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magnetism_lect23 - Magnetism and Magnetic Materials...

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Unformatted text preview: Magnetism and Magnetic Materials Chemistry 754 Solid State Chemistry Lecture #23 May 23, 2003 Magnetic Moment of an Electron Magnetism in solids originates in the magnetic properties of an electron. S = g[S(S+1)]1/2 [(eh/(4me)] [(eh/(4 B = (eh/(4me) (eh/(4 S = g[S(S+1)]1/2 B S = , the spin quantum number g ~ 2, the gyromagnetic ratio B = 9.2742 10-24 J/T, the Bohr magneton So that for a free electron S = 1.73 B 1 Magnetic Moments of Atoms & Ions Almost all atoms have multiple electrons, but most of the electrons are electrons paired up in orbitals with another electron of the opposite spin. When all of the electrons on an atom are paired the atom is said to be be diamagnetic. Atoms/ions with unpaired electrons are paramagnetic. paramagnetic. diamagnetic. Diamagnetic Ions = There is a very small magnetic moment associated with an electron traveling in a closed path around the nucleus. Paramagnetic Ions = The moment of an atom with unpaired electrons is given by the spin, S, and orbital angular, L and total momentum, J, quantum numbers. eff = gJ[J(J+1)]1/2 B Full treatment:Accurate for Lanthanides eff = [4S(S+1)+L(L+1)]1/2 B eff = 2[S(S+1)]1/2 B Neglecting spin-orbit coupling spinSpin only value Atomic Moments in Compounds Unpaired electrons and paramagnetism are usually associated with the presence of either transition metal or lanthanide (actinide) ions. In many ions. transition metal compounds the surrounding anions/ligands quench the orbital anions/ligands angular momentum and one needs only to take into account the spin only spin moment. Consider the following examples: Ion Ti4+ V2+ Cr3+ Fe3+ Ni2+ Cu2+ e- Config. Config. d1 d2 d3 d5 (HS) d8 (HS) d9 S 1/2 1 3/2 5/2 1 1/2 S(B) 1.73 2.83 3.87 5.92 2.83 1.73 S+L(B) 3.01 4.49 5.21 5.92 4.49 3.01 obs (B) 1.7-1.8 1.72.8-3.1 2.83.7-3.9 3.75.7-6.0 5.72.9-3.9 2.91.9-2.1 1.9- Deviations from the spin-only value can occur for the following reasons: spinOrbital (L) Contribution Can arise for partially filled (not full) t2g orbitals Spin-orbit Coupling Spin Increases the moment for d6, d7, d8, d9 Decreases the moment for d1, d2, d3, d4 2 Magnetic Ordering Paramagnetic Antiferromagnetic Ferromagnetic Ferrimagnetic Interaction between an Applied Magnetic Field and a Magnetic Material The interaction between an external magnetic field (H) and a material depends upon it's magnetic properties. Diamagnetic Repulsive Paramagnetic Attractive Magnetic Flux Lines B = H + 4I 4 B = Magnetic Induction (field strength within the sample) H = Applied magnetic field (field coming from external source) I = Magnetization Intensity (field originating from the sample) 3 Magnetic Susceptibility B = H + 4I 4 The permeability, P, is obtained by dividing the magnetic induction, B (total induction, field in the sample), by the applied field, H. P = B/H = 1 + 4I/H = 1 + 4 4 4 Where is the volume susceptibility (extrinsic property). To obtain the intrinsic material property, m, molar susceptibility we multiply by the Formula weight, FW, and divide by the density, . m = (FW)/ (FW)/ Typical molar susceptibilities are Paramagnetic Comp. ~ +0.01 B Diamagnetic Comp. ~ -110-6 B Ferromagnetic Comp. ~ +0.01-10 B +0.01Superconducting Comp. ~ Strongly negative, repels fields completely in some instances (Meisner effect) (Meisner In order for a material to be magnetically ordered, the spins on one atom must couple with the spins on neighboring atoms. The most common mechanism for this coupling (particularly in insulators) is through the semicovalent superexchange interaction. The spin information is transferred through covalent interactions interactions with the intervening ligand (say oxygen). Superexchange Fe3+ dx2-y2 x2 Filled O 2p Fe3+ dx2-y2 x2 Filled Fe3+ dx2-y2 x2 Filled O 2p Cr3+ dx2-y2 x2Empty The covalent interaction through the O 2p orbital stabilizes antiferromagnetic coupling. Here the oxygen based electron will spend some time on Cr3+ and due to Hund's rule polarize the t2g eleading to ferromagnetic coupling. 4 Magnetic Susceptibility vs. Temperature Paramagnetic substances with localized, weakly interacting electrons obey the Curie-Weiss law. Curie- m = C/(T+) C/(T+ m = Molar magnetic susceptibility C = Curie constant = Weiss constant A Curie-Weiss plot is a plot of 1/m vs. temperature. Ideally it Curie1/ should give a straight line if the C-W law is obeyed. From such a Cplot we can then extract the Curie constant from the inverse of the slope and the Weiss constant from the x-intercept. x- 1/m = (T+)/C = (1/C)T + /C (T+ 0.50 0.45 Curie-Weiss Plot Inverse Molar Susceptibility, (1/ m ) Fe 3+ (d5 ) = 5.9 m B = -50K 90.00 80.00 70.00 60.00 50.00 40.00 30.00 20.00 10.00 0.00 0 50 100 150 200 250 300 350 Fe 3+ (d5 ) = 5.9 m B Molar Susceptibility, m 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0 50 100 150 200 250 300 350 = 0 = -50K = 0 = 50K = 50K Temperature (K) Temperature (K) The Curie constant is equal to the inverse of the slope. It gives us the size of the moment per formula unit. C = (NA/3k)2 /3k) = (3kC/NA)1/2 = 2.84 C1/2 NA = Avogadro's Number k = Boltzman's constant The Weiss constant is equal to the xxintercept. It's sign tells us about the short range magnetic interactions. = 0 Paramagnetic > 0 Ferromagnetic Spins independent of each other Spins tending to align parallel < 0 Antiferromagnetic Spins tending to align antiparallel 5 Ferromagnets & Antiferromagnets 3.50 3.00 0.10 Ferromagnet Molar Susceptibility, m T C =100 K (T=0) = 3 B 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 TN Antiferromagnet TN =100 K Molar Susceptibility, m 2.50 2.00 1.50 1.00 0.50 0.00 0 100 200 TC 300 400 0 100 200 300 400 Temperature (K) Temperature (K) The susceptibility increases dramatically at the Curie temp. As the T decreases further the magnetic ordering and the susceptibility increase. Ferromagnet The susceptibility begins decreasing at the Neel temp. As the T decreases further the magnetic ordering increases and the susceptibility decreases. Antiferromagnet The Curie-Weiss Law characteristic of a pure paramagnet is typically only obeyed Curiewhen T 3TC in a ferromagnet. Deviations also occur in AFM systems above TN. ferromagnet. Other Classes of Magnetism Spin Glass A random orientation of frozen spin orientations (in a paramagnet the spin orientations are fluctuating.) Can occur when when the concentrations of magnetic ions are dilute or the magnetic exchange interactions are frustrated. Cluster Glass The spin orientations lock in with magnetic order in small clusters, but no order between the clusters (similar to a spin glass). Metamagnet There is a field-induced magnetic transition from a fieldstate of low magnetization to one of relatively high magnetization. magnetization. Typically the external field causes a transition from an antiferromagnetic state to a different type (such as a ferromagnet). ferromagnet). Superparamagnet A ferromagnet whose particle size is too small to sustain the multidomain structure. Thus the particle behaves as one large paramagnetic ion. 6 Magnetic Ordering in Rock Salt Oxides In the rock salt structure the primary mechanism for magnetic exchange is exchange the linear M-O-M superexchange interaction. In all of the compounds Mbelow the eg orbitals are filled, so the exchange interaction is AFM and overall magnetic structure is AFM as shown below. M-O Distance MnO FeO CoO NiO d5 d6 d7 d8 2.22 2.15 2.13 2.09 TN (K) 120 198 291 530 Moment (B) ~5 3.3 3.5 1.8 Spin down Ni Oxygen Spin up Ni AFM SE (M-O distance ) (Covalency ) (Superexchange ) (TN ) (MThe moments given in this table (and the ones that follow) were measured at low temperature using neutron diffraction. Under such circumstances the circumstances moment should roughly be equal to the number of unpaired electrons. electrons. AFM Magnetic Ordering in Perovskites The perovskite structure has even simpler magnetic interactions (in rock salt there is a competing interaction across the shared octahedral edge). octahedral Some magnetic data and the most common AFM structure for perovskite is perovskite shown below. The coloring of the magnetic structure is analogous to the analogous crystal structure of NaCl. It is called the G-type magnetic structure. NaCl. GM-X Distance LaCrO3 d3 1.97 1.90 1.99 2.00 TN (K) 282 110 750 275 Moment (B) 2.8 2.6 4.6 2.2 Oxygen AFM SE Spin down Fe Spin up Fe CaMnO3 d3 LaFeO3 KNiF3 d5 d8 Note that the superexchange that involves filled eg orbitals (LaFeO3) is much stronger than the corresponding interaction of filled t2g orbitals. Also orbitals. upon going from oxide (LaFeO3) to fluoride (KNiF3) the covalency decreases, which weakens the superexchange and lowers TN. 7 Magnetic Ordering in LaMnO3 La Mn O An interesting example that shows where the superexchange rules do not always lead to an AFM structure, with ferromagnetic nearest neighbor interactions is the magnetic structure of LaMnO3. Which contains HS Mn3+, a d4 ion with one electron in the eg orbitals. orbitals. This structure is called the A-type AAFM structure. 2.18 A 1.90 A Overlap of filled and empty eg orbitals gives FM coupling and stabilizes FM layers. filled dz2 type orbitals Double Exchange in Fe3O4 Fe2+ t2g4 eg2 eg t2g eg t2g Fe3+ t2g3 eg2 eg t2g eg t2g Oct. Sites Ferromagnetic Metallic Double Exchange eg eg t2g eg t2g Fe3O4 is an inverse spinel, with on the spinel, tetrahedral sites and a 1:1 mixture of Fe2+/Fe3+ on the octahedral sites. It is ferrimagnetic, with the octahedral sites ferrimagnetic, and the tetrahedral sites aligned in different directions. The ferromagnetic alignment of the octahedral sites is necessary for delocalized carrier transport of the minority spin t2g electron. This mechanism is called double exchange. Fe3+ t2g eg t2g Oct. Sites Antiferromagnetic Localized electrons/insulating 8 Magnetic Ordering in Solids Diamagnetism: No unpaired eDiamagnetism: Paramagnetism: Unpaired e-, disordered and fluctuating Paramagnetism: Ferromagnetism: All unpaired e- spins aligned parallel Ferromagnetism: Antiferromagnetism: Unpaired e- aligned antiparallel Antiferromagnetism: Ferrimagnetism: Unpaired e- aligned antiparallel but don't fully Ferrimagnetism: cancel out Magnetic Superexchange Unpaired electron spins couple through covalent interactions with with intervening ligand filled metal orbital filled metal orbital, AFM SE filled metal orbital empty metal orbital, FM SE Strength of superexchange interaction increases as covalency increases Double Exchange Spins on neighboring atoms must be aligned in a certain manner (usually ferromagnetically) in order to allow carrier delocalization ferromagnetically) Magnetism and electrical transport are intimately linked Summary 9 ...
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