Mass selective nuclear momentum distribution measurements: H, D, Li - - PowerPoint PPT Presentation

Mass selective nuclear momentum distribution measurements: H, D, Li and beyond

Nuclear momentum distribution in LiH and LiD

Motivation:

• Testing the Validity of the Born-

Oppenheimer Approximation in NCS;

• Benchmarking ab initio results from

CASTEP PW-DFT calculation;

• Testing whether nuclear momentum

distribution can be measured for lithium nuclei with accuracy beyond the mean kinetic energy.

Validity of the Born-Oppenheimer Approximation in NCS

Collisonally-induced non-adiabatic electronic excitation owing to ultra-short (i.e. attosecond) timescale of the NCS process. The value of the Watson scattering time: Gives us estimate of the energy scale involved. When the time becomes, we obtain which is of the order of the separation of electronic levels.

M q τ σ =

q q



16

10 s τ

− q 

/ 6.5 eV τq  

Validity of the Born-Oppenheimer Approximation in NCS

Two different mechanisms responsible for electronic excitation:

• the centre of mass recoil (CMR) effect;
• the non-adiabatic coupling (NAC) between the electrons and the nuclei

Validity of the Born-Oppenheimer Approximation in NCS

Adiabatic approximation to molecular systems reveals that it is insufficient for describing the electronic excitation

( )

( ) ( ) ( )

2

; ; , ;

t

i m m f i i f n n n

P E E e R r X R r R = Ψ Ψ Ψ = Φ

q R q 

( ) ( ) ( )

2 ' ' ' '

; '

i m m n m nm n n nn

P E E d X e X δ

⋅

= ∫

t

q R q

R R R

We therefore find that the adiabatic approximation does not allow for other than CMR type of electronic excitation Using the orthogonality condition for the electronic wavefunctions, we get:

Validity of the Born-Oppenheimer Approximation in NCS ( )

,

,t

ω

Ψq R

Represents a wave-packet composed of plane waves with a distribution of momenta around q and moving with recoil velocity. The energy spread of the average wavepacket:

2 2 2 2

4 3 2 2

q

q E M M σ τ

Ψ

    ∆ = =        

q

q

  

Validity of the Born-Oppenheimer Approximation in NCS

( ) ( ) ( )

2 2 1 2

, , , S S S ω α ω β ω = +

q q

q q q

( ) ( )

2 2 2 2 2 2 1 2

, and , are centred at and 2 2 E E q q S S E M M E E ω ω

Ψ Ψ

∆ ∆ − + +

q q

q q  

For high q, the scattering function splits in two distributions, S1 (main) and S2 (secondary), separated by

2

2 E E E

Ψ

∆ +

q

Validity of the Born-Oppenheimer Approximation in NCS

Expansion in the 1σg and 2σg basis

Testing the Validity of the Born-Oppenheimer Approximation in NCS by measuring nuclear momentum distributions in LiH and LiD

Comparison with ab initio predictions: LiH electronic band structure of ca. 2.99 eV at the Gamma point. PW-DFT GGA PBE 750 eV

Testing the Validity of the Born-Oppenheimer Approximation in NCS by measuring nuclear momentum distributions in LiH and LiD

Comparison with ab initio predictions: VDOS and PVDOS for H and Li in LiH PW-DFT GGA PBE 750 eV

Testing the Validity of the Born-Oppenheimer Approximation in NCS by measuring nuclear momentum distributions in LiH and LiD

Comparison with ab initio predictions: VDOS and PVDOS for D and Li in LiD PW-DFT GGA PBE 750 eV

Testing the Validity of the Born-Oppenheimer Approximation in NCS by measuring nuclear momentum distributions in LiH and LiD

Comparison with ab initio predictions: momentum distributions Theory Previous experiments

Testing the Validity of the Born-Oppenheimer Approximation in NCS by measuring nuclear momentum distributions in LiH and LiD

Comparison with ab initio predictions: momentum distributions Theory Values calculated from Experimental σ values using: On the basis of the above, the work estimates an upper conservative bound of 2-3 % for the effects of non-adiabatic dynamics on the second moment and Laplacian of the atomic momentum distributions in this benchmark system.

Testing the Validity of the Born-Oppenheimer Approximation in NCS by measuring nuclear momentum distributions in LiH and LiD

Idea: compare GCs in two isoelectric systems, e. g. LiH and LiD

100 200 300 400 500 600

• 60
• 40
• 20

20 40 60 TOF [us] y [inv. A] y y1 (main) y2 (secondary)

H in LiH

100 200 300 400 500 600

• 100
• 50

50 100 150 200 250 300 350 400 TOF [us] y [inv. A] y y1 (main) y2 (secondary)

D in LiD ( ) ( ) ( )

1 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2

4 2 3 2 2 4 2 3 2 2

gap gap gap

M J y J y J y q M q q y E q M M M M q q y E E q M M M σ ω σ ω = +       = − +       = − − −            

( ) ( )

1 2 2 2 2 2 2 2

, and , are centred at and 2 2 S S E E q q E M M E E ω ω

Ψ Ψ

∆ ∆ − + +

q q

q q  

Testing the Validity of the Born-Oppenheimer Approximation in NCS by measuring nuclear momentum distributions in LiH and LiD

Idea: compare GCs in two isoelectric systems, e. g. LiH and LiD Theory Simulation @ 300K Scattering angle 50 degrees Band gap = 3 eV Excitation probability 5% LiH LiD ( ) ( ) ( )

1 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2

4 2 3 2 2 4 2 3 2 2

gap gap gap

M J y J y J y q M q q y E q M M M M q q y E E q M M M σ ω σ ω = +       = − +       = − − −            

50 100 150 200 250 300 350 400 450 500 550 600 0.05 0.1 0.15 0.2 0.25 0.3 0.35 θ = 50 o; t0 = 0 µ s; L0 = 10.7 m; L1 = 0.7 m; E1 = 4908 meV; ELor

1

= 24 meV; EGauss

1

= 73 meV; TOF [us]

• arb. units

TOF spectrum simulation

50 100 150 200 250 300 350 400 450 500 550 600 0.02 0.04 0.06 0.08 0.1 0.12 θ = 50 o; t0 = 0 µ s; L0 = 10.7 m; L1 = 0.7 m; E1 = 4908 meV; ELor

1

= 24 meV; EGauss

1

= 73 meV; TOF [us]

• arb. units

TOF spectrum simulation

Testing the Validity of the Born-Oppenheimer Approximation in NCS by measuring nuclear momentum distributions in LiH and LiD

where and L = (H,D) for forward scattering or L = Li for backscattering. Furthermore, with M = Li in forward scattering and M = Al in both forward and backscattering.

Ockham’s razor: Forwad scattering

Ockham’s razor: Backward scattering

Model no 4 (forward scattering): LiH LiD

Model no 4 (backscattering): LiH LiD

Testing the Validity of the Born-Oppenheimer Approximation in NCS by measuring nuclear momentum distributions in LiH and LiD

Sums of normalised data and fits: LiH LiD Ockham's razor: Gaussian Approximation (GA) provides a satisfactory description of NMDs in LiH and LiD. Possible higher-order effects are sufficiently small such that their overall influence on the estimation of second moments for H, D, and Li is practically negligible in both LiH and LiD.

Going beyond lithium: simultaneous NMDs of H, C and O in the squaric acid across the paraelectric phase transition

h-SA in the paraelectric phase is characterised by an equal sharing of protons across adjacent oxygen atoms on the a-c plane, a situation where nuclear quantum effects (e.g., tunnelling through barriers) would be expected to dominate.

Going beyond lithium: simultaneous NMDs of H, C and O in the squaric acid across the paraelectric phase transition

Backscattering data @ 300K Forward scattering data @ 300K

Going beyond lithium: simultaneous NMDs of H, C and O in the squaric acid across the paraelectric phase transition

100 200 300 400 500 4.50 4.55 4.60 4.65 4.70 4.75 4.80 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0

width of proton momentum distributio (A

• 1)

temperature (K)

width of proton momentum distributio width of oxygen momentum distributio width of carbon

Heavy atom nuclear momentum distributions from Gamma resonances

10000 20000 30000 40000 50000

• 0.1

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

CsHSO4 @ 10K, Detector 135 C[t] Despite different values of lifetime widths at 5.5 and 22.5 eV the Doppler widths are equal within one st dev. They are: 471.9 meV (FWHM) = 200.4 (st dev) @ 5.5 eV 466.9 meV (FWHM) = 198.3 meV (st dev) @22.5 eV

C(E0) = A L(E0, dE0/dE1 dE1_lor) ⊗ G_th(E0, dE0/dE1 dE1_gauss) ⊗ G_x(E0,dE0/dx dx)

Cs Θ Cs Cs n n n n γ How to account for a resonance peak? dx = v0 h/wL

Normally, the resonance is described as Lorentzian in E0, centred at The Lorentzian with the width dE1L responsible for the lifetime broadening is convolved with a Gaussian term in E0 with the width dE1G responsible for thermal broadening: Using the Y scaling this is replaced by a J(y) of resonant nucleus M centred at: The Gaussian J(Y) is convolved with the resolution function R(y) In the Y space of the resonant nucleus M: Formulation of the resonance description using Y scaling

( )

1 1 1

2 1 cos m E E R E E E M θ − − = ⇔ − − − =    

( )

2 2 1 1

2 1 cos ,

M G eff eff

m dE T E T M M σ θ = − =     

[ ]

( )

1 1 1

2 1 cos

M

M M m y E E R E E E q q M θ   = − − = ⇔ − − − =        

M Θ n Cs Θ’ n γ

( ) ( )

( )

( )

( )

( )

, 1

' ' 0 , 1 , , ', , 1

' 1 , , ', , , ,

RES EN M RES

M M M R RES N R y L RES GEOM M RE ES M SR S

d d A e A N J y L E I E C t q R y L

θ π θ π θ θ θ θ

θ θ θ θ θ θ

= = = = − =

        −     ⊗ =      = ⊗  =

∑ ∫ ∑ ∫

First part of the problem: scattering plus resonances inside the sample

1

L → 10 L m ≈

100 200 300 400 500 600

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 TOF [us] Counting Rate tempcms_135.dat TOF spectrum fit

The Cs scattering – Cs resonance proce The sum of X scattering – Cs resonance processes With X = H, O, Al and S Ratio of the scattering – resonance To pure resonance is like SS/DS, i. e. 1%/10% = 1/10 First part of the problem: the scattering plus resonances inside the sample

n Cs Θ’ γ

1

L → 10 L m ≈

Second part of the problem: the pure resonances inside the sample

( ) ( )

( )

( )

( )

, 1

' , ', ' 0 , 1

' 1 , , ',

RES EN M RES

N R y L RES RES GEOM M RES

d A e E I E C t q R y L

θ π θ θ

θ θ θ

= − = =

             ⊗ = − = 

∑ ∫

100 200 300 400 500 600

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 TOF [us] g tempcms_135.dat TOF spectrum fit

The Cs scattering – Cs resonance process The sum of X scattering – Cs resonance processes With X = H, O, Al and S Ratio of the scattering – resonance To pure resonance is like SS/DS, i. e. 1%/10% = 1/10 Pure Cs resonance The fitted width is ca 67 inv A and fitted Nd sigma_0 is 0.88 H recoil peak fitted width is ca 2 inv A Second part of the problem: the pure resonances inside the sample

VESUVIO inverse geometry spectrometer

( ) ( ) ( ) ( )

( )

( )

( )

( )

( )

( )

( )

1 1

' , ', 1 ' 0 ' , , ' , 1 , 1 ' 0

' 1 , ', , , , , , ' 1 , , ',

RES EN RES EN R S RE E S

N R y L N R y L RES GEOM RES GEOM R M M M M RES ES RE M M M M RES M S RES

d d A N J y R y E I E C t d A e R y A e R y L q A N J L y L

θ θ π θ π π θ θ θ θ θ θ

θ θ θ θ θ θ θ θ θ θ

= − = = = − = = = =

            − ⊗ − ⊗ = =   + = = ⊗ + ⊗

∑ ∫ ∑ ∑ ∑ ∫ ∫

       

100 200 300 400 500 600

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 TOF [us] Counting Rate tempcms_135.dat TOF spectrum fit

sigma_H = 2.4387 inv. A Cs resonance at 5850 meV fitted with Nd sigma_01 of 6.7096 Cs resonance at 22500 meV fitted with Nd sigma_02 of 4.0530 Cs resonance at 47800 meV fitted with Nd sigma_03 of 29.7282 Cs resonance at 94800 meV fitted with Nd sigma_04 of 2.2204e-014

( )

( )

( )

1

' , ', 1 ' 0

' 1 , ',

RES EN RES

N dR y L RES GEOM RES RES

d A e R y L

θ π θ θ

θ θ

= − = =

− ⊗ = +

∑ ∫

The ratios of the Ndsigma_0 values for the two resonances can be constrained knowing that for a pair of (i,j) resonances: Ndsigma_0_i/ Ndsigma_0_j = sigma_0_i / sigma_0_j Thus, Ndsigma_0_1/ Ndsigma_0_2 was constrained at 7062.3/1477. The rationale behind this is that even if the values of sigma_0 will not be exactly true from ENDF-B6 their ratios may be;

100 200 300 400 500 600

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 TOF [us] Counting Rate tempcms_135.dat TOF spectrum fit

100 200 300 400 500 600

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 TOF [us] Counting Rate

Ndsigma_0_1/ Ndsigma_0_2 constrained at 7062.3/1477 leads to a fit converged after 13 iterations. The fitted value of sigma_Cs was 9.6 inv Angstrom. The fitted value of Sigma_H was 2.44 +/- 0.15 inv A within 1STD from ab initio predictions

CsHSO4: projected VDOS from CASTEP H O S Cs

Koppel et al. [PRC, vol. 1, issue 6 (1970), 2054]: A solid system can still be represented by a gas model with Teff under the so-called weak binding regime when the following condition is satisfied: wL/2 + 1/sqrt(2)*wG >> Debye temperature in meV The Debye temp for CsHSO4 is ca 15 meV from the projected Cs VDOS (our ab initio calculation with Castep) and ca 3.7 meV for pure Cs crystal. Thus, pure Cs crystal at 300K has almost the same kTeff and mom. distr as CsHSO4 at 300K!