3. magnetic properties (1) introduction magnetic materials are very important in electrical engineering • soft magnetic materials – the materials that can be easily magnetized and demagnetized applications: transformer cores, stator and rotor materials • hard magnetic materials – cannot be easily demagnetized (permanent magnets) applications: loud speakers, telephone receivers

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(2) magnetic fields and quantities (a) magnetic fields ferromagnetic materials such as Fe, Co and Ni – provide strong magnetic field when magnetized

magnetism is dipolar up to atomic level magnetic fields are also produced by currentcarrying conductors

magnetic field of a solenoid is 0.4π n i H = ──── unit: A/m l n : number of turns i : current l : length 1 A/m = 4π × 10-3 Oe (oersteds) 2

(b) magnetic induction if demagnetized iron bar is placed inside a solenoid, the magnetic field outside solenoid increases

the magnetic field due to the bar adds to that of solenoid - magnetic induction (B) intensity of magnetization (M) : induced magnetic moment per unit volume B = μ0 H + μ0 M = μ0 (H + M) μ0 : permeability of free space 4π × 10-7 T·m/A

in most cases μ0 M > μ0 H, therefore B ≈ μ0 M 3

(c) magnetic permeability and susceptibility magnetic permeability B μ = ── H for vacuum μ = μ0 = 4π × 10-7 T·m/A relative permeability μ μ r = ──

μ0 B = μ0 μ r H

relative permeability is measure of induced magnetic field magnetic materials that are easily magnetized have high magnetic permeability

magnetic susceptibility

M χm = ── H

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(3) types of magnetism magnetic fields and forces are due to intrinsic spin of electrons (a) diamagnetism – external magnetic field unbalances orbiting electrons causing dipoles that oppose applied field • very small negative magnetic susceptibility χm ≈ -10-6 (b) paramagnetism – materials exhibit small positive magnetic susceptibility χm ≈ 10-6 to 10-2 • paramagnetic effect disappears when the applied magnetic field is removed • produced by alignment of individual dipole moments of atoms or molecules • increasing in temperature decreases the paramagnetic effect

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(c) ferromagnetism – large magnetic fields that can be retained or eliminated as desired can be produced • ferromagnetic elements (such as Fe, Co, Ni) produce large magnetic fields it is due to spin of the 3d electrons of adjacent atoms aligning in parallel directions in microscopic domains by spontaneous magnetization • random orientation of domains results in no net magnetization • the ratio of atomic spacing to diameter of 3d orbit must be 1.4 to 2.7 for the parallel alignment to occur

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(d) magnetic moments of a single unpaired electron • each electron spinning about its own axis has dipole moment μB eh μB = ── 4πm m : electron mass e : electronic charge • in paired electrons positive and negative moments cancel ex. show that the numerical value for a Bohr magneton is 9.27 × 10-24 A·m2 eh (1.6 × 10-19 C)(6.63 × 10-34 J·s) μB = ── = ───────────── 4πm 4π (9.11 × 10-31 kg) = 9.27 × 10-24 A·m2 C·J·s (A·s)·(N·m)·s (A·s)·(kg·m/s2)·m·s ─── = ────── = ──────── = A·m2 kg kg kg

ex. calculate the saturation magnetization Ms and saturation induction Bs for pure Fe. assuming all magnetic moments due to 4 unpaired 3d electrons are aligned in a magnetic field Fe has BCC unit cell with a = 0.287 nm 7

(2 atoms)(4 Bohr magnetons/atom)(9.27 × 10-24 A·m2) MS = ───────────────────── (2.87 × 10-10 m)3 = 3.15 × 106 A/m Bs ≈ μo MS ≈ (4π × 10-7 T·m/A)(3.15 × 106 A/m) = 3.96 T

(e) antiferromagnetism Mn, Cr in presence of magnetic field, magnetic dipoles align in opposite directions (f) ferrimagnetism Fe3O4 ions of ceramics have different magnitudes of magnetic moments and are aligned in antiparallel manner creating net magnetic moments

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(4) effect of temperature on ferromagnetism • above 0 K, thermal energy causes magnetic dipoles to deviate from parallel arrangement • at higher temperature, (Curie temperature) ferromagnetism is completely lost and material becomes paramagnetic • on cooling, ferromagnetic domains reform • Curie temperature for Co 1123oC Ni 358oC Fe 770oC

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(5) ferromagnetic domains

magnetic dipole moments align themselves in parallel direction in small-volume regions called magnetic domains • when demagnetized by slowly cooling from above its Curie temperature, domains are rearranged in random order no net magnetic moment

• when external magnetic field is applied, the magnetic domains whose moments are initially parallel to the applied filed grow • when domain growth finishes, domain rotation occurs and domain rotation requires more energy than domain growth 10

(6) types of energies that determine the structure most stable structure is attained when overall potential energy is minimum total magnetic energy of a ferromagnetic material is the sum of the following energies: (a) exchange energy – potential energy within a domain is minimized when all atomic dipoles are aligned in single direction the alignment is associated with a positive exchange energy (b) magnetostatic energy – potential magnetic energy produced by its external field formation of multiple domain reduces magnetostatic energy

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(c) magnetocrystalline anisotropy energy magnetization with applied field for a single crystal varies with crystal orientation ex. for BCC Fe saturation magnetization occurs most easily for the direction saturation magnetization occurs with highest applied field for direction

for FCC Ni the easy directions of magnetization are and the hard direction grains at different orientations will reach saturation magnetization at different field strength magnetocrystalline anisotropy energy – the work done to rotate all domains to reach 12 saturation

(d) domain wall energy domain wall – the boundary between two domains whose overall moments are at different directions domain changes orientation gradually with a boundary about 300 atoms wide

large width of domain wall is due to balance between two forces: exchange force and magnetocrystalline anisotropy equilibrium wall width is width at which sum of two energies are minimum

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(e) magnetostrictive energy magnetostriction – magnetically induced reversible elastic strain (Δl/l) the order of 10-6

magnetostrictive energy – the energy due to mechanical stress created by magnetostriction it is due to change in bond length caused by rotation of electron-spin dipole moments equilibrium domain configuration is reached when sum of magnetostrictive and domain wall energies are minimum

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(7) magnetization and demagnetization hysteresis loop – magnetization loop of a ferromagnetic material

• magnetization and demagnetization do not follow same loop • saturation induction Bs • once magnetized, remnant induction Br remains even after demagnetization • negative field Hc (coercive force) must be applied to completely demagnetize • area inside the loop is a measure of work done in magnetizing and demagnetizing 15

(8) soft magnetic materials • easily magnetized and demagnetized • low coercive force and high saturation induction are desirable properties

• hysteresis energy losses – due to dissipated energy required to push the domain walls back and forth • impurities, crystalline imperfections and precipitates increase hysteresis energy losses • eddy current energy losses – induced electric current causes some stray electric currents resulting from transient voltage source of energy loss by electrical resistance 16 healing

(a) iron-silicon alloys • Fe–3 to 4% Si alloys are commonly used soft magnetic materials • Si increases electrical resistivity reduces the eddy-current losses • Si decreases magnetoanisotropy energy and increases magnetic permeability decreases hysteresis core losses • Si decreases magnetostriction and lower hysteresis energy losses and transformer noise • Si decreases saturation induction and Curie temperature (disadvantage) • laminated structure further reduces eddycurrent losses • decrease in energy loss is also achieved by using grain oriented silicon sheet

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(b) metallic glasses • noncrystalline domains • soft magnetic properties, have combination of ferromagnetic Fe, Co, Ni with metalloids B and Si • used in low-energy core-loss transformers, magnetic sensors and recording heads

• produced by rapid cooling (106 oC/s) as a thin film on a rotating copper surface mold a continuous ribbon of metallic glass (0.001 in. thick and 6 in. wide) is produced

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• strong, hard, flexible and corrosion resistant • easy movement of domain walls due to absence of grain boundaries can be magnetized and demagnetized easily • very narrow hysteresis loops and low hysteresis energy loss

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(c) nickel-iron alloys • higher permeability at lower field because of low magnetoanisotropy and magnetostrictive energy • used in highly sensitive communication equipments • 50% Ni alloy – moderate permeability, high saturation induction • 79% Ni alloy – high permeability, low saturation induction • initial permeability is increased by annealing in presence of magnetic field

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(9) hard magnetic materials – properties • high coercive force Hc and remanent magnetic induction Br • wide and high hysteresis loops and difficult to demagnetize • demagnetizing curves can be used for comparing the strength of permanent magnets 1. 2. 3. 4. 5. 6. 7. 8. 9.

Sm(Co, Cu)7.4 SmCo5 bonded SmCo5 alnico 5 Mn-Al-C alnico 8 Cr-Co-Fe ferrite bonded ferrite

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• magnetic potential energy is measured by maximum energy product (BH)max (BH)max is the area of largest rectangle that can be inscribed in the second quadrant of the hysteresis loop

ex. estimate the maximum energy product (BH)max for Sm(Co, Cu)7.4 trail 1 (0.8 T × 250 kA/m) = 200 kJ/m3 trail 2 (0.6 T × 380 kA/m) = 228 kJ/m3 trail 3 (0.55 T × 420 kA/m) = 231 kJ/m3 trail 4 (0.5 T × 440 kA/m) = 220 kJ/m3 the highest value is 231 kJ/m3 22

(a) alnico alloys alnico : aluminum + nickel + cobalt • high energy product (BH)max = 40~70 kJ/m3 high remnant induction Br = 0.7~1.5 T moderate coercivity Hc = 40~160 kA/m • compositions of the alnico alloys

• produced by casting or powder metallurgy • structure single phase BCC at 1250oC cooling to 750~850oC, α and α’ form α is rich in Ni and Al, is weakly magnetic α’ is rich in Fe and Co, is highly magnetic • if heat treated in magnetic field, α’ becomes elongated and hence is difficult to rotate –23 high coercivity

(b) rare earth alloys • very high maximum energy product (BH)max to 240 kJ/m3 and coercivity to 3200 kA/m due to unpaired 4f electrons • SmCo5 single phase magnets coercivity is based on nucleation and pinning down of domain walls at surfaces and grain boundaries high magnetic strengths with (BH)max in the range of 130~160 240 kJ/m3 • precipitation-hardened Sm(Co,Cu)2.5 alloy part of Co substituted by Cu precipitate produced at low temperatures and domain walls are pinned at precipitates addition of small amount of Fe and Zr promote the development of high coercivity

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(c) neodymium-iron-boron magnetic alloys • produced by powder metallurgy and rapid solidification melt-spun ribbon process • highly ferromagnetic Nd2Fe14B grains are surrounded by nonferromagnetic Nd rich intergranular phase • high coercivity and energy product due to difficulty in reverse nucleating • used in automotive starting motors

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(d) iron-chromium-cobalt magnetic alloys • structure and properties analogues to alnico 61% Fe, 28% Cr, 11% Co • single phase BCC structure forms at elated temperature (1200oC) • slow-cooling precipitates of Cr-rich α2 phase forms in a matrix of Fe-rich α1 phase below 650oC • particles are elongated by forming to increase coercivity

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• used in permanent magnets of modern telephone receivers

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(10) ferrites • magnetic ceramics made by mixing Fe2O3 with other oxides and carbonates in powdered form • domain structure and hysteresis loop similar to ferromagnets but low magnetic saturation (a) magnetically soft ferrites • exhibit ferrimagnetic behavior • composition: MO·Fe2O3 where M is Fe2+, Mn2+, Ni2+ or Zn2+ • inverse spinel structure – cubic unit cells with 8 subcells, each subcell has an FCC structure, only1/2 of octahedral sites and 1/8 of tetrahedral sites are occupied by metal ions

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• net magnetic moments in inverse spinel ferrites Fe2+ ions 4 unpaired 3d electrons. Fe3+ ions 5 unpaired 3d electrons. each unpaired 3d electron has one Bohr magneton

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ex. calculate the theoretical saturation magnetization M in A/m and the saturation induction Bs in tesla for ferrite FeO·Fe2O3 (lattice constant of unit cell is 0.893 nm, and neglect μH term for Bs) (4 Bohr magnetons) × 8 = 32 Bohr magnetons

32 Bohr magnetons -24 A·m2) M = ———————— × (9.27 × 10 (8.93 × 10-10 m)3 = 5.0 × 105 A/m Bs ≈ μoM = (4π × 10-7 T·m/A) × (5.0 × 105 A/m) = 0.63 T

• useful magnetic properties, good insulators high electrical resistivity low eddycurrent losses • applications: low-signal, memory-core, audiovisual and recording head applications • recording heads are made up of Mn-Zn and Ni-Zn spinel ferrites

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(b) magnetically hard ferrites • general formula: MO·Fe2O3 and hexagonal in crystal structure • the most important in this group: barium ferrite (BaO·6Fe2O3) and strontium ferrite (SrO·6Fe2O3) • low cost, low density, have a high coercive force • high magnetocrystalline anisotropy. • magnetization takes place by domain wall nucleation and motion. • applications: generators, relays, motors, loudspeakers and door closers

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