Total quadruple photoionization cross section of beryllium

Agapi EmmanouilidouITS, University of Oregon, Eugene, Oregon 97403-5203, USA�Received 10 September 2007; published 2 November 2007�

In a quasiclassical framework, we formulate the quadruple ionization by single-photon absorption of theCoulomb five-body problem. We present the quadruple photoionization total cross section of the ground stateof beryllium for energies up to 620 eV. Our results for energies close to threshold are in agreement with theWannier threshold law for four-electron escape. In addition, the agreement of our results with a shape formulaprovides support for the overall shape of our total quadruple cross section. Finally, we find that the photonenergy where the maximum of the total photoionization cross section occurs for single, double, triple, andquadruple photoionization of H, He, Li, and Be, respectively, seems to follow a linear relation with thethreshold energy for complete breakup of the respective element.

DOI: 10.1103/PhysRevA.76.054701 PACS number�s�: 32.80.Fb, 34.80.Dp

I. INTRODUCTION

Multiple ionization of atoms by single-photon absorptionis a process of great interest since it probes the correlatedmotion of many electrons. Quadruple photoionization of theground state of beryllium is the most fundamental atomicprocess involving four bound electrons. No other theoreticalor experimental study is, to our knowledge, currently avail-able regarding escape of four initially bound electrons.

The ionization processes involving two-electron escapecan now be treated quite accurately and significant progresshas been made concerning their understanding �1�. Regard-ing the three-electron escape by single-photon absorptionfrom the ground state of Li, significant advances have beenmade over the last few years in obtaining total triple ioniza-tion cross sections both experimentally �2,3� and theoreti-cally �4–6�. However, still great challenges remain, particu-larly concerning differential cross sections �7–10�.

Triple ionization of the ground state of Li by electronimpact, that is, a four-electron escape process, has been ad-dressed in very recent years. These experimental �11� andtheoretical �12� studies address very high energies of the im-pacting electron. The current theoretical study is the first, toour knowledge, work on four-electron escape by single-photon absorption for up to intermediate excess energies.Our study is based on an extension of the half collisionmodel. The latter is based on Samson’s �13� original idea fordouble photoionization: the ejection of the second electronresembles electron impact ionization by the primary electronthat absorbs the photon and leaves the atom. We formulatethe extension of the half collision model for the quadrupleionization in a quasiclassical framework. That is, we propa-gate classical trajectories using the classical trajectory MonteCarlo �CTMC� method �14,15� with a Wigner distribution asour initial state. This simple model has proven very success-ful in describing the double photoionization from the groundstate of He �16� as well as the singly excited states of He�17�. The same simple model has also accurately describedthe total photoionization cross section �6�, as well as singledifferential cross sections �9� and double energy differentialcross sections �10� for the three-electron escape from theground state of Li for up to intermediate excess energies. The

significance of studying quadruple photoionization using asimple model, which has already proven very successful indescribing several fragmentation processes, is twofold: First,given the challenges experimental as well as exact quantummechanical studies will face in the future, our results willserve as a benchmark calculation for these studies of the totalquadruple photoionization cross section. Second, given thatclose to threshold the quadruple ionization is a prohibitivelyrare process for ab initio studies with the current state of theart computational capabilities one can hope to gain physicalinsight only using simple but accurate models. Our currentstudy is a first step in this direction. In the future our studieswill address the shape of the interelectronic angular distribu-tions of the four escaping electrons and whether an unex-pected pattern arises for energies close to threshold. That wasfound to be the case for Li with the three electrons escapingin a surprising T-shape structure for excess energies close tothreshold �6�.

II. QUASICLASSICAL FORMULATION OF QUADRUPLEIONIZATION OF BERYLLIUM

We formulate the quadruple photoionization process fromthe Be ground state �1s22s2� as a two step process �see Refs.�4,6,13,16�� in the following way:

�4+ = �absP4+, �1�

where �abs is the total photoabsorption cross section and P4+

is the probability for quadruple ionization. For �abs we usethe experimental data from Ref. �19�. Our formulation ac-counts for the second step �half collision model�. Our con-struction of the initial phase space density ���� for the qua-druple photoionization of beryllium is similar to the doublephotoionization of He �16� and the triple photoionizationformulation of Li, which has been detailed in �6�. First,one electron absorbs the photon �photoelectron� at timet= tabs=0. Through electronic correlations, the energy is re-distributed, resulting in four electrons escaping to the con-tinuum. We first assume that the photon is absorbed by a 1selectron at the nucleus �r1=0�. This latter approximation be-comes exact in the limit of high photon energy �20�. For Be,

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at the energies we consider, the cross section for photon ab-sorption from a 1s orbital is much larger than from a 2sorbital �21�. Hence, we can safely assume that the photoelec-tron is a 1s electron, which significantly reduces the initialphase space to be sampled. We also neglect antisymmetriza-tion of the electrons in the initial state. We denote the pho-toelectron by 1, the other 1s electron by 2, and the two 2selectrons by 3 and 4, respectively. Following photon absorp-tion, we model the initial phase space distribution of theremaining three electrons, 1s and two 2s, by the Wignertransform �22� of the corresponding initial wave function��r1=0 ,r2 ,r3 ,r4�, where ri are the electron vectors startingat the nucleus. We approximate the initial wave function as asimple product of hydrogenic orbitals �i

Zi�ri�, with effectivecharges Zi, to facilitate the Wigner transformation. The Zi arechosen so that they reproduce the known ionization poten-tials Ii, namely, for one of the two 2s electrons Z4=1.656�I4=0.343 a.u.�, for the other 2s electron Z3=2.314�I3=0.669 a.u.�, and for the 1s electron Z2=3.363�I2=5.656 a.u.�. �We use atomic units throughout the paperif not stated otherwise.� The excess energy E is given byE=E�− I with E� the photon energy and I=14.67 a.u. theBe quadruple ionization threshold energy. Given the aboveconsiderations, the initial phase space density is given by

���� = N��r1���E1 + I1 − �� �i=2,3,4

W�iZi�ri,pi���Ei + Ii� ,

�2�

with normalization constant N. We determine the quadrupleionization probability P4+ by discretizing the initial phasespace. We then propagate the classical equations of motionusing the discretized phase space as initial conditions. Thefull Coulomb five-body Hamiltonian is used for the classicalpropagation. Regularized coordinates �23,24� are used toavoid problems with electron trajectories starting at thenucleus. We label as quadruply ionizing those trajectorieswhere the energies of all four electrons are positive, Ei�0with i=1,2 ,3 ,4, asymptotically in time.

III. RESULTS

In Fig. 1, we present our results for P4+. The very smallvalue of the quadruple photoionization probability clearlyshows the difficulties involved in obtaining theoretical re-sults or experimental measurements of the five-body breakup

process. Using Eq. �1� we present in Fig. 2 the results for thequadruple photoabsorption cross section �black circles�.Since there are no other results currently available we cannotcompare our results on an absolute scale.

However, a comparison of our results with a shape func-tion applicable to single-photon multiple ionization pro-cesses �25� provides some indication of how well our resultsreproduce the shape of the total cross section. This shapefunction reproduces, by construction, the correct behavior ofthe single-photon multiple-ionization cross section for excessenergies close to threshold �Wannier law �26�� and for largeexcess energies. It depends on two parameters, the positionEM and height �M of the cross section maximum and is givenby

��E� = �Mx� + 7/2

x + 7/2�+7/2

, �3�

where is the characteristic exponent in the Wannier thresh-old law and x is the excess energy E scaled by EM. For thebreakup of a four-electron atom this characteristic exponentwas found to be 3.331 for a plane configuration where theelectrons escape on the apexes of a square and 3.288 for athree-dimensional configuration where the electrons escapeon the apexes of a tetrahedron �18�. From the above twovalues for it is the smallest one, =3.288, that dominatesthe threshold behavior and thus the value we use in the shapeformula in Eq. �3�. Using the shape formula without anyfitting parameters but substituting for EM =120 eV and�M =0.0847 b, which are the location and height of themaximum cross section from our results, we obtain the solidcurve shown in Fig. 2. Given that there are no fitting param-eters the agreement between our numerical results and theshape formula is quite good, providing support that the shapeof the cross section we obtain is the correct one. The agree-ment becomes even better if instead we use the EM and �Mas fitting parameters to fit our results with the shape formula.Doing so, we obtain the values EM =134.5 eV and�M =0.0834 b and the agreement of the fitted shape formula�dashed curve in Fig. 2� with our results is indeed very good.

Also, for small values of the excess energy we have fittedour results for the quadruple cross section with the formula

100

10

10

10

10

−9

−8

−7

−6

excess energy (eV)

P4+

10

FIG. 1. Quadruple photoionization probability as a function ofexcess energy.

(b)

excess energy (eV)

σ4+

101

102

10−4

10−3

10−2

10−1

FIG. 2. Quadruple photoionization cross section as a function ofexcess energy. The black circles are our results. For the solid anddashed lines see the main text.

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��E�� �E�/I − 1�, �4�

with I the fragmentation threshold energy of beryllium andE� the photon energy. For energies from 4 to 9 eV our resultsyield a characteristic exponent of 3.17, which is quite closeto the analytical value of 3.288. The difference can probablybe attributed to the fact that our current results go only downto 4 eV. It is a well known fact that as the energy increasesthe value of decreases �6�. Thus, most probably, if we wereable to obtain results for energies below 4 eV would havebeen closer to the actual value of 3.288. Currently, however,the numerical difficulties involved in obtaining results below4 eV are prohibitive.

At this point, a few comments are in order. Our quasiclas-sical formulation is most accurate close to threshold. It candescribe the process accurately up to at most intermediateexcess energies while it cannot account for high energies.For high energies the process can only be described quantummechanically. In the case of triple ionization from the groundstate of Li our simple product initial state has proven to bevery good in describing not only the total cross section �6�but also single �9� and double energy differential cross sec-tions up to excess energies of 220 eV �see �10��, withI=203.5 eV for the triple escape process. In the current pa-per we consider only a total absorption cross section and notdifferential ones with the former being much less sensitivethan the latter to electron-electron correlation. The success ofour product initial state in describing single and double dif-ferential cross sections for double �16� and triple photoion-ization �9,10�, which are more sensitive to electron-electroncorrelation compared to the total cross section, strongly sug-gests the success of our model in describing accurately thetotal quadruple photoionization cross section up to interme-diate excess energies.

A comparison of the maximum of the single photoioniza-tion cross section for H, 6.3�106 b, with the maxima fordouble photoionization of He, 8.76�103 b �27�, triple

photoionization of Li, 7.47 b, and quadruple photoionizationof Be, 0.0847 b, clearly shows how rare the quadruple photo-ionization process is. For the double ionization cross sectionof He we use the results presented in Ref. �27� while for thetriple photoionization of Li we use the results presented inRef. �6�. Another interesting feature is the relation betweenthe ionization threshold energy and the photon energyE�,M = I+EM, where the maximum of the photoionizationcross section of the respective process occurs. For the singlephotoionization of H, EM =0 eV, for the double photoioniza-tion of He, EM =21.8 eV �27�, while for the triple and qua-druple photoionization of Li and Be, EM, is, respectively,52.7 and 134.4 eV. The values of 52.7 and 134.4 eV werefound by fitting Eq. �3� to our results for triple �6� and qua-druple photoionization, respectively. In Fig. 3, we plot E�,Mas a function of I for the above four processes. Interestingly,we find that for these four processes the maximum of thetotal photoionization cross section seems to follow a linearrelation with the threshold energy for complete breakup ofthe respective element. Whether there is a physical reasonunderlying this fact is an interesting question for future re-search.

A future study of interest for the four-electron escape inBe would be to investigate whether the collision processesthe four electrons follow to escape to the continuum can stillbe described as a sequence of three-body subsystems as isthe case for Li �9�. Another interesting aspect to be investi-gated is the shape of the interelectronic angular distributionsfor energies close to threshold. In the case of the three-electron escape from the ground state of Li we found that inan energy regime where the characteristic exponent =2.16�28� is recovered the angular distribution has a surprisingT-shape distribution. This T shape is different than what onewould expect from the three electrons escaping in the apexesof an equilateral triangle for excess energy zero �28�. At zeroexcess energy, the four electrons escape in the apexes of atetrahedron �18�. It remains to be seen whether a surprisingpattern will be found for the four-electron escape as was thecase for the three electrons.

In conclusion, we have presented results for the total crosssection for the quadruple photoionization of Be. With noother theoretical or experimental results currently availablewe hope that the current results will serve as a benchmarkcalculation for future studies of the Coulomb five-body prob-lem.

ACKNOWLEDGMENTS

The author is grateful to J. M. Rost, P. Wang, and to T.Pattard for helpful discussions and a critical reading of themanuscript. She is also indebted to Y. Smaragdakis withoutwhose computational expertise the current work would nothave been possible.

100 200 300 4000

100

200

300

400

500

(eV

)E

Threshold energy (eV)

phot

on,m

ax

FIG. 3. Photon energy where the maximum of the cross sectionfor single photoionization of H, double photoionization of He, triplephotoionization of Li, and quadruple photoionization of Be occursas a function of the threshold energy of the respective process.

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