Luigi Paolasini [email protected] Lecture 1 14 JANUARY 2014: “Why spins spin?” Lecture 2 28 JANUARY 2014: “Lonely atoms” Lecture 3 11 FEBRUARY 2014: “Local perturbations” Lecture 4 25 FEBRUARY 2014: “Magnetic interactions” Lecture 5 11 MARCH 2014: “Magnetic structures” Lecture 6 25 MARCH 2014: “Magnetic excitations” Lecture 7 8 APRIL 2014: “The dark side of magnetism: magnetic metals” Lecture 8 22 APRIL 2014: “Neutron magnetic scattering”
Lecture 9 6 MAY 2014: “X-ray magnetic scattering” Lecture 10 27 MAY 2014: “Other x-ray techniques for magnetism
L. Paolasini - LECTURES ON MAGNETISM- LECT.1 LECTURE 1: “WHY SPINS SPIN?”
Historical view. Classical and quantum mechanics. Self-rotating electron model. Spin algebra and coupling of two spins.
Reference books:
- Stephen Blundell: “Magnetism in Condensed Matter”, Oxford Master series in Condensed Matter Physics.
- Sin-itiro Tomonaga: “The story of spin”, University of Chicago press.
L. Paolasini - LECTURES ON MAGNETISM- LECT.1 Magnetism from the beginning
L. Paolasini - LECTURES ON MAGNETISM- LECT.1 The magnetic moment
The fundamental object in Magnetism is the elementary magnetic moment dμ (or magnetic dipole), which can be defined in classical electromagnetism as:
“ an electric current I circulating in a vanishingly small and oriented loop dS”
dS
The magnetic moment μassociated to a finite loop size is obtained by integrating the infinitesimal current loops.
NOTICE: the currents in the infinitesimal adiacent loops cancel each other, leaving only the periferical current I.
L. Paolasini - LECTURES ON MAGNETISM- LECT.1 The angular momentum
The magnetic moments are generated by rotating charged particles (such as the atomic electrons). The angular momentum L is due to the rotation of these massive particles, and is always connected with the magnetic moment: !
EINSTEIN-de HAAS-Effect (1915) Ampère's conjecture in 1820: ”Magnetism is caused by circulation of electric charges” A suspended ferromagnetic rod rotates when a magnetic field is applied. This is due to the conservation of angular momentum and the rotation is opposite to the magnetization direction.
L. Paolasini - LECTURES ON MAGNETISM- LECT.1 Definition of Bohr Magneton μB Hydrogen atom: - electron charge -e
- mass me - circular orbit at distance r
Electron current: I = −e/ τ Orbital period: τ = 2π r/v Angular momentum of electron: L = mevr = !
(= )
Angular momentum is oppositely direct with respect to the magnetic dipole moment because the negative electron charge.
Because µ=γ L =γ ! => γ = -e/2me and ωL= eB/2me
L. Paolasini - LECTURES ON MAGNETISM- LECT.1 Magnetic moment precession
The energy of a magnetic dipole in a static magnetic field is
A torque act perpendicularly to µ and B
and is equal to the rate of change of L, G = dL/dt
Equation of motion:
The moment µ precess around the magnetic field with constant modulus |µ| at Larmor’s precession frequency
… like a spinning top or a gyroscope!
L. Paolasini - LECTURES ON MAGNETISM- LECT.1 Magnetization and magnetic field
“Macroscopic” magnetization M= µ/v: magnetic moment per unit volume
Magnetic vector fields B or H (B is called also magnetic induction)
B is measured in Tesla, H in A/m
In the free space (vacuum): M=0
B= µ0 H -7 -1 where µ0= 4π 10 Hm is the vacuum permeability
In the solids: M≠0
B= µ0 (H + M) B could different from H in direction and magnitude.
In the linear materials: M= χ H where χ is the magnetic susceptibility
B= µ0 (1+χ)H = µ0µr H
where µr= (1+χ) is the relative permeability
L. Paolasini - LECTURES ON MAGNETISM- LECT.1 Magnetization of a system of electrons
Magnetization of a system of electrons: magnetic moment per unit volume that is induced by a magnetic field B is proportional to the rate of change of energy of the system.
Bohr-van Leeuwen theorem: “In a classical system there is no thermal equilibrium magnetization”
Lorentz force F=-e v x B: The magnetic field produces forces perpendicular to the particle velocity. No work is done!
As a consequence, the energy of the system does not depends on the applied magnetic field B
The magnetization must be zero in a classical system!
The classical mechanics cannot explain the existence of magnetic materials!!
. L. Paolasini - LECTURES ON MAGNETISM- LECT.1 Quantum mechanics
and quantum numbers
L. Paolasini - LECTURES ON MAGNETISM- LECT.1 Hydrogen spectrum
1913 Niels Bohr: “Theory for the spectrum of hydrogen atom, based on quantum theory that energy is transferred only in certain well defined quantities” n = principal quantum number (integral positive) size of electron orbit k = subordinate quantum number (integral positive) N. Bohr shape of the orbit (S, P, D … for k=1,2,3 …) denotes also the orbital angular momentum (unit ħ) m = magnetic quantum number (integral) components of the angular momentum vector k along the magnetic field (Zeeman split) Rules: “space quantization of angular momentum” n ≥ k m+1 ! k+1 ! k m # m -k ≤ m ≤k (2k+1 values) " k-1 " m-1
PS: k and m introduced by Sommerfield in 1916
L. Paolasini - LECTURES ON MAGNETISM- LECT.1 Inner quantum numbers
“Find the origin of the spectral multiplicity in the Zeeman effect”
A. Sommerfeld A. Landé W. Pauli To classify the variety of the levels in a multiplet term Sommerfeld introduced in 1920 the inner quantum number j. Now m specify the sublevels which are split for a level defined by n,k,j by a magnetic field, and -j ≤ m ≤j j+1 m+1 ! k+1 ! ! k j # j m # m " k-1 " j-1 " m-1
Because experimentally transitions occur only between the terms with the same multiplicity of differing by 2, Landé introduced a supplementary quantum number R: R=1/2 singlets R=multiplicity/2 R=1 doublets R=3/2 triplets
L. Paolasini - LECTURES ON MAGNETISM- LECT.1 Spectral terms of alkali atoms
Na Sodium have two closely spaced levels
Doublets: multiplicity=2
E (cm-1)
Inner quantum numbers for alkali doublet terms
L. Paolasini - LECTURES ON MAGNETISM- LECT.1 Spectral terms of alkaline earths
Singlets: multiplicity=1
Mg
Triplets: multiplicity=3
L. Paolasini - LECTURES ON MAGNETISM- LECT.1 Lande, Sommerfeld and Pauli Ersatzmodell
“A radiant electron moves in an electric field created by the core electrons and the atomic nucleus. In general, the core electrons are not spherical symmetric but have an angular momentum, which originates an internal Zeeman splitting. ” Landé Ersatzmodell:
The core is have an orbital momentum R and then a magnetic moment µR=-g0R. The total angular momentum of the system is J=K+R, where K=-µK is the angular momentum of radiant electron associated to it : K=k-1/2 |R-K|+1/2 ≤ J≤ |R+K|-1/2 -J+1/2 ≤ m ≤ J-1/2 Sommerfeld Ersatzmodell:
The core angular momentum is j0=(multiplicity-1)/2 and the total angular momentum j=ja+j0, where ja=k-1: ja=k-1 (=l) |j0-ja|≤ j≤ |j0+ja| -j ≤ m ≤ j
Pauli Ersatzmodell: The core angular momentum is r=(multiplicity+1)/2 and the total angular momentum JP=r+k, where ja=k-1: k=k |k-r|+1≤JP≤ |k+r|-1 -JP+1 ≤ m ≤ JP-1
L. Paolasini - LECTURES ON MAGNETISM- LECT.1 Model limitations.
The Landé model succeeded in deriving the rule for the multiplet splitting, confirming a large amounth of experimental results … apart for very light atoms like He and Li!! Splitting ratio in triplet terms
. ΔWmag= cost. (J-1/2)
Another inconsistency was found in the calculation of the values of ΔWmag in alkali doublets using the core-electron magnetic forces.
The idea that the origin of the multiplicity is not due to the electron core but the electron itself became to appears in 1924. Pauli says: “The very fact that the two electrons in He have to play entirely different roles – one for electron core and the other the radiant electron- is the failure of the model”. “The doublet structure of alkalis spectra and the breakdown of Larmor theorem is caused by the strange two-valuedness of quantum-theoretical properties of the radiant electron which cannot be described classically.”
L. Paolasini - LECTURES ON MAGNETISM- LECT.1 Self rotating electron model
Proposed originally by R. Kronig in 1925 but never published! “An electron rotating about its own axis and with an angular momentum of self-rotation of ½ and a g-factor g0=2”
R. Kronig Uhnlenbeck and Goudsmith The interaction between the magnetic moments generated by the self-rotation and the orbital motion could be derived through relativity … … and then calculate the interval between the multiplet terms. Landé empirical expression Kronig model and relativistic calculation
Because g0=2, these demonstration fail completely by a factor 2 and Pauli strongly reject this theory! Problems arise also for the necessity to have a fast rotation of the electron to have an orbital moment of ½ (surface speed 10 time c!). … 6 months later Uhnlenbeck and Goudsmith publish in Naturwissenschaften this “wrong” theory.
L. Paolasini - LECTURES ON MAGNETISM- LECT.1 The Thomas ½ factor
1926: Thomas introduce a correction to the spin-orbit interaction which take into account the relativistic “time dilatation” between electron and nucleus of an atom.
The electron rest frame is not inertial, but is accelerated, and it rotates with respect to the laboratory system.
As a result, the Larmor precession of µe of the electron in the magnetic field generate by its relativistic motion is not the same Llewellyn H. Thomas as seem in the laboratory system.
When this correction is taken into account, the calculation of level intervals became in accord with experiment and given by:
… this expression is half of that calculated by the Kronig and Uhnlenbeck/Goudsmith
L. Paolasini - LECTURES ON MAGNETISM- LECT.1 1926: a special year for quantum theories
Heisenberg Pauli Schrödinger Matrix mechanics Spin theory Wave mechanics
? Dirac Transformation theory
L. Paolasini - LECTURES ON MAGNETISM- LECT.1 Pauli spin theory
Definition of the spin angular momentum S
Pauli decided to use the component sz= ±½ as running variable for the spin degree of freedom and not the canonical momentum conjugate φ associated to the azimuthal rotation. Probability density to find the Electron wavefunction electron at x with spin up
Probability density to find the electron at x with spin down
… we need to write the Hamiltonian H which involve the external H1 and the internal H2 magnetic fields
L. Paolasini - LECTURES ON MAGNETISM- LECT.1 Operators
Schrödinger operators: => momentum => position … and than the orbital angular momentum operators l
Which spin operator? Pauli take the similarities with the angular momentum
matrices (mx, my, mz) used in matrix mechanics: Commutation relations
Eigenvalues
L. Paolasini - LECTURES ON MAGNETISM- LECT.1 Pauli matrices
Pauli propose to use the matrices (sx, sy, sz) direcly in the Schrödinger equation:
Pauli matrices
Properties:
This correspond to considering the spin state vector as a column
vector described by the two component wavefunction ψ(x,sz):
=>
L. Paolasini - LECTURES ON MAGNETISM- LECT.1 Limitation of Pauli equation
1) The Pauli equation contains the non-relativistic Hamiltonian H0, and so is non- relativistic! As a result cannot be used to calculate the doublet terms and the anomalous Zeeman effect!
2) Pauli introduces arbitrarily the spin ½ and g0=2 in the H1, and also the Thomas factor (g0-1) in H2, and judge his theory a tentative.
3) The choice of Pauli algebra have an enormous implication in the coupling of two spin:
Theorem. “When there are two particles of spin ½, the wave function for which the sum of two spins equal 1 does not change its value when the spin variable of the electrons are exchanged (symmetric). The wave function for which the sum of the two spin becomes 0 change sign when the spin variables are interchanged (antisymmetric)”
L. Paolasini - LECTURES ON MAGNETISM- LECT.1 Two electron states I
Let we consider two electrons with spin matrices:
2 The square of the magnitude of the total spin |s1+s2| is:
and because:
=>
Let we assume that the wave function describing the two spin states is a
column vector (we omit the variables coordinate x1 and x2) and we
L. Paolasini - LECTURES ON MAGNETISM- LECT.1 Two electron states II
Then we obtain: … because:
and:
and
… because:
L. Paolasini - LECTURES ON MAGNETISM- LECT.1 Two electron states III
TOTAL SPIN 1
=>
=> Symmetric wave function
TOTAL SPIN 0
=>
=> Anti-Symmetric wave function
L. Paolasini - LECTURES ON MAGNETISM- LECT.1