MEASUREMENT OF ATOMIC MOMENTUM DISTRIBUTIONSBY HIGH ENERGY NEUTRON SCATTERING

J Mayers (ISIS)

Lectures 1 and 2. How n(p) is measured

The Impulse Approximation. Why high energy neutron scattering measures the momentum distribution n(p) of atoms.

The VESUVIO instrument.Time of flight measurements. Differencing methods to determine neutron energy and momentum transfersData correction; background, multiple scatteringFitting data to obtain sample composition, atomic kinetic energiesand momentum distributions.

Lectures 1 and 2. Why n(p) is measured

What we can we learn from measurements of n(p)(i) Lecture 3 n(p) in the presence of Bose-Einstein condensation.(ii)Lecture 4 Examples of measurements on protons.

1

2

The “Impulse Approximation” states that at sufficiently high incident neutron energy.(1) The neutron scatters from single atoms.(2) Kinetic energy and momentum are conserved in the collision.

M

q

q

My

2ˆ.

2

qp

M

p

M 22

)( 22

qpEnergy transfer

Mpi 2/2Initial Kinetic Energy

Mf 2/)( 2qp Final Kinetic Energy

Momentum transfer

Gives momentum componentalong q

3

The Impulse Approximation

pqp

pq dMM

pnS

2

)(

2)(),(

22

),(),,(

0

12

1

102

qSE

Eb

dEd

EEd

Kinetic energy and momentum are conserved

4

Why is scattering from a single atom?

If q >> 1/Δr interference effects between

different atoms average to zero.

Incoherent approximation is good for q such that;Liquids S(q) ~1 q >~10Å-1

Crystalline solids – q such that Debye Waller factor ~0.

. . . . . . ±Δr

5

Why is the incoherent S(q,ω) related to n(p)?

)()(),(2

EEANS ff

f qq

rrrqrq diA ff )().exp()()( *

E = Initial energy of particle

ω= energy transfer

Ef = Final energy of particle

q = wave vector transfer

Single particleIn a potential

6

)exp()( rkr ff iC

IA assumes final state of the struck atoms is a plane wave.

rrrqr diA ff )().exp()(*

ME f

f 2

22k

rrrkqr diCA ff )(]).(exp[)(*

2

].exp[)()( rrprp din

)(22

ff nCA kq

)2

()(),(2

Em

knS f

ff kqq

momentum distribution

8

qkp f

fdVf

kk

32

dpEM

nS

2

)()(),(

2qppq

p in Å-1 throughout - multiply by ħ to get momentum

9

E

MnS

2

)()(),(

2qppq

p

pqp

pq dM

p

MnS

22

)()(),(

22

Final stateis plane wave

Impulse Approximation

Difference due to “Initial State Effects”Neglect of potential energy in initial state.Neglect of quantum nature of initial state.

q→∞ gives identical expressions

10

Infinite Square well

rrrqr diA ff )().exp()(*

12

T=0

ER=q2/(2MED)

13

T=TD

ER=q2/(2MED)

14

Can be shown that (V. F. Sears Phys. Rev. B. 30, 44 (1984).

......)(

72

)(

36)()(

4

4

24

22

3

3

2

2

dy

yJd

q

FM

dy

yJd

q

VMyJyJ IAIA

IA

Thus FSE give further information on binding potential(but difficult to measure)

All deviations from IA are known as Final State Effectsin the literature.

15

Increasing q,ωDensity of States

16

FSE in Pyrolytic GraphiteA L Fielding,, J Mayers and D N Timms Europhys Lett 44 255 (1998)

Mean width of n(p) V2

17

q=40.8 Å-1

q=91.2 Å-1

FSE in ZrH2

18

Measurements of momentum distributions of atoms

Need q >> rms p

For protons rms value of p is 3-5 Å-1

q > 50 Å-1, ω > ~20 eV required

Only possible at pulsed sources such as ISIS UK, SNS USA

Short pulses ~1μsec at eV energies allow accurate measurement of energyand momentum transfers at eV energies.

Lecture 2

• How measurements are performed

L0

L1

v0

v1

θ

Source

Detector

Sample

1

1

0

0

v

L

v

Lt

Time of flight measurements

cos2 2122

21

2 kkkkq

0

0

mvk

10 EE

Time of flight neutron measurements

202

10 mvE

Wave vectortransfer

Energytransfer

1

1

0

0

v

L

v

Lt 1vFixed v0 (incident v)

(Direct Geometry)

1

1

0

0

v

L

v

Lt 0vFixed v1 (final v)

(Inverse Geometry)

1

1

mvk 2

121

1 mvE

22

The VESUVIO Inverse Geometry Instrument

23

VESUVIO INSTRUMENT

Foil out

Difference

Foil in

E M Schoonveld, J. Mayers et al Rev. Sci. Inst. 77 95103 (2006)

Foil cycling method

25

Cout=I0 A

C=Cout-Cin= I0 [ 1-A2]

Foil out

Cin=I0 (1-A)A

Foil in

26

Filter Difference Method

Cts = Foil out – foil in

6Lidetector

gold foil “in”

27

Blue = intrinsic width of lead peak

Black = measurement using Filter difference method

Red = foil cycling method

28

Foil cyclingFilter difference

YAP detectors give

Smaller resolution width

Better resolution peak shape

100 times less counts on filter in and filter out measurementsThus less detector saturation at short times

Similar count rates in the differenced spectra

Larger differences between foil in and foil out measurementstherefore more stability over time.

30

Energy Transfer (eV)

-20 0 200.0

0.2

0.4

0.6

0.8

1.0MARI

Ei=100 eV

MARIEi=40 eV

MARIEi=20 eV

VESUVIO

Comparison of chopper and resonance filter spectrometers at eV energiesC Stock, R A Cowley, J W Taylor and S. M. Bennington Phys Rev B 81, 024303 (2010)

31

Θ =62.5º

Θ =~160º

32

YAPdetector

PrimaryGold foil

Secondarygold foil “in”

Secondarygold foil "“out”

new

Gamma background

Pb

old Pb

33

YAPdetector

PrimaryGold foil

Secondarygold foil “in”

Secondarygold foil "“out” old

new

ZrH2

ZrH2

34

Need detectors on rings

Rotate secondary foils keeping the foil scattering angle constant

Should almost eliminate gamma background effects

35

Nearest

Furthest

corrections

for gamma

Background

Pb

36

corrections

for gamma

Background

ZrH2

37

p2n(p) without a background correction

with a background correction

ZrH2

38

Multiple Scattering

Total scattering

Multiple scattering

J. Mayers, A.L. Fielding and R. Senesi, Nucl. Inst. Methods A 481, 454 (2002)

39

Back scattering ZrH2

A=0.048. A=0.092, A=0.179, A=0.256.

Forward scattering ZrH2

Multiple Scattering

40

Forwardscattering

Backscattering

41

Correction for Gamma Background and Multiple Scattering

30 second input from user

Correction procedure runs in ~10 minutes

Automated procedure. Requires;

Sample+can transmission

Atomic Masses in sample + container

Correction determined by measured data

42

Uncorrected

Corrected

Data AnalysisImpulse Approximation implies kinetic energy and momentum are conserved in the collision between a neutron and a single atom.

M

q

q

M

2ˆ.

2

qpMomentum along q

M

p

M

qp

22

)( 22

Energy transfer

Mpi 2/2 Mqpf 2/)( 2

Initial Kinetic Energy Final Kinetic Energy

Momentum transfer

Y scaling

M

q

q

My

2ˆ.

2

qp

In the IA q and ω are no longer independent variables

Any scan in q,ω space which crosses the line ω=q2/(2M) gives the same information in isotropic samples

Detectors at all angles give the same information for isotropic samples

45

M

MM d

dEd

dNEDEI

L

E

mtC

1

2

100

2/30

2/1

)()(2

2)(

)(0

12

1

2

MMMM yJ

q

M

E

Eb

dEd

d

M

q

q

MyM 2

2

Data Analysis

2

2

2 2exp

2

1)(

M

M

M

MMw

y

wyJ

Strictly valid only if(1) Atom is bound by harmonic forces(2) Local potential is isotropic

Spectroscopy shows that both assumptions are well satisfied in ZrH2

Spectroscopy implies that wH is 4.16 ± 0.02 Å-1

VESUVIO measurements give

47

wtd mean width= 4.141140 +- 7.7802450E-03 mean width = 4.134780 st dev= 9.9052470E-03

48

WH

3356 Sep 2008 4.15

3912 Nov 2008 4.13

4062 Dec 2008 4.11

4188 May 2009 4.16

4642 Nov 2009 4.15

5026 Jul 2010 4.13

Expected ratio for ZrH1.98 is 1.98 x 81.67/6.56 =24.65

Mean value measured is 21.5 ± 0.2

Intensity shortfall in H peak of 12.7 ± 0.8%

AH/AZr

21.8

21.3

21.9

20.8

21.5

21.5

ZrH2 Calibrations

Momentum Distribution of proton

50

Sum of 48detectors atforward angles

M

q

q

My

2

2

y in Å-1

51

Sep 2008Dec 2008May 2009Nov 2009

Measured p2n(p) for ZrH2

Lecture 3.

What can we learn from a measurement

of the momentum distribution n(p).

Bose-Einstein condensation

53

Bose-Einstein CondensationT>TB 0<T<TB T~0

D. S. Durfee and W. Ketterle Optics Express 2, 299-313 (1998).

ħ/L

54

Kinetic energy of helium atoms. J. Mayers, F. Albergamo, D. Timms Physica B 276 (2000) 811

T.R. Sosnick, W.M SnowP.E. Sokol Phys Rev B 41 11185 (1989)

3.5K 0.35K

BEC in Liquid He4

f =0.07 ±0.01

55

Interference between separately prepared condensates

of ultra-cold atoms

Quantised vortices in 4He and ultra-cold trapped gases

http://cua.mit.edu/ketterle_group/

Macroscopic Quantum Effects

56

Superfluid helium becomesmore ordered as the temperatureIncreases. Why?

57

Line width of excitationsin superfluid helium is zero as T → 0. Why?

58

Basis of Lectures

J. Mayers J. Low. Temp. Phys 109 135 (1997) 109 153 (1997)

J. Mayers Phys. Rev. Lett. 80, 750 (1998) 84 314 (2000)

92 135302 (2004)

J. Mayers, Phys. Rev.B 64 224521, (2001) 74 014516, (2006)

J Mayers Phys Rev A 78 33618 (2008)

59

2

11212 ).exp(),..,(,..)( rrprrrrrp diddn NN

Quantum mechanical expression for n(p) in ground state

What are implications of presence of peak of width ħ/L for properties of Ψ?

1rr Nrrs ,..2

2

).exp(),()( rrpsrsp didnħ/L

Ground state wave function

60

)(/),()( ssrrS P

rsrs dP2

),()(

spsp S dnPn )()()(

23

3)/.exp()(

1)( rrprp SS din

61

2),( sr is pdf for N coordinates r,s

2)(rS is conditional pdf for r given N-1 coordinates s

1)(2

rrS d

ΨS(r) is “conditional wave function”

rsrs dP2

),()( is pdf for N-1 coordinates s

62

ħ/L

What are implications of presence of peak of width ħ/L for properties of ψS?

spsp S dnPn )()()(

23

3)/.exp()(

1)( rrprp SS din

Δp Δx ~ħ

ψS(r) must be delocalised over length scales ~L

Delocalized, BEC

Localized, No BEC

64

Feynman - Penrose - Onsager Model

Ψ(r1,r2, rN) = 0 if |rn-rm| < a a=hard core diameter of He atom

Ψ(r1,r2, rN) = C otherwise

65

f ~ 8%O. Penrose and L. Onsager Phys Rev 104 576 (1956)

J. Mayers PRL 84 314, (2000) PRB64 224521,(2001)

24 atoms

192atoms

Periodic boundary conditions.Line is Gaussian with same mean and standard deviation as simulation. rrS d)(

Has same value for allpossible s to within terms~1/√N

2)(1 rrSS d

Vf

f

f

N/1

Δf

66

Macroscopic Single Particle Quantum Behaviour (MSPQB)

N

nnN

121 )(),...,( rrrr

η(r) is non-zero over macroscopic length scales

Coarse grained average

)(

2

21)( 1

2

21 ),...,(..1

),...,(1 N

NNNN ddrr

rrrrrrrr

Smoothing operation – removes structure on length scales ofinter-atomic separation. Leaves long range structure.

Coarse Grained

average over r

Nfd /1~)()(

1)(

)(rrrr

r SS

volume Ω

containing

NΩ atoms

NFdFF /1~)()]([

1)]([

)(rrrr

r SS

)(

22)(

1)(

r SS rrr d

N/1~)()(22

rrS

2)()( rssr S dPN Density at r

N/1~)()(2

rr

)(

2

)( 21)(

1..

1r Sr

rrrrs dddPN

NN

)(

2

21)( 1 ),...,(..1

1 NNNN

ddrr

rrrrr

2)()( rssr S dPN Density at r

NddPN

NN/1~)(..

1)( 21

rrrsr

70

2

21 ),...,( Nrrr NP N /1~)()..( 12 rrr

N

nnN

1

2

21 )(),...,( rrrr

Provided only properties which are averages over regions of spacecontaining NΩ particles are considered |Ψ|2 factorizes to ~1/√NΩ

NP N /1~)()..,( 231 rrrr

NP N /1~)()..,,( 3421 rrrrr

71

)()()()()(

2 2

22

nn rrrr

r

r

nnneff

n

Vm

)]([)( rr effV

η(r) is macroscopic function. Hence MSPQB.

Quantised vortices, NCRI, macroscopic density oscillations.

2)()()]([ rrr cc

Weak interactions

Gross-PitaevskiEquation

)()(2

rr

Schrödinger Equation (Phys Rev A 78 33618 2008)

72

Depends only upon

(a) ψS(r) is delocalized function of r – must be so if BEC is present

(b) ψS(r) has random structure over macroscopic length scales – liquids and gases.

NOT TRUE IN ABSENCE OF BEC, WHEN ψS(r) IS LOCALIZED

Summary T=0

BEC implies ψS(r) is delocalized function of r non-zero over macroscopic length scales

Delocalization implies integrals of functionals of ψS(r) over volumescontaining NΩ atoms are the same for all s to within ~1/√N Ω

Hence BEC implies MSPQB

73

Finite T

At T=0 only ground state is occupied. Unique wave function Ψ0(r1,r2…rN)

At Finite T many occupied N particle states

Measured properties are average over occupied states

i

ii ETBTU )()(

)/exp()( TETB ii

NNiNii dddHE rrrrrrrrr ...)...,(ˆ)...,( 212121*

74

Delocalisation implies MSPQB

ψS(r) for occupied states cannot be delocalized at T=TB

Consider one such “typical” occupied state with wave function Ψ(r,s)

But typical occupied state is delocalized as T→0

MSPQB does not occur for T=TB

Typical occupied state Ψ(r,s) must change from localised to delocalised function as T is reduced below TB.

75

Delocalized Localized

),( rsD ),( rsL

),()(),()(),( rsrsrs LD TT

α(T) =1 at T=0 α(T) =0 at T=TB

76

)()()()()( rrr SSSSS LD TbTa

1)()()(222 rrrrrr SSS ddd LD

1)( TaS 0)( TaS0T BTT

)()( 0 rr SS D),(),( 0 rsrs D

(a) r space

(b) p space

VD ~)(~ pS

NVL /~)(pS

Vp /1~

VD /1~)(rS

VNL /~)(rS

NVr /~

Overlap region ~1/√N

V

(a) r space

p space

VN /~

VD ~)(~ pS~V~-V

NVL /~)(pS

Vp /1~

VD /1~)(rS

VN /~

NVr /~

Overlap region ~1/√N

N/V

79

CTdbdad LD )(

22

)(

22

)(

2)()()(

r SSr SSr S rrrrrr

CCdbaCT LD )(

** )()(r SSSS rrr

)(rSL Localised within )(rNd

CT

L

1~

)()(

2

r S rr

)(rSL Localised outside )(r 0CT

Contribution of CT is at most ~1/√N

Two fluid behaviour

ssrr S dPN )()()(2

NLD /1~)()()( rrr

Fluiddensity

)(

)(1

)(r

rrr d

NLD /1~)()()( rFrFrF

srrsrF SS dm

P )()()()(2

Fluidflow

Flow of delocalised component is quantised

No such requirement for flow of localised component

Localised component is superfluid

Delocalized component is normal fluid

),()(),()(),( rsrsrs LD TT

NTT LD /1~)()()()()(22 rrr

)()(2

TT S )()(2

TT N

Superfluid fraction Normal fluid fraction

More generally true that

in any integral of Ψ(r1,r2…rN) over (r1,r2…rN)

overlap between ΨD and ΨL is ~1/√N

E=ED +EL

n(p)=nD(p)+nL (p)

S(q,ω)=SD(q,ω)+SL(q,ω)

S(q)=SD(q)+SL(q)

84

)()()( rrr LNDS EEE

)()()( rrr LNDS PPP

J. Mayers Phys. Rev. Lett. 92 135302 (2004)

)0(

)(

T

85

)0()()( fTTf s

)0(

)(

f

Tf

Superfluid fractionJ. S. Brooks and R. J. Donnelly, J Phys. Chem. Ref. Data 6 51 (1977).

Normalised condensate fraction

o o T. R. Sosnick,W.M.Snow and P.E. Sokol Europhys Lett 9 707 (1989).x x H. R. Glyde, R.T. Azuah and W.G. Stirling Phys. Rev. B 62 14337 (2000).

J. Mayers Phys. Rev. Lett. 92 135302 (2004)

),(),( 0 rsrs D

86

More spaces give smaller pair correlations

)()()( 0 qqq LNS SSS

As T increases, superfluid fraction increases, pair correlations reduce

)()()( 0 qqq LNS SSS

1)(

1)()(

qS

qST

B

T

SB-1

ST -1

)]0(1)[(1)( TT S

α(T)

V.F. Sears and E.C. Svensson, Phys. Rev. Lett. 43 2009 (1979).

J. Mayers Phys. Rev. Lett. 92 135302 (2004)

α(0)

Lattice model

Fcc, bcc, sc all give same dependence on T as that observed

Only true if N/V and diameter d of He atoms is correct

Change in d by 10% is enough to destroy agreement

J. Mayers PRL 84 314, (2000) PRB64 224521,(2001)

Seems unlikely that this is a coincidence

90

)()(),(2

EEAS ff

f qq

srsrrqsrq ddiNA ff ),().exp(),()( *

),()(),()(),( 0 rsrsrs LTT

Identical particles

),(),(),( 0 qqq LNS SSS

Only S0 contributes to sharp peaks

91

Anderson and StirlingJ. Phys Cond Matt (1994)

92

New prediction

)()()( 0 rrr LNS

)(0 r Has density oscillations identical to gnd state

)(rL Has no density oscillations

Measure density oscillations close to gnd state

Measure superfluid fraction wD before release of traps

Simple prediction of visibility of density oscillations

Summary

Most important physical properties of BE condensedsystems can be understood quantitatively purely from the form of n(p)

Non classical rotational inertia – persistent flow

Quantised vortices

Interference fringes between overlapping condensates

Two fluid behaviour

Anomalous behaviour of S(q)

Anomalous behaviour of S(q,ω)

Amomalous behaviour of density

Lecture 4.

What can we learn from a measurement

of the momentum distribution n(p).

Quantum fluids and solids Protons

95

Measurement of flow without viscosity in solid helium

E. Kim and M. H. W. ChanScience 305 2004

96

can

He4

Focussed data Fitted widths on Individual detectors

97

liquid

O single crystal high purity He4 X polycrystal high purity He4□ 10ppm He3 polycrystal

solid

Focussed data after subtraction of can. Dotted line is resolution function

98

T (K) (101) (002) (100)

0.115 2.759 (7) 3.1055 (300) 0.400 2.759 (7) 3.1055 (300) 0.150 2.758 (7) 3.1056 (300) 0.070 2.758 (7) 3.1055 (300)

0.075 2.758 (2) 2.934 (4) 3.131 (2)

0.075 2.757 (3) 2.940(3) 3.128 (2)

Measured hcp lattice spacings

99

• No change in KE, no change in vacancy concentration through SS transition.

• Implies SS transition quite different to SF transition in liquid.

• Probably not BEC of atoms• What is cause??

100

3He mean kinetic energy, EK as a function of the molar volume: experimental values (solid circles), diffusion Monte- Carlo values (open circles).self-consistent phonon method (dashed line).

R. Senesi, C. Andreani, D. Colognesi, A. Cunsolo, M. Nardone,

Phys. Rev. Lett. 86 4584 (2001)

Kinetic Energy of He3

101101

n(p) is the “diffraction pattern” of the wave function

2

).exp()()( rrprp din

Measurements of protons

101n(p) in Å-1Position in Å

If n(p) is known ψ(r) can be reconstructed in a model independent way

In principle ψ(r) contains all the information which can be known about themicroscopic physical behaviour of protons on very short time scales.

pp

ppr

rp

rp

de

dMp

eVE

i

i

)(~

)(~2)(.

2.

rrp rp dei .)()(~

Potential can also be reconstructed

103

VESUVIO Measurements on Liquid H2

J Mayers (PRL 71 1553 (1993)

2

2

2

)(exp)(

Rr

Cr

2R

2

).exp()()( rdrpirpn

104

2

).exp()()( rdrpirpn

105

Puzzle

QM predicts R=bond length not ½ bond length

Fit to data givesR=0.36 σ=5.70

Spectroscopy givesR=0.37 σ=5.58

R should be 0.74!

106

Red data H2 1991Black YAP 2008

J(y)

p2n(p)

107

R=0.37R=0.74

108

Heavy Atoms

Single crystals

H

109

Reconstruction of Momentum Distribution from Neutron Compton Profile

mln

mllnmln qYyHay

yqJ,,

,2,,

2

)ˆ()()exp(

),ˆ(

Spherical HarmonicHermite polynomial

)ˆ()()1(!2exp(

)( 22/1,,

,,

22/3

2

pYpLpanp

pn lmln

lmln

n

mln

ln

Laguerre polynomial

an,l,m is Fitting coefficient

110

111

112

Nafion

113

114

115

116

In press PRL (2010)

Measurements of n(p) give unique information

on the quantum behaviour of protons in a wide

range of systems of fundamental importance