5 Magnetic materials I Magnetic materials I ● Diamagnetism ● Paramagnetism Diamagnetism - susceptibility All materials can be classified in terms of their magnetic behavior falling into one of several categories depending on their bulk magnetic susceptibility χ. without spin M⃗ 1 χ= in general the susceptibility is a position dependent tensor M⃗ (⃗r )= ⃗r ×J⃗ (⃗r ) H⃗ 2 ] In some materials the magnetization is m / not a linear function of field strength. In A [ such cases the differential susceptibility M is introduced: d M⃗ χ = d d H⃗ We usually talk about isothermal χ susceptibility: ∂ M⃗ χ = T ( ∂ ⃗ ) H T Theoreticians define magnetization as: ∂ F⃗ H[A/m] M=− F=U−TS - Helmholtz free energy [4] ( ∂ H⃗ ) T N dU =T dS− p dV +∑ μi dni i=1 N N dF =T dS − p dV +∑ μi dni−T dS−S dT =−S dT − p dV +∑ μi dni i=1 i=1 Diamagnetism - susceptibility It is customary to define susceptibility in relation to volume, mass or mole: M⃗ (M⃗ /ρ) m3 (M⃗ /mol) m3 χ= [dimensionless] , χρ= , χ = H⃗ H⃗ [ kg ] mol H⃗ [ mol ] 1emu=1×10−3 A⋅m2 The general classification of materials according to their magnetic properties μ<1 <0 diamagnetic* χ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ B=μ H =μr μ0 H B=μo (H +M ) μ>1 >0 paramagnetic** ⃗ ⃗ ⃗ χ μr μ0 H =μo( H + M ) → μ r=1+χ μ≫1 χ≫1 ferromagnetic*** *dia /daɪə mæ ɡˈ n ɛ t ɪ k/ -Greek: “from, through, across” - repelled by magnets. We have from L2: 1 2 the force is directed antiparallel to the gradient of B2 F = V ∇ B i.e. away from the magnetized bodies 20 - water is diamagnetic χ ≈ − 10 − 5 (see levitating frog - Ignoble) −5 −3 ** para- Greek: beside, near; for most materials χ ≈ 10 − 10 [1]. ***susceptibility ranges from several hundred for steels to 100,000 for soft magnetic materials (Permalloy) In electronics only ferromagnetic materials are relevant (superconductors, with χ=-1 are used in some devices too) Diamagnetism - susceptibility It is customary to define susceptibility in relation to volume, mass or mole: M⃗ (M⃗ / ρ ) m3 (M⃗ /mol) m3 χ = [dimensionless] , χ = , χ = H⃗ ρ H⃗ [ kg ] mol H⃗ [ mol ] The general classification of materials according to their magnetic properties μ<1 <0 diamagnetic* χ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ B=μ H =μr μ0 H B=μo( H +M ) μ>1 >0 paramagnetic** ⃗ ⃗ ⃗ χ μr μ0 H =μo( H + M ) → μ r=1+χ μ≫1 χ≫1 ferromagnetic*** image source: С. В. Бонсовский, Магнетизм, Издательство ,,Наука", Москва 1971 [5] room temperature atomic magnetic susceptibility Bohr–van Leeuwen Theorem Leeuwen Bohr–van 1921). again, and compensate terms diamagnetic and paramagnetic Thus walls. the hit not do that to electrons due current surface the exactlycancels and isopposite density it. on This current that bounce to electrons due theboundary along develops also acurrent inabox, isconfined electrons ofseveral If results. motion the clouds, larger and for larger vanishes to total number the their ratio since field,but diamagnetic this feed electrons to H. outer Only the orthogonal current asurface build but “ ● [1] see – words other in still or magnetization beno can there mechanics, In classical other words in or identically vanishes equilibrium of of collection a magnetization net the fields, ormagnetic electric applied finite all in and temperature, At anyfinite The orbits of a cloud of electrons in space give no net current density in bulk, bulk, in density netcurrent no give space in of electrons of cloud a orbits The independent of independent of to that opposite direction with moment magnetic ofsmall source is a field inmagnetic orbiting electron that single a Note ” [2] B in thermodynamic limit thermodynamic in [3] no magnetization survives magnetization no classical electrons classical no magnetization per unit volume volume unit per magnetization no (van Leeuwen’s theorem, theorem, (vanLeeuwen’s B is moment [2] this but in thermal thermal in image source: (WP:NFCC#4), Fair use, https://en.wikipedia.org/w/index.php?curid=42038503 image source: www.wikipedia.org Bohr–van Leeuwen Theorem ● Note that a single electron orbiting in magnetic field is a source of small magnetic moment with direction opposite to that of B [2] but this moment is independent of B [3] ● For the electron orbiting in circle under the influence of central force adiabatic switching of the field (slow increase of the field, taking much longer than orbital period) does induce the additional moment that is opposite to the B direction [3] There are two cases to consider: ● the influence of magnetic field on a free (not bound) electron ● the same for an electron under the action of central force from nucleus instantaneous electron velocity Bohr–van Leeuwen Theorem ● Note that a single electron orbiting in magnetic field is a source of small magnetic moment with direction opposite to that of B [2] but this moment is independent of B [3] ● For the electron orbiting in circle under the influence of central force adiabatic switching of the field (slow increase of the field, taking much longer than orbital period) does induce the additional moment that is opposite to the B direction [3] Faraday law gives emf induced in the “circular current loop”* period of one tour 2 dB emf =−π r dt 2 2 2π r Magnetic moment of a loop is μ=I A , I - loop current, A - loop area , μ=π r I=π r q/ ( v ) 1 1 μ= r q v and angular momentum is J =mv r →μ= (q/m) J 2 2 An induced emf force induces torque τ acting on the electron giving it an acceleration er 2 dB emf r dB τ=⃗r⋅F⃗ =E q r= e istead of q ← E cirle= =− cirle 2 dt 2 π r 2 dt and a torque is equal to time derivative of angular momentum l dJ er2 dB er2 “This is the extra angular momentum from the twist = → dJ= dB given to the electrons as the field is turned on” - dt 2 dt 2 Feynman [4] and the corresponding change of magnetic moment is (remember q=-e) is e2 r2 e2 r 2 d μ=− dB and because the field starts from zero → Δμ=− B 4 m 4m *please read the appropriate Feynman lecture – they are now free to read (www.feynmanlectures.caltech.edu/II_34.html) Digression – time independent non-degenerate perturbation method We have a system described by a Hamiltonian H(0) for which we know the solutions [6,7] H (0) ψ=E ψ → E(0) , E(0) , E(0) ,... ψ(0) , ψ(0) , ψ(0) ,... the eigenfunctions are orthonormal and a form 1 2 3 1 2 3 complete set The system is now perturbed so that the new Hamiltonian differs only slightly from H(0) (H (0)+H (1))ψ=E ψ → (H(0)+λ V^ )ψ=E ψ (1) We assume that the solutions of the perturbed equation can be expressed as a power series relative to λ (0) (1) 2 (2) Em=Em +λ Em +λ Em +... (0) (1) 2 (2) ψm=ψm +λ ψm +λ ψm +... Inserting the above solutions to (1) we get (0) ^ (0) (1) 2 (2) (0) (1) 2 (2) (H +λV −Em −λ Em −λ Em −...)(ψm +λ ψm +λ ψm +...)=0 Collecting terms according to the powers of λ we obtain [7] (0) (0) (0) (H −Em )ψm =0 unperturbed equation (0) (0) (1) (1) ^ (0) (H −Em )ψm =(Em −V )ψm (2) (0) (0) (2) (1) ^ (1) (2) (0) contains zeroth and first order terms (H −Em )ψm =(Em −V )ψm +Em ψm Digression – time independent non-degenerate perturbation method We want to find the first order correction to the energy [7] (H (0)−E(0))ψ(0)=0 (0)* m m To that end we multiply (2) with ψ to get (0) (0) (1) (1) ^ (0) m (H −E m ) ψm =(E m −V )ψm (2) (0)* (0) (0) (1) (0)* (1) (0) (0) (0) (2) (1) ^ (1) (2) (0) ^ (H −Em ) ψm =(Em −V ) ψm +E m ψm ψm (H −E m )ψm =ψm (E m −V )ψm Integrating the above expression we have (0)* (0) (0) (1) (0)* (1) (0) (0)* ^ (0) (0)* ^ (0) ∫ ( ψm (H −E m )ψm ) dV =∫ ψm E m ψm dV −∫ ψm V ψm dV V lm :=∫ ψl V ψm dV (0)* (0) (0) (1) (0)* (0) (1) (0)* (0) (1) (1) (0) (0) * (0)* (0) (1) ∫ ( ψm (H −E m ) ψm ) dV =∫ ψm H ψm dV −∫ ψm E m ψm dV =∫ ψm (H ψm ) dV −∫ ψm E m ψm dV = (1) (0) (0)* (0)* (0) (1) ∫ ψm E m ψm dV −∫ ψm E m ψm dV =0 becaues energy is a real number: Hermitian operator is defined (1) (E(0))*=E(0) 0=Em −V mm m m by the condition [8] ψ* x^ ψ dV = ψ x^ ψ * dV and finally ∫ 1 2 ∫ 2 ( 1 ) (1) (0)* ^ (0) (0) (1) Em =V mm=∫ ψm V ψm dV E m=E m +λ V mm E m=E0 +<0 m| H |0 m> The correction to the eigenvalues in the first order approximation is the equal to the average energy of the perturbation in the unperturbed state (0)* (1) (0) (1) (0) * (0 ) (1 ) (1) ∫ ψm Em ψm dV =Em ∫ ψm ψm dV = Em , because eigenfunctions are normalized to 1 and E m is just a number Digression – time independent non-degenerate perturbation method To find eigenfunctions in the first order of perturbation we express them as the superpositions of the functions of the unperturbed Hamiltonian [7] ψ(1)= ψ(0) c m ∑ l lm and insert them to Eq.