ʘ exchange Forces

1. ʘ exchange Forces For the hydrogen molecule For a particular pair of atoms, situated at a certain distance apart, there are certain electrostatic attractive forces and repulsire forces which can be calculated by Coulomb’s law. But there is still another force, entirely non-classical, which depends on the relative orientation of the spins of the two electrons -> exchange force. If the spins are antiparallel, the sum of all the forces is attractive and a stable molecule is formed; the total energy of the atoms is then less for a particular distance of separation than it is for smaller or larger distance. If the spins are parallel, the two atoms repel one another. The exchange force is a consequence of the Pauli exclusion principle, applied to the two atoms as a whole. The consideration introduces an additional term, the exchange energy, into the expression for the total energy of the two atoms.

2. The exchange energy forms an important part of the total energy of many molecules and of the covalent bond in many solids. If two atoms i and j have spin angular momentum Sih/2π and Sjh/2π, respectively, the exchange energy between them is given by Jex: exchange integral θ: the angle between the spins (cos θ=1) If Jex> 0, Eex is a minimum, When the spins are parallel Eex is a maximum, When the spins are anti-parallel (cos θ=-1) If Jex< 0, Eex is a minimum, When the spins are anti-parallel Eex is a maximum, When the spins are parallel Ferromagnetism is due to the alignment of spin moments on adjacent atoms, -> Jex> 0 Jexis commonly negative, as in the hydrogen molecules, Jex< 0

3. Bethe-Slater curve Bethe-Slater curve, shows the postulated variation of the exchange integral with ratio ra/r3d the radius of its 3d shell of electron. When is large, Jex is small and positive. When is small, a further decrease in the interatomicdistance brings the 3d electron so close together that their spins must becomes anti-parallel (Jex < 0) -> antiferromagnetism When Jex> 0, it magnitude is proportional to Tc, because spins which are held parallel to each other by strong exchange forces can be disordered only by large amounts of thermal energy. Jex(Co)> Jex(Fe)> Jex(Ni) Co ,Tc=1131oC > Fe ,Tc=770oC > Ni,Tc=358oC

4. Band theory The pauli principle: eachenergy level in an atom can contain a maximum of two electrons and they must have opposite spin. The 2P subshell is actually composed of 3 sub-subshells of almost the same energy, each capable of holding two electrons. 3d and 4s level have nearly the same energy and they shift their relative positive positions almost from atom to atom. The transition elements, those in which an incomplete 3d shell is being filled are the ones of most interest to us because they include 3 ferromagnetic metals. When atoms are brought close together to from a solid, the positions of the energy level are profoundly modified.

5. When two atoms approach so closely that their electron clouds begin to overlap. • In the transition elements, the outermost electrons are the 3d and 4s; these electron clouds are the first to overlap as the atoms are brought together, and the corresponding level are the first to split. • When the interatomic distance d→do, the 3d levels are spread into a band extending from B → C, and 4s levels are spread into a much wider band from A → D. ↓(because the 4s electrons are farther from the nucleus) • However, the inner core electrons (1s and 2s) are too far apart to have much effect on one another, and the corresponding energy levels show a negligible amount of splitting.

6. N(E) is not constant but a function of the energy E. ↑density of state • The product of the density N(E) and any given energy range gives the number of levels in that range; thus N(E)dE is the number of levels between E and E+dE. • Since the 3d and 4s bands overlap in energy. ↓the corresponding density curve as shown as following.

7. The density of 3d levels far greater than that of 4s levels, because there are five 3d levels per atom, with a capacity of a capacity of 2 electrons. • Filled energy levels can’t contribute a magnetic moment, because the two electrons in each level have opposite spin and thus cancel each other out. • Suppose that 10 atoms are brought together to form a “crystal”.Then the single level in the free atom will split into 10 levels, and the lower 5 will each contain 2 electrons. • If one electron reverses its spin, as in (b), then a spin imbalance of 2 is created, and the magnetic moment, μH=2/10 μB/atom. • The force creating this spin imbalance in a ferromagnetic is just the exchange force.

8. The ferromagnetism of Fe, Co, and Ni is due to spin imbalance in the 3d band. • The maximum imbalance in 3d (the saturation magnetization), is achieved when one half-band is full of 5 electrons. • Suppose we let n = no. of (3d+4s) electron per atom x = no. of 4s electron per atom n - x = no. of 3d electron per atomAt saturation, five 3d electrons have spin upand (n - x - 5) have spin down → μH= [5 - (n - x - 5)]μB = [10 - (n - x)] μB • This eq. Shows that the max spin imbalance is equal to the no. of unfilled electron states in the 3d band.(The 4s electrons are assumed to make no contribution) • For Ni, n = 10 and the experimental value of μH= 0.6μB insert μH= [10 - (n - x)]μB, we found x = 0.6To assume that the no. of 4s electrons is constant at 0.6 for elements near Ni, we have μH= (10.6 - n)μB.

9. The magnetic moments per atom predicted by this eq. agree well with Fe, Co, Ni, and that the predicted negative moment for Cu has no physical meaning, since 3d band of Cu is full. • In Fe, we have assumed • Since the observed spin imbalance in Fe is about 20% less than this predicted value, and in Mn actually zero, it appears that the exchange force can’t keep one half-band full of electrons if the other half-band is less than about half full.

10. ◎Magnetic ceramics Each electron in an atom contributes a quantized amount of magnetic moment form its For the transition metal ions used in most ferrites, the contribution from the orbital angular momentum is negligible, and the magnetic moment of an ion is determined by the no. of unpaired electron spins. Each unpaired electron contributes a moment of one Bohr magneton (μ0) In the transition metal series where 3d-shells are partially filled, the moment is determined by the net no. of unpaired spins. ◎ The resulting magnetic moments for the transition metal series.

11. Ferrimagnetism refers to the condition where the moments of ions on type of site are partially offset by antiparallel interaction with ions of another site, but there remains a net magnetization. Where a metal oxide containing magnetic ions is ferromagnetic, antiferromagnetic, or ferrimagnetic depends on (1) the magnitude of indiridual moments, (2) the type and no. of sites that are occupied, (3) the nature of the interaction between sites. The exchange interaction between any two cations is mediated by the intervening oxygen ions, and is known as a superexchange interaction. The superexchange interaction involves the temporary transfer of an electron from one of the oxygen ions dumbbell-shaped 2P orbitals to one of the adjacent cations, leaving behind an unpaired 2P electron interacting with the opposing cation. For cations with more than half-filled d levels, this interaction generally results in antiparallel spin between the cations.

12. The interaction is stronger for more closely separated cations and for metal-oxygen-metal angles closer to 180 o , in spinel structure the a-b interaction > b-b interaction > a-a interaction. Ex. Transition metal monoxides, MnO, FeO, CoO, NiO, in each the cation has at least a half-filled d-level, there are no a cations, we expect an antiparallel b-b interaction to domain. These oxides are anti ferromagnetic, ordered cations within a (111) plane have parallel spins adjacent (111) planes have parallel spins.

13. Ferrimgneticspinels are generally those with some degree of inverse structure. Fe3O4 is an inverse spinel with cation ordering Fe3+ (Fe2+ Fe3+)O4 ↑↑↑↑↑ (↓↓↓↓　　↓↓↓↓↓) a b The antiparallel a-b interaction dominates , the net magnetization or saturation magnetization per formula formula unit is 4 μB. Inverse spinel ferrite of formula M2+ Fe2O4 will have saturation magnetizations determined by M2+ ion, since the Fe3+ ions appear in equal no. on a and b sites and cancel. Ferrites with fractional degree of inversion can be computed by simple simulation as long as the site occupancy is known.

14. The initial increase is due to the substitution of nonmagnetic Zn2+ onto a sites, displacing Fe3+ ions to the b sites where they contribute to the net moment. ◎saturation magnetization We can calculate the saturation magnetization of a ferrite at 0 ok, knowing (a) the moment in each ion, (b) the distribution of the ions between A and B (c)the fact that the exchange interaction between A and B sites is negnative.

15. Ex1 Ni ferrite is inverse spinel, with are the Ni2+ ions in B sites and Fe3+ ions evenly divided between A and B sites. The moment of the Fe3+ ions cancel, and the net moment is simply that of the Ni+ ion, which is 2 μB. Ex2 Zn ferrite the normal spinel, and Zn2+ ions of zero moment fill the A sites , there are no A-B interaction, The negative B-B interaction, then comes into play: the Fe3+ ions B sites then fin to have antiparallelmoments, and there is no net moment.

16. Ex3 If Mg ferrite was completely inverse spinel, its net moment would be zero, because the moment of Mg2+ ions is zero. But 0.1 of Mg2+ ions are on a Sites, displacing an equal no. of Fe3+ ions, A-site moment becomes 0.9x(5) = 4.5 μB and B-site moment 1.1x(5) = 5.5 μB , giving an net moment of 1.0μB . Ex4 A mixed ferrite containing 10 mol% Zn-ferrite in Ni-Ferrite . The Zn2+ ions of zero moment go to the A sites as in pure Zn-ferrite, thus weakening the A-site moment, and the Fe3+ ions from the Zn-ferrite, have parallel moment in B sites, because of the strong A-B interaction. The expected net moment increases from 2.0 μB for pure Ni-ferrite to 2.8 μB for the mixed ferrite. ◎ exchange energy in terms of electron spin (can we separate the exchange energy and make it dependent only on spin?) The exchange energy dictates a lower energy, when J > 0, ferromagnetic order J < 0, ferromagnetic alignment The exchange Hamiltonian can be written simply in terms of the spins on two electrons Heisen have moment of ferrimagnetic

17. For a particular pair of atoms, situated at a certain distance apart, there are certain electrostatic attractive forces (between the electrons and protons ) and repulsive force (between the two electrons and between the two protons ) which can be calculated by coulomb’s law. But there is still another force, entirely non-classical, which depends on the relative orientation of the spins of the two electrons. exchange force If the spins are antiparallel, the sum of all the force is attractive and stable molecule is formed. If the spins are parallel, the atoms repel one another. The exchange force is a consequence of the Pauli exchange principle applied to the two atoms. two electrons can have the same energy only if they have opposite spins. The exchange energy forms an important part of the total energy of many molecules and of the covalent bond in many solids. If two atoms i and j have spin angular momentum Sih/2π and Sjh/2π, respectively, the exchange energy between them is given by Jex: exchange integral, φ:the angle between the spins

18. If two atoms of the same kind are brought closer but without any change in the radius r3d, the ratio ra/r3d will decrease from large to small value. When ratio is large, Jex is small and positive. As the ratio decreases and 3d electrons approach one another more closely, the positive exchange interaction, favoring parallel spins, becomes stronger and decreases to zero. A further decrease in interatomicdistance, bring 3d electrons so close together, that their spins must become antiparallel, (Jex< 0). This condition is called antiferromagnetism. The Bethe – Slater curve can be applied to a series of different elements if we compute ra/r3dfrom their known atom diameters and shell radii. Mn < 100 okantiferromagnetic Cr < 37 ℃ antiferromagnetic When Jex > 0, its magnitude Tc.

19. If Jex < 0, Eex is a min, when the spins are parallel (cosφ=1) Eex is a max, when the spins are anti parallel (cosφ= -1) If Jex < 0, Eex is a min, when the spins are antiparallel. Ferromagnetism is due to the alignment of spin moments on adjacent atoms. For ferromagnetism, Jex > 0. According to the weiss theory ferromagnetism is caused by a powerful “ molecular field “ which aligns the atomic moments. Exchange forces decrease rapidly with distance, so that some simplification is possible by restricting the summation to the nearst-neighbor paris. The Bethe – Slater curve, shows the postulated variation of the exchange integral with the ratio ra/r3d, where ra: radius of an atom and r3d : radius of its 3d shell of electron.

20. Similarly for a nearest-neighbor coupling, the interaction parameter J can be found from the Tcusing the eq.⊙Antiferromagnetism(Is it also possible to explain antiferromagnetic order by a Weiss interaction?) • In antiferromagnetism, the nearest-neighbor moment are alighed anti parallel can also be interpreted on the basis of the weiss model. • 1. The material is divided into two sublattices A and B, with the moments on their own sublattice interacting with the moment on the other sublattice with a negative coupling coefficient, but interacting with the moments on their own sublattice with a positive coupling coefficient.(the magnetic moments on the two sublattice point in different directions)

21. Antiferromagnet appear a straight line intercepting the T axis at –Tc, but above 0 K known as the Neel temperture. ↑(-Tc) T • 2. With a negative interaction between nearest neighbor. • The Curie-Weiss law also applies to antiferromagnets above their ordering Tc (critical temperture). • The plot • Ex. Cr, TN = 37OC Mn, TN = 100 K

22. ⊙ Ferrimagnetism (How can the properties of ferrites be explained?) • Ferrimagnetism is a particular case of antiferromagnetism in which the magnetic moments on the A and B sublattice while still pointing in opposite direction have different magnitudes. • They have a spontaneous magnetization below Tc and are organized into domains. • 1. The most familiar ferrimagnetic is Fe3O4, other magnetic ferrites with the general formula MO · Fe2O3 with cubic and have spinel structure.(M: a transition metal, ex: Mn, Ni, Co, Zn…) • 2. Hexagonal ferrite, such as BaO · 6(Fe2O3), SrO · 6(Fe2O3).These are magnetically hard and have been extensively used as permanent-magnet materials. • They have high anisostropy with the moments lying along the c axis.Tc≒500~800OC • 3. garnets have the chemical formula: 5Fe2O3 · 3R2O3 (R: rare earth ion) • These materials have a complicated cubic crystal structure. Their order-disorder transition temperatures are around 550OC • 4. Another ferrimagnetic materials is γ-Fe2O3, which is widely used as a magnetic recording medium. This is obtained by oxidizing Fe3O4.