Diamagnetism and paramagnetism
• Langevin diamagnetism
• paramagnetism
• Hund’s rules
• Lande g-factor
atom
• Brillouin function
• crystal field splitting
• quench of orbital angular momentum
• nuclear demagnetization
• Pauli paramagnetism and
Landau diamagnetism
free electron
gas
Dept of Phys
M.C. Chang
B=(1+χ)H
Curie’s law χ=C/T
important
Basics
( E → F = E − TS if T ≠ 0)
• System energy
E(H)
• magnetization density
M (H ) = −
• susceptibility
∂M
1 ∂2E
=−
χ≡
∂H
V ∂H 2
1 ∂E
V ∂H
Atomic susceptibility
⎛ pi2
⎞
e2
e
2
+ Vi ⎟ + μ B L + gS ⋅ H +
=
,
μ
H = ∑⎜
A
∑ i B 2mc
2mc i
i ⎝ 2m
⎠
= H 0 + ΔH
(
)
Order of magnitude
•
μ B ( L + gS ) ⋅ H ≈ μ B H ≈ ωc
≈ 10−4 eV
•
Ai =
when H = 1 T
H
(− yi , xi , 0)
2
2
2
e2
⎛ eH ⎞
2
2
∑ Ai ≈ ⎜⎝ mc ⎟⎠ ma0 , a0 ≡ me2
2mc i
2
ωc )
(
≈
2
e / a0
≈ 10−5 of the linear term at H =1 T
important
• Perturbation energy (to 2nd order)
ΔEn = n ΔH n + ∑
n ΔH n '
n '≠ n
2
En − En '
2
e
= μ B n L + gS n ⋅ H +
n
2mc 2
∑A
2
i
i
n + ∑'
n'
(
)
n μ B L + gS ⋅ H n '
En − En '
• Filled atomic shell
(applies to noble gas, NaCl-like ions…etc)
Ground state |0〉:
L 0 =S 0 =0
e2
2
2
H
0
ri 2 0
∴ ΔE =
∑
2
8mc
3 i
(for spherical charge dist)
For a collection of N ions,
N ∂ 2 ΔE
e2 N
χ =−
=−
0
2
2
6mc V
V ∂H
∑r
i
2
0 <0
i
Larmor (or Langevin) diamagnetism
2
An atom with many electrons
• Without SO coupling
⎛ pi2
⎞ 1
H = ∑⎜
+ Vi ⎟ + ∑ Vij
i ⎝ 2m
⎠ 2 ij
• Maximally mutually commuting set
H , L2 , S 2 , Lz , S z
• single electron ground states
Degeneracy D=d
…
• N-electron ground states
• Eigenstates (including ground states)
α , L, S , mL , mS
• With SO coupling (weak)
⎛ pi2
⎞ 1
+ Vi ⎟ + ∑ Vij + ∑ λi Si ⋅ Li
H = ∑⎜
i ⎝ 2m
i
⎠ 2 ij
• Without e-e interaction
Degeneracy D=CdN
…
• With e-e interaction
• Maximally mutually commuting set
H , L2 , S 2 , J 2 , J z
…
• Eigenstates (including ground states)
α , L , S , J , mJ
many-electron
levels
Ground states
w/o SO: labeled by L,S D=(2L+1)(2S+1)
w/ SO: labeled by L,S,J D=(2J+1)
important
non-interacting
Ground state of an atom with unfilled shell (no H field yet!):
n, l , ml , ms
• Atomic quantum numbers
• Energy of an electron depends on
• Degeneracy of electron level
n, l ( no ml , ms )
ε n ,l : 2(2l+1)
• If an atom has N (non-interacting) valence electrons, then the
degeneracy of the “atomic” ground state (with unfilled ε n ,l shell) is
C N2(2l +1)
interacting
e-e interaction will lift this degeneracy partially, and then
• the atom energy is labeled by the conserved quantities L and S, each is
(2L+1)(2S+1)-fold degenerate
• SO coupling would split these states further, which are labeled by J (later).
What’s the values of S, L, and J for the
atomic ground state?
Use the Hund’s rules (1925),
1. Choose the max value of S that is consistent
with the exclusion principle
2. Choose the max value of L that is consistent
with the exclusion principle and the 1st rule
To reduce Coulomb repulsion,
electron spins like to be parallel,
electron orbital motion likes to be in
high ml state. Both help disperse the
charge distribution.
Example: 2 e’s in the p-shell (l1 = l2 =1, s1 =s2 =1/2)
(a) (1,1/2)
(b) (0,1/2)
(c) (-1,1/2)
(a’) (1,-1/2)
(b’) (0,-1/2)
(c’) (-1,-1/2)
C62 ways to put these 2 electrons in 6 slots
• Spectroscopic notation:
2 S +1
1
X J ( X = S , P, D...)
Energy levels of Carbon atom
S0 ,3 P0,1,2 ,1 D2 are o.k.; 3 S0 ,1 P1 ,3 D3 are not.
(It's complicated. See Eisberg and Resnick
App. K for more details)
S=1
ml = 1
0
-1
L=1
3
• Ground state is P0,1,2 ,
physics.nist.gov/PhysRefData/Handbook/Tables/carbontable5.htm
(2L+1)x(2S+1)=9-fold degenerate
• There is also the 3rd Hund’s rule related to SO coupling (details below)
important
v
Review of SO coupling
• An electron moving in a static
E field feels an effective B field
Beff = E ×
v
c
E
• This B field couples with the electron spin
H SO = − μ ⋅ Beff
v⎞
⎛ q ⎞ ⎛
= −⎜
S ⎟⋅⎜ E × ⎟,
c⎠
⎝ mc ⎠ ⎝
⎛ q dφ ⎞
=⎜ 2 2
⎟S ⋅L
⎝ m c rdr ⎠
≡ λS ⋅ L
=
λ
(J
2
2
− L2 − S 2 )
E = − rˆ
dφ
e
for central force, φ =+
dr
r
(x 1/2 for Thomas precession, 1927)
λ> 0 for less than half-filled (electron-like)
λ< 0 for more than half-filled (hole-like)
Quantum states are now labeled by L, S, J
(2L+1)x(2S+1) degeneracy is further lifted to become
(2J+1)-fold degeneracy
Hund’s 3rd rule:
• if less than half-filled, then J=|L-S| has the lowest energy
3
• if more than half-filled, then J=L+S has the lowest energy
previous example
P0 is the ground state in
Paramagnetism of an atom with unfilled shell
1) Ground state is nondegenerate (J=0)
ΔE = μ B 0 L + gS 0 ⋅ H +
2
e
0
2 mc 2
∑ Ai2 0 + ∑ '
i
(
)
0 μ B L + gS ⋅ H n
2
E0 − En
n
(A+M, Prob 31.4)
Van Vleck PM
2) Ground state is degenerate (J≠0)
Then the 1st order term almost always >> the 2nd order terms.
(
)
(
m = − μ B L + 2S = − μ B J + S
)
• Heuristic argument: J is fixed, L and S rotate around J,
maintaining the triangle. So the magnetic moment is
given by the component of L+2S parallel to J,
J ⋅S
J
=
J
J 2 − L2 + S 2 )
2
2 (
J
2J
J
=
[ J ( J + 1) − L( L + 1) + S ( S + 1)]
2 J ( J + 1)
S // =
∴ meff = − g J μ B J
Lande g-factor
(1921)
J ( J + 1) − L( L + 1) + S ( S + 1)
gJ = 1+
2 J ( J + 1)
L
H
J
S
• ΔE (mJ ) = g J μ B mJ H , so χ= 0?
No! these 2J+1 levels are closely
packed (< kT), so F(H) is
nonlinear (next page).
Langevin paramagnetism
Z=
J
∑
mJ =− J
e − E ( mJ )/ kBT , ΔE (mJ ) = g J μ B mJ H ( ∼ 1K at H = 1 T )
F = E − TS = − k BT ln Z
⎛ g μ JH ⎞
N ∂F N
= g J μ B JBJ ⎜ J B
⎟
V ∂H V
k
T
B
⎝
⎠
2J +1
⎛ 2J +1 ⎞ 1
⎛ x ⎞
where BJ ( x) ≡
coth ⎜
x⎟ −
coth ⎜
⎟
2J
⎝ 2J
⎠ 2J
⎝ 2J ⎠
M =−
• k BT << g J μ B JH
( x >> 1)
N
g J μB J , χ = 0
V
• k BT >> g J μ B JH ( x << 1)
Brillouin
function
M=
M=
BJ ( x) ∼
N
2 J ( J + 1)
H
( g J μB )
3k BT
V
J +1
x
3J
χ (T )
• at room T, χ(para)～500χ(dia) calculated earlier
• Curie’s law χ=C/T (note: not good for J=0)
N ( μB p )
, where p = g J J ( J + 1)
C=
V 3k
2
effective Bohr
magneton number
f-shell (Lanthanides)
鑭系元素
In general (but not always), energy from low to high:
1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d …
Due to low-lying
J-multiplets
(see A+M, p.657)
• Before ionization, La: 5p6 6s2 5d1; Ce: 5p6 6s2 4f2 …
3d-shell (transition metal ions)
?
• Curie’s law is still good, but p is mostly wrong
• Much better improvement if we let J=S
Crystal field splitting
In a crystal, crystal field may be more
important than the LS coupling
• Different symmetries would have
different splitting patterns.
淬滅
Quench of orbital angular momentum
• Due to crystal field, energy levels are now labeled by L (not J)
• Orbital degeneracy not lifted by crystal field may be lifted by
1) LS coupling, or 2) Jahn-Teller effect, or both.
Spontaneous
lattice
distortion
• The stationary state ψ of a non-degenerate level can be chosen
as real
when t → −t ,
ψ → ψ * (= ψ if nondegenerate)
•
ψ Lψ = ψ r× ∇ψ
is purely imaginary
i
but ψ L ψ has to be real also
∴ψ Lψ =0
(ψ
L2 ψ can still be non-zero )
• for 3d ions, crystal field > SO interaction
• for 4f ions, SO interaction > crystal field (because 4f is hidden inside
5p and 6s shells)
• for 4d and 5d ions that have stronger SO interaction, the 2 energies
maybe comparable and it’s more complicated.
• Langevin diamagnetism
• paramagnetism
• Hund’s rules
• Lande g-factor
• Brillouin function
• crystal field splitting
• quench of orbital angular momentum
• nuclear demagnetization
• Pauli paramagnetism and
Landau diamagnetism
Adiabatic demagnetization (proposed by Debye, 1926)
• Without residual field
• The first method to reach below 1K
J
Z=
∑
mJ =− J
e − E ( mJ )/ kT , assume E (mJ ) ∝ H
⎛H ⎞
F = − kT ln Z ⎜
⎟
⎝ kT ⎠
∂F
⎛H ⎞
S =−
= S⎜
⎟
∂T
⎝ kT ⎠
• If S=constant, then kT～H
絕熱去磁
T f = Ti
∴ We can reduce H to reduce T
Freezing is
effective
only if spin
specific heat
is dominant
(usually
need T<<TD)
增
加
磁
場
Hf
Hi
• With residual field
(due to spin-spin int,
crystal field… etc)
Can reach 10-6 K (dilution refrig only 10-3 K)
analogy
增
加
磁
場
→ 環境
絕
熱
去
磁
← 樣品
wikipedia
Pauli paramagnetism for free electron gas (1925)
• Orbital response to H neglected, consider only spin response
• One of the earliest application of the exclusion principle
N = N↑ + N↓
1
N↑ − N↓ ) μB
(
V
For T << TF ,
M=
n↑ − n↓ ≅ g (ε F ) μ B H , g = g↑ + g↓
∴ M = g (ε F ) μ B2 H
1
⎛α ⎞
−6
= g (ε F ) μ B2 = ⎜
k
a
(
)
⎟ F 0 ∼ 10
⎝ 2π ⎠
2
⇒ χ Pauli
g (ε ) =
m
π2
e
μB =
2mc
• unlike the PM of magnetic ions, here the magnitude ～ DM’s
(supressed by Pauli exclusion principle)
3
2mε
a0 =
2
me 2
e2
α=
c
Landau diamagnetism for free electron gas (1930)
• The orbital response neglected earlier gives slight DM
• The calculation is not trivial. For free electron gas,
e2 k F
χ Landau = −
12π 2 mc 2
1
= − χ Pauli
3
• So far we have learned PM and DM for a free electron gas.
How do we separate these contributions in experiment?
X-ray magnetic circular dichroism (XMCD)
Hoddeson, Our of the crystal maze, p.126