SLIDE 1
Neutron Compton Scattering as a Probe Of Hydrogen Bonded Systems
George Reiter Jerry Mayers Collaborators
- P. Platzman
- J. Noreland
Neutron Compton Scattering as a Probe Of Hydrogen Bonded Systems - - PDF document
Neutron Compton Scattering as a Probe Of Hydrogen Bonded Systems - - PDF document
Neutron Compton Scattering as a Probe Of Hydrogen Bonded Systems George Reiter Jerry Mayers Collaborators P. Platzman J. Noreland J.C.Li Outline How do You Extract the Momentum Dis- tribution? How Much Detail Can One See? How Does
SLIDE 2
SLIDE 3
Neutron Scattering in the Impulse Approximation Limit Deep Inelastic Neutron Scattering
SM( q, ω) =
n(
p)δ(ω − ¯
hq2 2M − p. q M )d
p ¯ hω is the energy transfer, M is the mass of the tar- get particle, q=| q| is the magnitude of the wavector transfer and n( p) is the momentum distribution. SM( q, ω) = M
q
n(
p)δ(y − p.ˆ q)d p = M
q J(ˆ
q, y) y=M
q (ω − ¯ hq2 2M ), ˆ
q= q/q
SLIDE 4
A Very Useful Theorem
If we express J(ˆ q, y) as
J(ˆ q, y) = e−y2
π
1 2
- n,l,m
an,l,mH2n+l(y)Ylm(ˆ q)
then
n( p) = e−p2
π
3 2
- n,l,m
22n+ln!(−1)nan,l,mplL
l+1
2
n
(p2)Ylm(ˆ p)
SLIDE 5
Statistical Errors in Measured n( p)
δn( p) =
- i
δn( p) δρi δρi
< δn( p)2 >=
- i,j
δn( p) δρi δn( p) δρj
< δρiδρj >
SLIDE 6
5 10 15 20 25
pz-inv. Angstroms
0.001 0.002 0.003 0.004 0.005 0.006 0.007
n(p)
momentum distribution error limits anharmonicity
Momentum Distribution Along Bond
SLIDE 7
KHC2O4
SLIDE 8
Momentum Distribution for Two Displaced Gaussians
- f Relative Weight r
n(px, py, pz)=(1+r2+2rcos(2pza))
(1+r2+2re
− a2 2σz2)
- i
e
− p2 i 2σ2 i
(2πσi)
1 2
If r=1,then n(px, py, pz)=2cos2(pza)
(1+e
−a2 2σz2)
- i
e
− p2 i 2σ2 i
(2πσi)
1 2
SLIDE 9
SLIDE 10
−0.002 0.002 0.004 0.006 0.008
n(p)
10 20 30 Momentum (Inv. Angstroms)
KDP Momentum Distribution Along Bond
T=130K T=90K 10 20 30 Ab−initio Experiment
SLIDE 11
Direct Measurement of the Born-Oppenheimer Potential Φ( p) = ±
- n(
p) Ψ( r) =
ei
p· rΦ(
p)d p E − V ( r) =
p2
2Mei p· rΦ(
p)d p
ei
p· rΦ(
p)d p
SLIDE 12
−0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 Angstroms −500 500 1000 1500 2000 MILLI−EV
KDP Potential and Wavefunction
Potential Wave Function(arb. units)
SLIDE 13
SLIDE 14
10 20 30 momentum−inv. angstroms 0.002 0.004 0.006 0.008 0.01 radial momentum distribution p**2 n(p)
hydrogen momentum distribution in water
ice −4C 23C 1bar 300C 100 bar 400C 750 bar errors errors
SLIDE 15