Neutron Optics and Neutron Interferometry• Neutron Optics ⇒Analogy to Light Optics • Neutron...
Transcript of Neutron Optics and Neutron Interferometry• Neutron Optics ⇒Analogy to Light Optics • Neutron...
Neutron Optics andNeutron Interferometry
Helmut KaiserIUCF
Outline: IntroductionRefractionReflectionDiffractionInterferometry
Appeared in “The New Yorker”, 1940
Message
• Neutron Optics ⇒ Analogy to Light Optics
• Neutron is a de Broglie Matter Wave
• Neutron Interferometry ⇒ Unique technique for Probing and Elucidating Fundamental Quantum Mechanical Principles on a Macroscopic Scale
hp
λ⎛ ⎞
=⎜ ⎟⎝ ⎠
INTRODUCTION
Properties of the Neutron
Bound Coherent Scattering Lengths
b : phenomenological constant, determined by experiment
REFRACTIONStrong nuclear interaction:
2 2 2
2
2 , 2
1 12 2
n c
n c
kV Nb Em mV Nb
nE
π
λπ
= =
= − = −
Including absorption:2
0 1
222
22
1 ,2 2 4
n c r
r rc
r a i
V V iV Nb i Nm
NNn b i
with
π σ
σ λ σλπ λ π
σ σ σ
→ − = −
⎛ ⎞− − +⎜ ⎟⎝ ⎠
= +
Snell’ law:0
0
0 0
sin ,
sinsin
, sin
K vn neutronsk v
vK cn lightk v v
φφ
φφ
= =
= = =
0
: exp( )tITransmission T N dI
σ= = −
Magnetic interaction:
REFRACTION (cont.)
2
2
( ) ,2
,
(0)1 1 ,
2
(0)2 2
m Z S F
m Z
n m c
Ae
B
V V V V B E k divEmc mc
V V B
V V b pn N
Ewith
mBp p rN
μμ μ
μσ
λπ
μμ γπ μ
±
= + + = − ⋅ − ⋅ × −
= = − ⋅
+ ±= − = −
= = − = −
: 9.54 5.98 fm: 2.54 4.64 fm: 10.31 1.62 fm
cb pFeCoNi
±
±±±
:
: ( )
: 1.913 :
2
N
N
p average magnetic scatteringin forward direction
B mean magnetization magneticfield of the unpaired electrons
neutron magnetic dipole momentgyromagnetic ratio
e
μ γμγ
μ
=
= −
=
2
2
:
2.818 fm :
: 2
: 4
ee
Be
A
nuclear magnetonmc
er classical electron radiusm c
e Bohr magnetonm cB average magnetic dipole momentN
per atom
μ
μπ
= =
=
=
Examples:
REFLECTION
Rewriting Snell’s law
Reflectivity and Transmissivity:
Glancing incidence : θ → 0
Θ = 0 ↔ θ = θ‘
2
cos cos , cos
:
1
=
c
c
c
cc
n
critical angle of total reflection
n
Nb
θ θ
θ
θ
λ θ θπ
= =Θ
= −
⇒ ≤
0 0
2
22
2
, with 1
sin sinsin sin
1 ,
1 1 ( ) ,
1 1 ( )
1 :
r t
c
c
c
I RI and I TI R T
nRn
R forSince n
R f
is close to
or
θθ
θ θ
θ θ
θ θ
= = + =
− Θ=
+ Θ= ≤
− −=
+ −cθ θ>
• Diffraction from macroscopic objects
• Diffraction from perfect crystals
DIFFRACTIONBranches of neutron scattering theory
Diffraction ⇒ Interference of coherent waves
Diffraction from macroscopic objects• Single and double slit diffraction (Frauenhofer)• Edge diffraction (Fresnel)• Grating diffraction• Diffraction by planar structures → reflectometry• Fresnel zone plates and supermirrors• ............
Laue diffraction
Bragg diffraction
Diffraction from perfect crystals
Bragg planes
Brag
g pl
anes
Kinematic Bragg Diffraction
Kinematic Bragg Diffraction
Kinematic Bragg Diffraction
Kinematic Bragg Diffraction
Dynamical Diffraction Theory (Laue case)
0 forward scattered waveinternal
external forward scattered waveH
K
K
=
=
Bragg vector2
HnH
dπ
=
=
Bragg condition:
0HK K H− =
Solve Schrödinger Eqn. inside crystal:
( )2 20 ( ) ( ) ( )k r v r r∇ + Ψ = Ψ
( )with ( ) 4 n
n
iH ri i H
i nv r b r r v eπ δ ⋅= − =∑ ∑
Dynamical Diffraction Theory
0 00 0internal wave function: ( ) H HiK r iK riK r iK r
H Hr e e e eβ βα αα β α βψ ψ ψ ψ⋅ ⋅⋅ ⋅Ψ = + + +
Dynamical Diffraction Theory
0 00 0internal wave function: ( ) H HiK r iK riK r iK r
H Hr e e e eβ βα αα β α βψ ψ ψ ψ⋅ ⋅⋅ ⋅Ψ = + + +
ψ 0α =
12
1−y
1+ y2
⎡
⎣⎢⎢
⎤
⎦⎥⎥A0
ψ 0β =
12
1+y
1+ y2
⎡
⎣⎢⎢
⎤
⎦⎥⎥A0
ψ Hα = −
12
11+ y2
⎡
⎣⎢⎢
⎤
⎦⎥⎥A0
ψ Hβ = +
12
11+ y2
⎡
⎣⎢⎢
⎤
⎦⎥⎥A0
y =k0 sin 2θB
2νH
δθ
misset parameter
Dynamical Diffraction Theory
Dynamical Diffraction Theory (Laue case)
Transmitted wave: 0trans tr 0 tr H( ) Hik r ik rr e eψ ψ⋅ ⋅Ψ = +
1 0
1 0
( )tr 0 02
( )tr H 02
cos sin1
sin1
i
i
iy e Ay
iy e Ay
φ φ
φ φ
ψ
ψ
−
− +
⎡ ⎤= Φ − Φ⎢ ⎥
⎢ + ⎥⎣ ⎦⎡ ⎤−
= Φ⎢ ⎥⎢ + ⎥⎣ ⎦
00 1
2
,cos cos
1cos1
H
B B
HB
D D
Dy
ν νφ φθ θ
νθ
= =
⎛ ⎞⎜ ⎟Φ =⎜ ⎟+⎝ ⎠
with
Transmitted intensities:
22 2 2 2
0 tr 0 0 2
2 2 2tr H 0 2
cos sin1
1 sin1H
yI Ay
I Ay
ψ
ψ
⎡ ⎤= = Φ + Φ⎢ ⎥+⎣ ⎦
⎡ ⎤= = Φ⎢ ⎥+⎣ ⎦
“Pendellösung interference”
Transmitted Intensities
For the ⟨111⟩ reflection in Siat λ=2.70 Å:
y = 1 ⇒ 0.9 arcsec
Angle Amplification
For small δ (~10-3 arcsec): Ωδ
≈ 106
Dynamical Diffraction Theory (Laue case)
Dynamical Diffraction Theory (Bragg case)
Dispersion Equation:
approximate:
2 cos( ) /
( ) sin( )
e B
e
Be w
c
yPendelloesung length
extinction length
Nb e
π θ θ λ
π θλ
= − Δ Δ
= Δ
Δ =
Phase of the neutron wave function is directly accessible to experiment
NEUTRON INTERFEROMETRY
Applications:Measurement of bc
Optics experimentsQuantum mechanics experiments
Basic Principle
NEUTRON INTERFEROMETRY
2 2 22
:1 d d
I II I II I III
Phase shift
p s k s
ψ ψ ψ ψ ψ ψ
χ
+ = + +
= =∫ ∫
∼ Perfect-crystal interferometry:
X-ray interferometry: Bonse and Hart (1965)
Neutron interferometry: Rauch, Treimer and Bonse (1974)
Michelson Interferometer
Mach-Zender Interferometer
NEUTRON INTERFEROMETRY
Practical Neutron Interferometer
Perfect Crystal LLL Neutron Interferometer
Bragg condition: 2 sinn dλ θ=
lattice spacingd =
Perfect Crystal LLL Neutron Interferometer
Perfect Crystal LLL Neutron Interferometer
Perfect Crystal LLL Neutron Interferometer
Nuclear Phase Shift
Nuclear Phase Shift
2
index of refraction: 12
Nbn λπ
= −
0 0
relative phase shift:
cosDk nk Nbχ λ
θΔ = − =
Interferogram
Interferogram
[ ]2 1
2 1
O beam: 1 cos( )
H beam: cos( )
O
H
I A f
I B A f
χ χ
χ χ
= + −
= − −
max min
max min
contrast C CfC C
−=
+
IH
IO
Precision Phase Shift Measurement
cosDNbχ λ
θΔ =
Example:
aluminum sample, λ = 2.70 Å, ⟨111⟩ reflection
D = 100 μm ⇒ Δχ = 2π
Non-Dispersive Geometry
path length sin
Dθ
=
2Nb d DχΔ =
independent of λ
Perfect Crystal LLL Neutron Interferometer
NIST perfect crystal silicon interferometers
Scattering length bc measurements
4π Rotational Symmetry of Spinors
Rotation operator:ˆ
ˆ ( )i n S
nR eα
α− ⋅
=
Spin-1/2 particle:ˆ1 2
ˆ2 so ( )i n
nS R eα σ
σ α− ⋅
= =
Rotations about z-axis: / 2
/ 2
0( )
0
i
z i
eR
e
α
αα
−⎛ ⎞= ⎜ ⎟
⎝ ⎠
Symmetry:(2 )
(4 )
z
z
R
R
π χ χ
π χ χ
= −
=
4π spinor symmetryQuantum mechanical principle:
B Bμ μσ= − ⋅ = − ⋅H
/ 2½( ) ( ) (0) (0) (2 ) (0)
(4 ) (0)
ispin e σ αψ α α ψ ψ ψ π ψ
ψ π ψ
− ⋅= ℜ = = −
=
⇒
/ / / 2( ) (0) (0) (0) ( )i t i Bt it e e eμ σ αψ ψ ψ ψ ψ α− − ⋅ − ⋅= = = =H
Larmor precession angle:
2 2B dt B dsv
μ μα = ≅∫ ∫
Experimental result:α = (715.87±3.8)°
Rauch et al., Phys.Lett. 54A, 1975
Werner et al., PRL 35, 1975
Larmor precession phase:
22 /n nm Bφ πμ λΔ = ±
Spin SuperpositionQuantum mechanical principle:
Quantum mechanical spin superposition(Wigner, Am J. Phys. 31 (1963)
Results:
Summhammer at al., PRA27, 1983
Quantum Phase Shift Due To Gravity (COW Experiments)
in grav2
2
area of parallelogram
gA m mh
A H
πλφΔ =
= =
min = neutron inertial mass
mgrav = neutron gravitational mass
test of weak equivalence principle at the quantum limit
Gravitationally induced quantum interference
2
0 2
( , )2
ˆ( ) -
1
g
i
m Mpr p G Lm r
L r pGMg r rr
p dr
p i WKB p k
= − − Ω ⋅
= ×
=
ΔΦ = ⋅
→ − ∇ → → =
∫
H
Quantum mechanical principle:
Neutron moving in the gravitational field of the rotating Earth
Phase shift due to gravity
022 sini gCOW II I
m mgH S
hπλ βΔΦ = Φ − Φ = −
β
α
Colella, Overhauser, Werner, PRL 34, 1975Staudenmann et al., PRA21, 1980
0
( ) ( ) ( )
( ) (2 / ) sin
grav bend
bend L
α α α
α π λ α
ΔΦ = ΔΦ + ΔΦ
ΔΦ = − Δ
Results:2 2 1/ 2exp
2 2 1/ 2
( ) sin( )
(exp) (60.12 1.45 ) 1. 58.72 0.03rad
59.2 0.1ra
2
( d
4
)
grav Sagnac bend
grav
grav
qq q q q
q
q theory
α αΔΦ =
= − −
=
±
− =
=
±−
Werner, Kaiser, et al., Physics B151, 19880.8%
Floating COW Experiment
D2O+ZnBr2
ρ(Si)=2.33g/cm3
ρ(D2O)=1.11g/cm3ρ(ZnBr2)=4.20g/cm3
References to Neutron Optics:
• Neutron Optics, Varley F. Sears, Oxford University Press, Oxford (1989).
• Neutron Interferometry – Lessons in Experimental Quantum Mechanics, Helmut Rauch and Samuel A. Werner, Oxford University Press, New York (2000).
• “De Broglie wave optics: neutrons, atoms and molecules,” Helmut Kaiser and Helmut Rauch, in Optics (ed. H. Niedrig), Walter de Gruyter, Berlin (1999).