SME
Inorganic Chemistry · Problem type
Plane spacing and Bragg’s law
Use this for “Tungsten is cubic with a = 316.5 pm. Find the spacing of its (110) planes and the angle 2θ of their peak with 154.2 pm X-rays.”
Diagram
The
(111)
plane
cuts
the
axes
at
a
h
,
a
k
and
a
l
(never,
for
a
0);
the
next
plane
is
d
=
a
÷
√
h
2
+
k
2
+
l
2
farther
on.
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.
d (pm)
=
a (pm)
÷
h
2
+
k
2
+
l
2
λ (pm)
=
2
×
d (pm)
sin
θ (°)
2θ
Detector angle
calculated · °
Formulas
d
=
a
÷
√
h
2
+
k
2
+
l
2
208.7
=
361.5
÷
√
1
2
+
1
2
+
1
2
λ
=
2
×
d
×
sin(
θ
)
154.2 = 2 × 208.7 × sin(21.68°)
θ
=
2
θ
2
21.68
=
43.36
2
Assumptions
•
A cubic cell: the planes (hkl) are a ÷ √(h² + k² + l²) apart.
•
First order: the extra path 2d sin θ is one wavelength, λ = 2d sin θ.
•
The detector reads 2θ, the angle between the beam in and out; copper Kα X-rays have λ = 154.2 pm.
•
The picture draws a face-centered cell like copper’s, with the plane and its rows.
Step-by-step
WE KNOW
a = 361.5 pm, h = 1, k = 1, l = 1, λ = 154.2 pm
FIND
plane spacing (d), Bragg angle (θ), detector angle (2θ)
Step 1 · Find plane spacing (d)
Use d = a ÷ √(h² + k² + l²)
Higher indices cut the cell more often, so their planes sit closer together.
d
=
361.5
÷
√
1
2
+
1
2
+
1
2
d
=
361.5
÷
√
1 + 1 + 1
d
=
361.5
÷
√
3
d
=
361.5
1.732
d
=
208.7
pm
Step 2 · Find Bragg angle (θ)
Use λ = 2 × d × sin(θ)
The
angle
whose
sine
is
λ
2
d
;
there
is
no
peak
when
that
is
more
than
1.
θ
=
sin⁻¹(
λ
2 ×
d
)
θ
=
sin⁻¹(
154.2
2 × 208.7
)
θ
=
21.68°
Step 3 · Find detector angle (2θ)
Use θ = (2θ) ÷ 2
The
beam
turns
through
twice
the
Bragg
angle
θ
.
2
θ
=
2
×
θ
2
θ
=
2
×
21.68
2
θ
=
43.36°
Check
Put the numbers back in. Both sides should match.
208.7
=
361.5
÷
√
1
2
+
1
2
+
1
2
✓
154.2 = 2 × 208.7 × sin(21.68°) ✓
21.68
=
43.36
2
✓
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Radius ratio and the rock-salt cell
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Lattice energy from the Born–Landé equation
Topic check
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Solid-state structures
Main lesson
Radius ratio and the rock-salt cell
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Use this for “MgO has the rock-salt structure with r₀ = 210 pm and Born exponent 7. Estimate its lattice energy.”