Magnetic Phenomena and Their Interpretation-Classical Approach 12

12.1 Overview KINDS OF MAGNETISM Different types of magnetism are characterized by the magnitude and the sign of the susceptibility(k) (magnetic permeability) KINDS OF MAGNETISM

12.1.1 Diamagnetism What is diamagnetism? Lenz’s law : Current is induced in a wire loop whenever a bar magnet is moved toward(or from) this loop. The induced current will appear in such a direction that it opposes change that produce it. Diamagnetism: explained by postulating that the external magnetic field induces a change in the magnitude of inner-atomic currents in order that their magnetic moment is in then opposite direction from the external magnetic field. A more accurate and quantitative explanation of diamagnetism replaces the induced currents by precessions the electron orbits about the magnetic field direction (Larmor precession)- bounded electrons For metals with free electrons- circulating free electrons can cause a magnetic moment

Larmor frequency External field induce a change in the magnitude of inner-atomic currents, either accelerate or decelerate the orbiting electrons (Torque: tendency of a force to rotate an object about an axis. t = r x F) Again, for metals with free electrons- circulating free electrons can cause a magnetic moment It has been observed that superconducting materials expel the magnetic flux lines when in the superconducting state (Meissner effect)

12.1.2 Paramagnetism What is paramagnetism? Paramagnetism in solids is attributed, to a large extent, to a magnetic moment that results from electrons which spin around their own axes. (a) Spin paramagnetism: (in most solids) (b) Electron-orbit paramagnetism: not effective in most solids since the electron-orbits are essentially coupled to the lattice. (free atoms(diluted gas) as well as rare earth elements which have “deep lying” 4f-electrons. In Curie Law, susceptibility is inversely proportional to the absolute temperature T In Curie-Weiss Law,

Hund’s rule: In materials with partially filled bands, the electrons are arranged to have the total spin moment to be maximized. Ref. Modern Magnetic Materials ( R.C. O’Handley) The smallest unit (or quantum) of the magnetic moment is called one Bohr magneton.

12.1.3 Ferromagnetism -Spontaneous magnetization Hysteresis Loop -transition metals : Fe, Co, Ni, rare-earth : Gd, Dy -alignment of an appreciable fraction of molecular magnetic moment in some favorable direction in crystal -related to the unfilled 3d and 4f shells -ferromagnetic transition temperature (Curie) Hysteresis Loop Mr = remanent magnetization Ms = saturation magnetization Hc = coercive field

TEMPERATURE-DEPENDENCE OF SATURATION MAGNETIZATION Figure 15.7 (a) Temperature dependence of the saturation magnetization of ferromagnetic materials. (b) Enlarged area near the Curie temperature showing the paramagnetic Curie point (see Fig. 15.3) and the ferromagnetic Curie temperature . Above Curie Temperature Tc, ferromagnetics become paramagnetic. For ferromagnetics, the Curie temperature Tc and the constant θ in the Curie-Weiss law are nearly identical. However a small difference exists because the transition from ferromagnetism to paramagnetism is gradual.

Piezomagnetism The magnetization of ferromagnetics is also stress dependent. A compressive stress increases M for Ni, while tensile stress reduces M

Magnetostriction When a substance is exposed to a magnetic field, its dimensions change. This effect called magnetostriction. (inverse of piezomagnetism) M orientation => change in dimension

DOMAIN Minimization of magnetostatic energy with changing domain shape Approximately half magnetostatic energy Closed path within crystal to reduce the magnetostatic energy Domain growth Domain wall or Bloch wall: Barkhausen effect: Kerr effect: 11

12.1.4. ANTIFERROMAGNETISM What is antiferromagnetism? Antiferromagnetic materials exhibit just as ferromagnetics, a spontaneous alignment of moments below a critical temperature. However, the responsible neighboring atoms are aligned in an antiparallel fashion. Consider to be divided into two interpenetrating sublattices A and B.

Antiferromagnetic ordering A site B site A B Thus, the magnet moments of the solid cancel each other and the material as a whole has no net magnetic moment.

TEMPERATURE-DEPENDENCE OF ANTIFERROMAGNETIC MATERIAL

ferromagnetic Antiferromagnetic

The Neel temperature is often below room temperature. Most antiferromagnetics are found among ionic compounds – insulators or semiconductors.

Non-cooperative (statistical) behavior Diamagnetism Paramagnetism Ferromagnetism Antiferromagnetism Ferrimagnetism Kinds of magnetism Non-cooperative (statistical) behavior Cooperative behavior

12.1.5 FERRIMAGNETISM Different elements, different moments, for an ions situated in crystal lattice, and thus be described as imperfect antiferromagnetics. Ionic bonding, localized field theory Cubic: MO•Fe2O3 M = Mn2+, Ni2+, Fe2+, Co2+, Mg2+, Zn2+, Cd2+ etc. (ferrite) soft magnet except CoO•Fe2O3 Hexagonal: BaO•6Fe2O3 hard magnet

Ferrimagnetic substances consist of self-saturated domains, and they exhibit the phenomena of magnetic saturation and hysteresis. Their spontaneous magnetization disappears above a certain critical temperature, also called Curie temperature and they become paramagnetic There are two types of lattice sites which are available to be occupied by the metal ions. A sites: tetrahedral site B sites: octahedral sites

3d level: Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn (from at. number 21) They all have ----- 4s2 3dn electrons

More rapid suppression Non Curie-Weiss behavior Thermal variation of the magnetic properties of a typical ferrimagnetic (NiO •Fe2O3) Example : NiO •Fe2O3 (Nickel Ferrite) 12 B if ferromagnetic ordering (5 Bohr magnetrons for Fe+3 and 2 for Ni+2) but experimental value is 2.3 B (56 emu/g) at 0 K Why? Fe3+ occupies both A site(tetrahedral) and B site (octahedral) and cancelled out. Ni2+ occupies B site.

Tetrahedral A site Octahedral B site

Oxygen 8 molecules Oxygen B site: 16b A site: 8a

Normal spinel: AO(B2O3) A2+ on tetrahedral sites, B3+ on octahedral sites (e.g.) ZnFe2O4, CdFe2O4, MgAl2O4, CoAl2O4, MnAl2O4 MO•Fe2O3 (M = Zn, Cd) non-magnetic (paramagnetic) 8M2+ : A site, 16Fe3+ : B site Inverse spinel: B(AB)O4 half of the B3+ on tetrahedral sites, A2+ and remaining B3+ on octahedral sites (e.g.) FeMgFeO4, FeTiFeO4, Fe3O4, FeNiFeO4 MO•Fe2O3 (M = Fe, Co, Ni) Ferrimagnetic 8M2+ : B site, 16Fe3+ : A,B sites (disordered state) Mixed ferrite: NiO•Fe2O3 + ZnO•Fe2O3   (Ni,Zn)O•Fe2O3

12.2 Langevin Theory of Diamagnetism Classical model Applied field r v Induced field 26

r Lenz’s law Induced field 27

r A change in the magnetic field strength from 0 to H yields a change in the velocity of the electrons : Induced field Magnetic moment per electron: 28

Average value of magnetic moment per electron So far, we tacitly assumes that the magnetic field is perpendicular to the plane of orbiting electron. In reality, the orbit plane varies constantly in direction with respect to the external field. R H  r 29

Average value of magnetic moment per atom Magnetization caused by this change of magnetic moment Diamagnetic Susceptibility The susceptibility is temperature-independent. 30

12.3 Langevin Theory of (Electron orbit) Paramagnetism Not treat spin paramagnetism, which is responsible for the paramagnetic behavior of many metals and which is only slightly temperature dependent. Assumptions: no interaction, only m , H interaction, and thermal agitation In a state of thermal equilibrium at temperature T The probability of an atom having an energy E follows the Boltzmann distribution vectors representing the magnetic moments m0H mm a In general, the potential energy is, If R=1 ( unit sphere ) 31

This function can be brought into a standard form by setting Intergrating “dn” to calculate the total magnetic moments in a unit volume The magnetization (M) is the magnetic moment per unit volume – sum of all individual magnetic moment in the field direction. This function can be brought into a standard form by setting Langevin function, L(z) 32

33

Because is usually small, (Curie’s law) 34

12.4 Molecular Field Theory: So far, we implied that the magnetic field, which tries to align the magnetic moments, stems from the external source only. Weiss observed that some materials obey a somewhat modified Curie law. Postulate that the magnetic moment of individual electrons interact each other. Molecular field Ms: spontaneous magnetization by a molecular field 35

Weiss postulated that the above-mentioned internal or molecular field is responsible for the parallel alignment of spins and considered ferromagnetics to be essentially paramagnetics having a very large molecular field. Molecular field is essentially the exchange force. For instance, when there is no external magnetic field, T1 T2 T3 Langevin function Curie temp Magnetization is a linear function of z with a temperature as a proportionality factor.

At Curie temperature, there is no spontaneous magnetization, thus, the slope is identical to the slope of the Langevin function. By rearranging it,

Saturation magnetization & Curie temperature

Spontaneous magnetization Spin states Short range order Spontaneous magnetization Spin fluctuation due to thermal agitation