CITE THIS NOTEBOOK: Three-dimensional model of a split-crystal X-ray and neutron interferometer by Giovanni Mana. Wolfram Community MAR 9 2023.
ORIGINAL ARTICLE: Sasso, C. P., Mana, G. & Massa, E. (2022). Three-dimensional model of a split-crystal X-ray and neutron interferometer. J. Appl. Cryst. 55, 1500–1513. https://doi.org/10.1107/S1600576722008962.
Article Abstract: The observation of neutron interference using a crystal interferometer having a separate analyser opens the way to the construction and operation of interferometers with vast arm separation and length. Setting the design specifications requires a three-dimensional dynamical theory model of their operation. This paper develops the required three-dimensional mathematical framework, which also comprises coherent and incoherent illuminations; it is applied to study the visibility of the interference fringes.

Introduction

The observation of neutron interference using a split-crystal interferometer opens the way to realising interferometers with vast arm separation and length. The “Three-dimensional model of a split-crystal X-ray and neutron interferometer” paper aims to develop a mathematical framework by which the interferometers’ operation could be modelled and analysed and design specifications provided. These goals require knowing the effect of the limited spatial coherence of the source and of three-dimensional misalignments between the crystals splitting and recombining the neutron wave function. Therefore, we extended the dynamical theory of neutron and x-ray diffraction to model crystal interferometers operating both with coherent and partially coherent x-rays and neutrons, in three dimensions.

2.3 Laue diffraction in a symmetrically cut crystal

The
χ
0,h
→ 0 limit of
U
0
(p,z) reproduces the free propagation F(p;z)
Clear["Global`*"];Remove["Global`*"]​​R=
νSinζ
2
η
+
2
ν

2
η
+
2
ν
/.ζ
πz
Δ
e
,η
Δ
e
Tan[
Θ
B
]p
π
;​​T=Cosζ
2
η
+
2
ν
+
ηR
ν
/.ζ
πz
Δ
e
,η
Δ
e
Tan[
Θ
B
]p
π
;​​(*the
χ
0,h
→0limitisthesameasthe
Δ
e
->∞one*)​​Print["lim(R,χh→0)=",Limit[R,
Δ
e
->Infinity]]​​Limit[T,
Δ
e
->Infinity];​​Print["lim(T,χh→0)=",FullSimplify[%]]
lim(R,χh → 0) = 0
lim(T,χh → 0) = Cos[pzTan[
Θ
B
]]+Sin[pzTan[
Θ
B
]]

2.5 Coherent source

Clear["Global`*"];Remove["Global`*"];​​
ψ
in
[x_]=
-
2
(x-x0)
2
l0
+p0x

;(*Eq.(7)*)​​PowerExpand[Simplify[FourierTransform[
ψ
in
[x],x,p,FourierParameters{0,-1}]]];​​Print"

ψ
in
​[p] ∝ ",Simplify
%
2
l0
-p0x0

(*Eq.(8)*)​​

ψ
in
​[p] ∝
-
1
4
2
l0
2
(p-p0)
-px0


2.6 Partially coherent source.

Direct-space representation of the density matrix
Clear["Global`*"];Remove["Global`*"]​​
ψ
in
[x_,x0_,ϕ0_,p0_]=
-
2
(x-x0)
2
l0
+ϕ0+p0x

;(*Eq.(9)*)​​(*tmpisthedirect-spacerepresentationoftheargumentofE(.)inEq.(10)*)​​tmp=
ψ
in
[x1,xa,ϕa,pa](
ψ
in
[x2,xb,ϕb,pb]/.->-);
ϕa!=ϕb
terms; ensenble average over
ϕa
and
ϕb
FullSimplifyIntegratetmp
-
2
ϕa
2
2
σϕ

-
2
ϕb
2
2
σϕ

,{ϕa,-Infinity,+Infinity},{ϕb,-Infinity,+Infinity},Assumptionsσϕ>0​​(*ifσϕ->∞,theϕa!=ϕbtermscanbeomitted*)​​Limit[%,σϕ->Infinity]
2
-
2
(x1-xa)
+
2
(x2-xb)
+
2
l0
-pax1+pbx2+
2
σϕ

2
l0

π
2
σϕ
0
ϕa=ϕb
,
pa!=pb
, and
xa!=xb
terms; ensenble average over
pa
and
pb
Integrate(tmp/.ϕb->ϕa)
-
2
pa
2
2
σp

-
2
pb
2
2
σp

,{pa,-Infinity,+Infinity},{pb,-Infinity,+Infinity},Assumptionsw0>0&&σp>0​​Integrate%
-
2
2
xa
2
w0

-
2
2
xb
2
w0

,{xa,-Infinity,+Infinity},{xb,-Infinity,+Infinity},Assumptionsl0>0&&w0>0&&σp>0​​(*ifσp->∞,theϕa=ϕbandpa!=pbtermscanbeomitted*)​​Limit[%,σp->Infinity,Assumptionsl0>0&&
2
x1
>0&&
2
x2
>0]
2
-
2
(x1-xa)
+
2
(x2-xb)
2
l0
-
1
2
(
2
x1
+
2
x2
)
2
σp

π
2
σp
2
-
(
2
x1
+
2
x2
)4+2
2
l0
2
σp
+
2
w0
2
σp

22
2
l0
+
2
w0


2
π
2
σp
1
2
l0
+
2
2
w0
0
ϕa=ϕb
,
pa=pb
, and
xa!=xb
terms; ensenble average over
pa
FullSimplifyIntegrate(tmp/.{ϕbϕa,pb->pa})
-
2
pa
2
2
σp

,{pa,-Infinity,+Infinity},Assumptionsw0>0&&σp>0​​FullSimplifyIntegrate%
-
2
2
xa
2
w0

-
2
2
xb
2
w0

,{xa,-Infinity,+Infinity},{xb,-Infinity,+Infinity},Assumptionsl0>0&&w0>0&&σp>0​​(*ifσp->∞,theϕa=ϕb,pa=pb,andxa!=xbtermscanbeomitted*)​​Limit[%,σp->Infinity,Assumptionsl0>0&&
2
x1
>0&&
2
x2
>0]
-
2
(x1-xa)
+
2
(x2-xb)
2
l0
-
1
2
2
(x1-x2)
2
σp

2π
σp
2
-
2(
2
x1
+
2
x2
)
2
2
l0
+
2
w0
-
1
2
2
(x1-x2)
2
σp

2
l0
3/2
π
2
w0
σp
2
2
l0
+
2
w0
0
xa=xb
,
ϕa=ϕb
, and
pa=pb
terms; ensenble average over
xa
and
pa
jin[x1_,x2_]=Simplify
2
2
l0
+
2
2
w0
2
πσp
Integrate(tmp/.{xbxa,ϕbϕa,pb->pa})
-
2
2
xa
2
w0

-
2
pa
2
2
σp

,{xa,-Infinity,+Infinity},{pa,-Infinity,+Infinity},Assumptionsw0>0&&l0>0&&σp>0/.
2
l0
+2
2
w0
->2
2
w0
(*Eq.(12a)*)
-
2
2
l0
2
x1
+
2
w0
2
(x1-x2)
+2
2
l0
2
x2
+
2
l0

2
l0
+
2
w0

2
(x1-x2)
2
σp
2
2
l0

2
l0
+
2
w0


FullSimplify[Part[jin[x1,x2],2]/.
2
l0
+
2
w0
->
2
w0
]​​Collect[%,{
2
(x1-x2)
,
2
x1
+
2
x2
},Simplify];​​Print["
j
in
​[x1,x2] = ",ReplacePart[jin[x1,x2],2->%]](*Eq.(12a)*)​​
1
2
-
2
(x1-x2)
2
l0
-
2(
2
x1
+
2
x2
)
2
w0
-
2
(x1-x2)
2
σp
j
in
​[x1,x2] =
-
2
x1
+
2
x2
2
w0
-
2
(x1-x2)
1+
2
l0
2
σp

2
2
l0

reciprocal-space representation of the density matrix
Clear["Global`*"];Remove["Global`*"]​​
j
in
[x1_,x2_]=
-
2
x1
+
2
x2
2
w0
-
2
(x1-x2)
2
2
ℓ0

;(*Eq.(12a)*)​​
2π
2
2
w0
+
1
2
ℓ0
InverseFourierTransform[
j
in
[x1,x2],{x2},{p2},FourierParameters{1,-1}];​​
1
2
w0
+
1
2
w0
+2
2
ℓ0
π
FullSimplify[FourierTransform[%,{x1},{p1},FourierParameters{1,-1}]];​​%/.
2
w0
+
2
ℓ0
->
2
w0
;​​Print["

j
in
​[p1,p2] = ",FullSimplify[%]](*Eq.(12b)*)​​

j
in
​[p1,p2] =
-
1
8
2
(p1-p2)
2
w0
-
1
4

2
p1
+
2
p2

2
ℓ0


2.6.1 first order correlations

Clear["Global`*"];Remove["Global`*"]​​
j
in
[x1_,x2_]=
-
2
(x1-x2)
2
2
ℓ0
-
2
x1
+
2
x2
2
w0

;(*Eq.(12a)*)​​Print"
(1)
g
in
= ",SimplifyPowerExpand
j
in
[x1,x2]
Sqrt[
j
in
[x1,x1]
j
in
[x2,x2]]

(1)
g
in
=
-
2
(x1-x2)
2
2
ℓ0


2.6.2 free-space propagation

Imaginary part of the exponent
Real part of the exponent

2.7 Interference signal: coherent source

Definitions
Integrations over p2, q2 and p1, q1
Definitions

2.7.3 Moiré fringes

Moiré pattern spacing

Fringe contrast

2.7.4 Travelling fringe

2.8.2 Moiré fringes

joo12[y] exponent; real part
joo12[y] exponent; imaginary part

Fringe contrast

Appendix C - Laue diffraction in a displaced crystal

Rotation matrix