Zone Axis
Lattice planes (hkl) of the zone [mnp]
Weiss Zone Law:
mh + nk + pl = 0
(ZOLZ)
g•r=0
[hkl] • [mnp] = 0
|hkl| x |mnp| cosθ = 0
(θ = 90°)
Zone axis [mnp]
MSE 421/521 Structural Characterization
Zone Axis Identification
iˆ
ˆj
kˆ
det h1 k1 l1
h2 k2 l2
u = k1l2 - k2l1
v = h2l1 - h1l2
w = h1k2 - h2k1
= k1l2 - k2l1, h2l1 – h1l2, h1k2 - h2k1
Example: (111) and (111)
iˆ
jˆ
kˆ
det 1
1
1
1
1 = 1 + 1, 1 + 1, - 1 + 1
1
= [ 2 2 0 ] || [ 1 1 0 ]
MSE 421/521 Structural Characterization
Nd2Hf2O7
Zone Axis = [110]
Indexing Patterns
g1
g2
45°
220
400
45°
Nd2Hf2O7
g3
Rd = Lλ = 11.277 cmÅ (calibrated)
220
r1 = 3.0 cm ∴ d1 = 3.8 Å {220}
r2 = 4.2 cm ∴ d2 = 2.7 Å {400}
r3 = 3.0 cm ∴ d3 = 3.8 Å {220}
Rule: g1 + g3 = g2
r1/r2 = d2/d1 = 0.71 ... d400/d220 = 0.71
∴ Zone Axis = [001]
MSE 421/521 Structural Characterization
Indexing Patterns
g1
224
30°
g2
60°
202
Nd2Hf2O7
g3
Rd = Lλ = 11.277 cmÅ (calibrated)
220
r1 = 5.2 cm ∴ d1 = 2.2 Å {224}
r2 = 3.0 cm ∴ d2 = 3.8 Å {202}
r3 = 3.0 cm ∴ d3 = 3.8 Å {220}
Rule: 2g2 - g3 = g1
r2/r1 = d1/d2 = 0.58 ... d224/d220 = 0.58
∴ Zone Axis = [111]
MSE 421/521 Structural Characterization
Convergent Beam
USES:
Crystallographic information inside
disks - space group determination
Incident
Electrons
Local thickness measurements
Specimen
Strain analysis
REQUIREMENTS
Defect-free region of crystal
Very high vacuum
Low temperature
Crystal stable against intense beam
Diffraction Discs
MSE 421/521 Structural Characterization
Diffraction Errors
A
A’
specimen
β
objective lens
Error = MCsβ3
B
C
B’
C’
SA aperture
Spherical aberration in the objective lens, Cs, can result in off-axis rays being
focused nearer to the lens than are axial rays.
Information from AA’ for off-axis rays goes to CC’ instead of BB’ as occurs for
axial rays near the centre of the objective lens.
MSE 421/521 Structural Characterization
Convergent Beam
HOLZ lines
in
central disk
of CBED
pattern from
α-Al2O3
viewed along
[0001]
[111] CBED
pattern from Cu
Kebbede
and Carim,
Penn State
MSE 421/521 Structural Characterization
Thickness Determination
The fringes present in diffracted discs can be used to determine local sample thickness.
They arise because Ig is a function of thickness, t.
As electron beam goes through specimen, the parts at various s interfere to give fringes.
The width of the fringes depends on t according to:
So plotting (si / ni)2 vs -(1 / ni)2 results in a line whose
slope is (1/ξg)2 and whose y-axis intercept is (1 / t)2.
where ni is an integer and si is:
t = 134 nm
ξg = 54.6 nm
accurate to about 1 - 2%
MSE 421/521 Structural Characterization