Transcript Slide 1

MEASUREMENT OF ATOMIC MOMENTUM DISTRIBUTIONS
BY 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 transfers
Data correction; background, multiple scattering
Fitting data to obtain sample composition, atomic kinetic energies
and 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
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.
Initial Kinetic Energy
 i  p 2 / 2M
Final Kinetic Energy
 f  (p  q)2 / 2M
Momentum transfer
Energy transfer
Gives momentum
component
along q
(p  q) 2 p 2


2M
2M
M
y  p.qˆ 
q

q2
  
2M




2
The Impulse Approximation
d 2 ( E0 , E1 , )
E1
2
b
S ( q,  )
ddE1
E0

p 2 (p  q) 2 
dp
S (q,  )   n(p)   

2M
2M 

Kinetic energy and momentum are conserved
3
Why is scattering from a single atom?
. . . . . .
±Δr
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.
4
Why is the incoherent S(q,ω) related to n(p)?
S (q,  )  N  A f (q)  (  E f  E )
2
f
Single particle
In a potential
A f (q)   (r ) exp( iq.r ) f (r )dr
*
E = Initial energy of particle
Ef = Final energy of particle
ω= energy transfer
q = wave vector transfer
5
A f   (r ) exp( iq.r ) f (r ) dr
*
IA assumes final state of the struck atoms is a plane wave.
 f (r)  C exp(ik f  r)
2
Ef 
 kf
2
2M
6
A f  C  * (r ) exp[ i (q  k f ).r ] (r )dr
n(p)   (r) exp[ip.r]dr
Af
2
2
momentum distribution
 C n(q  k f )
2
S (q,  )   n(q  k f ) ( 
f
kf
2
2m
 E)
p  k f q

kf
3

2 

V
 dk
f


(p  q) 2
S (q,  )   n(p)  
 E dp
2M


p in Å-1 throughout - multiply by ħ to get momentum
8


(p  q) 2
S (q,  )   n(p)   
 E 
2M
p



(p  q) 2 p 2
S (q,  )   n(p)   

2M
2M

Final state
is plane wave

dp

Impulse
Approximation
q→∞ gives identical expressions
Difference due to “Initial State Effects”
Neglect of potential energy in initial state.
Neglect of quantum nature of initial state.
9
A f   * (r ) exp( iq.r ) f (r ) dr
Infinite Square well
10
T=0
ER=q2/(2MED)
12
T=TD
ER=q2/(2MED)
13
All deviations from IA are known as Final State Effects
in the literature.
Can be shown that (V. F. Sears Phys. Rev. B. 30, 44 (1984).
M  2V d 3 J IA ( y ) M 2 F 2 d 4 J IA ( y )
J ( y )  J IA ( y ) 

 ......
2
3
4 2
4
36 q
dy
72 q
dy
Thus FSE give further information on binding potential
(but difficult to measure)
14
Increasing q,ω
Density of States
15
FSE in Pyrolytic Graphite
A L Fielding,, J Mayers and D N Timms Europhys Lett 44 255 (1998)
Mean width of n(p)
 2V
16
FSE in ZrH2
q=40.8 Å-1
q=91.2 Å-1
17
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 energy
and momentum transfers at eV energies.
18
Lecture 2
• How measurements are performed
Time of flight measurements
Sample
L0
θ
v0
Source
v1
L0 L1
t 
v0 v1
L1
Detector
Time of flight neutron measurements
Fixed v0 (incident v)
(Direct Geometry)
L0 L1
t 
v0 v1
v1
Fixed v1 (final v)
(Inverse Geometry)
L0 L1
t 
v0 v1
v0
mv
k0  0

E0  mv0
1
2
2
mv1
k1 

E1  12 mv1
2
q2  k12  k22  2k1k2 cos
Wave vector
transfer
  E0  E1
Energy
transfer
The VESUVIO Inverse Geometry Instrument
22
VESUVIO INSTRUMENT
23
Foil cycling method
Foil out
Foil in
Difference
E M Schoonveld, J. Mayers et al
Rev. Sci. Inst. 77 95103 (2006)
Cout=I0 A
Foil out
Cin=I0 (1-A)A
Foil in
C=Cout-Cin= I0 [ 1-A2]
25
Filter Difference Method
6Li
detector
gold foil “in”
Cts = Foil out – foil in
26
Blue = intrinsic width of lead peak
Black = measurement using Filter difference method
Red = foil cycling method
27
Foil cycling
Filter difference
28
YAP detectors give
Smaller resolution width
Better resolution peak shape
100 times less counts on filter in and filter out measurements
Thus less detector saturation at short times
Similar count rates in the differenced spectra
Larger differences between foil in and foil out measurements
therefore more stability over time.
Comparison of chopper and resonance filter
spectrometers at eV energies
C Stock, R A Cowley, J W Taylor and S. M. Bennington
Phys Rev B 81, 024303 (2010)
1.0
MARI
Ei=100 eV
0.8
MARI
Ei=40 eV
0.6
0.4
0.2
MARI
Ei=20 eV
VESUVIO
0.0
-20
0
20
Energy Transfer (eV)
30
Θ =62.5º
Θ =~160º
31
Gamma background
Secondary
gold foil "“out”
Pb
old
YAP
detector
Primary
Gold foil
Secondary
gold foil “in”
Pb
new
32
Secondary
gold foil "“out”
old
ZrH2
YAP
detector
Primary
Gold foil
Secondary
gold foil “in”
ZrH2
new
33
Need detectors on rings
Rotate secondary foils keeping the foil scattering angle constant
Should almost eliminate gamma background effects
34
Nearest
corrections
for gamma
Background
Pb
Furthest
35
corrections
for gamma
Background
ZrH2
36
ZrH2
p2n(p)
without a background correction
with a background correction
37
Multiple Scattering
J. Mayers, A.L. Fielding and R. Senesi, Nucl. Inst. Methods A 481, 454 (2002)
Total scattering
Multiple scattering
38
Multiple Scattering
Back scattering ZrH2
A=0.048. A=0.092,
A=0.179, A=0.256.
Forward scattering ZrH2
39
Forward
scattering
Back
scattering
40
Correction for Gamma Background and Multiple Scattering
Automated procedure. Requires;
Sample+can transmission
Atomic Masses in sample + container
Correction determined by measured data
30 second input from user
Correction procedure runs in ~10 minutes
41
Uncorrected
Corrected
42
Data Analysis
Impulse Approximation implies kinetic energy and momentum
are conserved in the collision between a neutron and a single atom.
Initial Kinetic Energy
 i  p 2 / 2M
Final Kinetic Energy
  2
 f  ( p  q) / 2M
Momentum transfer
Energy transfer
Momentum along q
  2
( p  q)
p2


2M
2M
M
p.qˆ 
q

q2
  
2M




Y scaling
M
y  p.qˆ 
q

q2
  
2M




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
Data Analysis
1/ 2
2
C (t )  2 
m
E0
d 2 M
I ( E0 ) D( E1 ) N M
d
L0
ddE1
M
3/ 2
d 2 M
E1 M
2
 bM
J M ( yM )
ddE1
E0 q
M
yM 
q

q2 
 

2M 

45
J M ( yM ) 
1
2wM
2
  yM 2
exp
2
2
w
 M




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
wtd mean width= 4.141140 +- 7.7802450E-03
mean width = 4.134780 st dev= 9.9052470E-03
47
ZrH2 Calibrations
WH
AH/AZr
3356
Sep 2008
4.15
21.8
3912
Nov 2008
4.13
21.3
4062
Dec 2008
4.11
21.9
4188
May 2009
4.16
20.8
4642
Nov 2009
4.15
21.5
5026
Jul 2010
4.13
21.5
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%
48
Momentum Distribution of proton
M
y
q
Sum of 48
detectors at
forward angles

q2
  
2M




y in Å-1
50
Measured p2n(p) for ZrH2
Sep 2008
Dec 2008
May 2009
Nov 2009
51
Lecture 3.
What can we learn from a measurement
of the momentum distribution n(p).
Bose-Einstein condensation
Bose-Einstein Condensation
T>TB
0<T<TB
T~0
ħ/L
D. S. Durfee and W. Ketterle Optics Express 2, 299-313 (1998).
53
BEC in Liquid He4
3.5K
0.35K
T.R. Sosnick, W.M Snow
P.E. Sokol
Phys Rev B 41 11185 (1989)
f =0.07 ±0.01
Kinetic energy of helium atoms.
J. Mayers, F. Albergamo, D. Timms
Physica B 276 (2000) 811
54
Macroscopic Quantum Effects
http://cua.mit.edu/ketterle_group/
Quantised vortices in 4He and
ultra-cold trapped gases
Interference between
separately prepared condensates
of ultra-cold atoms
55
Superfluid helium becomes
more ordered as the temperature
Increases. Why?
56
Line width of excitations
in superfluid helium is
zero as T → 0. Why?
57
Basis of Lectures
J. Mayers
J. Low. Temp. Phys
109 135
109 153
(1997)
(1997)
J. Mayers
Phys. Rev. Lett.
80, 750
84 314
92 135302
(1998)
(2000)
(2004)
J. Mayers,
Phys. Rev.B
64 224521,
74 014516,
(2001)
(2006)
J Mayers
Phys Rev A
78 33618
(2008)
58
Quantum mechanical expression for n(p) in ground state
n(p)   dr2 ,..drN  (r1 , r2 ,..rN ) exp(ip.r1 )dr1
r  r1
s  r2 ,..rN
2
Ground state wave function
n(p)   ds  (r, s) exp(ip.r)dr
2
ħ/L
What are implications of presence of
peak of width ħ/L for properties of Ψ?
59
 S (r)  (r, s) / P(s)
P(s)    (r, s) dr
2
n(p)   P (s)nS (p)ds
2
1
3
nS (p)  3  S (r ) exp( ip.r /  )dr

60
 (r , s )
2
is pdf for N coordinates r,s
P(s)    (r, s) dr
2
 S (r )

2
is pdf for N-1 coordinates s
is conditional pdf for r given N-1 coordinates s
(r ) dr  1
2
S
ΨS(r) is “conditional wave function”
61
n(p)   P (s)nS (p)ds
2
1
3
nS (p)  3  S (r ) exp( ip.r /  )dr

What are implications of presence of
peak of width ħ/L for properties of ψS?
ħ/L
Δp Δx ~ħ
ψS(r) must be delocalised over length scales ~L
62
Delocalized,
BEC
Localized, No BEC
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
64
J. Mayers PRL 84 314, (2000)
PRB64 224521,(2001)
f ~ 8%
O. Penrose and L. Onsager
Phys Rev 104 576 (1956)
24
atoms
Δf
f
f
192
atoms
fS 
Periodic boundary conditions.
Line is Gaussian with same mean
and standard deviation as
simulation.

S

1
 S (r )dr

V
(r )dr

1/ N
2
Has same value for all
possible s to within terms
~1/√N
65
Macroscopic Single Particle Quantum Behaviour (MSPQB)
N
(r1 , r2 ,...rN )   (rn )
n 1
η(r) is non-zero over macroscopic length scales
Coarse grained average
1
 (r1 , r2 ,...rN )  N

2

 ( r1 )
dr1..
 ( rN )
drN  (r1, r2 ,...rN )
Smoothing operation – removes structure on length scales of
inter-atomic separation. Leaves long range structure.
2
66
Coarse Grained
average over
r
volume Ω
containing
NΩ atoms
1
 S (r )    S (r)dr  f (r ) ~ 1 / N 
 (r )
1
F [ S (r )]   F [ S (r)]dr  F (r ) ~ 1 / N 
 (r )
1
2
 S (r )    S (r) dr
  (r )
2
 S (r )   (r)  ~ 1 / N 
2
2
 (r)  N  Ps ds S (r)
2
Density at r
 (r )   (r ) ~ 1 / N 
2
1
N


 ( r1 )
dr1..
 ( rN )
drN  (r1, r2 ,...rN )
2
1
1
2



P
s
d
r
..
d
r

(
r
)
dr
2
N
S
N 1  ( r )


(
r
)
N


 (r)  N  Ps ds S (r)
1
 N 1 
Ps dr2 ..drN  (r ) ~ 1 / N 
 ( rN )

2
Density at r
 (r1 , r2 ,...rN )
2
 P(r2 ..rN ) (r1 ) ~ 1/ N
 P(r1, r3..rN ) (r2 ) ~ 1/ N
 P(r1, r2 , r4 ..rN ) (r3 ) ~ 1/ N
N
 (r1 , r2 ,...rN )    (rn )
2
n 1
Provided only properties which are averages over regions of space
containing NΩ particles are considered |Ψ|2 factorizes to ~1/√NΩ
70
Schrödinger Equation (Phys Rev A 78 33618 2008)
 2  2 (rn )
 Veff (rn )  (rn )  (rn )   (rn )
2
2m rn

 (r )   (r )
2

Veff (r)   [ (r)]
Weak interactions
 [  (r )]  c (r )  c  (r )
2
Gross-Pitaevski
Equation
η(r) is macroscopic function. Hence MSPQB.
Quantised vortices, NCRI, macroscopic density oscillations.
71
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 volumes
containing NΩ atoms are the same for all s to within ~1/√N Ω
Hence BEC implies MSPQB
72
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
U (T )   Bi (T ) Ei
i
Bi (T )  exp(Ei / T )
*
Ei   i (r1 , r2 ...rN ) Hˆ i (r1 , r2 ...rN )dr1 dr2 ...drN
73
Consider one such “typical” occupied state with wave function Ψ(r,s)
Delocalisation implies MSPQB
MSPQB does not occur for T=TB
ψS(r) for occupied states cannot be delocalized at T=TB
But typical occupied state is delocalized as T→0
Typical occupied state Ψ(r,s) must change from localised
to delocalised function as T is reduced below TB.
74
D (s, r)
Delocalized
L (s, r)
Localized
(s, r)   (T )D (s, r)   (T )L (s, r)
α(T) =1 at T=0
α(T) =0 at T=TB
75
 S (r)  aS (T ) DS (r)  bS (T ) LS (r)
 DS (r)   0S (r)
D (s, r)  0 (s, r)

T 0
(r ) dr    DS (r ) dr    LS (r ) dr  1
2
S
aS (T )  1
2
2
T  TB aS (T )  0
76
 LS (r) ~ N / V
(a) r space
r ~ V / N
 DS (r) ~ 1 / V
(b) p space
V
~DS (p) ~ V
p ~ 1 / V
Overlap region ~1/√N
 LS (p) ~ V / N
r ~ V / N
 DS (r) ~ 1 / V
~-V
~V
~DS (p) ~ V
p space
p ~ 1 / V
 LS (p) ~ V / N
~ N /V
~ N /V
N/V
Overlap region ~1/√N

 S (r ) dr   aS
2
(r )
CT  aS bS 
*
2

(r )
 DS (r ) dr   bS
2
2

 LS (r ) dr   CT
2
(r )
 D* S (r ) LS (r )dr   CC
(r )
 LS (r)
Localised within
(r )
CT

2
(r )
 LS (r)
Localised outside
(r )
 LS (r) dr
~
1
N
CT  0
Contribution of CT is at most ~1/√N
79
Two fluid behaviour
 (r )  N   S (r ) P(s)ds
2
1
 (r )    (r)dr
 (r )
 (r)  D (r)  L (r) ~ 1/ N
Fluid
density

2
F(r )   P (s)  S (r )  S (r )ds
m
F(r)  FD (r)  FL (r) ~ 1/ N
Flow of delocalised component is quantised
No such requirement for flow of localised component
Localised component is superfluid
Delocalized component is normal fluid
Fluid
flow
(s, r)   (T )D (s, r)   (T )L (s, r)
 (r )   (T )  D (r )   (T )  L (r ) ~ 1 / N
2
 (T )   S (T )
2
Superfluid fraction
2
 (T )   N (T )
2
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)
E(r)  S ED (r)   N EL (r)
P(r)  S PD (r)   N PL (r)
 (T )
 (0)
J. Mayers Phys. Rev. Lett.
92 135302 (2004)
84
f (T )  s (T ) f (0)
D (s, r)  0 (s, r)
J. Mayers Phys. Rev. Lett.
92 135302 (2004)
Superfluid fraction
J. S. Brooks and R. J. Donnelly, J Phys. Chem.
Ref. Data 6 51 (1977).
Normalised condensate fraction
f (T )
f (0)
oo
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).
85
S (q)  S S0 (q)   N SL (q)
More spaces give smaller pair correlations
As T increases, superfluid fraction increases, pair correlations reduce
86
SB-1

ST ( q )  1
 (T ) 

S B (q )  1
ST -1
S (q)  S S0 (q)   N SL (q)
 (T )  1  S (T )[1   (0)]
V.F. Sears and E.C. Svensson,
Phys. Rev. Lett. 43 2009 (1979).
J. Mayers Phys. Rev. Lett.
92 135302 (2004)
α(T)
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
S (q,  )   A f (q)  (  E f  E )
2
f
A f (q)  N   * (r, s) exp( iq.r ) f (r, s)drds
Identical particles
(s, r)   (T )0 (s, r)   (T )L (s, r)
S (q, )  S S0 (q, )   N SL (q, )
Only S0 contributes to sharp peaks
90
Anderson and Stirling
J. Phys Cond Matt (1994)
91
New prediction
 (r)  S 0 (r)   N  L (r)
0 (r)
 L (r)
Has density oscillations identical to gnd state
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
92
Summary
Most important physical properties of BE condensed
systems 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
Measurement of flow without viscosity in solid helium
E. Kim and M. H. W. Chan
Science 305 2004
95
can
He4
Focussed data
Fitted widths on
Individual detectors
96
liquid
solid
Focussed data after subtraction of
can. Dotted line is resolution function
O single crystal high purity He4
X polycrystal high purity He4
□ 10ppm He3 polycrystal
97
Measured hcp lattice spacings
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)
98
• 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??
99
Kinetic Energy of He3
R. Senesi, C. Andreani, D. Colognesi, A. Cunsolo, M. Nardone,
Phys. Rev. Lett. 86 4584 (2001)
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).
3He
100
Measurements of protons
n(p) is the “diffraction pattern” of the wave function
n(p)   (r) exp(ip.r)dr
Position in Å
2
n(p) in Å-1
101
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 the
microscopic physical behaviour of protons on very short time scales.
Potential can also be reconstructed
p2 ~
 e 2M (p)dp
E  V (r) 
ip.r ~
e
  (p)dp
ip.r
~ (p)   (r )eip.r dr
VESUVIO Measurements on
Liquid H2
J Mayers (PRL 71 1553 (1993)

2R


  2
n( p)   (r ) exp(ip.r )dr
  (r  R) 2 

 (r )  C exp

2
 2

103


  2
n( p)   (r ) exp(ip.r )dr
104
Puzzle
Fit to data gives
R=0.36 σ=5.70
Spectroscopy gives
R=0.37 σ=5.58
QM predicts R=bond length not ½ bond length
R should be 0.74!
105
J(y)
Red data H2 1991
Black YAP 2008
p2n(p)
106
R=0.37
R=0.74
107
Single crystals
Heavy
Atoms
H
108
Reconstruction of Momentum Distribution
from Neutron Compton Profile
Hermite polynomial
J (qˆ, y) 
exp( y 2 )

Spherical Harmonic
a
n ,l , m
H 2 nl ( y)Yl ,m (qˆ )
n ,l , m
an,l,m is Fitting coefficient
 exp( p 2
n( p) 
3/ 2

2
2 n l
n!(1) n an,l ,m p l Lln1/ 2 ( p 2 )Ylm ( pˆ )
n ,l , m
Laguerre polynomial
109
110
111
Nafion
112
113
114
115
In press PRL (2010)
116
Measurements of n(p) give unique information
on the quantum behaviour of protons in a wide
range of systems of fundamental importance