Date post: | 19-Jul-2015 |
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Fig. 4.1 The geometry involved in calculating the integrated intensity from a small single crystal, which is rotated at constant angular velocity ω about an axis normal to the paper.
32
22
122
max)( NNNFII Tep =
∫∫ ∫∫∫ γβ2 dddtRIdtdAIE pp ==
ω/)α(ddt=
∫∫∫ γβαω
2
dddIR
E p=
11211
12
112 Δ
λ
πsin)
λ
Δ(πsin)
λ
Δ(πsin aNS
aNShNaN
SH hkl =+=+
)(λ
Δ332211 bpbpbp
S++=
112
113322112
112 πsin)(πsin
λ
Δπsin pNaNbpbpbpaNS
=++=
∫∫∫ γβαπsin
πsin
πsin
πsin
πsin
πsin
ω 32
332
22
222
12
112
22
dddp
pN
p
pN
p
pNF
RIE Te=
α)Δ( α dSd = β)Δ( β dSd = γ)Δ( γ dSd =
,
,
γβαθ2sin)Δ()Δ()Δ( αγβ dddSdSdSddV =×=
∫∫∫3
233
2
22
222
12
11222
πsin
πsin
πsin
πsin
πsin
πsin
θ2sinωdV
p
pN
p
pN
p
pNFRIE
T
e=
3213
3213
332211 )/λ(λλλλ dpdpdpvdpdpdpvdpbdpbdpbdV ab ==×=
∫ ∫ ∫∞
∞
∞
∞
∞
∞ 323
332
222
222
121
112232
)π(
πsin
)π(
πsin
)π(
πsin
θ2sinω
λdp
p
pNdp
p
pNdp
p
pN
v
FRIE
a
T
e ×=
321
232
θ2sinω
||λNNNN
v
FNRIE
a
Te ==
+=
2
θ2cos1 2
242
4
Rcm
eII oe
+
=
θθδλ
ω 2sin22cos1|| 2
2
23
42
40
a
T
v
FV
cm
eIE
……Total energy collected from a small crystal of volume δV
Fig. 4.2 The geometry involved in calculating the integratedintensity for an extended face mosaic crystal
∫∞
0
θsin/μ22
2
23
42
40
θsinδθ2sin2
θ2cos1δλ
ω =
+=
z
oz
a
T
V
dzAe
v
VF
cm
eIE
+=
θ2sin2
θ2cos1
μ2
λ
ω
2
2
23
42
4
a
To
v
F
cm
epE
+
=
θθ
µλ
ω 2sin2
2cos1
2
2
2
23
42
4
a
To
v
F
cm
epE
……Total energy collected from a mosaic crystal witha linear absorption coefficient of µ
In a powder sample , only a fraction (φ) of the irradiated crystals is effective in intensity contribution.
Δαθcos2
Δαθcosπ2π4
Φ 22
pH
H
p==
p = multiplicity factor
Fig. Representation of the number of crystal in powder sample whose hkl planes make angles between θ+α and θ+α+dα with the primary beam.
α∆
Total intensity collected from a powder sample by a pinhole camera:
+=
+=
θsin2
θ2cos1
4
λ
μ2
Δα
ω
θ2sin2
θ2cos1
μ2
λ
ωΦ
2
2
23
42
4
2
2
23
42
4
a
Too
a
To
v
pF
cm
eAI
v
F
cm
epE
μ2θsin→ ∫
∞
0
θsin/)μ2( o
z
oz AdzAeV =
=
With effective volume V:
+
=
θθλ
sin22cos1
4
|| 2
2
23
42
4
a
To
v
FpV
cm
eIP
θ2sinπ2 R
PP =′
+
=′
θθθλ
π 2sinsin2cos1||
16
2
2
23
42
4
a
To
v
FpV
cm
eR
IP
Lorentz-polarization factor LP=(1+cos²2θ)/(sinθsin2θ)
Integrated intensity collected from a powder sample by a pinhole camera:
Integrated intensity collected from a powder sample by a diffractometer:
Extinction• Primary extinction: after diffracted at A an B,
K2 has a phase shift of π and thus interfere destructively with Ko (happens in perfect crystal).
• Secondary extinction: In mosaic crystal, if mis-orientations among domains are not large enough, incident beam on an interior domain has been weakened by previous correctly oriented domains due to diffraction
Integrated intensity from a thick perfect crystal with negligible absorption, N = number of unit cells per unit volume
Reflectivity = 100% over a range of 2s in the rocking curve