A Time-dependent density matrix equations

# Effects of laser polarization on photoelectron angular distribution through laser-induced continuum structure

## Abstract

We theoretically investigate the effects of laser polarization on photoelectron angular distribution through laser-induced continuum structure. We focus on a polarization geometry where the probe and dressing lasers are both linearly polarized, and change the relative polarization angle between them. We find that the total ionization yield and the branching ratio into different ionization channels change as a function of the relative polarization angle, and accordingly the photoelectron angular distribution is altered. We present specific results for the - and - systems of the K atom, and show that the change of the polarization angle leads to the significant modification of photoelectron angular distribution.

###### pacs:
32.80.Qk, 42.50.Hz, 32.80.Rm
1

## I Introduction

The similarity between the autoionizing structure (AIS) and the laser induced continuum structure (LICS) is very well known. In an AIS process a discrete state lying above the ionization threshold is coupled to the continuum through configuration interaction, a resonance structure being created. In LICS two bound states are coupled to the common continuum through two laser fields (the probe and dressing lasers). The bound state coupled to the continuum through a strong laser (dressing laser) can also create a resonance structure having AIS-like properties, but compared to the AIS resonance its position and width are now controllable by the frequencies and intensities of the lasers. The first experimental observation of LICS was successfully reported in Refs. (1); (2). More comprehensive information on LICS can be found in a review paper by Knight et al (3).

Through LICS not only the ionization yield but also a number of some other processes can be altered: Several works based on LICS investigated non-linear optical effects such as the enhancement of third-high harmonic generation (4). In Ref. (5), effects of LICS on spin-polarization were studied for heavy alkali atoms. Recently, LICS for multiple continua was experimentally and theoretically investigated (6). In Ref. (7), the control of ionization products in LICS was suggested for the case of decay into multiple continua.

It is well known that the photoelectron angular distribution (PAD) provides more information about the ionization process than the angle-integrated ionization signal (8). PAD’s of Na by the two linearly polarized lasers with a variable polarization were reported in Ref. (9), and the phase difference between the continua with same parity was extracted. By measuring PAD’s of an alkali atom in a bichromatic laser field, a theoretical method was proposed in Ref. (10) in order to extract the phase difference of the continua with opposite parities.

Most recently we have theoretically investigated how LICS affects PAD (11), and specific results have been presented for the K atom. In Ref. (11), however, we have assumed that the probe and dressing lasers are linearly polarized along the same direction. A natural question would be how PAD is modified, through LICS, by changing the relative polarization angle between the probe and dressing lasers.

The aim of the present paper is to generalize our previous work (11). We now vary the relative polarization angle and analyze the modifications of the ionization yield and the photoelectron angular distribution through LICS, and see how the enhancement or suppression of a particular ionization channel takes place. Since the photoelectrons ejected into different involved continua have different angular distributions, and those angular distributions depend on the relative polarization angle, we expect important modifications in terms of the ionization yields, branching ratios, and photoelectron angular distributions.

The paper is organized as follows. In Sec. II we present the theoretical model: The time-dependent amplitude equations which describe the dynamics of the LICS process are derived, and the ionization yield and photoelectron angular distribution are calculated. The consistency of our results has been checked using an alternative approach based on the density matrix equations. The theoretical results obtained using these two formalisms are of course identical. Further details about the density matrix formalism are provided at the beginning of Sec. III, and in Appendix A. Sec. III is mainly devoted to the discussions on the numerical results for the total and partial ionization rates, branching ratios, and PAD’s at different polarization angles. The atomic parameters needed for the LICS calculation are given in Appendices B-D.

## Ii Theory

The system we consider in this paper consists of an initially occupied state, initially unoccupied state, and the continuum states of the K atom together with the linearly polarized probe and dressing lasers that couple and to the continuum states, respectively. This implies that, prior to the interaction of the system with the probe and dressing lasers, K atoms in the ground state have been excited to the state by a linearly polarized auxiliary laser. By choosing an appropriate frequency of the auxiliary laser, we can selectively excite either () or (), which will serve as an initial state in this work. For simplicity, we assume that the polarization axis of the dressing laser is parallel to that of the auxiliary laser, while the polarization axis of the probe laser can be arbitrary.

Here we are interested in a particular geometry where the polarization vector of the dressing and auxiliary lasers are along the axis and that of the probe laser is assumed to lie in the -plane, as shown in Fig. 1. For such a case, the polarization vector of the probe laser is defined as , where , and are the unit vectors along the and axes, respectively, and represents the polarization angle of the probe laser with respect to that of the dressing laser. Defining the frequencies of the probe and dressing lasers as and , respectively, the total electric field vector can be written as,

 E(t)=∑α=p,dE0α(t)e(α)cos(ωαt). (1)

A Gaussian temporal profile was employed for the amplitude of the laser fields: , where represents the temporal width for the full width at half maximum (FWHM) of the probe or dressing pulse with or , indicating the probe and the dressing pulses. is the polarization vector of the laser pulse .

Based on the above assumptions, the level scheme we consider in this paper is now described in Figs. 2(a)-2(c) for the K - system, at , , and in between, i.e., . If both polarization axes of the probe and dressing lasers are parallel, i.e. as shown in Fig. 2(a), due to the selection rule (where the prime index is used for quantum numbers of the continuum) the entire - system with can be decomposed into the two independent subsystems which consist of (), (), and the continua () and (). The ionization yields for these subsystems are obviously symmetric to each other, and for simplicity we can consider only one of them, as already explained in our previous paper (11). Similarly, at , because of the selection rule , the entire - system with can be decomposed into the two independent subsystems consisting of (), (), and the continua () and () with additional incoherent channels (), as shown in Fig. 2(b). Again, both subsystems are completely symmetric, and it is sufficient to study only one of the two subsystems. However, for the intermediate values of the polarization angle, as shown in Fig. 2(c), because of the selection rules (from the perpendicular component of the probe polarization vector with respect to the quantization axis, ) and (from the parallel component, ), the entire system - cannot be decomposed into the two independent subsystems anymore, and the entire system, , , and the continua with all possible magnetic sublevels, have to be taken into account at the same time. The continuum states , , not presented in Fig. 2, have the same quantum numbers as the continuum states , (, for ), but they correspond to a different value of energy because of the incoherent one-photon ionization from by the probe laser. A similar level scheme, taking into account appropriate dipole selection rules, has been considered for the K - system.

In order to observe LICS, it is necessary that the initially occupied (or ) state (denoted as for the magnetic sublevel having and for the magnetic sublevel having ), and the initially unoccupied (or ) state (denoted as for and for ) are coupled by the probe and dressing lasers whose frequencies nearly satisfy the two-photon resonance, i.e., . As long as we use a ns laser with appropriate intensities and detunings, it is perfectly valid to treat each and system separately, as it was explained in our previous paper (11).

It should be mentioned that we expect a different behavior of the two systems: For the K system the initial state, , is an isotropic mixture of all possible magnetic sublevels (recall that the magnetic sublevels are equally populated by the auxiliary laser), implying that the initial state is spherically symmetric. It is obvious that the PAD from the spherically symmetric initial state orientates along the polarization axis of the probe laser if the dressing laser is off. PAD changes neither its shape nor magnitude (12). That is not the case for the K system, since not all possible magnetic sublevels are excited with the same probability, and accordingly the initial state, , is non-spherical (polarized). Therefore, we expect quite different modification of PAD for the and systems through LICS, by varying the relative polarization angle, as we have already seen for (11). Further details will be provided in Sec. III.

### ii.1 Time-dependent amplitude equations

In order to study the temporal evolution of the atomic system in laser field, we have used the standard procedure as described in our previous paper (11). Briefly, we solve the following set of time-dependent amplitude equations:

 ˙u1=−12˜Γ1u1−iΩ13(1−iq13)u3−iΩ14(1−iq14)u4, (2) ˙u2=−12˜Γ2u2−iΩ23(1−iq23)u3−iΩ24(1−iq24)u4, (3) Missing or unrecognized delimiter for \right (4) Missing or unrecognized delimiter for \right (5)

where ’s () represent the probability amplitudes of states . Note that all the probability amplitudes for the continuum states have already been adiabatically eliminated in the Eqs. (2)-(5). is a two-photon detuning defined by , where the static detuning is defined by , and is a total dynamic ac Stark shift defined by . In all the numerical results presented in this work the zero point of the detuning has been chosen such that , since the ac Stark shifts simply translate the LICS resonance on the detuning scale. The superscript of means that the ac Stark shift is calculated at the peak value of laser intensity. ’s are the bound-free matrix elements by laser ( or ) from the bound state to the continuum , which are connected to the partial ionization widths through the relation . represents the total ionization width of state , i.e., (for ) and (for ), where is the phenomenologically introduced spontaneous decay width of state . In the above equations the two-photon Rabi frequency, , can be written as a sum of the partial two-photon Rabi frequencies into the coherent and continua of energy , i.e.,

 Ωij(1−iqij)=∑β=ϵs,ϵdΩβij⎛⎝1−iqβij⎞⎠, (6)

where and represent the total and partial asymmetry parameters, respectively. It is very well known that, for the light alkali-metals such as Li, N, and K, the dependence of radial matrix elements and phase shifts on the total angular momentum quantum number is very small and it can be neglected (13). Now, the following relations are satisfied by the angle-integrated atomic parameters: , , , , , , , and . Details about the calculation of the atomic parameters such as Rabi frequencies, ionization widths, ac Stark shifts and asymmetry parameters are given in the Appendices B-D.

Since the behavior of the population dynamics in the continuum is of our interest, we also need the following set of amplitude equations for the continua:

 ˙uca=−iδcauca−iD(p)ca1u1−iD(p)ca2u2−iD(d)ca3u3−iD(d)ca4u4, (7) ˙ucb=−iδcbucb−iD(p)cb3u3−iD(p)cb4u4. (8)

Here represents the probability amplitude of the coherent continuum state (), and represents the probability amplitude of the incoherent continuum state (). As already explained at the beginning of Sec. II the incoherent continuum states are not presented in Fig. 2 to avoid the complexity of the figure. They have the same quantum numbers as the coherent continuum states , but located at different energies.

Having solved Eqs. (2)-(5) and Eqs. (7-8), we can now calculate the total (angle-integrated) ionization yield, , from the relation,

 R(t)=8∑a=1Rca(t), (9)

where the partial photoelectron yields , into each coherent and incoherent continuum state , () can be calculated through the following formulae:

 Rc1(t) = ∫t−∞dt′Γ(p)1c1|u1|2, (10) Rc2(t) = ∫t−∞dt′{Γ(p)1c2|u1|2+Γ(p)2c2|u2|2+Γ(d)3c2|u3|2 (11) + 4Im[Ωc213(1+iqc213)]Re(u1u∗3)+4Im[Ωc223(1+iqc223)]Re(u2u∗3)}, Rc3(t) = ∫t−∞dt′{Γ(p)1c3|u1|2+Γ(p)2c3|u2|2+Γ(d)4c3|u4|2 (12) + 4Im[Ωc314(1+iqc314)]Re(u1u∗4)+4Im[Ωc324(1+iqc324)]Re(u2u∗4)}, Rc4(t) = ∫t−∞dt′Γ(p)2c4|u2|2, (13) Rc5(t) = ∫t−∞dt′Γ(p)3c5|u3|2, (14) Rc6(t) = ∫t−∞dt′[Γ(p)3c6|u3|2+Γ(p)4c6|u4|2], (15) Rc7(t) = ∫t−∞dt′[Γ(p)3c7|u3|2+Γ(p)4c7|u4|2], (16) Rc8(t) = ∫t−∞dt′Γ(p)4c8|u4|2. (17)

Since the total ionization yield is a sum of ionization into the coherent and incoherent continua, it might be rewritten as,

 R(t)=∑β=ϵs,ϵd[4∑a=1Rβca(t)+8∑b=5Rβcb(t)]=∑β=ϵs,ϵd[Rβ(t)+Rincohβ(t)]. (18)

To see the effects of LICS, it is useful to calculate the branching ratio, , defined as the ratio between the partial ionization yield into each coherent continuum and :

 B=RϵdRϵs. (19)

The total ionization yield given by Eq. (9) and the partial ionization yields given by Eqs. (10)-(17) are calculated at the end of the pulses.

### ii.2 Photoelectron angular distribution

For the purpose of calculating photoelectron angular distribution we need equations before angle integration. In order to simplify the calculation of the bound-free dipole matrix elements we use a partial wave expansion for the continuum of an alkali-metal atom in a coupled basis:

 |k;ms′⟩=∑l′,ml′,j′al′ml′(−1)l′−1/2+ml′+ms′√2j′+1(l′1/2j′ml′ms′−mj′)|{k};(l′s′)j′mj′⟩, (20)

where k represents the wave vector of photoelectron, , and is the phase shift which is a sum of the Coulomb phase shift and the scattering phase shift; recall that the prime index indicates a quantum number for the continuum state.

We are interested in PAD as a function of polarization angle . If the final spin state of the photoelectron is not detected, we have to incoherently sum over the final spin projection, . The partial photoelectron yield into a solid angle, , defined by the polar angle , and the azimuthal angle , can be written as,

 dR(Θ,Φ)dtdΩk∣∣∣mj=±1/2=0.589π∑ms′=±1/2 ⎡⎢⎣∣∣ ∣∣2∑j=1√Γ(p,ms′)j(Θ,Φ)uj+4∑j=3√Γ(d,ms′)j(Θ,Φ)uj∣∣ ∣∣2 (21) + ∣∣ ∣∣4∑j=3√Γ(p,ms′)j(Θ,Φ)uj∣∣ ∣∣2⎤⎥⎦,

where is a conversion factor for the appropriate normalization, and the formula is valid for both and systems, for of the initial state. After the considerable angular momentum algebra, we obtain expressions for the differential ionization widths from states and for the system:

 Γ(α,+1/2)j(Θ,Φ) = ∣∣∣−13R(α)jϵseiδsY00(Θ,Φ)e(α)0+23√5R(α)jϵdeiδdY20(Θ,Φ)e(α)0 (22) + 1√15R(α)jϵdeiδdY21(Θ,Φ)e(α)1+1√15R(α)jϵdeiδdY2−1(Θ,Φ)e(α)−1∣∣∣2Iα, Γ(α,−1/2)j(Θ,Φ) = ∣∣∣−√23R(α)jϵseiδsY00(Θ,Φ)e(α)−1−√23√5R(α)jϵdeiδdY20(Θ,Φ)e(α)−1 (23) − 2√15R(α)jϵdeiδdY22(Θ,Φ)e(α)1−√2√15R(α)jϵdeiδdY21(Θ,Φ)e(α)0∣∣∣2Iα,

and for the differential ionization widths , from states and :

 Γ(α,+1/2)j(Θ,Φ) = ∣∣∣√23R(α)jϵseiδsY00(Θ,Φ)e(α)1+√23√5R(α)jϵdeiδdY20(Θ,Φ)e(α)1 (24) + √2√15R(α)jϵdeiδdY2−1(Θ,Φ)e(α)0+2√15R(α)jϵdeiδdY2−2(Θ,Φ)e(α)−1∣∣∣2Iα, Γ(α,−1/2)j(Θ,Φ) = ∣∣∣13R(α)jϵseiδsY00(Θ,Φ)e(α)0−23√5R(α)jϵdeiδdY20(Θ,Φ)e(α)0 (25) − 1√15R(α)jϵdeiδdY21(Θ,Φ)e(α)1−1√15R(α)jϵdeiδdY2−1(Θ,Φ)e(α)−1∣∣∣2Iα,

where and represent the radial bound-free matrix elements from state ( ) to the continua and , respectively, by laser ( or ), evaluated in atomic units. Here , with , are the spherical components of the polarization vector of laser , namely , and . The laser intensities, and , are expressed in W/cm. For the coherent continuum, the relevant phase shifts are and , which are the sum of the Coulomb phase shifts, and , and the scattering phase shifts, and with () being the quantum defects estimated from the linear extrapolation of the bound Rydberg and series of the K atom to the continuum energy of interest. Eq. (21) together with Eqs. (22)-(25) gives PAD for the system with appropriate normalization, so that the angle-integrated quantity becomes identical to the total ionization yield calculated with Eq. (9).

Similarly the differential ionization widths from states and for the system, are given by,

 Γ(α,+1/2)j(Θ,Φ) = ∣∣∣√23R(α)jϵseiδsY00(Θ,Φ)e(α)0−2√23√5R(α)jϵdeiδdY20(Θ,Φ)e(α)0 (26) − √2√15R(α)jϵdeiδdY21(Θ,Φ)e(α)1−√2√15R(α)jϵdeiδdY2−1(Θ,Φ)e(α)−1∣∣∣2Iα, Γ(α,−1/2)j(Θ,Φ) = ∣∣∣−13R(α)jϵseiδsY00(Θ,Φ)e(α)−1−13√5R(α)jϵdeiδdY20(Θ,Φ)e(α)−1 (27) − √2√15R(α)jϵdeiδdY22(Θ,Φ)e(α)1−1√15R(α)jϵdeiδdY21(Θ,Φ)e(α)0∣∣∣2Iα,

and from states and they are derived as,

 Γ(α,+1/2)j(Θ,Φ) = ∣∣∣−13R(α)jϵseiδsY00(Θ,Φ)e(α)1−13√5R(α)jϵdeiδdY20(Θ,Φ)e(α)1 (28) − 1√15R(α)jϵdeiδdY2−1(Θ,Φ)e(α)0−√2√15R(α)jϵdeiδdY2−2(Θ,Φ)e(α)−1∣∣∣2Iα, Γ(α,−1/2)j(Θ,Φ) = ∣∣∣√23R(α)jϵseiδsY00(Θ,Φ)e(α)0−2√23√5R(α)jϵdeiδdY20(Θ,Φ)e(α)0 (29) − √2√15R(α)jϵdeiδdY21(Θ,Φ)e(α)1−√2√15R(α)jϵdeiδdY2−1(Θ,Φ)e(α)−1∣∣∣2Iα.

The Eqs. (22)-(29) are applicable for both probe and dressing lasers. Recalling that the polarization vector of the dressing laser is parallel to the quantization axis (), the relative polarization angle between the probe and dressing lasers becomes identical to . Because of the symmetry properties of spherical harmonics, we can show that and is equal to and , respectively, by interchanging and .

## Iii Numerical Results and Discussion

In this section we present numerical results and discussions. All the necessary single and effective two-photon dipole matrix elements needed for our schemes have been obtained using quantum defect theory and Green‘s function technique. The calculated atomic parameters such as Rabi frequencies, asymmetry parameters, ac Stark shifts, for the and systems, are listed in Tables I and II, respectively.

For the K system only Rabi frequencies are dependent, ionization widths and ac Stark shifts by the probe laser do not present any dependence since the initial state is isotropic (12); (14) (recall that by using a linearly polarized auxiliary laser the ground state, , of the K atom is excited to the state, thus all the magnetic sublevels for the state are occupied with the same probability). In contrast, for the K system the atomic parameters such as Rabi frequencies, and ionization widths and ac Stark shifts by the probe laser depend on the polarization angle . This is due to the fact that not all the magnetic sublevels for the initial state, , are occupied with the same probability. The total and partial asymmetry parameters are independent on the laser fields (3), and, obviously, do not depend on the polarization angle of the probe laser for both K and systems.

When we solve the set of amplitude equations a special care has to be taken: Note that the probability amplitudes of the initially occupied states, and , have arbitrary phases. However, if we use the amplitude equations, once the initial conditions are given for the probability amplitudes , , initial coherence inevitably exists due to nonzero value of (with ). We have to avoid any coherent interference between the ionization paths starting from and . Therefore, we should separately solve the set of amplitude equations Eqs. (2)-(5) with either and (for ), or and (for ), and average the photoelectron angular distribution given by Eq. (21) over of the initial state:

 dR(Θ,Φ)dtdΩk=12⎡⎢⎣dR(Θ,Φ)dtdΩk∣∣∣mj=+1/2+dR(Θ,Φ)dtdΩk∣∣ ∣∣mj=−1/2⎤⎥⎦. (30)

Pulse durations and peak laser intensities are chosen to be ns (FWHM) and MW/cm for the probe laser, and ns ns (FWHM) and MW/cm MW/cm for the dressing laser. If the probe and the dressing pulse durations are comparable the LICS resonance profile is going to be smeared out (11) due to the ac Stark shifts. In order to circumvent this problem the pulse duration of the dressing laser was chosen to be much longer than that of the probe laser, since, under the condition that , atomic states are quasi-statistically Stark-shifted by the strong dressing pulse during the interaction with the probe pulse. By substituting the atomic parameters listed in Tables I and II into Eqs. (2)-(5), we can easily solve those equations for the given peak intensities, detunings, and temporal profile of the lasers. Once the solution is obtained for (), the total and partial ionization yields can be calculated from Eq. (9) and Eqs. (10)-(17). The radiative lifetimes of and , and and levels, are about 26 ns and 345 ns, respectively, and are included in the numerical calculations.

In order to check the consistency of our results an alternative formalism based on the density matrix equations was used to calculate the dynamics of the system. The amplitude equations approach has the advantages of dealing with fewer number of differential equations and obtaining more compact formulae for the ionization yield. On the other hand, the density matrix equations approach has the advantage of its capability to control the coherence through the off-diagonal density matrix elements (that are proportional to , with ), and therefore it is suitably used for mixed states (such as or ) when at least two levels with arbitrary phase are initially occupied. Details about the density matrix approach are given in Appendix A, and the numerical results are, of course, identical to the ones obtained by using the amplitude equations. In the following subsections we present numerical results for the K and systems.

### iii.1 K 4p1/2−6p1/2 system

Figs. 3(a) and 3(b) show the variation of the total ionization yield and branching ratio as a function of two-photon detuning, , at four different values of the polarization angle, , , and . Note that the polarization angle of the dressing laser is fixed to . Pulse durations and peak laser intensities are chosen to be ns and MW/cm, and ns and MW/cm, for the probe and dressing lasers, respectively. Clearly, the profile of the LICS resonance and the branching ratio as a function of detuning for the system changes by varying the polarization angle. The position of the LICS resonance is also altered by varying , specifically the maximum of the LICS profile shifts toward larger values of the detuning as increases and its minimum vanishes completely when . At the value of the asymmetry parameter, which is connected to the resonance profile, , is much larger compared to the case when both lasers are linearly polarized in the same direction, and the asymmetry parameter takes the value, . That particular value of the asymmetry parameter for is due to the fact that the corresponding angular coefficients for the and ionization channels are equal, and the radial matrix elements have opposite signs.

Figs. 4(a) and 4(b) show the variation of the total ionization yield and branching ratio as a function of detuning at three different dressing laser intensities , , and MW/cm, with the probe laser intensity and the pulse durations fixed to be MW/cm, ns, and ns. The polarization angle is . As we have already seen in Figs. 3(a) and 3(b), the ionization yields and branching ratios vary significantly near resonance. The LICS structure is naturally broadened as the dressing laser intensity is increased.

For the particular polarization geometry shown in Fig. 1 the azimuthal angle dependence, , of the photoelectron signal, through the spherical harmonics , does not vanish as it happens when the polarization axes of both lasers are parallel to each other (11), and the cylindrical symmetry of PAD is broken. This is due to the presence of the spherical harmonics with in the differential ionization widths formulae Eqs. (22)-(29). Before studying PAD for LICS it would be instructive to give an answer to the following question: What is the modification of PAD for one-photon ionization from the initial state through the variation of the polarization angle of the probe laser without the dressing laser, i.e., ? Since the initial state, , is spherically symmetric (recall that both sublevels are equally populated), one could intuitively guess that the magnitude of PAD does not change, and PAD just aligns along the polarization axis of the probe laser. Under the condition of , three-dimensional (3D) PAD is plotted in Figs. 5(a)-5(c) as a function of photoelectron angles and , for the three different values of the polarization angle, , , and . As expected, PAD changes its orientation along the polarization direction of the probe laser. The fourfold rotational symmetry (, and , at ) that exists when both lasers are linearly polarized along the quantization axis, breaks into a twofold symmetry () when the polarization of the probe laser varies (15); (16).

Now we consider the case of LICS, i.e., the dressing laser is turned on. 3D PAD’s of the K system at are shown in Fig. 6(a)-6(c) for the three representative detunings corresponding to the far-off resonance ( GHz), maximum ( GHz), and minimum ( GHz) of the branching ratio (see Fig. 3(b)). The view point of all 3D plots in this paper is from the -plane with the Cartesian coordinates (2,2,0), if not otherwise stated. At far-off resonance (Fig. 6(a)), the 3D PAD again tends to follow the change of the polarization angle, , with some small distortion due to the dressing laser. The distortion, however, is almost invisible, since the interference effect through LICS is negligible at far-off resonance. In Figs. 6(b) and 6(c), we see that the 3D PAD’s are significantly modified. Especially in Fig. 6(c), a maximum distortion is observed in PAD due to the strong destructive interference between the and partial waves.

### iii.2 K 4p3/2−6p3/2 system

We now turn to the case of the