SME
Mineralogy
Crystallography
Use this for “Copper Kα X-rays (1.5406 Å) give quartz a peak at 2θ = 26.64°. How far apart are its planes?”
Diagram
The wave off the next plane travels 2d sin θ farther: a whole number of wavelengths, nλ = 2d sin θ, and the two add.
This is an example. Type your own values to start a new problem.
n
×
λ (Å)
=
2
×
d (Å)
×
sin(
θ
°)
2θ
Detector angle
example · °
Formulas
θ
=
2
θ
2
13.32
=
26.64
2
n
×
λ
=
2
×
d
×
sin(
θ
)
1 × 1.5406 = 2 × 3.3435 × sin(13.32°)
Assumptions
•
Diffractometers report 2θ, the angle between the beam in and out; the Bragg angle θ is half of it.
•
A peak needs the extra path 2d sin θ to be a whole number n of wavelengths: nλ = 2d sin θ.
•
Copper Kα X-rays have λ = 1.5406 Å; most peaks are read in first order (n = 1).
Step-by-step
WE KNOW
2θ = 26.64°, λ = 1.5406 Å, n = 1
FIND
Bragg angle (θ), plane spacing (d)
Step 1 · Find Bragg angle (θ)
Use θ = (2θ) ÷ 2
The
detector
reads
2
θ
,
the
angle
between
the
beam
in
and
out.
θ
=
26.64
2
θ
=
13.32°
Step 2 · Find plane spacing (d)
Use n × λ = 2 × d × sin(θ)
The
extra
path
2
d
sin
θ
holds
n
wavelengths,
so
divide
them
by
2
sin
θ
.
d
=
n
×
λ
2 × sin(
θ
)
d
=
1
×
1.5406
2 × sin(13.32°)
d
=
1.5406
2 × 0.2304
d
=
1.5406
0.4608
d
=
3.3435
Å
Check
Put the numbers back in. Both sides should match.
13.32
=
26.64
2
✓
1 × 1.5406 = 2 × 3.3435 × sin(13.32°) ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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