Magnetic effects and the peculiarity of the electron spin in Atoms
Pieter Zeeman Hendrik Lorentz
Nobel Prize 1902
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
Otto Stern
Nobel 1943
Wolfgang Pauli
Nobel 1945
The orbital angular momentum of an electron in orbit
Semiclassical approach
r
μ
Hence
e r
μ =−
L
2me
r
Magnetic interaction:
-e
+Ze
2πν
r
L
Angular velocity
ω = 2πν
Magnetic moment
μ = IA = πr 2eν
Angular momentum
L = mvr = mωr 2 = 2πmvr 2
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
r r
VM = − μ ⋅ B = − μ z Bz
From classical electrodynamics:
-A magnetic dipole moment μ undergoes
no force in a magnetic field B
-A magnetic dipole undergoes a torque,
is oriented in a B field
-A magnetic dipole μ undergoes a force
in an inhomogeneous B-field, i.e.
a field gradient (Stern-Gerlach experiment)
The energy of a magnetic dipole in an external B-field
r
r
r
e
L
L = −μ B
μ =−
2me
h
Dipole
r r
VM = − μ ⋅ B = − μ z Bz
r
r
L
for the magnetic dipole moment: μ = − gμ B
h
eh
with: μ B =
2me
the “Bohr magneton”, the atomic
unit for a magnetic dipole moment
~ 5 x 10-9 eV/c ~ 9 x 10-24 Am2
This may be generalized for any
orbiting charge Q of mass M
Q r
μ=g
L
2M
r
Where the g-factor
for a classical motion
Magnetic interaction:
Expectation value:
VM = − μ z B =
Because:
gμ B B
Lz = gμ B Bm
h
Ψnlm Lz Ψnlm = mh
for any atomic quantum state
g =1
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
So a B-field breaks the degeneracy for m
Normal Zeeman effect
Energy splitting for an nl state
For an 1S Æ 1P-transition
m=l
m = l −1
nl
Ψnlm
m = −l
Energy splitting:
δEm = gμ B M
σ− σz σ+
Recall selection rules for m
and polarization of light
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
Normal Zeeman effect
Energy splitting for an nl state
For an 1P Æ 1D-transition
m=2
m =1
m=0
m = −1
m = −2
l=2
m=l
m = l −1
nl
Ψnlm
m = −l
Energy splitting:
δEm = gμ B
m =1
m=0
m = −1
l =1
Δm=-1
Δm=1
Δm=0
9 transitions
3 spectral lines (in hydrogen)
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
Wave functions in a magnetic field
Schrödinger equation;
⎡ h2 2
⎤
∇ + V ' (r )⎥ Ψnlm = En Ψnlm
⎢−
2
μ
⎣⎢
⎦⎥
V ' (r ) = VCoulomb (r ) + VZeeman
= VC (r ) +
Note:
gμB
gμB h ∂
Lz = VC (r ) +
h
h i ∂φ
r
Ψnlm (r , t ) = Rnl (r )Υlm (θ , φ )e −iEn t / h
Are simultaneous eigenfunctions of H0 and Lz
⎡ h2 2
gμ B B ∂ ⎤
∇ + VC (r ) +
⎢−
⎥ Ψnlm = (Enl + gμ B Bm )Ψnlm
∂
μ
φ
2
i
⎢⎣
⎥⎦
Eigenvalues of the Zeeman Hamiltonian:
E = Enl + gμ B Bm
Selection rules remain the same
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
Electron spin
No classical analogue for this phenomenon
s=
1
h
2
Origin of the spin-concept
-Stern-Gerlach experiment;
space quantization
Pauli:
There is an additional “two-valuedness”
in the spectra of atoms, behaving like
an angular momentum
Goudsmit and Uhlenbeck
This may be interpreted/represented
as an angular momentum
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
-Theory: the periodic system requires
an additional two-valuedness
Electron spin
In analogy with the orbital angular momentum
of the electron
r
r
L
μL = −gLμB
h
1
s= h
2
Spin is an angular momentum, so it
should satisfy
S 2 s, ms = h 2 s(s + 1) s, ms
S z s, ms = hms s, ms
1
1
s = , ms = ±
2
2
gL = 1
A spin (intrinsic) angular momentum can be
r
defined:
r
μS = − g S μB
S
h
a) in relativistic Dirac theory
gS = 2
b) in quantum electrodynamics
g S = 2.00232...
Note: the spin of the electron cannot be explained
from a classically “spinning” electronic charge
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
Electron spin and the atom
Additional quantum number
nlm → nlsml ms
Wave functions
r
Ψnlm (r , t ) = Rnl (r )Υlm (θ , φ )e −iEnt / h ↑, ↓
No effect on energy levels,
except for magnetic coupling to
other angular momenta
Degeneracy
2(2l + 1)
2n 2
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
Selection rules
s is not a spatial operator,
so no effect on dipole selection rules
Spin-orbit interaction
Frame of nucleus:
r
v
Frame of electron:
-e
+Ze
+Ze
-e
r
−v
The moving charged nucleus induces
a magnetic field at the location of the
electron, via Biot-Savart’s law
r μ0 Ze(− vr )× rr
B=
4π
r3
1
r
r r
=
μ
ε
0 0
Use L = mr × v ;
c2
r
r
Ze
L
Bint =
Then
4πε 0 me c 2 r 3
Spin of electron is a magnet with dipole
r
μS = − ge
μB r
h
The dipole orients in the B-field with energy
2
r r
r r
VLS = − μ S ⋅ B =
Ze
4πε 0 me2c 2 r 3
S ⋅L
A fully relativistic derivation
r r
(Thomas Precession) yields VLS = ζ (r )S ⋅ L
with
2
ζ (r ) =
Ze
1
8πε 0 me2c 2 r 3 nl
Use:
1
r
3
=
2
a n l(l + 1 / 2)(l + 1)
3 3
2
⎛ Zαmc ⎞
⎜
⎟ 3
⎝ hn ⎠ n l(l + 1 / 2)(l + 1)
3
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
S
=
Fine structure in spectra due to Spin-orbit interaction
In first order correction to energy
for state lsjm j
Then the full interaction energy is:
⎧⎪ j ( j + 1) − l(l + 1) − s(s + 1)⎫⎪
ESO = α Z hcR⎨
⎬
⎪⎩ 2n3l(l + 1 / 2)(l + 1) ⎪⎭
2 4
ESO = lsjm j VSL lsjm j
r r
= lsjm j ζ nl L ⋅ S lsjm j
Evaluate the dot-product
r r2
r r
2
2
J = L + S = L + S + 2L ⋅ S
2
S-states
l = 0, j = s
P-states
l = 1, j = l ± 1 / 2
ESO = 0
Then
(
)
r r
1
L ⋅ S lsjm j = J 2 − L2 − S 2 lsjm j
2
1
= h 2 { j ( j + 1) − l(l + 1) − s(s + 1)} lsjm j
2
ESO =
α 2 Z 4 hcR
2n3
Show that the “centre-of-gravity”
does not shift
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
Relativistic effects in atomic hydrogen
Relativistic kinetic energy
rel
Ekin
=
mc
2
2 2
2 4
2
p c + m c − mc =
2
2 2
2
1 + p / m c − mc =
⎡
⎤
p2
p4
mc ⎢1 +
−
+ L⎥
2 2
4 4
8m c
⎢⎣ 2m c
⎥⎦
K rel = Ψnljm −
−
Z 4α 2
2n 3
K rel = −
Ψnljm =
1
3⎞
− ⎟
⎝ 2l + 1 8n ⎠
Of the same order as Spin-orbit coupling
Relativistic energy levels:
p
8me3c 2
To be used in perturbation analysis:
r
hr
p=− ∇
i
8me3c3
(hc )R⎛⎜
2
First relativistic correction term
4
p4
operator does not
change wave function
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
Enj = En −
Z 4α 2
2n 3
2
3 ⎞
− ⎟⎟
⎝ 2 j + 1 4n ⎠
(hc )R⎛⎜⎜
So j=1/2 levels would be degenerate
Relativistic energy diagram for atomic hydrogen
For each configuration (electrons)
Several terms with term symbols
2S+1L
P.A.M. Dirac
Nobel 1933
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
J
Shortcomings of quantum-mechanical model of hydrogen
1.
2.
Spontaneous emission from
stationary states
g-factor of electron ≠ 2
Quantum Electro Dynamics (QED)
vacuum polarisation
e+
self-energy
e-
e-
3.
Lamb shift
ep
n
(n = 2)2 S1 / 2
Δ = 1060 MHz
Nobel
1965
(n = 2)2 P1 / 2
Sin-Itiro
Tomonaga
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
Julian
Schwinger
Richard
Feynman
Hyperfine structure in atomic hydrogen
Nucleus has a spin as well, and therefore
a magnetic moment
r
I
;
μI = gI μN
h
r
μN =
eh
2M p
Interaction with electron spin, that may have density
at the site of the nucleus (Fermi contact term)
(
r r r r 1 2
I ⋅S = I ⋅J = F − J2 − I2
2
)
Splitting : F=1 ↔ F=0 1.42 GHz
F=1
or λ = 21 cm
F=0
Magnetic dipole transition
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
Hyperfine structure in atomic hydrogen
1420.405 751 768 MHz
Search in space for H-atoms
via 21 cm radiation
Map of the Milky Way in H
Westerbork Telescope Array
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
Anomalous Zeeman effect
Occurs if S and L play a role
r
r
r
μ = μL + μS = −
=−
μB r
h
r
( L + 2S )
μB
h
r
r
(gL L + gS S )
Approximation: Paschen-Back effect
for VZeeman>>VLS
Ψnlml ms
proper wave functions
μ B
VM = B Lz + 2 S z = μ B B(ml + 2ms )
h
But in general:
Ψnljm j
proper wave functions
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
At weaker fields:
r
μ=−
μB r
h
r
( L + 2S ) and
r
r r
J = (L + S )
So J and μ non-colinear
Work with projections onto an axis B
r r r r
r
⎛
r
μ ⋅ J ⎞⎛ J ⋅ B ⎞
⎜
⎟⎜
⎟
VM = − μ ⋅ B = ⎜ −
⎟
⎜
J ⎠⎝ J ⎟⎠
⎝
r r
r
e L + 2S ⋅ J J z B
=
J
J
2m
r r
r
r r
2
2
L + 2 S ⋅ J = L + 2 S + 3L ⋅ S
With
r r 3 2 2
3L ⋅ S = J − L − S 2
2
(
)
(
)
(
)
Anomalous Zeeman effect; Lande factor
Result
(
⎛ 2
2 3 2
2
2
⎜ L + 2S + J − L − S
2
VZ = μ B B⎜
2
⎜
J
⎜
⎝
)⎞⎟
⎟ JZ
⎟
⎟
⎠
With the Lande g-factor dependent on J, S, L.
⎛ (J ( J + 1) − L( L + 1) + S ( S + 1) ) ⎞
⎟⎟
g L = ⎜⎜1 +
2 J ( J + 1)
⎝
⎠
Zeeman effect
VZ = μ B Bg Landem j
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs