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

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 = + + +


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


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 (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


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


Practical Neutron Interferometer

Perfect Crystal LLL Neutron Interferometer

Bragg condition: 2 sinn dλ θ=

lattice spacingd =

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

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).

neutron optics and neutron interferometry• neutron optics ⇒analogy to light optics • neutron...

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