ʘ exchange Forces

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.

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 Jex is commonly negative, as in the hydrogen molecules, Jex < 0

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 interatomic distance 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

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.

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.

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.

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.

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.

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.

◎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.

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 interactionbetween 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.

⊙Molecular field theory We expect that exchange forces between the metal ions in a ferrimagnetic will act through the O ions by means of the indirect exchange (super exchange) mechanism, just as in antiferromagnetic. The exchange interaction in antiferromagnetic ionic solid takes place by the mechanism of indirect exchange (super exchange). The positive metal ions, which carry the magnetic moment are too far apart for direct exchange forces. Instead, they act indirectly through the neighboring anions (negative ions). For example, two Mn2+ ions being brought up an O2- ion from a large distance as in (a). The moment on these two ions are at first only randomly related and O2- has no net moment. However, the outer electrons of the O2- ion constitute two superimposed orbits, one with net spin up, the other with net spin down.

When a Mn2+ ion with an up spin is brought close to O2- ion the up-spin part will be displaced as in (b), because parallel spins repel one another. If another Mn2+ ion is brought up from the right, it is forced to have a down-spin when it comes close to the up-spin side of the “unbalanced” O2- ion. The strength of the antiparallel coupling between metal ions M depends on the band angle AOB and is generally greatest when this angle is 180O (M-O-M colinear). --------------------------------------------------------------------------------------- However, molecular field theory for a ferrimagnetic is inherently more complicated than for an antiferromagnetic. The AA interaction in a ferrimagnetic will differ from the BB interaction, even though the ions involved are identical. The basic reason is that an ion on an A site has a different no. and arrangement of neighbors than the same ion on a B site.

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-site 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.

Ferrimgnetic spinels 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+ Fe2O4will 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.

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 negative.

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.

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.

Summary: kinds of magnetism circle: an atom or ion (open and solid circles represent atoms of different valence or chemical species) arrow: through that circle represent its net magnetic moment. • The fire kinds of magnetism can be divided into two categories: • Diamagnetism and ideal paramagnetism (Curie law), characterized by non-cooperative behavior of the individual magnetic moments. • Nonideal paramagnetism (Curie-weiss law), ferromagnetism, antiferromagnetism and ferrimagnetism, which are all example of cooperative phenomena.

◎Ferrimagnetism The fractional magnetization σs/ σ0 of a typical ferrite decreases rather rapidly with increasing T, whereas the value of σs/ σ0 for iron, remains large unit T/Tc exceeds about 0.8. evidence that the ferrite are not ferromagnetic Ferrimagnetic substances exhibit a substantial spontaneous spontaneous magnetization at R.T. and they consist of self-saturated domains. exhibit the phenomena of Neel made the basis assumption that exchange force acting between ions at A site and B site was negative as in an antiferromagnetic . However, in ferrimagnetic, the magnitudes of the A and B sublattice magnetizations are not equal, the two opposing m do not cancel, and a net spontaneous magnetization results.

◎structure of cubic ferrites Spinel structure is complex, in that there are 8 “molecules” or a total of 8x7=56 ions per unit cell. O ions are packed quite close together in fcc arrangement, and the much smaller metal ions occupy the spaces between them. These spaces are of two kind: (1) a tetrahedral (A) site, it is located at the center of tetrahedron, whose corners are occupied by O ions. (2)an octahedral (B) site, the O ions occupy the corners of an octahedron.

By no means all of the available sites are actually occupied by metal ions. Only of the A sites and of the B sites are occupied. In the mineral spinel, MgO·Al2O3, the Mg2+ ions are in A sites, and the Al3+ ions are in B site. Some ferrites MO·Fe2O3 have exactly this structure, with M2+ in A sites and Fe3+ in B sites, which call the normal spinel structure, such as M2+=Zn2+, Cd2+, with paramagnetism. Manny other ferrites, have the inverse spinel structure, in which the 2+ ions are in B sites, and the 3+ ions are equally divided between A and B sites, such as M2+=Fe2+,Co2+,Ni2+… with ferrimagnetism.

◎Saturation magnetization The discrepancies between theory and experiment evident in above Figure, are generally ascribed to one or both of the following: Orbital moments may not be completely quenched; ie, there may be an orbital moment, not allowed for in the theory, besides the spin moment, as shown of the Co2+ ion. The structure may not be complete inverse, the degree of inversion can sometimes be changed by heat treatment. The saturation magnetization then becomes a structure-sensitive

◎Saturation magnetization The expect net moment increase from 2.0 μB for pure Ni Ferrite, to 2.8 μB for containing 10 mole% Zn-ferrite, we would expect pure Zn ferrite to have 10 μB . However, this can’t occur because A moments will soon become too week to affect the B moments, and the net moment must sooner or later begin to decrease. The experimental curve does begin with a slope very close to the theoretical as shown in Fig 6.4. The way which the experimental curve depart from the straight lines has explained in terms of superparamagnetism.

⊙ 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. 39

Similarly for a nearest-neighbor coupling, the interaction parameter J can be found from the Tc using the eq.⊙Antiferromagnetism(Is it also possible to explain antiferromagnetic order by a Weiss interaction?) • In antiferromagnetism, the nearest-neighbor moment are aligned 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)

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

⊙Domain wall structure Domain wall are interfaces between regions in which the spontaneous magnetization has different directions. At or within the wall, the magnetization must change direction. A row of atoms is shown, parallel to x, with the 180O domain wall lying in the yz plane; the easy axis is ±y. This exchange energy can be decreased, if we allow the 180O change in spin direction to take place gradually over N atoms. 1. The exchange energy tries to make the wall as wide as possible, in order to make the angle Φ between adjacent spins as small as possible. 2.The anisotropy energy tries make the wall thin, in order to reduce the no. of spins pointing in noneasy direction.For 1. and 2., this competition, the wall has a certain finite width and a certain structure. 38

The angle Φ between adjacent spins, equal to π/n, as in Fig.9.2, which is drawn for Φ = 30O. The spins within the wall of Fig.9.2 are pointing in noneasy directions, so that the crystal anisotropy energy within the wall is higher than it is in the adjoining domains. The first theoretical examination of the structure of a domain wall was made by Bloch in 1932, and the domain walls are accordingly often called Bloch walls. 39

↑ (independent of angle and has the same value within a domain as within a wall, and it can therefore be dropped) 40 ⊙an approximate calculation of domain wall width ⊙Exchange energy (Eex) For a pair of atoms of the same spin S is The extra exchange energy existing within the wall is JS2Φ2, per spin pair. The exchange energy per unit area of domain wall. We assume simple cubic, with an atom at each corner of a cell of edge a, and the place of the wall parallel to the cubic face {100}. The wall is N atoms thicks, the extra exchange energy per unit area of wall is γex = (JS2Φ2)(N)(1/a2) If Φ = π/N for a 180O wall,

⊙anisotropy energy (Ean) The anisotropy energy is of the order of the anisotropy constant K times the volume of the wall.[a2(Na)] Ean=Kˑa2(Na) The anisotropy energy per unit area of wall The total energy per unit area, for a wall of thickness δ=Na The energy has a min for a particular value of δ, 41

The smaller the anisotropy constant, the thicker the wall; therefore, wall thickness in creases with temperature, because K almost always decrease with rising temperature. (The min in the total energy occurs when the exchange and anisotropy energies are equal) Calculate δ and γ for Ni and Fe : J≈0.3kTc ( k: Boltzmannis constant = 1.38 х10-16 erg/oK) S=1/2 For Fe, Tc=1043oK, K=4.8 х 105 ergs/cm3, a=2.48Å Å The angle Ø between adjacent spins is 180o/120=1.5o The wall energy 42

For Ni, Tc=631 oK, K=0.5x105 ergs/cm3, a=2.49Å The general conclusion: (1) domain walls are several hundred A6 thick (2) wall energies are a few ergs/cm2 47

⊙ Theories of ordered magnetism (What types of ordered magnetic structures exist and how do they differ?) Different types of magnetic order in solids including ferromagnetism antiferromagnetism ferrimagnetism helimagnetism Some materials( heavy rate earth), exhibit more than one ordered magnetic states. Ferromagnetism T > Curie T paramagnetic antiferromagnetism T > Neel T paramagnetic Some solids( Tb, Dy, Ho) have Curie T Neel T ⊙Ferromagnetism (What cause the transition from paramagnetic to ferromagnetism?) In ferromagnetic solid at T < Tc, the magnetic moments within domains are aligned parallel.( This can be explained by the Weiss interaction field.) Transition metal ferromagnets Fe, Tc= 770oC, 2. Ni, Tc= 358oC, 3. Co, Tc= 1131oC rare earth metals Gd, Tc=293K, 2. Dy, Tc=85oC, 3. Tb, Tc=219, 4. Ho, Tc=19K 5. Er, Tc=19.5K, 6. Tm, Tc=32K

The alignment of magnetic moments in various ordered ferromagnetic solid. At a critical T, the randomizing effect of thermal energy overcomes the aligning effect of the interaction energy, and above this T, the magnetic state becomes disordered.

⊙ Weiss theory of ferromagnetism (How can the Weiss interaction be used to explain magnetic order in ferromagnets? ) If the unpaired electronic magnetic moments which are responsible for the magnetic properties are localized on the atomic sites. The interaction between the unpaired moments, leads to the existence of a critical T be low which the thermal energy of the electronic moments is insufficient to cause ranolom paramagnetic alignment. The effective field He can be used to explain the alignment of magnetic moments within domains for T < Tc. Theme for the interatomic interaction (exchange field): (1)the mean-field approximation -> used for the paramagnetic region (2) a nearnest-neighbor interaction -> used for the ferromagnetic region Suppose that any atomic magnetic moment mi experiences an effective field Heij due to another moment mj. ( If we assume that this field is also in the direction of mj) Jij

The total exchange interaction field at the moment mi wil be the vector sum of all interaction with other moments. ⊙Mean-field approximation(Is there a simple explanation of the Weiss interaction?) If the interactions between all moments are identical and hence independent of displacement between the moments, then all of the Jij are equal. within a domain: The interaction energy of the moment under these conditions:

If we consider the case of a zero external field, then the only field operating within a domain will be the Weiss field Htot = He If we apply the mean-field model, the interaction field will be proportional to the spontaneous magnetization Ms within a domain. Following an analogous argument to that given by Langevin for paramagnetism As T ↑, the spontaneous M within a domain ↓. The energy of a moment within a domain can be generalized to include the effect of a magnetic field H as follows.

Magnetization within a domain This eq. is not encountered very often, because αMs >>H in ferromagnet (in iron, Ms = 1.7x106 A/m, so αMs can up to 6.8x108 A/m, while H will rarely exceed 2x106 A/m) Consequently within the body of a domain, the action of the H field is not very significant when compared with the interaction field. Moderate magnetic field (H ≒8x103 A/m) can cause significant changes in the bulk magnetization M in ferromagnet. These changes occur principally at the domain boundaries where the exchange interaction is competing with the anisotropy energy to give an energy balance. Under these conditions, the additional field energy can just tip the balance and result in change in the direction of magnetic moments within the domain wall. ↓ domain wall motion

The magnetic moment in the domain wall do not couple to the spontaneous magnetization of the domain. The net interaction field per moment is different in this case because field per moment is different in this case because the domains on either side are dligh in different directions. ◎ Nearest – neighbor interactions ( can the Weiss model be interpreted on the basis of localized interaction only? ) In the nearest – neighbor approximation, the electronic moments interact only with those of its Z nearest neighbors. For a simple cubic lattice, Z = 6 body – centered cubic, Z = 8 face – centered cubic, Z = 12 hexagonal lattice, Z = 12 The nearest – neighbor approach is particularly useful for considering magnetic moment in the domain wall. In this case, the moments do not couple to the magnetization within the body of the domain simple because they lie between domains with different magnetic directions and the direction of magnetization changes within the wall.

In this approximation, the exchange interaction field We assume that each neares – neighbor interaction is identical and equal to J. J = 0, non – interacting J = 0, each moment interacts equally with each of its nearest neighbors.

Corresponos to J > 0 ferromagnetic alignment J < 0 antiferromagnetic alignment Interaction energy of magnetic moment Summing over the Z nearest neighbor Weiss interaction it is possible to provide a description of ferromagnets which is similar to the Langevin model of Paramagnetism. Weiss interaction model is only correct for ferromagnets in which the moments are localized on the atomic cores. Thus it applies to La series, because 4f electrons are tightly bound to the nuclei. The model also works reason ably well for Ni, which obeys the curie – Weiss law.

Curie Constant Curie Temperature ◎Curie temperature on the basis of the mean – field model ( How does the Weiss interaction explain the existence of a critical temperature?) Weiss presented that the existence of an internal or atomic field proportional to the magnetization M, led to a modified from of curies law known as the Curie – Weiss law From the Curie’s law From the Curie – Weiss law From above eq., it is possible to determine the mean-field coupling or weiss constant α, from Tc which provides the is known.

ʘ exchange Forces