic   ^  v  ^  v  ^   antiparallel, equal
 Ferrimagnetic       ^  v  ^  v  ^   antiparallel, unequal
                    (long ^, short v)

Ferromagnetism

• Positive exchange interaction makes all neighbouring moment vectors parallel.
• Their sum gives a large spontaneous magnetization inside each domain; susceptibility $\chi \approx 10^2$–$10^5$.
• Shows hysteresis and saturation. Above the Curie temperature it becomes paramagnetic: $\chi = C/(T - T_C)$.
• Examples: Fe, Co, Ni.

Antiferromagnetism

• Negative exchange interaction makes neighbouring moment vectors antiparallel and equal in size.
• The two sublattice magnetizations cancel, so net magnetization is zero and $\chi$ is small and positive.
• $\chi$ increases with temperature up to the Néel temperature $T_N$, then decreases: $\chi = C/(T + \theta)$.
• Examples: MnO, FeO, NiO, Cr.

Ferrimagnetism

• Two sublattices A and B with moment vectors antiparallel but unequal ($M_A > M_B$).
• Net magnetization $M = M_A - M_B \ne 0$, so a spontaneous magnetization, domains and hysteresis exist, as in ferromagnets but smaller.
• They are ceramic oxides with very high resistivity, so eddy-current loss is low.
• Example: magnetite Fe₃O₄ (Fe³⁺ on A and B sites cancel; the Fe²⁺ moments remain), MnZn and NiZn ferrites; used in high-frequency cores.

A domain wall (Bloch wall) is the thin boundary layer, typically a few hundred atomic spacings thick, between two magnetic domains magnetized in different directions. Inside the wall the atomic moments turn gradually, a little from atom to atom, from the direction of one domain to that of the next. A wall has energy (exchange + anisotropy), and its width is set by the balance between them: exchange energy favours a wide wall; anisotropy energy favours a narrow one.

 domain A        wall           domain B
 ^ ^ ^ ^ |  ^  /  ->  \   v  | v v v v
         |<- spins rotate ->|

How domain wall motion occurs

When an external field $H$ is applied, the domains magnetized close to the direction of $H$ have lower energy ($-\mu_0 M\cdot H$) than the others. The material lowers its energy by enlarging these favourable domains. This happens by moving the walls: spins at the edge of the wall rotate one after another towards the favourable direction, so the wall shifts into the unfavourable domain.

 H = 0         H ->              H -> (larger)
 [ ^ | v ]     [ ^ ^ | v ]       [ ^ ^ ^ |v]
  wall at       wall moves        favourable
  centre        right             domain grows

Stages

1. Reversible wall motion (low field): walls bulge or shift slightly but stay held by defects. Removing $H$ returns them; the initial part of the B–H curve is gentle.
2. Irreversible wall motion (medium field): walls break free from impurities, voids, dislocations and grain boundaries and jump suddenly to new positions. These jumps (Barkhausen effect) can be heard as clicks with a coil and amplifier. B rises steeply, and after $H$ is removed the walls stay in new positions, causing remanence and hysteresis loss.
3. Domain rotation (high field): after the walls have swept through, the remaining domain rotates into the field direction, giving saturation.

Importance: easy wall motion (pure, annealed, large-grain material) gives soft magnetic behaviour; walls pinned by fine particles and defects give hard magnetic behaviour.

For the core, yoke and armature of an electromagnetic relay a soft magnetic material is chosen, such as soft iron, low-carbon steel, silicon steel, or Ni-Fe alloys (permalloy) for sensitive relays.

How a relay works

       spring
   +----/\/\--+
   |    armature (soft iron)
   |  ====*===========  contacts
   |  |   ^ pulled when coil on
   |  |  +--+
   |  |  |##| coil on soft-iron core
   +--+--+--+--- yoke

When current flows in the coil, the core is magnetized and pulls the armature, closing (or opening) the contacts. When the current stops, the core must lose its magnetism immediately so the spring can pull the armature back.

Justification

1. High permeability: a small coil current (low power) produces enough flux to pull the armature; fast pick-up.
2. Low coercivity and low retentivity: on switching off, almost no residual magnetism remains, so the armature releases at once and does not stick. A hard material would keep the armature held.
3. Narrow hysteresis loop / low hysteresis loss: suitable for frequent operation and for AC relays; less heating.
4. High saturation flux density: strong pull from a small core.
5. Good mechanical workability and low cost: soft iron is easy to shape into cores and armatures.

A small non-magnetic shim (residual pin) is often fixed on the armature to keep a tiny air gap, further preventing sticking due to any residual magnetism.

Magnetic behaviour depends on whether atoms carry a permanent magnetic moment and on how the magnetization vectors of neighbouring atoms are arranged. Magnetization $M = \chi H$ is the net moment per unit volume.

 Diamagnetic:   no moments; induced m opposes H
                H ->    m <-  m <-  m <-
 Ferromagnetic: ^  ^  ^  ^   parallel
 Ferrimagnetic: ^  v  ^  v   antiparallel, unequal
               (long ^, short v)

Diamagnetism

• Atoms have no permanent magnetic moment (all electron orbits and spins paired).
• An applied field changes the orbital motion of electrons; by Lenz's law, the induced magnetization vector points opposite to $H$.
• $\chi$ is small and negative ($\approx -10^{-5}$), $\mu_r$ slightly below 1, and independent of temperature.
• Field lines are pushed out; the material is repelled from strong-field regions.
• Examples: Cu, Bi, Ag, Au, water, graphite. A superconductor is a perfect diamagnet, $\chi = -1$ (Meissner effect).

Ferromagnetism

• Atoms have permanent moments (unpaired 3d electrons in Fe, Co, Ni).
• Strong positive exchange interaction aligns neighbouring magnetization vectors parallel, giving spontaneous magnetization in domains.
• A small $H$ grows favourable domains and rotates others, so $M$ becomes very large: $\chi \approx 10^2$–$10^5$.
• Shows hysteresis, saturation, remanence. Above the Curie temperature $T_C$ (Fe 770 °C) it becomes paramagnetic: $\chi = C/(T - T_C)$.
• Examples: Fe, Co, Ni, Gd and their alloys.

Ferrimagnetism

• The crystal has two sublattices (A and B sites) whose magnetization vectors are antiparallel but unequal.
• Net spontaneous magnetization $M = M_B - M_A \ne 0$; domains and hysteresis exist, but saturation is lower than in ferromagnets.
• In magnetite Fe₃O₄, Fe³⁺ ions on A and B sites cancel and the Fe²⁺ moments give the net magnetization.
• Ferrites are ceramic oxides with very high resistivity, so eddy-current loss is small; ideal at high frequency.
• Above the Curie (Néel) temperature they become paramagnetic.
• Examples: Fe₃O₄, MnZn and NiZn ferrites, barium ferrite, YIG.

Deperming method of demagnetization

Deperming is the process of removing the permanent (remanent) magnetism of large steel bodies, mainly ships and submarines. A steel hull becomes magnetized by the earth's field during construction (welding, riveting, hammering) and in service. This magnetism can trigger magnetic mines and disturb compasses.

Procedure

1. Heavy cables are wrapped round the hull (or the ship enters a deperming station with built-in coils).
2. Large DC current pulses of alternating polarity and decreasing magnitude are passed: +I₁, −I₂, +I₃, … with I₁ > I₂ > I₃ ….
3. Each pulse drives the steel round a smaller hysteresis loop, so the B–H point spirals towards the origin.
4. The magnetic signature is measured and the process is repeated until the remanent field is nearly zero.

   B
   ^  /\      loops shrink
   | /  \  /\
 --+------\/--\/-> H
   |  ending near B = 0

It is the same principle as AC demagnetization with a decaying field, applied with DC pulses to a very large object. (Degaussing, by contrast, uses coils carrying a steady current on board to cancel the remaining field.)

Graphite in a non-uniform magnetic field

Graphite is diamagnetic, and strongly so (one of the largest diamagnetic susceptibilities, $\chi \approx -10^{-4}$, especially pyrolytic graphite).

• In a field it develops a magnetization opposite to $H$.
• Its energy is $U = -\tfrac{\chi V B^2}{2\mu_0}$, which is positive for $\chi < 0$, so it is lowest where $B$ is weakest.
• Hence in a non-uniform field, graphite feels a force $F = \dfrac{\chi V}{2\mu_0}\nabla(B^2)$ directed from the stronger field region towards the weaker region: it is repelled by the magnet's poles.
• A thin piece of pyrolytic graphite can even levitate stably above an array of strong NdFeB magnets.
• In a uniform field it feels no net force, only a weak magnetization opposing the field.

The difference is explained by the domain theory of ferromagnetism.

Why a (small) iron crystal shows a magnetic field

• In iron, the strong exchange interaction aligns the spins of neighbouring atoms parallel, giving spontaneous magnetization (about 2.2 Bohr magnetons per atom).
• A small, perfect single crystal of iron, below a critical size, can exist as a single domain (or with its domains mostly along one easy axis). All atomic moments point the same way, so the moments add up and the crystal has poles and produces an external magnetic field around it, like a tiny bar magnet.

Why an ordinary piece of iron does not

1. Domain formation: a bulk piece would have a very large magnetostatic energy if it were one domain. To lower this energy it splits into many domains with closure domains at the ends, so the flux is closed inside the material.
2. Random orientation: an ordinary piece of iron is polycrystalline: it has many grains, each with its own crystal axes and many domains. The magnetization directions of all these domains are random.
3. Cancellation: the vector sum of the magnetization of all domains is zero, so no free poles appear at the surface and no external field is produced, even though each domain is saturated inside.

  Small crystal            Piece of iron
  (single domain)          (many random domains)
  +-----------+           +--+--+--+--+
  | --> --> --> |           |^ |->|v |<-|
  | --> --> --> |           +--+--+--+--+
  +-----------+           |<-|v |^ |->|
   N         S             +--+--+--+--+
  external field          net M = 0, no field

Making the piece magnetic

If the piece of iron is placed in an external field, favourable domains grow by domain wall motion and others rotate, so the domains line up and the piece becomes a magnet. Soft iron loses this alignment when the field is removed, which is why it is used as an electromagnet core.

Ferromagnetic materials have all atomic moments aligned parallel; ferrimagnetic materials have two sublattices whose moments are antiparallel but unequal, leaving a net moment.

 Ferromagnetic:  ^  ^  ^  ^  ^
 Ferrimagnetic:  ^  v  ^  v  ^   (long ^, short v)

Both show domains, hysteresis, saturation and become paramagnetic above their critical temperature.

Magnetic susceptibility is $\chi = M/H$, the magnetization produced per unit applied field; $\mu_r = 1 + \chi$. Based on the sign and size of $\chi$, magnetic materials are classified as follows.

 1/chi
   ^        para    ferro      antiferro
   |       /        /          /
   |      /        /       \  /
   |     /        /         \/
   +----/--------/-----------+----> T
       0        Tc          TN

Brief description

• Diamagnetic: no permanent atomic moment; induced moment opposes the field (Lenz's law); weakly repelled.
• Paramagnetic: permanent but non-interacting moments; partly aligned by the field; weakly attracted.
• Ferromagnetic: moments aligned parallel in domains by exchange interaction; very strong magnetization, hysteresis.
• Antiferromagnetic: equal antiparallel moments cancel; weak response.
• Ferrimagnetic: unequal antiparallel moments; net spontaneous magnetization.

Basic difference between ferromagnetic and ferrimagnetic materials

The basic difference is the arrangement of spins:

• In a ferromagnetic material all neighbouring atomic moments are parallel, so the net magnetization is the full sum of the moments.
• In a ferrimagnetic material the moments sit on two sublattices that are antiparallel but unequal, so the net magnetization is only their difference.

 Ferro:  ^  ^  ^  ^       Ferri:  ^  v  ^  v

Consequences: ferromagnets (metals) have higher saturation but low resistivity and high eddy loss; ferrimagnets (ceramic ferrites) have lower saturation but very high resistivity, so they are used at high frequencies.

The choice depends on the shape of the B–H (hysteresis) loop.

       B   soft (narrow)         B   hard (wide)
       ^   __                    ^   ______
       |  / /                    |  /     /
  -----+-/-/-----> H     --------+-/-----/----> H
      -/-/ |               -Hc  / |    / +Hc
      /_/  |                   /__|___/
   small Hc, small area       large Br, Hc, area

Hard magnetic material for permanent magnets

A permanent magnet must keep strong magnetism indefinitely without a supply and resist demagnetization.

• High retentivity $B_r$: a large flux density remains after the magnetizing field is removed, so the magnet is strong.
• High coercivity $H_c$: a large reverse field is needed to destroy the magnetism; stray fields, vibration, shocks and temperature changes cannot easily demagnetize it.
• Large $(BH)_{max}$ (area of the second-quadrant curve): more magnetic energy stored per unit volume, so a smaller magnet is enough.
• The wide loop (large hysteresis loss) does not matter because the magnet is magnetized only once.
• Examples: Alnico, NdFeB, SmCo, barium/strontium ferrite; used in loudspeakers, PMDC motors, meters.

Soft magnetic material for high-frequency applications

In transformers, inductors and chokes at high frequency, the flux reverses thousands to millions of times per second.

• Narrow loop, low $H_c$: hysteresis loss per cycle (loop area) is small. Since $P_h = \eta B_{max}^{1.6} f V$ is proportional to $f$, a small loop area is essential at high $f$.
• High permeability: little magnetizing current is needed and the material follows the rapidly changing field.
• Low remanence: no residual magnetism to oppose the next half-cycle.
• High resistivity (soft ferrites, MnZn and NiZn): eddy-current loss $P_e \propto f^2$ is kept small, which is critical at high frequency.
• Examples: soft ferrites in SMPS transformers, RF coils, antenna rods; permalloy and amorphous alloys in pulse transformers.

Soft magnetic materials are those that can be magnetized and demagnetized easily.

Properties

• Low coercivity ($H_c$ < about 1000 A/m) and low retentivity.
• High permeability ($\mu_r$ of thousands to $10^5$).
• Narrow, tall hysteresis loop, so low hysteresis loss.
• High saturation flux density.
• High resistivity (in Si-steel and ferrites) for low eddy-current loss.
• Few impurities and defects, so domain walls move easily.

      B
      ^  __
      | / /   narrow loop
  ----+/-/----> H
     /-/ |

Uses

Magnetic materials are divided into soft and hard according to the shape of their hysteresis loop.

        B   soft                   B   hard
        ^   ___                    ^   _______
        |  / Bs                    |  /      /
     Br |-/  /                 Br  |-/      /
   -----+/--/----> H       -------+/------/----> H
       /|  / Hc small       -Hc   /|     /  Hc large
      / | /                      / |    /
     /__|/                      /__|___/
   narrow, tall loop          wide, fat loop

Summary: a soft material follows the field with little loss, so it suits alternating fields; a hard material holds its magnetism, so it suits permanent magnets.

Ferrimagnetic materials are materials whose crystal has two magnetic sublattices with atomic moments aligned antiparallel but of unequal size. The moments do not cancel, so a net spontaneous magnetization exists. They show domains, hysteresis and saturation like ferromagnets, and are mostly ceramic oxides (ferrites) with very high resistivity. Examples: magnetite Fe₃O₄, MnZn and NiZn ferrites, YIG.

 Ferrimagnetic:     ^  v  ^  v   (long ^, short v)
 Antiferromagnetic: ^  v  ^  v   (equal lengths)

Difference from antiferromagnetic materials

Based on the shape and size of the hysteresis loop, magnetic materials are classified into soft and hard magnetic materials (with square-loop materials as a special group).

        B   soft                   B   hard
        ^   __                     ^   _______
        |  / /                     |  /      /
   -----+-/-/----> H       --------+-/------/----> H
       /-/  |  Hc small      -Hc  / |      /  Hc large
      /_/   |                    /__|_____/
   narrow, tall loop           wide, fat loop

1. Soft magnetic materials

• Narrow loop, low coercivity ($H_c$ < $10^3$ A/m), low retentivity, high permeability, small hysteresis loss.
• Easily magnetized and demagnetized; domain walls move freely.
• Examples: soft iron, silicon steel, permalloy, Mumetal, soft ferrites, amorphous alloys.

Applications

• Cores of power transformers, motors and generators (silicon steel, CRGO): low loss under alternating flux.
• Relays and electromagnets (soft iron): quick release after the coil is switched off.
• High-frequency transformers, SMPS, RF inductors, antenna rods (soft ferrites): high resistivity, low eddy loss.
• Magnetic shielding and current transformers (permalloy, Mumetal): very high permeability.
• Recording and reading heads (permalloy, ferrite).

2. Hard magnetic materials

• Wide loop, high coercivity ($H_c$ > $10^4$ A/m), high retentivity, large $(BH)_{max}$, large hysteresis loss.
• Hard to magnetize and demagnetize; domain walls are pinned by impurities and fine particles.
• Examples: carbon/tungsten steel, Alnico, NdFeB, SmCo, barium/strontium ferrite.

Applications

• Permanent magnets in loudspeakers, microphones and headphones.
• PMDC motors, small generators, magnetos, alternators in bikes.
• Moving-coil meters, energy meters (braking magnet).
• MRI and magnetic separators, magnetic holders and locks.

3. Square-loop (rectangular-loop) materials

• Nearly rectangular loop with two stable states $\pm B_r$ and sharp switching at $H_c$.
• Examples: γ-Fe₂O₃, CrO₂, Mg-Mn ferrites, Co-alloy thin films.
• Applications: magnetic tapes, hard disks, magnetic cards, old core memories and magnetic switching devices, where $+B_r$ and $-B_r$ represent binary 1 and 0.

Ferromagnetic vs antiferromagnetic

 Ferro: ^ ^ ^ ^     Antiferro: ^ v ^ v

Magnetostriction

Magnetostriction is the change in the dimensions (length) of a ferromagnetic material when it is magnetized. It is measured by the strain $\lambda = \Delta l/l$, typically $10^{-6}$–$10^{-5}$.

• Cause: as domains rotate towards the field, the spin–orbit coupling changes the spacing of atoms along the magnetization direction.
• Nickel contracts ($\lambda < 0$); iron expands at low field.

  H = 0:   |<----- l ----->|
  H -> :   |<-- l - dl -->|     (Ni contracts)
          coil around rod

Effects/uses: hum of transformers at 100 Hz; sonar and ultrasonic transducers; sensors (Terfenol-D).

Demagnetization is the process by which the net magnetization of a ferromagnetic material is reduced to zero. It occurs naturally when the material divides into magnetic domains.

Demagnetization through domain formation

A single domain has free poles and a large external field, i.e. high magnetostatic energy. The crystal lowers this energy by splitting:

 (a) single   (b) two      (c) closure
 +-------+   +-------+    +-------+
 | --->  |   | --->  |    |\ <-- /|
 |N    S |   | <---  |    |^     v|
 +-------+   +-------+    |/ --> \|
 big field   half field   +-------+  zero field

1. (a) One domain: large magnetostatic energy.
2. (b) Two antiparallel domains: energy about halved; more domains reduce it further, until the wall energy balances the saving.
3. (c) Closure domains at the ends close the flux inside; no free poles, net $M = 0$.

In practice a material is demagnetized by an AC field of slowly decreasing amplitude or by heating above $T_C$; the domains then return to this random, flux-closed arrangement.

Material for storing digital information

Hard (semi-hard) ferrimagnetic or ferromagnetic materials with a square hysteresis loop, e.g. γ-Fe₂O₃, CrO₂, barium ferrite, Co-Cr/Co-Pt thin films (hard disks).

Why: the square loop gives two stable remanent states $+B_r$ and $-B_r$ for 1 and 0; high remanence gives a strong read signal; adequate coercivity keeps data safe from stray fields (non-volatile) yet allows writing by the head; sharp switching and fine grains allow fast, dense storage.