lecture 7 slides - mit opencourseware · lecture 7 slides author: cappellaro, paola created date:...

Neutron Interferometry

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NIST Center for Neutron ResearchHome to a 20 MW reactor that provides neutrons for scien>fic researchDozens of instruments (most for Solid State applica>ons)Some instruments for the study of Fundamental Physics

Public domain image

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The Neutron Interferometer and Optics Facility

Isolated 40,000 Kg room is supported by six airspringsActive Vibration Control eliminates vibrations less than 10Hz Temperature Controlled to +/- 5 mK

Public domain image

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Inside the NCNRReactor Core

LH2

7 Neutron Guides

NIOF

Fuel Elements

Guide Hall

Public domain images4

Wavepacket

p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 10 p x

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 k

Neutron coming out of the reactor is a wavepacket:Sum of many plane waves with different wavenumber k[not a stationary state: evolves (moves!) in time]

Fourier Transform Pair

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Monochromator

Monochromator selects a small range of momenta��

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Wavepacket

p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 10 p x

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 k

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3-blade interferometer from single Si crystal

Neutron Interferometer

�!� �"��# � �#$�%��&�

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5-blade interferometer from single Si crystal

��

'��"( ��)��

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Wavepacket ➙ Plane wave

p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 10 p x

Wavepacket ∆x ≫ Interferometer ➞ consider ∆x =∞

or neutron = plane wave |ki = '

k

(x) =1p2⇡

e

ikx

�� '��"( ��)��

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Momentum eigenfunctions

We can analyze the neutron interferometer looking only at the momentum eigenfunctions:STATIONARY SOLUTION (no time evolution)

|ki|ki

|-ki

�� '��"( ��)��

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The neutron is a plain wave with k>0. The first blade is a beam splitter (50/50% probability of going up or down)

Interference(Calculations: 1)

| (0)i = |ki

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After the first blade, the state is a superposition.


| 1i =1p2(|ki+ |-ki)

t1

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The second blade is a mirror, exchanging neutrons with positive and negative k


| 2i =1p2(|-ki+ |ki)

t2

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Neutrons in the upper path (with negative momentum) go through the phase flag (an object)


| 3i =1p2(ei' |-ki+ |ki)

t3

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The third blade recombines the beams and allows them to interfere.

' | i = cos' |ki+ sin' |-ki


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The detector measure the neutron flux intensity (number of neutrons per unit time).

Interference

Detector

'

P (+k) = cos

2(')

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Interference

�� '��"( ��)��

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Flux of particles

• Plane wave wavefunction is not properly normalized

• It is difficult to interpret as as the probability of finding a particle at position x.

• Interpret as a flux of particles

set

(x) = Ae

ikx

| (x)|2

v| (x)|2 = I

A =

rmI

~k 19

Scatteringof Waves and Particles

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Transmission

Energy > Potential Step

H

E=T+V ➜ mv02/2 > mgH

Region I Region II21

Transmission

Energy > Potential Step

H

E=T+V ➜ mv02/2 > mgH


Reflection

Energy < Potential Step

H

E=T+V ➜ mv02/2 < mgH


Reflection/Transmission

Incoming wave

Reflected wave Transmitted wave

eikx

e�ikx eikx

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Reflection/Transmission

Incoming wave

Reflected wave Transmitted wave

eikx

e�ikx eikx

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lecture 7 slides - mit opencourseware · lecture 7 slides author: cappellaro, paola created date:...

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