Multiplicity Factor
MSE 321 Structural Characterization
Polarisation Factor
Intensity of scattered x-ray scattered by stationary charged particle:
μ
4
sin
µ0 = 4πx10-7 m∙kg∙C-2
K = 7.94x10-30m2 for electron
α = angle between scattered photon & particle
sin
Coherent
scattering…
Photon traveling down x
but on average
so:
and since E ≡ A and I ∝ A2 then
Collision at O: Iy accelerates e- down y axis:
Iz accelerates e- down z axis:
…from unpolarised beam
∴
sin 90°
sin 90°
and
2!
cos $2!%
cos 2!
2
2
cos 2!
1
cos 2!
2
MSE 321 Structural Characterization
Thomson
Equation
∴
∝
1
cos 2!
2
Lorentz Factor
Part 1: Geometrical Factor
Crystals can produce non-zero diffracted intensity even for angles not exactly equal to the Brag angle.
1
2
1’
C
C D
θ1
a
D
2’
θ2
A
θ2
A
B
θ1
a
B
θ2 = θB - ∆θ
Na
a
A
B
θ1 = θB + ∆θ
δ = AD – CB = acosθ2 – acosθ1 = a[cos(θB – ∆θ) – cos(θB + ∆θ)]
= 2asinθΒsin∆θ ~ 2a∆θsinθB (sin∆θ ~ ∆θ)
λ = 2Na∆θsinθB
∆2θ = λ / (NasinθB) ∝ 1/sinθB
I = ½Im∆2θ ∝ ½(1/cosθB)(1/sinθB) = 1/sin2θB
(triangular peak shape)
Im ∝ 1/cosθB (see Scherrer formula)
MSE 321 Structural Characterization
Lorentz Factor
Part 2: Orientation Factor
The number of crystals oriented near Bragg angle is not constant at all angles.
Radius of circle is hypotenuse: r sin(90 – θB)
Perimeter of strip: 2πr sin(90 – θB)
Thickness of strip: r ∆θ
(r∆θ) 2πr sin(90 – θB)
4πr2
∴ I ∝ cosθB
MSE 321 Structural Characterization
=
∆θcosθB
2
Lorentz Factor
Part 3: Diffraction Geometry Factor
The detector will see a greater proportion of a diffraction cone when reflections are in forward/backward direction.
Circumference of diffraction cone: 2π(r sin2θB)
∴Intensity per length ∝ 1/sin2θB
MSE 321 Structural Characterization
Lorentz Polarization Factor
Lorentz Factor, L =
1
1
cos !
sin2!
sin 2!
Polarisation Factor, p =
'(cos
cos !
sin 2!
1
4sin ! cos !
)
Lorentz Polarisation Factor, Lp =
MSE 321 Structural Characterization
sin
'
) *+, )
'(cos
)
'(cos )
,-. ) *+, )