time--dependent simulations of electron emission in

Time–dependent simulations ofelectron emission in grazingion–surface collisions

Darıo MitnikInstituto deAstronomıay Fısica del Espacio

Departamento deFısicaUniversidad deBuenos Aires

Argentina

Preliminars Outline Track Schrodinger Track expansion Coupled–Channels Results 2p Relaxation Continuum Projection Propagation

Darıo MitnikJorge MiragliaMarıa GravielleIAFE – UBA

Andres ArnauInigo AldazabalUPV – EHU

Vıctor PonceCAB

Darıo Mitnik (IAFE – UBA) Time–dependent ion–surface collisions ISIACXX

Grazing ion–surface collision

Capture to the Continuum:The Convoy Peak

Continuum Capture: The Convoy Peak

G.B. Crooks and M.E. Rudd,

Phys. Rev. Lett. 25, 1599

(1970).

The Convoy Peak is (de)accelerated?

M.S. Gravielle and J.E.

Miraglia, Phys. Rev. A 67,

042901 (2003)

The Convoy Peak is (de)accelerated?

G.R. Gomez et al., Phys.

Rev. B 58, 7403 (1998)

Outline

I Development of theoretical strategies for ab initio calculationsof ion–surface collisions

I Development of numerical methods for solving theTime–dependent Schrodinger Equation

I Study of the Convoy Peak Shift

Outline

The track potential

I The incident projectil is at the origin of the laboratory frameof reference, at t = 0

The track potential

I The charge density when the ion is in position {0,Y (Z ),Z (t)}

n(z , t) = n0e−α|Y (Z)|e−

β|z−Z(t)|vz

The track potential

I The track potential is

V (x , y , z , t) =

∫ Z(t)

−∞du

n(u, t)√x2 + y2 + (u − z)2

The track potential

I Change of coordenates: The origin at the ion

x ′ = x

y ′ = y − Y (Z )

z ′ = z − Z (t)

V (x ′, y ′, z ′, t) =

∫ Z(t)

−∞du

n(u, t)√x ′2 + (y ′ + Y )2 + (u − (z ′ + Z ))2

The track potential

-40 -20 0 20 40-0.5

The Track Potential

The Track Potential + Coulomb

The time–dependent Schrodinger equation

idΨ(r, t)

dt= H(r, t)Ψ(r, t)

has the solution

Ψ(r, t + ∆t) = exp[−i∆tH]Ψ(r, t)

Simple case: radial potential

if the potential V = V (r) → variable separation

ψnlm(r) = Rnl(r)Yml (r) ≡ Pnl(r)

The radial part of the Hamiltonian becomes

H(r) = −1

∂r2+

l(l + 1)

2r2+ V (r)

≡ T + V (r)

and the time–independent Schrodinger equation is

H(r)Pnl(r) = EnlPnl(r)

the time evolution of this solution is given by

ψnlm(r, t) = exp[−i∆tH]Rnl(r , t0)Yml (r)

= exp[−i∆tEnl ]Pnl(r , t0)

The Schrodinger equation

In our case, the track potential

I Is a time–dependent potential

I Is not a central potential

Expansion of the track potential inspherical harmonics

V (r, t) = V (r , θ, φ, t) =∑lm

Vlm(r , t)× Y ml (θ, φ)

Vlm(r , t) ≡∫ 2π

0∂φ

∫ π

0∂θ sin θ Y m

l∗(θ, φ)V (r, t)

Expansion of the track potential inspherical harmonics

V (r, t) = V (r , θ, φ, t) =∑lm

Vlm(r , t)× Y ml (θ, φ)

Vlm(r , t) ≡∫ 2π

0∂φ

∫ π

0∂θ sin θ Y m

l∗(θ, φ)V (r, t)

Coupled–ChannelsUsing the expansion

ψ(r) =∑l1,m1

Pl1,m1(r)

l1(r),

the time–independent Schrodinger equation becomes

H(r)ψ(r) = [T + V (r)]∑l1,m1

Pl1,m1(r)Ym1l1

= E∑l1,m1

Pl1,m1(r)Ym1l1

Coupled–ChannelsUsing the expansion

ψ(r) =∑l1,m1

Pl1,m1(r)

l1(r),

the time–independent Schrodinger equation becomes

H(r)ψ(r) = [T + V (r)]∑l1,m1

Pl1,m1(r)Ym1l1

= E∑l1,m1

Pl1,m1(r)Ym1l1

Coupled–Channelsand using the potential expansion

V (r) =∑l2,m2

Vl2,m2(r)Ym2l2

the Schrodinger equation is

V (r) =∑l2,m2

Vl2,m2(r)Ym2l2

[T + V (r)]∑l1,m1

Pl1,m1(r)Ym1l1

= E∑l1,m1

Pl1,m1(r)Ym1l1

V (r) =∑l2,m2

Vl2,m2(r)Ym2l2

[T +∑l2,m2

Vl2,m2(r)Ym2l2

(r)]∑l1,m1

Pl1,m1(r)Ym1l1

= E∑l1,m1

Pl1,m1(r)Ym1l1

Coupled–Channels

The term ∑l2,m2

Vl2,m2(r)Ym2l2

(r)∑l1,m1

Pl1,m1(r)Ym1l1

is expanded as∑l1,m1

Pl1,m1(r)∑l2,m2

Vl2,m2(r)Ym1l1

(r)Y m2l2

=∑l1,m1

∑l2,m2

Pl1,m1(r)Vl2,m2(r)∑l3,m3

Sm3,m1,m2

l3,l1,l2Y m3

Coupled–Channels

∑l1,m1

∑l2,m2

Sm3,m1,m2

l3,l1,l2Y m3

Sm3,m1,m2

l3,l1,l2= (−1)m3

√(2l1 + 1)(2l2 + 1)(2l3 + 1)

l3 l1 l20 0 0

) (l3 l1 l2−m3 m1 m2

Coupled–Channels

[T +∑l2,m2

Vl2,m2(r)Ym2l2

(r)]∑l1,m1

Pl1,m1(r)Ym1l1

= E∑l1,m1

Pl1,m1(r)Ym1l1

Coupled–Channels

T∑l1,m1

Pl1,m1(r)Ym1l1

+∑l1,m1

∑l2,m2

Sm3,m1,m2

l3,l1,l2Y m3

l3(r) =

= E∑l1,m1

Pl1,m1(r)Ym1l1

Coupled–Channels

multiplying and integrating

∫∂ rY m3

l3(r) T

∑l1,m1

Pl1,m1(r)Ym1l1

∫∂ rY m3

∑l1,m1

∑l2,m2

Sm3,m1,m2

l3,l1,l2Y m3

l3(r) =

∫∂ rY m3

l3(r) E

∑l1,m1

Pl1,m1(r)Ym1l1

Coupled–Channels

we obtain coupled equations for every channel

TPl3,m3(r) +∑l1,m1

∑l2,m2

Vl2,m2(r)Sm3,m1,m2

l3,l1,l2Pl1,m1(r) =

= E∑l1,m1

Pl3,m3(r)

ResultsStarting from the 1s state of hydrogen

Replay Movie

Results

Starting from the 2p1 state of hydrogen

Results

The relaxation method

Propagation of the Schrodinger equation in imaginary time

τ ≡ it

∂Φ(r, τ)

∂τ= −HΦ(r, τ)

The solution of this equation

Φ(r, τ) = e−HτΦ(r, 0)

is expanded in the stationary eigenvectors

Φ(r, τ) =∞∑

aqe−Eqτψq(r).

In the limit τ →∞, Φ tends to the ground state:

limτ→∞

Φ(r, τ) = ψ1(r).

Starting from the 1s hydrogenic function, the relaxed ground stateis obtained by an imaginary time propagation

Results

Starting from the relaxed ground state

Results

Capture in the Continuum

Assuming that the electron is captured in a low–energy continuumstate

Φ(r , 0) =1

(ωπ)1/4exp

[−(r − a)2

]exp [−iki r ]

Propagation with Coulombic potential

Replay Movie

Projection to continuum states

Propagation: Coulombic + Track potential

Replay Movie

Projection to continuum states

Conclusions

1. We are trying to simulate a grazing ion–surface scatteringprocess

2. We developed numerical solutions of the time–dependentSchrodinger equation

3. As a first step, we assume an electron captured on acontinuum state of the projectil

4. A track potential represents the effect of the surfaceinteraction on the electronic state

5. We analyze the effect of the track potential on the electronvelocity distribution

6. Our method is able to detect a deceleration of theconvoy–electron peak, due to the surface interaction, inquantitative agreement with the experimental data

Conclusions1. We are trying to simulate a grazing ion–surface scattering

process

time--dependent simulations of electron emission in

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