Radial Amplitude and Phase Extraction

0 1 2 Delay [fs]

1 1.2 1.4 1.6 h 0()

0 1 2

Delay [fs]

1.5 2 2.5

h 2()

0 1 2

Delay [fs]

-1.5 -1 -0.5

h 4() (a)

(b)

Figure 3.9: (a) Fit of equation 3.16 (red line) to the photoelectron angular distribution (black dots), which is repeated for all recorded delay steps. Each fit results in one point in (b), showing the delay dependence of the obtained coefficients h_i(τ ), i = 0, 2, 4 for sideband 18. The error bars correspond to one standard deviation returned from the fit.

This can be further expanded by replacing the products of spherical harmonics by an expansion in associated Legendre polynomials, using the following relation:

Y_Lm(θ, ϕ) = (−1)^m s

(2L + 1)(L− m)!

4π(L + m)! P_L^m(cosθ)e^imϕ (3.18) The absolute square in equation 3.17 results in squared and cross terms between odd Legendre polynomials, which can be replaced by the following identities:

P₁⁰
₂

= 1

3 P₀⁰+ 2P₂⁰
P₁⁰P₃⁰ = 1

7 3P₂⁰+ P₄⁰
P₃⁰
2

= 1

7 P₀⁰+ 4

21 P₂⁰+18

77 P₄⁰+100 231P₆⁰ P₁¹
2

= 2

3 P₀⁰− P2⁰

P₁¹P₃¹ = 12

7 P₂⁰− P4⁰

P₃¹
2

= 12

7 P₀⁰+ P₂⁰
+36

77 P₄⁰−300 77 P₆⁰

Note that the associated Legendre polynomials are equal to the regular ones for m = 0 and thus P_L⁰ = P_L for L = 0, 2, 4. The angular coefficients C_l,l+1^m are derived by calculating the angular transition matrix element between an initial state with angular momentum ℓ and a final state with angular momentum ℓ + 1 as:

C_ℓ,ℓ+1^m = (−1)^ℓ+m−1(ℓ + 1)^1/2

 ℓ + 1 1 ℓ

−m 0 m

 .

Finally, only Legendre polynomials of the 0^th, 2^ndand 4^thorder are left, as all sixth­

order terms vanish. Hence, we recover the general form: ISB(θ, τ ) = h₀(τ ) + h2(τ )P2(cos θ) + h4(τ )P4(cos θ), which is identical to equation 3.16. The three

parameters hi, i = 0, 2, 4can thus be identified as:

h₀(τ ) = 25 9

h

(σ⁺₁₀₁)²+ (σ⁻₁₀₁)² i

+34 9

h

(α⁺₂σ₃₂₁⁺ )²+ (α⁻₂σ₃₂₁⁻ )² i

+ 4 h

(σ₃₂₁⁺ )²+ (σ₃₂₁⁻ )² i

+40 9

h

α⁺₂σ₁₀₁⁺ σ₃₂₁⁺ cos φ⁺₀₁− φ⁺₂₁+ ϕ⁺₁₀− ϕ⁺₁₂
+ α⁻₂σ₁₀₁⁻ σ₃₂₁⁻ cos φ⁻₀₁− φ⁻₂₁+ ϕ⁻₁₀− ϕ⁻₁₂
i

+50

9 σ⁺₁₀₁σ₁₀₁⁻ cos 2ωτ + ∆Φ2q+ φ⁺₀₁− φ⁻₀₁+ ϕ⁺₁₀− ϕ⁻₁₀
+40

9 h

(α⁻₂σ₁₀₁⁺ σ₃₂₁⁻ cos 2ωτ + ∆Φ2q+ φ⁺₀₁− φ⁻₂₁+ ϕ⁺₁₀− ϕ⁻₁₂
+ α⁺₂σ₃₂₁⁺ σ₁₀₁⁻ cos 2ωτ + ∆Φ2q+ φ⁺₂₁− φ⁻₀₁+ ϕ⁺₁₂− ϕ⁻₁₀
i +68

9 α⁺₂α⁻₂σ₃₂₁⁺ σ₃₂₁⁻ cos 2ωτ + ∆Φ2q+ φ⁺₂₁− φ⁻₂₁+ ϕ⁺₁₂− ϕ⁻₁₂
+ 8 σ₃₂₁⁺ σ₃₂₁⁻ cos 2ωτ + ∆Φ2q+ φ⁺₂₁− φ⁻₂₁+ ϕ⁺₃₂− ϕ⁻₃₂

h₄(τ ) = 24 7

h

(σ⁺₃₂₁)²+ (σ⁻₃₂₁)² i

+16 7

h

α⁺₂(σ₃₂₁⁺ )² cos ϕ⁺₃₂− ϕ⁺₁₂
+ α⁻₂(σ₃₂₁⁻ )² cos ϕ⁻₃₂− ϕ⁻₁₂
i

+80 7

h

σ⁺₁₀₁σ⁺₃₂₁cos φ⁺₀₁− φ⁺₂₁+ ϕ⁺₁₀− ϕ⁺₃₂
+ σ⁻₁₀₁σ₃₂₁⁻ cos φ⁻₀₁− φ⁻₂₁+ ϕ⁻₁₀− ϕ⁻₃₂
i

+48

7 σ⁺₃₂₁σ₃₂₁⁻ cos 2ωτ + ∆Φ2q+ φ⁺₂₁− φ⁻₂₁+ ϕ⁺₃₂− ϕ⁻₃₂
+16

7 h

α⁻₂σ⁺₃₂₁σ⁻₃₂₁cos 2ωτ + ∆Φ2q+ φ⁺₂₁− φ⁻₂₁+ ϕ⁺₃₂− ϕ⁻₁₂
+ α⁺₂σ₃₂₁⁺ σ₃₂₁⁻ cos 2ωτ + ∆Φ2q+ φ⁺₂₁− φ⁻₂₁+ ϕ⁺₁₂− ϕ⁻₃₂
i +80

7 h

σ₃₂₁⁺ σ⁻₁₀₁cos 2ωτ + ∆Φ2q+ φ⁺₂₁− φ⁻₀₁+ ϕ⁺₃₂− ϕ⁻₁₀
+ σ⁺₁₀₁σ₃₂₁⁻ cos 2ωτ + ∆Φ2q+ φ⁺₀₁− φ⁻₂₁+ ϕ⁺₁₀− ϕ⁻₃₂
i

h₂(τ ) = 50 9

h

(σ₁₀₁⁺ )²+ σ⁻₁₀₁)² i

+ 14 9

h

(α⁺₂σ₃₂₁⁺ )²+ (α₂⁻σ₃₂₁⁻ )² i

+32 7

h

(σ₃₂₁⁺ )²+ (σ₃₂₁⁻ )² i

+80 9

h

α⁺₂σ₁₀₁⁺ σ₃₂₁⁺ cos φ⁺₀₁− φ⁺₂₁+ ϕ⁺₁₀− ϕ⁺₁₂
+ α₂⁻σ⁻₁₀₁σ⁻₃₂₁cos φ⁻₀₁− φ⁻₂₁+ ϕ⁻₁₀− ϕ⁻₁₂
i +96

7 h

α⁺₂(σ⁺₃₂₁)²cos ϕ⁺₃₂− ϕ⁺₁₂

+ α⁻₂(σ₃₂₁⁻ )²cos ϕ⁻₃₂− ϕ⁻₁₂
i +60

7

σ₁₀₁⁺ σ₃₂₁⁺ cos φ⁺₀₁− φ⁺₂₁+ ϕ⁺₁₀− ϕ⁺₃₂

+ σ₁₀₁⁻ σ₃₂₁⁻ cos φ⁻₀₁− φ⁻₂₁+ ϕ⁻₁₀− ϕ⁻₃₂
i +100

9 σ₁₀₁⁺ σ⁻₁₀₁cos 2ωτ + ∆Φ2q+ φ⁺₀₁− φ⁻₀₁+ ϕ⁺₁₀− ϕ⁻₁₀
+80

9 h

α⁻₂σ₁₀₁⁺ σ₃₂₁⁻ cos 2ωτ + ∆Φ2q+ φ⁺₀₁− φ⁻₂₁+ ϕ⁺₁₀− ϕ⁻₁₂
+ α₂⁺σ⁺₃₂₁σ⁻₁₀₁cos 2ωτ + ∆Φ2q+ φ⁺₂₁− φ⁻₀₁+ ϕ⁺₁₂− ϕ⁻₁₀
i +60

7 h

σ₃₂₁⁺ σ₁₀₁⁻ cos 2ωτ + ∆Φ2q+ φ⁺₂₁− φ⁻₀₁+ ϕ⁺₃₂− ϕ⁻₁₀
+ σ₁₀₁⁺ σ₃₂₁⁻ cos 2ωτ + ∆Φ2q+ φ⁺₀₁− φ⁻₂₁+ ϕ⁺₁₀− ϕ⁻₃₂
i +28

9 α⁺₂α⁻₂σ⁺₃₂₁σ⁻₃₂₁cos 2ωτ + ∆Φ2q+ φ⁺₂₁− φ⁻₂₁+ ϕ⁺₁₂− ϕ⁻₁₂
+64

7 σ₃₂₁⁺ σ₃₂₁⁻ cos 2ωτ + ∆Φ2q+ φ⁺₂₁φ⁻₂₁+ ϕ⁺₃₂− ϕ⁻₃₂
+96

7 h

α⁻₂σ₃₂₁⁺ σ₃₂₁⁻ cos 2ωτ + ∆Φ2q+ φ⁺₂₁− φ⁻₂₁+ ϕ⁺₃₂− ϕ⁻₁₂
+ α₂⁺σ⁺₃₂₁σ⁻₃₂₁cos 2ωτ + ∆Φ2q+ φ⁺₂₁− φ⁻₂₁+ ϕ⁺₁₂− ϕ⁻₃₂
i

As is clear from these equations, each h_i(τ )oscillates with the delay τ at frequency 2ω. The values for α⁽_λ^±) and ϕ⁽_Lλ^±) are taken from figures 3.3 and 3.4. The nine unknown quantities are now determined through a simultaneous fit of the three analytical expressions of hi(τ )(equations above), to the experimentally measured coefficients hi(τ )in figure 3.9. The result of this global fit is shown in figure 3.10 for sideband 18. The extracted phases are however not completely independent, since they only appear as differences in the analytical expressions above, which means that they carry a global offset and are not absolute. However, if one of the phases is locked, e.g. φ⁻₀₁, all of the others can be determined. To map out the

0 0.5 1 1.5 2 2.5 Delay [fs]

-2 -1 0 1 2 3 h₂

h₀

h₄

Figure 3.10: Global fit results of sideband 18 for all three asymmetry parameters h_i, where i = 0, 2, 4 are shown in red, green and blue respectively.

energy dependence of the one­photon phases, the global fit is repeated for each sideband. Using the fact that φ⁻_λ1(SB_n) = φ⁺_λ1(SB_n+2), the evolution of the one­photon phases as a function of energy is iteratively retrieved. Consequently, the superscript (±) can be neglected, since in the following the phases are shown as a function of the kinetic energy.

Figure 3.11 displays the extracted phases for λ = 0 (green) and λ = 2 (blue). The solid lines correspond to calculated values, based on the angular­channel­resolved many­body perturbation theory. As explained above, the procedure only enables us to determine the phases up to a global phase offset. After adjusting this phase offset, the theoretical and experimental results are in excellent agreement: both the energy dependence and the difference between the s­ and the d­channel is well reproduced by the experimental results. The deviation of the highest energy might be due to low statistics caused by the comparably low intensity of the 25^th harmonic.

The phase φ_λℓresults from the radial part of the final scattering state, as introduced in equation 3.5 and is a sum of the Coulomb phase ςλ, a contribution δλfrom the short­range potential and a contribution from the centrifugal barrier−πλ/2. In order to investigate the influence of each term, figure 3.11 also shows the Coulomb phases ςλ (dashed lines), as well as ςλ + δ_λ− πλ/2 (dotted lines), where δλ are taken from Kennedy and Manson [94]. The calculated as well as the experimen­

tally retrieved phases are very close to ς_λ+ δ_λ− πλ/2 (cf. the solid and dotted lines). The difference between φ01and ς0is due to the short range potential con­

tributing with∼ 1.2π, while that between φ02 and ς2 is mainly a result of the effect of the centrifugal barrier, leading to a π phase shift. Both phase shifts are indicated by black arrows. For λ = 2, short range effects are small because of

0 2 4 6 8 10 12 14 16 18 20 -2

0 2 4 6

~1.2

Figure 3.11: Extracted one-photon scattering phases φ01(green) and φ21(blue) for experimental (dots) and simulated data (solid lines). The dashed lines show the contributions from the Coulomb phase ςλ, where for the dotted lines the effect of the short-range potential and the centrifu-gal effect is added: ςλ+ δ_λ− πλ/2. For λ = 0 this shift is dominated by the short range potential (δ0≈ 1.2π) and for λ = 2 by the centrifugal effect (π), as indicated by the arrows.

the centrifugal barrier that prevents the d­electron from coming close to the core.

Since the effect of the short­range potential for the s­electron is comparable to that of the centrifugal barrier for the d­electron, the two contributions almost cancel each other out. Hence, the difference between φ01and φ21is primarily due to the difference in Coulomb phases. The increase of the phases for low energies and the asymptotic behavior for higher energies also follows the behavior of the Coulomb phases.

Finally, in order to fully characterize single photon ionization, the radial ampli­

tudes σλℓare determined as described above. Again, equation 3.13 is rewritten in terms of Legendre polynomials (equation 3.15) where the coefficients h0 and h2

result in:

h₀ = 1 12π

h

σ₀₁² + σ²₂₁ i

, (3.19)

h2 = 1 3π

h1

2σ²₂₁+ σ01σ21cos(φ01− φ21) i

. (3.20)

The coefficients h0 and h2 are extracted from the experimental data. Using the single­photon phases obtained previously, the relative radial amplitudes of the λ = 0and λ = 2 channel are determined from equation 3.19 and 3.20. The absolute values of the amplitudes depend on the experimental parameters, in particular the harmonic intensity, which is why figure 3.12 displays the ratio between them. The black solid line corresponds to the calculated values. As can be seen, the ratio is above one for all energies, which agrees well with Fano’s propensity rule for one­

photon absorption [95].

0 2 4 6 8 10 12 14 16 18 20 1

1.5 2 2.5

Figure 3.12: Ratio between one-photon amplitudes σ21and σ01extracted from XUV only data (dots). The black line shows the result of the calculations.

The above mentioned extraction of amplitude and phase of the s and d ionization channels fully characterizes the individual matrix elements and thus the photoion­

ization dynamics. In particular, the study of the phases of the different angular mo­

mentum channels, unravels the interplay between short­range, correlation and/or centrifugal effects. The method is general and can be applied to other atomic shells (e.g. d).

Chapter 4

Photodissociation

The following chapter focuses on the dissociation dynamics of carbon­based molecules upon ionization by XUV pulses. Paper Iv sets out how the Intense XUV Beamline was used to study the diamondoid adamantane based on corre­

lated ion and electron spectroscopy using the double­sided VMIS. Characterizing the charged fragments enabled a detailed investigation of the individual dissocia­

tion channels.

Further, as described in papers vI and vII, experiments were carried out at Free­

Electron­Lasers (FELs) using pump­probe schemes to resolve dissociation dynam­

ics on a femtosecond timescale.

In document Atomic and Molecular Dynamics Probed by Intense Extreme Ultraviolet Attosecond Pulses Peschel, Jasper (Page 82-90)