SME
Physical Chemistry II: Quantum & Spectroscopy · Problem type
Hyperfine lines in an EPR spectrum
Use this for “The unpaired electron of a radical couples to 2 equivalent ¹⁴N nuclei (I = 1). How many EPR lines appear?”
Diagram
Row 3: 1, 3, 3, 1; they add to 2³ = 8.
This is an example. Type your own values to start a new problem.
lines
=
2
×
n
×
I
+
1
Formulas
lines
=
2
×
n
×
I
+
1
4 = 2 × 3 × 0.5 + 1
Assumptions
•
The nuclei are equivalent: each couples to the unpaired electron with the same hyperfine constant.
•
One nucleus of spin I splits each line into 2I + 1 lines of equal height, so n of them give 2nI + 1 lines.
•
The triangle gives the heights for spin-½ nuclei (¹H, ¹³C, ¹⁹F); its rows stop at 12.
Step-by-step
WE KNOW
n = 3, I = 0.5
FIND
number of lines (lines)
Step 1 · Find number of lines (lines)
Use lines = 2 × n × I + 1
Each
nucleus
can
line
up
2
I
+
1
ways;
the
equivalent
ones
add
their
mI,
giving
2nI
+
1
totals.
lines = 2 × 3 × 0.5 + 1
lines = 6 × 0.5 + 1
lines = 3 + 1
lines
=
4
→
With
spin-
1
2
nuclei
the
line
heights
follow
row
3
of
Pascal’s
triangle:
1:3:3:1.
Check
Put the numbers back in. Both sides should match.
4 = 2 × 3 × 0.5 + 1 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.