1.6 Angular dependence of the atomic time delay
1.6.2 Argon
In the case of argon, the same shape and field amplitude of the electromagnetic perturbation was used. To improve the RPAE approximation we substitute the HF energy eigenvalues with the experimental ionization thresholds [16]. Therefore, the energy difference between the 3sand 3psubshells is 13.48 eV [83]. To avoid some accidental photoionization of both initial subshells (3sand 3p), we have to adjust the spectral width which has to be narrower.
Therefore, we use an XUV pulse with a longer duration in comparison to the neon case.
The FHWM of 300 as is sufficient.
The partial cross sections corresponding to the photoionization process of the 3sand 3p subshells are depicted in Fig.1.6(a). Theoretically, they are calculated for a specific subshell with the quantum numbersni, ℓiaccording to
σniℓi(ω) =2.689 ωNi 2ℓi+1
∑
ℓ=ℓi±1
|dℓ,niℓi(k)|2, (1.47)
where k=p
2(ω+εniℓi). Some integral features can be observed. First, the 3ppartial cross section reveals a Cooper minimum, i.e. a minimum of the photoionization probability around 50 eV, while it does not exist in the case of the nodeless 2porbital of neon [89].
Second, the correlation correction due to the RPAE changes the shape of the 3spartial cross section completely and evidencing a deep Cooper minimum around 42 eV which cannot
1.6 Angular dependence of the atomic time delay 26
×
×
×
×××××××××× ×
20 40 60 80 100
0.01 0.10 1 10 100
Photon Energy ℏωXUV[eV]
Crosssection[Mb]
(a)
×
RPAE(3s) RPAE(3p) HF(3s) HF(3p) Expt.(3s) Expt.(3p)
20 40 60 80 100
-2 -1 0 1 2 3 4
Photon EnergyℏωXUV[eV]
Phases[unitsofπ]
(b)
ℓ=0, RPAE ℓ=1, RPAE ℓ=2, RPAE ℓ=0, HF
ℓ=1, HF ℓ=2, HF
Fig. 1.6 (a) Photoionization cross sections corresponding to the 3sand 3psubshells in argon.
(b) Scattering phases of argon of the different ionization channels within HF approximation and RPAE.
be captured by the HF approximation or other single-active electron approximations. In the case of both partial cross sections, the RPAE calculations reproduce the experimental measurement to a fair agreement. Therefore, we forgo the full numerical simulations for argon.
A look at the scattering phases in Fig.1.6(b) reveals strong differences in comparison with neon. According to the Levinson theorem in the absence of the Coulomb potential the HF phase corresponding to the transition 3s→k pwould tend to 2π atk→0 because two p subshells are occupied. Taking the Coulomb potential into account the quantum defect is characterized by δℓ=1HF(k→0)−σℓ=1(k→0) =1.75π, i.e. µℓ=1(∞) =1.75. The RPAE phases are very different in comparison to neon because the phaseδℓ=1RPAE(k)makes a jump ofπ when the cross section goes through the Cooper minimum at 42 eV. The explanation is very simple: Imagine the photoionization amplitudeµi(εk,Ωkkk)is real and had a node (Cooper minimum). In this case, it would change the sign which is similar to adding a phase factor ofπ in the complex number representation [46]. A similar situation can be observed in the case of the 3psubshell. For the transition 3p→kd we find the quantum defect µℓ=2(∞) =0 since for argon no d orbital is occupied. The corresponding RPAE phaseδℓ=2RPAE(k)as well asδℓ=0RPAE(k)(3p→kstransition) also make a substantial jump of
−π andπ, respectively, when the partial cross section passes the Cooper minimum at 50 eV.
The prominent features in the RPAE phases have a significant impact on the resulting time delays. In panel Fig.1.7(a) the time delay corresponding to the photoionization process of the 3ssubshell is shown. It is not surprising that theτW3sshows no angular dependence since we have only one photoionization channel (3s→k p). The comparison between the results in HF approximation and within RPAE show the large impact of the correction due to intershell correlation effects. While the HF result is relatively flat and is comparable to the characteristics of 2stime delay of neon, we find a very pronounced peak of the RPAE time delay for photon energies around the Cooper minimum. The explanation is given by δℓ=1RPAE(k) which makes a sudden jump ofπ at ¯hωXUV =42 eV. Therefore, according to
1.6 Angular dependence of the atomic time delay 27
40 50 60 70 80
0 100 200 300 400 500
Photon Energy ℏωXUV[eV]
TimedelayτW3s [as]
HF
HF+RPAE(θk=0∘) HF+RPAE(θk=30∘)
(a)
40 50 60 70 80 90 100
-100 -50 0 50
Photon Energy ℏωXUV[eV]
TimedelayτW3p [as] τW
3p(θk=0°) τW3p
(θk=15°) τW3p(θk=30°)
τW3p(θk=45°) τW, HF3p (θk=0°)
-π4 0 π4
-120-100-80-60-40-20200
Angleθk[rad]
TimedelayτW3p[as]
ℏωXUV=51 eV ℏωXUV=60 eV ℏωXUV=75 eV
θmax=45°
θmax=20°
(b)
Fig. 1.7 (a) Time delay corresponding to the photoionization process of the 3ssubshell in argon. (b) Time delayτW3pfor different asymptotic directionsϑkkkof the liberated photoelec-tron. The inset shows the angular dependence of the full averaged delays in the case of three different photon energies around the Cooper minimum.
Eq. (1.26) the time delayτW3s= ∂
∂ εk
δℓ=1HF(k) +δℓ=1RPAE(k)
shows a very distinctive positive peak at theπ-jump of the RPAE scattering phase.
In Fig.1.7(b) the whole time delay of the photoionization of the 3psubshell is depicted.
The contributions of all magnetic substates are included according to Eq. (1.27). Substantial variation with the photon energy ¯hωXUV can be observed. Furthermore, in contrast to the results for neon, we find a pronounced angular modulation around 50 eV which is correlated with the Cooper minimum. The origin is the angular dependence of τWℓi=1,mi=0 due to the existence of two ionization channels (3p→ksand 3p→kd). The time delays of the other magnetic substates τWℓi=1,mi=±1≡ ∂
∂ εk
δℓ=2HF(k) +δℓ=2HF(k)
do not show any angular modulation since only the transition 3p→kdis possible.
At an angle ofϑkkk=0◦ the probability of the photoionization processes corresponding to the initial statesℓi=1,mi=±1 are zero, i.e. in this case τW3p=τWℓi=1,mi=0. The negative sign ofτW3paround the Cooper Minimum despite the positive energy derivative of the HF scattering phaseδℓ=2HF corresponding to the dominant transition 3p→kd can be explained by the RPAE phaseδℓ=2RPAE, which makes a sudden jump of−π. However, the peak of the corresponding time delay at the Cooper minimum is not as pronounced and sharp as in the case of the photoionization of the 3ssubshell. The reason is the interference between both possible photoionization channels which on the other hand, leads to the substantial angular dependence. The usually weak transition 3p→ks becomes stronger near the Cooper minimum, i.e. it is of the same magnitude as the otherwise dominant transition 3p→kd. As a consequence, the negative delay peak induced by the RPAE phaseδℓ=2RPAE is damped by the 3p→kstransition where the corresponding RPAE phaseδℓ=0RPAEmakes a positive jump ofπ (cf. Fig.1.6(a)). Therefore, the resulting time delay does not fall below
−100 as. Nevertheless, we can observe a local and pronounced negative time delay due to the intershell correlation correction.
1.6 Angular dependence of the atomic time delay 28
35 40 45 50 55 60 65
-60 -40 -20 0 20 40 60
Photon Energy ℏωXUV[eV]
TimedelayτW3p [as] RPAE
Expt.
●
(a)
35 40 45 50 55 60 65 70
-100 0 100 200 300 400 500
Photon Energy ℏωXUV[eV]
TimedelayτW3s-3p [as] τ
W3s-3p
(θk=0°) τW3s-3p(θk=15°) τW3s-3p
(θk=30°) τW3s-3p
(θk=45°) τW,HF3s-3p
(θk=0°)
SB 26
×
□
SB 24
□×
SB 22
×
□
(b)
46 48 50 52 54 56 58 60 0
20 40 60 80 100
Fig. 1.8 (a) Comparison of the angle-integrated time delayτW3pcorresponding to the photoion-ization of the 3psubshell calculated within the RPAE with the experimental measurement [66]. (b) Full relative time delayτW3s−3pin dependence on the photon energy and different asymptotic directionsϑkkkof the photoelectron. For comparison, the experimental results by the RABBIT method are included (closed circles, Ref. [5]; open squares, Ref. [4]).
For anglesϑkkk>0◦and photon energies around the Cooper minimum, the transition 3p→ks begins to dominate the photoionization process. Now, it marks the primary contribution to τWℓi=1,mi=0with the consequence that the resulting full time delayτW3p (with contributions from all possible initial magnetic substates) tending to increase due to the positiveπ-jump of the RPAE phaseδℓ=0RPAE. The reason is that the Legendre polynomialP20(cosϑkkk)decreases for larger anglesϑkkkwhileP00(cosϑkkk)is constant (cf. Eq. (1.24)). For photon energies far away from the Cooper minimum, the effect due to angular dependence becomes very subtle.
In this regime, the photoionization process is entirely dominated by the transition 3p→kd [86] and the same characteristics as for neon can be observed. In addition, the RPAE phase δℓ=2RPAEbecomes very flat which means the angular dependence of the time delay is of the same magnitude as in the case of neon.
The small inset of Fig.1.7(b) shows the time delay of the 3pphotoionization process angle-resolved for three different photon energies. Again large differences between the atomic systems neon and argon can be observed due to the existence of the Cooper minimum in the second case. The time delay corresponding to photoionization of neon shows nearly no angular modulation in the range between −45◦ and 45◦ while in the case of argon, we find a substantial variation with the angle for photon energies around the Cooper minimum. Especially for the photon energy ¯hωXUV=51 eV, a strong dependence ofτW3pup toϑkkk=±45◦can be observed.
Recently, the angular dependence of the time delay was addressed experimentally [66].
The comparison between the angle-integrated measurement of the 3ptime delayτW3pand the RPAE result in Fig.1.8(a) reveals a reasonably well agreement. Therefore, the RPAE prediction regarding the angular dependence of the time delay seems to be qualitatively correct. However, a more accurate angle-resolved experimental measurement is needed
1.7 Conclusion 29