1. Lecture: Basics of Magnetism: Magnetic reponse Hartmut Zabel Ruhr-University Bochum Germany
Lecture overview 1. Lecture: Basic magnetostatic properties 2. Lecture: Paramagnetism 3. Lecture: Local magnetic moments 2 H. Zabel, RUB 1. Lecture: Magnetic Response
Content 1. Definitions 2. Electron in an external field 3. Diamagnetism 4. Paramagnetism: classical treatment of 3 H. Zabel, RUB 1. Lecture: Magnetic Response
1. Classical magnetic moments Loop current has an angular Loop current generates a momentum magnetic field q ω q = = I π T 2 Magnetic dipole moment = current × enclosed area q ω 1 q 1 q γ πr ω 2 = = = = = m IA A m L L π π e 2 2 m 2 m e e γ = gyromagnetic ratio, m e = electron mass 4 H. Zabel, RUB 1. Lecture: Magnetic Response
Torque and precession Zeeman energy of magnetic moment in an external magnetic field: ⋅ E = - m B Energy is minimized for m || B. B is the magnetic induction or the magnetic field density. Applying B, a torque is exerted on m: = × T m B If m were just a dipole, such as the electric dipole, it would be turned into the field direction to minimize the energy. However, m is connected with an angular momentum, thus torque causes the dipole to precess: d L = = × γ T L B dt Assuming B = B z , the precessional frequency is: B z ω = γ B L z ω L is called the Lamor frequency . See also EPR, FMR, MRI, etc. 5 H. Zabel, RUB 1. Lecture: Magnetic Response
Bohr magneton An electron in the first Bohr orbit with a Bohr radius r Bohr has the angular momentum: ω 2 L = m e r = Bohr Then magnetic moment is: 1 q 1 e μ = L = = m - - Bohr B 2 m 2 m e e Because of negative charge, L and m are opposite. L 1 e = = γ μ µ B B 2 m e µ B is the Bohr magneton. [ µ B ] = 9.274 x 10 -24 Am 2 . [m] = A m 2 Magnetic moment: 6 H. Zabel, RUB 1. Lecture: Magnetic Response
Electron spin L Orb S Spin S of the electon contributes to the magnetic moment: q = m S spin m e The missing factor ½ is of quantum mechanical origin and will be discussed later. Including orbital and spin contributions, the magnetic moment of an electron is: 1 q = + = − + γ( m ( L 2 S ) L 2 S ) 2 m e 7 H. Zabel, RUB 1. Lecture: Magnetic Response
Magnetic field and magnetic induction I H = Oersted field H due to dc current: πr 2 Any time variation of the magnetic flux Φ = BA through the loop ( ) d = − ⋅ causes an induced voltage: U ind B A dt Therefore B is called the magnetic induction or the magnetic flux density B = Φ /A. In vacuum both quantities are connected via the permeability of the = μ B H vacuum: 0 ⋅ = ⋅ ⋅ ⋅ V s V s [ ] [ ] I V s A V s = = π = ⋅ = μ 4 10 A m B μ 4 -7 1 1 T 10 G × = = T ⋅ ⋅ 0 πr 0 2 2 2 A m m m m 8 H. Zabel, RUB 1. Lecture: Magnetic Response
Definitions 1. Magnetization is the sum over all 1 ∑ = M m magnetic moments in a volume element i V i normalized by the volume element: N = M m 2. Thermal average of the magnetization: V ∂ M 3. Magnetic susceptibility: = = χ χ M H , ∂ mag mag H 4. Magnetic Induction: . ( ) = + = + = = μ μ χ μ μ μ B H M H (1 ) H H 0 0 mag 0 r H = magnetic field, usually externally applied by a magnet. µ 0 = magnetic permeability of the vacuum. µ r = relative magnetic permeability µ r = (1+ χ ) (tensor, or a number for collinearity) 9 H. Zabel, RUB 1. Lecture: Magnetic Response
Potential Energy and Derivatives Potential energy (Zeeman – term): ⋅ = − E Zeeman m B 1. Derivative → magnetic moment: ∂ E = − Zeeman m ∂ B 2. Derivative → Susceptibility: ∂ ∂ 2 M N E χ = = − µ Zeeman ∂ ∂ mag 0 2 H V B The susceptibility is the response f 10 H. Zabel, RUB 1. Lecture: Magnetic Response
What is more fundamental, H or B? ( ) [ ] = × F = N F q v B Lorentz force: [ ] = • E = J = VAs = Ws E - m B Zeeman energy: [ ] 2 = × B = T = Vs m B A / Vector potential: I [ ] = H = A m H Oersted field: πr 2 χH [ ] M = A m M = Magnetization: 11 H. Zabel, RUB 1. Lecture: Magnetic Response
Classification Application of an external field: a. Paramagnetism: χ>0 und µ r >1 Magnetic moments align parallel to external field, field lines are more dense in the material than in vacuum. b. Diamagnetism: χ< 0 und µ r <1 External field is weakend by inducing screening currents according to Lenz rule. Field lines are less dense than in vacuum. Ideal diagmagnetism, realized in superconductors with M and B antiparallel, for χ = − 1 and µ r =0. c. Ferromagnetism: Spontaneous Magnetization without external field due to the interaction of magnetic moments µ r attaines very high values for ferromagnets, > 10 4 -10 5 12 H. Zabel, RUB 1. Lecture: Magnetic Response
2. Electron in an external field Consider a non-relativistic Hamilton operator for electrons in an external magnetic field: ( ) 2 + 1 = H p q A 2 m e = B × A The vector potential: is defined by the Coulomb gauge: A and using ( ) B = , ,B 0 0 z 2 2 p e = + µ + 2 2 * H B L B a B z z z 2 m 12 m e e paramagnet ism orbital kinetic diamagneti sm ~ B energy 2 z ~ B z *L z is here a dimensionless quantum number Where we assumed an average over the electron orbit perpendicular to the magnetic field: 2 a + = 2 2 2 x y 3 13 H. Zabel, RUB 1. Lecture: Magnetic Response
Hamiltonian for electron with spin Considering the electron spin in the external field with a Zeeman energy: e = ⋅ = µ ⋅ = ⋅ E - m B - g S B S B Zeeman s s B m = g 2 Landé factor s e µ = = × - 24 2 - 9 . 27 10 Am Bohr magneton B 2 m Hamilton operator for spin and orbital contributions of a single bond electron then is: 2 2 ( ) p e = + µ + + 2 2 H B L 2 S B a B z z z z 2 m 12 m e e paramagnet ism + spin orbital kinetic diamagneti sm ~ B energy 2 z ~ B z The g S =2 for the electron is put into the Schrödinger equation by „hand“ but would occur naturally using the Dirac equation. The exact value of 2.0023 is determined by QED. 14 H. Zabel, RUB 1. Lecture: Magnetic Response
Response functions 2. derivative 1. derivative ∂ 2 N H ∂ χ = µ H - = m - mag 0 ∂ 2 V B ∂ B z z 2 Ze 2 Diamagnetic response Ze − µ 2 a 2 a B 0 6 for Z electrons z m 6 m e e ( ) µ + > 0 * L 2 S 0 Paramagnetic response B z z *For single atom we can not define a paramagnetic susceptibility. This is only possible for an ensemble of atoms. 15 H. Zabel, RUB 1. Lecture: Magnetic Response
3. Properties of the Langevin diamagnetism With Z electrons in an atom and an effective radius of <a> 2 2 N e N Ze ∑ χ µ µ 2 2 = = a - r - Langevin 0 i 0 V m V m 6 6 i e e χ Langevin is constant, independent of field strength; χ Langevin is induced by external field; χ Langevin < 0, according to Lenz‘ rule; χ Langevin is alway present, but mostly covered by bigger and positive paramagnetic contribution; χ Langevin the only contribution to magnetism for empty or filled electron orbits; yiels 〈 a 〉 and the symmetry of the electron distribution; χ Langevin χ Langevin is proportional to the area of an atom perpendicular to the field direction, important for chemistry; χ Langevin is temperature independent. 16 H. Zabel, RUB 1. Lecture: Magnetic Response
Examples for Diamagnetism χ Langevin at RT Material -1.9 ⋅ 10 -6 cm 3 /mol He -43 ⋅ 10 -6 cm 3 /mol Xe -16 ⋅ 10 -6 cm 3 /g Bi -1.06 ⋅ 10 -6 cm 3 /g Cu -2.2 ⋅ 10 -6 cm 3 /g Ag -1.8 ⋅ 10 -6 cm 3 /g Au ( χ is normalized to the magnetization of 1 cm 3 containing one 1 Mol of gas at 1 Oe) • All noble metals and noble gases are diamagnetic. In case of the nobel metals Ag, Au, Cu mainly the d-electrons contribute to the diamagnetism. • In 3d transition metals the diamagnetismus is usually exceeded by the much bigger paramagnetic response. 17 H. Zabel, RUB 1. Lecture: Magnetic Response
Anisotropy of diamagnetismus for Li 3 N Levitation of diamagnetic materials 18 H. Zabel, RUB 1. Lecture: Magnetic Response
4. Paramagnetism of free local moments: classical treatment (free = without interactions) Orientation of permanent and isolated magentic moments in an external field B z = µ 0 H z parallel to the z-axis (orientational polarization) N H z = θ M m cos( ) V θ N m H = z z m L z V k T m B 2 N m H ≈ z z V 3 k T − Hohe T B 1 ( ) ( ) = − Langevin function L x coth x x 19 H. Zabel, RUB 1. Lecture: Magnetic Response
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