SME
Quantum Mechanics · Problem type
Zeeman splitting and spin resonance
Use this for “An atom in a p state (ℓ = 1) sits in a 0.5 T field. How far apart are its levels, how many are there, and at what frequency does a free electron’s spin flip?”
Diagram
3 levels, mℓ = −1 to +1, each ΔE = 0.0001158 eV apart.
W = 2 × ΔE = 0.0002315 eV
Magnetic field B = 2 T
Orbital quantum number ℓ = 1
Spin-flip energy Eflip = 2.3152 × 10⁻⁴ eV
Spin-flip frequency f = 55.977 GHz
This is an example. Type your own values to start a new problem.
ΔE
=
5.788
×
10⁻⁵
eV/T
×
B (
)
=
ΔE (eV)
ℓ
Orbital quantum number
example
levels
Number of levels
calculated
W
Spread from mℓ = −ℓ to ℓ
calculated · eV
Eflip
Spin-flip energy
calculated · eV
f
Spin-flip frequency
calculated
Formulas
ΔE
=
5.788 ×
10
-5
×
B
1.1576 ×
10
-4
=
5.788 ×
10
-5
×
2
levels
=
2
×
ℓ
+
1
3 = 2 × 1 + 1
W
=
(levels
−
1)
×
ΔE
2.3152 ×
10
-4
=
(3
−
1)
×
(1.1576
×
10
-4
)
Eflip
=
2
×
5.788 ×
10
-5
×
B
2.3152 ×
10
-4
=
2
×
5.788 ×
10
-5
×
2
f
=
Eflip
4.136 ×
10
-15
÷
10
9
55.977
=
2.3152 ×
10
-4
4.136 ×
10
-15
÷
10
9
Assumptions
•
Normal Zeeman effect: level mℓ shifts by mℓ × μB × B, so neighbouring levels sit ΔE = μB × B apart; μB = 5.788 × 10⁻⁵ eV/T.
•
A level with orbital number ℓ splits into 2ℓ + 1 evenly spaced levels, mℓ = −ℓ to ℓ, spread over W = 2ℓΔE.
•
An electron’s spin (g = 2) has two levels, ±μB × B; a photon of hf = gμB × B flips it. h = 4.136 × 10⁻¹⁵ eV·s.
Step-by-step
WE KNOW
B = 2 T, ℓ = 1
FIND
splitting between levels (ΔE), number of levels (levels), spread from mℓ = −ℓ to ℓ (W), spin-flip energy (Eflip), spin-flip frequency (f)
Step 1 · Find splitting between levels (ΔE)
Use ΔE = 5.788 × 10⁻⁵ × B
Each step of mℓ moves the level by one Bohr magneton times the field.
ΔE
=
5.788 ×
10
-5
×
2
ΔE
=
1.1576 ×
10
-4
eV
Step 2 · Find number of levels (levels)
Use levels = 2 × ℓ + 1
One
level
for
each
mℓ
from
−
ℓ
to
ℓ
.
levels = 2 × 1 + 1
levels = 3
Step 3 · Find spread from mℓ = −ℓ to ℓ (W)
Use W = (levels − 1) × ΔE
From
the
lowest
level
to
the
highest
are
2
ℓ
steps
of
ΔE:
one
fewer
than
the
levels.
W
=
(3
−
1)
×
(1.1576
×
10
-4
)
W
=
2
×
(1.1576
×
10
-4
)
W
=
2.3152 ×
10
-4
eV
Step 4 · Find spin-flip energy (Eflip)
Use Eflip = 2 × 5.788 × 10⁻⁵ × B
Spin
up
sits
at
+μB
×
B
and
spin
down
at
−μB
×
B
(g
=
2):
the
gap
is
twice
μB
×
B
.
Eflip
=
2
×
5.788 ×
10
-5
×
2
Eflip = 0.00011576 × 2
Eflip
=
2.3152 ×
10
-4
eV
Step 5 · Find spin-flip frequency (f)
Use f = Eflip ÷ (4.136 × 10⁻¹⁵) ÷ 10⁹
The
photon
carries
the
gap:
f
=
E
h
,
in
Hz,
then
10
9
for
GHz.
f
=
2.3152 ×
10
-4
4.136 ×
10
-15
÷
10
9
f
=
2.3152 ×
10
-4
4.136 ×
10
-15
÷
10
9
f
=
5.598 ×
10
10
10
9
f
=
55.977
GHz
Check
Put the numbers back in. Both sides should match.
1.1576 ×
10
-4
=
5.788 ×
10
-5
×
2
✓
3 = 2 × 1 + 1 ✓
2.3152 ×
10
-4
=
(3
−
1)
×
(1.1576
×
10
-4
)
✓
2.3152 ×
10
-4
=
2
×
5.788 ×
10
-5
×
2
✓
55.977
=
2.3152 ×
10
-4
4.136 ×
10
-15
÷
10
9
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Related lessons
Angular momentum and spin
Main lesson
Measuring spin along a tilted axis
Use this for “An electron leaves a Stern–Gerlach magnet spin up along z and enters a second magnet tilted 120° from z. Find the chances of up and down and the average reading.”
Adding two angular momenta
Use this for “A d electron has ℓ = 2 and spin s = ½. Which total j can it have, how many states does each hold, and how many states are there in all?”