Chapter Eleven Diamagnetism and Paramagnetism The story of magnetism begins with a mineral called magnetite “Fe3O4”. The first truly scientific study of magnetism was made by William Gilbert. His book “On the magnet” was published in 1600. In 1820 Hans Christian Oersted discovered that an electric current produces a magnetic field. In 1825 the first electromagnet was made by Sturgeon. Each stores the same magnetic energy ~0.4J. 1 paramagnetism ferromagnetism ) ) 3 3 diamagnetism M (emu/cm M M (emu/cm M H (Oe) H (Oe) Magnetic susceptibility per unit volume M χ ≡ M : magnetization, magnetic moment per unit volume H B : macroscopic magnetic field intensity CGS In general, paramagnetic material has a positive susceptibility (χ >0). diamagnetic material has a negative susceptibility (χ<0). 2 P. Curie 1859~1906 1895 Curie proposed the Curie law 1903 1905 Langevin proposed the theory of diamagnetism and paramagnetism 1906 Wiess proposed the theory of ferromagnetism 1920’s The physics of magnetism was developed with theories involving electron spins and exchange interactions – the beginnings of QM Three principal sources for the magnetic moment of a free atom : Electrons’ spin Electrons’ orbital angular momentum about the nucleus Change in the orbital moment induced by an applied magnetic field First two sources give paramagnetic contributions. The last source gives a diamagnetic contribution. 3 Reviews, in CGS B = H + 4π M = µ H N M = m m : magnetic dipole moment V [emu/cm3] e e m = − ∑ ri × pi = − L = −µBL Angular 2m i 2m momentum m = µBgo ∑ si = µBgoS Spin i e Bohr magneton µ = = 9.27×10-24 J/T; 1T =1Vs/m2 B 2m Electronic g-factor go=2.0023 µ electron I v L and S are quantum operators with no basis in classical physics. 4 Theory of diamagnetism was first worked out by Paul Langevin in 1905. Negative magnetism present in an applied field even though the material is composed of atoms that have no net magnetic moment. Motion of the electron is equivalent to current in the loop. Faraday’s law dΦ Emf = − B Paul Langevin [volt] dt 1872~1946 Consider a circular loop with radius r, the effect of applied field H is to change µ by an amount ∆µ in negative direction. 2 NA ρ e Z 2 I r χ = − R A 2 µ 6mc [emu/Oe-cm3] 5 H The change in moment of a single orbit : Emf A dH π r 2 dH r dH E = = − = − = − dt 2π r dt 2 dt dv eE er dH er a = = = − dv = − dH dt mc 2mc dt ∫ ∫ 2mc er ∆v = − H 2mc er e2r 2 ∆µ = ∆v = − H the plane of the orbit is perpendicular 2c 4mc 2 to the applied field All possible orientations in the hemisphere θ sin 2 θ dA R r 2 = R2 sin 2 θ = R2 ∫ A R2 π / 2 2 = sin 2 θ (2πRsin θ )(Rdθ ) = R2 2 ∫ 2πR 0 3 6 A single electron with consideration of all possible oriented orbits, eR e2 R2 ∆µ = ∆v = − H 2c 6mc2 For an atom that contains Z electrons, z e2 R2 e2H z ∆µ = − i = − 2 R is the radius of the ith orbit ∑ 2 H 2 ∑ Ri i i=1 6mc 6mc i=1 Volume density of mass The number of atoms per unit volume : N ρ/A A Atomic mass Avogadro’s number On the volume basis, 2 2 z NA ρ e Z 2 NA ρ e 2 χ = − ∆µ = − R H Susceptibility, 2 R 2 ∑ i A 6mc A 6mc i=1 Classical Langevin result Carbon (radius =0.7Å) 23 3 −10 2 6.02×10 (2.22g/cm ) (4.80×10 esu) 6 −8 2 χ = − 2 (0.7×10 cm) 12.01g 6(9.11×10−28g)(3×1010cm/s ) -6 3 7 = −1.5×10 emu/Oe − cm Experimental value : -1.1×10-6 emu/Oe-cm3 The agreement between calculated and experimental values for other diamagnetic materials is generally not that good, but it is at least within an order of magnitude. This classical theory of Langevin is a good example of the use of a simple atomic model to quantitatively explain the bulk properties of a material. Negative susceptibility magnitude : 10-5 ~10-6 cm3/mole The other consistency is that χ is independent of temperature for most diamagnetic materials. Quantum theory of diamagnetism results in the same expression. 8 Diamagnetic materials Pierre Weiss Closed-shell electronic structures result in no net moment The monatomic rare gases He, Ne, Ar, … Polyatomic gases, H2, N2, … Some ionic solids, NaCl, … Some covalent solids, C, Si, Ge, … Superconductors : Meissner effect 1865~1940 The first systematic measurements of susceptibility of a large number of substances over an extended range of temperature were made by Pierre Curie and reported by him in 1895. -- mass susceptibility is independent of temperature for diamagnetics, but that it vary inversely with temperature for paramagnetics. C C' χ = χ = in general T T −θ Curie law Curie-Weiss law (1907) 9 Quantum theory of diamagnetism 2 1 Q Hamiltonian H = p − A + QV 2m c Terms due to the applied field B = ∇× A ie e2 H '= (∇ • A + A•∇)+ A2 2mc 2mc 2 In the uniform magnetic field B = Bkˆ ∂ ∂ ∂ ˆ ˆ ˆ ˆ ˆ ˆ ∇× A = i + j+ k× (Ax i + Ay j+ Azk) ∂x ∂y ∂z ∂ ∂ ∂ ∂ ∂A ∂ Az Ax ˆ Ax Az ˆ y Ax ˆ = − i + − j+ − k ∂y ∂z ∂z ∂x ∂x ∂y yB xB One solution A = − ˆi + ˆj+ 0kˆ 2 2 ieB ∂ ∂ e2B2 = − + 2 + 2 H ' x y 2 (x y ) 4mc ∂y ∂x 8mc ~ 10 Lz z-component of angular momentum ~ ∂ ∂ paramagnetism Lz = −i x − y ∂y ∂x For diamagnetic materials with all electrons in filled core shells, the sum ∑ L z + 2S z over all electrons vanishes 2 For a spherically symmetric system x 2 + y2 = r 2 3 2 2 e2B2 e r the additional energy E'= x 2 + y2 = B2 8mc 2 12mc 2 2 2 ∂E' e r the associated magnetic moment µ = − = − B ∂B 6mc 2 in agreement with the classical Langevin result 11 Vibrating Sample Magnetometer (VSM) →→ measuring sample’s magnetic moment oscillate sample up and down measure induced emf in coils A and B compare with emf in coils C and D from known magnetic moment12 13 Classical theory of paramagnetism In 1905, Langevin also tried to explain paramagnetism qualitatively. He assumed that paramagnetic materials have molecules or atoms with the same non-zero net magnetic moment µ. In the absence of magnetic field, these atomic moments point at random and cancel one another. M=0 When a field is applied, each atomic moment tends to align with the field direction; if no opposing force acts, complete alignment would be produced and the material demonstrates a very large moment. But thermal agitation of atoms tends keep the atomic moments pointed at random. Partial alignment and small positive χ. As temperature increases, the randomizing effect of thermal agitation increases and hence, χ decreases. This theory leads naturally into the theory of ferromagnetism. 14 A unit volume material contains n atoms with a magnetic moment µ each. H E = −µ • H = −µHcosθ Let the direction of each moment be presented by a vector, θ and let all vectors be drawn thru. the center of the sphere. µ E µH cosθ − kBT kBT dn = KdA e = K 2π sinθdθ e # at angle θ θ+dθ µH cosθ π k T n = ∫dn = K 2π ∫dθ sinθ e B total # 0 µH cosθ n π M = ∫dnµ cosθ = K 2πµ∫dθ cosθ sinθ e kBT 0 0 π nµ ∫ dθ cosθ sinθ ea cosθ 0 Let a = µH/kBT, M = π ∫ dθ sinθ ea cosθ 0 −1 ax nµ ∫ dx xe a −a e + e 1 = 1 = µ − 1 −1 n a −a = nµcoth(a)− ax e − e a 15 ∫ dx e a 1 Langevin function L(a)=coth(a)-1/a L(a) can be expressed in series, a a3 2a5 L(a) = − + −... o 3 45 945 M/M When a is small (<0.5), L(a) is a straight line w/. a slope 1/3. When a is large, L(a) → 1. a=µH/kBT Two conclusions : (1) Saturation will occurs at large a implying that at large H and low T the alignment tendency of field is going to overcome the randomizing effect of thermal agitation. (2) At small a, L(a) depends linearly on a implying that M varies linearly with H. 16 The Langevin theory also leads to the Curie law. nµa nµ µH For small a, M = = 3 3 k BT 2 2 nµ M nµ C C = χ = = = 3k B H 3k BT T Oxygen is paramagnetic and obeys the Curie law w/. χ=1.08×10-4emu/gOe. 3k T χ 3(32g/mole )(1.38×10−16erg/K )(1.08×10−4 emu/gOe ) µ = B = n 6.02×1023 = 2.64×10-20erg/Oe − mole = 2.85µB Even in heavy atoms or molecules containing many electrons, each with orbital and spin moments, most of the moments cancel out and leave a magnetic moment of only a few Bohr magnetons. µH (2.64×10−20erg/Oe )(10000Oe) = = = Typically, a −16 0.0064 <0.5 k BT (1.38×10 erg/K )(300K) smaller enough L(a)=a/3 17 The Langevin theory of paramagnetism which leads to the Curie law, is based on the assumption that individual carrier of magnetic moment do not interact with one another, but are acted on only by applied field and thermal agitation.