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PHYSICAL REVIEW A 73, 062712 2006

Triple differential cross section for sequential double photoionization of atoms by ultrashort pulses
FakultÄt fÝr Physik, UniversitÄt Bielefeld, D-33615 Bielefeld, Germany Fock Institute of Physics, State University of Sankt Petersburg, Sankt Petersburg 198504, Russia 3 Institute of Nuclear Physics, Moscow State University, Moscow 119992, Russia Received 4 February 2006; published 15 June 2006
1

A. K. Kazansky1,2 and N. M. Kabachnik1,3

2

A time-dependent approach for calculation of the triple differential cross section of sequential double photoionization of atoms by femto- and attosecond pulses is developed. The method involves the numerical solving of a system of coupled nonstationary SchrÆdinger equations which describe the inner-shell photoionization and the following Auger decay. The shape of the spectra and the angular distributions of photoelectrons are considered in the near-threshold region for the case of photoelectron­Auger electron coincidence experiments as well as for noncoincidence measurements. As examples, the cross sections for Ar 2 p and Kr 3d sequential double photoionization are calculated and discussed. It is shown that the cross sections strongly depend on the duration of the ionizing pulse. In contrast, the asymmetry parameters, characterizing the angular distribution of photoelectrons, are practically independent of the characteristics of the pulse. DOI: 10.1103/PhysRevA.73.062712 PACS number s : 32.80.Fb, 32.80.Hd

I. INTRODUCTION

Photoionization of inner atomic shells and the following core relaxation with emission of x rays and/or Auger electrons is one of the most developed parts of atomic collision physics 1 . Especially quick progress in this field in the last two decades is associated with the use of dedicated synchrotron radiation sources. Recently a new direction of investigation has emerged, namely inner-shell ionization of atoms by ultrashort femto- and attosecond electromagnetic pulses with duration being less than or comparable with the lifetime of the produced ions. Availability of the ultrashort extreme ultraviolet xuv pulses 2­ 4 makes it possible to study the time evolution of the inner-shell processes and, in particular, of the Auger decay 5,6 . The new type of experiments requires the development of corresponding nonstationary theoretical methods capable to describe the time evolution of the unstable atomic system produced in the short-pulse photoionization. A few theoretical papers devoted to this problem have appeared recently 7­11 . We concentrate our attention on the near-threshold innershell photoionization by short pulses with the following emission of fast Auger electrons. The process results in the formation of the doubly charged ions, and thus it is often referred to as sequential double photoionization. In the nearthreshold area a specific type of electron correlations, the so-called post-collision interaction PCI , plays an important role 12,13 . It manifests itself in shifts of the photoelectron and Auger electron spectral lines, in distortion of their shape, in such phenomena as electron recapture after photoionization, etc. All these effects are very well studied experimentally 13,14 and well understood theoretically within the conventional stationary approaches 15­21 . On the other side, the time evolution of the near-threshold processes is much less investigated. In particular, it is of interest to study those phenomena with the ultrashort pulses, which might give a new insight into their nature. Recently we suggested a nonstationary method of treating the inner-shell near1050-2947/2006/73 6 /062712 12

threshold ionization with the subsequent Auger decay 10 . The method is based on the numerical solving of a system of time-dependent SchrÆdinger equations and is suitable for studying the time evolution of the photoelectron wave packets created by photoionization and influenced by the Auger decay. We have shown that the position and the shape of the photoelectron line strongly depend on the duration of the ionizing pulse. We have tested the method by comparing the calculated probabilities of PCI-induced photoelectron recapture in Ne 1s near-threshold photoionization with recent experimental data 22,23 . Good agreement between theory and experiment has been observed. Later on we applied our approach to the description of coincidence experiments 11 . The photoelectron­Auger electron coincidence measurements 24 ­30 proved to be a valuable source of information concerning the tiny details of the ionization dynamics. In particular, such experiments were successfully used for studying the PCI effects in nearthreshold photoionization 31,32 . Our first calculations 11 demonstrated that the probability of sequential double photoionization that can be measured in coincidence experiments with ultrashort ionizing pulses is strongly influenced by the pulse properties. This opens an additional dimension in studying the process. In the papers 10,11 we considered a single open-channel problem, and applied the method to the calculation of characteristics of the 1s photoionization of Ne atom where only p photoelectron can be emitted. In this case, only the energy spectra are of interest, since the angular distribution of photoelectrons is trivial and does not depend on the dynamics of the process, at least within the standard dipole approximation. In the general case of multichannel ionization, the angular distributions can be affected also. The influence of the PCI effects on the angular distributions of emitted particles, or in other words the angular dependence of the PCIdistorted shape of spectral lines, was studied for the Auger electrons in stationary approaches in Refs. 33­35 . It was shown that if energies of photo- and Auger electrons are
©2006 The American Physical Society

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strongly different, the influence of the PCI effects is rather small unless both electrons are emitted in close directions. This theoretical conclusion was confirmed by experiments with synchrotron radiation sources in which the angleresolved photo- and Auger electrons were detected in coincidence 28 . If only one electron is detected then the influence of the PCI effects on the angular distribution of the emitted electron is small 34 . It is of interest to investigate this problem for the shortpulse photoionization. In order to implement this task we extend our model to a multichannel case including channels, characterized by different orbital angular momenta of photoelectrons. This gives us the possibility to analyze the angular distribution of photoelectrons and study its dependence on the parameters of the ultrashort pulses. For the case of innershell photoionization and the following Auger decay, we consider the triple differential cross section TDCS which is differential with respect to the energies of the both emitted electrons and the angle of the photoelectron emission. Thus we are able to analyze the angle-dependent shape of the energy distributions as measured in coincidence experiments. We shall show that the energy distribution strongly depend on the duration of the ionizing pulse. On the other hand, the angular anisotropy is independent of the pulse parameters. As examples, we calculate the TDCS for photoionization of the 2 p shell in Ar and 3d shell in Kr. Both these examples were well studied experimentally with quasi monochromatic photons and well described theoretically within different models 36 ­38 . In the next section we give a description of our model. The derivation of the basic equations and the main approximations have already been published 10,11 . Therefore here we only remind them briefly in order to introduce all necessary notations. The second part of Sec. II deals with the partial amplitudes Sec. II B , the cross sections Sec. II C , and the angular distributions Sec. II D in a multichannel case. Section II E provides necessary computational details. The results of calculations for Ar 2 p and Kr 3d photoionization are presented and discussed in Sec. III. Concluding remarks are given in Sec. IV. Atomic units are used throughout unless otherwise indicated.
II. THEORY A. Basic equations of the nonstationary approach

amplitude ¯ 0 and the pulse shape function t with a unit E amplitude. Initially, for t - , i.e., before the atom is illuminated by the radiation pulse, the system is in the ground state g which, within the independent electron model, may be presented as
g

^ =A

0

r

h

.

3

Here 0 r is the wave function of the core electron to be excited, h is the wave function of all other electrons the ^ wave function of the atom with the vacancy , and A is the operator of antisymmetrization. In Eq. 3 denotes all variables which characterize all other electrons except the active one. Here we assume that one may consider only one vacancy, others are assumed to be energetically far away and may be neglected. Time evolution of the system is described by the nonstationary SchrÆdinger equation i t t ^ =H t t 4

with the initial condition t =- = g. In order to solve Eq. 4 we define the basis of the states relevant to the process considered. The radiation excites the core electron to an unoccupied Rydberg or continuum state so that a core-excited atomic state d is formed. Within the frozen core approximation, the wave function of this state may be presented as
d

^ =A

d

r

h

5

with d r being the single-electron wave function of the excited electron. The state d is unstable against Auger decay autoionization to the final state containing doubly ionized atom and emitted Auger electron. Since we are not interested in the angular distribution or spin polarization of the fast Auger electron, we average over its emission angle and its spin direction. Then the only parameter characterizing the Auger electron is its energy. We denote the final states of where index j the system after the Auger decay by j identifies the final ionic state and is the energy of the Auger electron. Thus
j

^ =A

j

r

j

,

6

We consider a multielectron atom interacting with a short electromagnetic pulse. In the long-wavelength dipole approximation the Hamiltonian of the system can be presented in the following form: ^ ^ H t = He - D · E t . 1

where j r is the one-electron wave function of the photois the corresponding function of excited electron, and j all other electrons in the channel which contains doubly charged ion core state j with two vacancies in outer shells and one electron with energy in continuum. A solution of Eq. 4 can be presented as an expansion in terms of the above functions: t = Cg t
g

^ Here He is the electronic Hamiltonian, D is the operator of the dipole moment, and E t is the electric field of the radiation pulse E t = eE0 t cos
0

+ Cd t

d

+
j

d Cj t

j

.

7

t = e¯ E

0

t cos

0

t

2

with the carrier frequency 0, the polarization vector e, and the envelope E0 t which is presented as a product of the

The initial conditions are Cg - = 1 and Cd - = C j - = 0. Inserting this function into the SchrÆdinger equation 4 , multiplying the latter from the left in turn by and j and integrating over , we obtain the folh lowing basic system of equations for the combined coordi-

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PHYSICAL REVIEW A 73, 062712 2006

nate functions of the excited electron and corresponding time-dependent coefficients see Ref. 10 for details : i
d

r, t ^ = H1 r t +
j

one can show that within these approximations the system of equations 8 and 9 can be reduced to a simpler system of equations with a complex Hamiltonian: i
d

d

r, t - r · E t ^ dV

0

r exp - i 0t 8 r, t ^ = H1 r - i t 2
d

j

j

r, t ,

1 r, t - zE0 t 2

0

r 12

-

exp - iEexct ,
j

i

j

r, t ^ = H2 r + t

^ r, t + V

* j

d

r, t .

9 i

j

Here d r , t is a wave function of the excited electron in the ^ intermediate decaying state, H1 r is the corresponding single-particle Hamiltonian which includes the field of the ^ vacancy, created by the photon, H2 r is the Hamiltonian which contains the field of the doubly ionized state of the final ion, and j r , t is a wave function of the photoexcited electron after the Auger decay to the final state j with emission of the Auger electron of energy . The latter is counted from the position of the corresponding diagram Auger line EAj0 , i.e., =E
j A

r, t ^ = H2 r + t

j

r, t + V

* j

d

r, t .

13

-E

j A0

=E

j A

- Eh - E

j hh

,

10

where EAj is the Auger electron energy as measured in an j experiment, Eh and Ehh are the energies of the singly charged and doubly charged ions in the decay channel j, respectively. Thus the value = 0 corresponds to the energy of the diagram line. In deriving Eqs. 8 and 9 it was supposed that the states j are prediagonalized and orthonormal, i.e.,
j

^ H

e

j

^ = H2 r +

j, j

-

.

11

Here the subscript means that integration is made over the variables describing all electrons except the excited one. The second term in the right-hand side of Eq. 8 describes the interaction of the active electron, being initially in the state 0 r with energy 0, with the electric field of the pulse. It is assumed that the electric field is weak and, in the spirit of the perturbation theory, we consider only linear terms with respect to the field and disregard "recoupling" of the ground ^ and the excited state. Finally, V j in Eqs. 8 and 9 is the interaction responsible for the decay of the intermediate state ^ ^ He h . to the final continuum channel j, V j j We consider the case where the Auger electron is fast the typical energy is in 100-eV range . In contrast, the nearthreshold photoelectron is slow a few eV . Thus the interaction of the Auger electron with the photoelectron is small and may be ignored in comparison with the abrupt change of the interaction of the photoelectron with the core. Conse^ quently, the interaction V j can be considered as independent of photoelectron coordinates r. To proceed further, we apply two additional approximations. The interaction of the pulse electric field with the electron is described within the rotating wave approximation. We also suppose that the decay amplitude is independent of the ^ Auger electron energy , V j V j but it is still different for different channels. Similarly to the derivation in Ref. 10 ,

Here = j j =2 j V j 2 is the total width of the decaying state, j being the partial width of the decay to the jth final state, Eexc = 0 - 0 is the excess energy, i.e., the difference between the photon energy and the threshold for the innershell ionization. We have assumed that the photon beam is 1 linearly polarized along the z axis. The factor 2 in Eq. 12 appears within the rotating wave approximation due to the cos 0t factor in Eq. 2 . In what follows, we apply the above formalism to innershell photoionization of noble-gas atoms Ar and Kr. To simplify the formulas and the discussion, we describe the ground and excited states of the atoms within the LS coupling approximation. Besides, since the dipole operator does not act on spins, we suppress the spin variables and characterize all atomic ionic and electron states only by the orbital angular momenta. If necessary, the spin can be easily incorporated, moreover, the formalism can be easily adopted to any coupling scheme. Thus the ground state of the atom is g L0 =0 , the core-excited state created in photoabsorption is Po state, i.e., d L =1 . If the ground atomic state is described by a single configuration then in the initial state the active electron possesses a certain orbital angular momentum 0 r = 0 r . Due to the dipole selection rules only states with = 0 ± 1 are populated by absorption of the electromagnetic pulse. In general, the single-particle Hamil^ tonians Hi r , i =1,2 are matrices on the basis of the orbital angular momentum eigenfunctions. However, in the majority of practical methods of calculation of atomic structure, angle-average single-electron potentials are used and the corresponding Hamiltonians are supposed to be diagonal with respect to . In this case the system of equations 12 and 13 splits into independent systems for each particular partial wave. Using the standard technique of calculating the multielectron matrix elements 39 , one can obtain equations similar to Eqs. 12 and 13 but written for the singleelectron wave functions with particular . Separating the angular and the radial parts of the single-particle Hamiltonians and averaging over the angular coordinates, one can reduce the system to the following system of equations for the radial parts of the single-electron wave functions: i u
d

r, t t

^ =h

1

r -i
0

2

u

d

1 r, t - rE0 t C , 2

0

u

0

r exp - iEexct ,

14

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PHYSICAL REVIEW A 73, 062712 2006

i

u

j

r, t t

^ =h

2

r+

u

j

r, t + V

* j

u

d

r, t .

15

¯~ Eu

^ r, ¯ = h E

2

r+ ~ u

r, ¯ + V E r, t ,

* j -

dt 17

Here the functions u r , t are the radial parts of the correr , t = r-1u r , t sponding partial wave functions Y m , , where Y m , are spherical harmonics. The ^ radial parts of the Hamiltonian Hi r i =1,2 are denoted ^ hi r . They are determined as 1 d2 r =- + 2 dr2 +1 +v 2r2

¯ exp iEt u where ~ u transform of 17 can be GE r , r of ~ u r , ¯ = dt exp E the continuum written using ^ the operator h r, ¯ =- E
0

d

¯ iEt u r , t is the Fourier wave function. A solution of Eq. the stationary Green's function 2 r as follows:
* j -

^ h

l i

dr V G
E

¯ dt exp iEt r ,t , 18

i

r,

16

r, r u

d

where vi l r is the effective potential acting on the photoelectron. For simplicity, we assume that this potential is local and approximate the exchange interaction of the photoelectron with the core electrons by a local Slater-type potential see below although, in principle, the nonlocal exchange interaction can be included, at least within the Hartree-Fock approach. The coefficient C , 0 in Eq. 14 is the angular part of the dipole matrix element C , 0 = 2 0 +1 /3 00,10 0 , where the last factor is the Clebsch-Gordan coefficient. The solution ud r , t of Eq. 14 with the above discussed initial conditions is unique; it represents a radial wave packet which is pumped by the electromagnetic pulse and at the same time is subjected to the decay due to the coupling with open channels. Because of the decay, the function ud r , t disappears at t + and therefore it has no physical meaning. On the contrary, the solution of the second equation 15 , u j r , t , describes the photoelectrons with a certain outgoing in a certain channel j. This wave packet is pumped by the decay process and its time development is determined ^ by the Hamiltonian h2 r with the potential of the doubly charged ion. In the following we discuss the double and triple differential cross sections for photoionization with Auger decay to a particular channel j. Since in our model the decay channels are independent, to simplify the formulae we suppress the index of channel j everywhere except in the notation V j in order to remind the reader that in general V j determines the partial width and not the total width in our equations.
B. Partial amplitudes

where E = ¯ - . As we shall see below, the chosen Green's E function corresponds to the emission of an electron with the kinetic energy E. The solution 18 contains full information about the photoelectron of the given energy E and angular momentum in the selected channel j when the Auger electron escapes with the energy . To achieve our goal, we should evaluate this wave function at the far-away "screen" large r . We apply the well-known representation for the stationary Green's function GE r , r see, for example, Ref. 40 : G r, r = 1 W
E + -

E;r E;r

- +

E;r E;r

r r

r r

E

. 19

Here WE is a function, related to the Wronskian determinant see Eq. 23 below . The functions ± E ; r are the solutions of the homogeneous SchrÆdinger equation ^ h
2

r -E

±

E ; r =0

20

with different boundary conditions which we choose as follows:
-

E ; r Ar 2 sin kr - k

+1

at r 0, at r 21

-

E;r

ln 2kr +

E

k = 2E,
+

= Z / k, here Z =2 , and E ; r exp i kr - ln 2kr at r . 22

Basically, the time-dependent wave functions, obtained from Eqs. 14 and 15 , allow one to calculate all of the physical observables which can be measured in the process. However, the method of extraction of information from those wave packets deserves a more detailed consideration. Usually, in a photoemission experiment, the yield of the electrons within a certain energy range at a given solid angle is measured, and we have to define the corresponding quantity in the theory. As a first step, we perform a Fourier transformation of Eq. 15 :

E =- /2+ E + E is the phase shift The value E . The boundary which includes the Coulomb phase conditions 21 ensure that the solution - E ; r is the con^ tinuum eigenfunction of the Hamiltonian h2 r corresponding to the energy E and normalized over the energy scale. Using the asymptotic forms 21 and 22 , one can evaluate the function WE as

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PHYSICAL REVIEW A 73, 062712 2006

W

E

=- =

1d 2

+

E;r dr

-

E;r -

d

-

E;r dr

+

E;r 23

i

dA

,E

t

dt

= E+

A

,E

t + exp i

EV

* j

C

E

t. 29

k exp - i 2

E.

The solution of this equation,
t

Since we are interested in the values of the function ~ r , ¯ at large r, and on the other side, the function E u ud r , t is localized due to decay, we should take the upper line in the representation 19 . Taking into account Eq. 23 , E we obtain the asymptote of the solution ~ r , ¯ at large r: u ~ u r, ¯ E 2 exp i kr - k ln 2kr A ,E . 24 Here we have introduced the partial amplitudes A , E = exp i E
-

A

,E

t = exp i

E
-

dt exp i E +

tV

* j

C

E

t, 30

dt exp i E +

tV

* j

C

E

t 25

with C

E

t being an overlap integral: C
E

t=
0

dr

* -

E;r u

d

r, t .

26

Early we have obtained similar expression for the modulus of the amplitude in a single-channel case see Ref. 11 , Eq. 9 . Expression 25 is an extension of that expression which allows one to calculate the angular distributions and can be applied for a multichannel case. We note here that according to Eqs. 25 , 26 , and 14 , the partial amplitudes E A , E are proportional to the electric-field amplitude ¯ 0 /2. In calculations we always set this overall factor to unity, taking it into account in the final normalization of the cross sections see below . One can easily check see Ref. 11 that the amplitude 25 can be presented in another equivalent form A , E = exp i E
0

coincides with the amplitude 25 at t . Expressions 25 and 27 provide two alternative ways for evaluation of the amplitudes 10,11 . One way is to propagate Eq. 15 together with solving numerically Eq. 14 for few selected values of Auger electron energy and then to project the solutions u r , t at large t onto the continuum function * E ; r . Another way is to project the solution ud r , t - onto - * E ; r in order to evaluate CE t Eq. 26 at several values of the photoelectron energies E and then to perform the Fourier transformation Eq. 30 . For future discussions it is useful to present the amplitude A , E in yet another form. Following the reasoning in Ref. 11 , one easily proves that the amplitude A , E can be presented as a product of two factors: the spectral function of the pulse, f , and some universal function S , E which contains the entire information about the particular photoionization and Auger transition but which is independent of the properties of the pulse: A ,E = =f where S ,E = C , ^ h
1 0

df

-E

exc

S

,E ,E ,

+E- 31

+E-E

exc

S

exp i r -i 2

E

-

E;r V
-1

* j

- -E

zu

0

0

r,

32

dru

r, t +

* -

E;r , 27

and f + E - Eexc is the spectral function, i.e., the Fourier transform of the pulse: f =
-

dt exp i t

t.

33

i.e., as a projection of the function u r , t taken at infinitely large time onto the continuum wave function - E ; r , ^ which is an eigenfunction of Hamiltonian h2 r normalized on the energy scale. In order to prove this, let us introduce the functions A
,E

C. Cross sections

t = exp i

E
0

dru

r, t

* -

E;r .

28

Knowing the amplitudes, one can calculate the cross section for the process. First, taking into account the axial symmetry of the problem with respect to the electric-field vector z axis , one can write the Fourier transform of the total wave r , t as function ~ r, E = r-1~ u r, ¯ Y E
0

On the one side, the amplitude 27 is the limit tion A ,E t at large t . On the other side, this to satisfy the following equation which may be multiplication of Eq. 15 with - * E ; r from by integration over r:

of the funcfunction has obtained by the left and

.

34

According to Eq. 24 , the asymptotic form of this function is

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PHYSICAL REVIEW A 73, 062712 2006

~ r

,E =

2 -1 r exp i kr - k A ,E Y
0

ln 2kr . 35

d2 =2 d dE

0

K

-1

A

,E

2

,

41

where =1/ c = 1 / 137 is the fine-structure constant. Using representation of the amplitudes given by Eq. 31 , we can rewrite Eq. 41 in the following form: d2 =2 d dE
0

Applying the inverse Fourier transformation one obtains the asymptotic form of the wave function r,t : r ,t = dE 2 k A
-1/2 -1

K

-1

f

+E-E

exc

2

S

,E

2

.

42

r exp i kr -
0

ln 2kr - Et 36

,E Y

.

Suppose a detector is placed at large r at the angle with respect to the electric-field vector and it registers electrons with energy E, while another detector registers Auger electrons with energy . Then, according to the general quantummechanical approach 41 , from the asymptotic form 36 it follows that the number of registered photoelectrons electron yield per one pulse can be calculated as follows: d 3Y , E , d dEd = A ,E Y
2 0

.

37

It is convenient to normalize the yield to one photon of the carrier frequency in the pulse. In the following we call this normalized yield a "cross section," although it becomes the usual cross section only at the limit of a very long pulse. For a short pulse this value depends on the pulse properties. In fact, the total number of the photons passing through a unit square during the pulse is N= c 4
0 T/2

One clearly sees that the DDCS measured in coincidence experiments depends only on the spectral distribution of the 2 pulse, f . It is not sensitive to any kind of phase relations within the ionizing pulse in the frequency domain. The same is valid for the TDCS Eq. 40 also. This is a wellknown fact see, for example, Refs. 42,43 which is a consequence of ionization by a weak field. Formally it follows from the use of the first-order perturbation theory for the electric-field­atom interaction. In the case of quasi monochromatic light t =1 in Eq. 2 , one should take the limit of Eq. 42 at large T T . The factor f + E - Eexc 2 turns out to be 2 T + E - Eexc 41 , and the factor K T. Then Eq. 42 can be reduced to the usual stationary expression for the cross section of photoionization with the following Auger decay see, for example, Refs. 16 or 18 . Indeed, by expanding the Green's function in S , E see Eq. 32 in ^ terms of eigenfunctions of Hamiltonian h1 r , we obtain the following expression: d2 =2 d dE
0

+E-E C,

exc

j

0

dq

-

q qzu 2

0

0

r

2

,

dt E t
-T/2

2

= c¯ K/ 8 E
2 0

0

,

38

q-i

-E

exc

43 which is equivalent to Eq. 5 of Ref. 18 . With the help of the energy function one can integrate Eq. 43 over or over E, to obtain the line profile in the spectrum of the photoelectrons or the Auger electrons, respectively, which includes the effects of PCI. Finally, if the excess energy is large, the line shape becomes Lorentzian. Then the total yield of the particular Auger line, W j, can be obtained by integration over the line profile giving the obvious expression: W j =4 =
2 0

where c is the light velocity, T characterizes the pulse duration interval T should be large enough to cover the pulse and
T/2

K=
-T/2

dt

t

2

39

with t being the power-independent envelope of the field see Eq. 2 . Taking into account the overall factor ¯ 2 / 4, the E0 normalized triple differential yield the cross section can be written as d
3

C,
j

0

-

zu

0

0

r

2

j

/ 44

photo

/

,

, E, d dEd

=

2 cK

0

A

,E Y

2 0

.

40

where photo is the total photoionization cross section for the considered inner shell 36 .
D. Angular dependence of the cross sections

By integration over angles of the emitted photoelectron, one obtains the partial double differential with respect to the energy of photo- and Auger electrons cross section DDCS for the transition to the certain final state of the doubly charged ion,

The TDCS, Eq. 40 , can be written in a more familiar form. We note that the amplitudes A , E have a joint factor f + E - Eexc which may be separated out of the sum

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PHYSICAL REVIEW A 73, 062712 2006

over , similar to Eq. 42 . Thus different photon frequencies do not interfere also in the angular distributions of photoelectrons which may be presented in the following standard form: d3 d dEd = 1 d2 1+ 4 d dE , E P2 cos , 45

where d2 / d dE is the DDCS, given by Eq. 41 . This angular dependence follows directly from the axial symmetry of the problem and the dipole character of the photon-atom interaction 44,45 . We remind that we have averaged over the emission angle of the Auger electron and consider the angular distribution of photoelectrons with respect to the only symmetry axis, which is the electric-field vector of linearly polarized light. , E can be written in the The asymmetry parameter following form compare, for example, with Eq. 2.136 in Ref. 45 : ,E = 2 1 A ,E
2 -1

6
,

-1

0

2

+1

+1 1

1/2

0, 0 20
0

2

A

,E A

*

,E ,

46

where the standard notation for the Wigner 6 j symbol is used. The TDCS, Eq. 45 , describes the energy and angular distribution of photoelectron excited by the short pulse and measured in coincidence with the energy selected Auger electron. As we have shown previously, the shape of the line in the spectrum of photoelectrons, as well as of Auger electrons, depends on the pulse duration 10 . In contrast, the asymme, E is independent of the properties and, in try parameter particular, of the duration of the pulse. Indeed, the function , E is determined by the ratio of the amplitudes, therefore the pulse spectral function cancels out. If in an experiment only photoelectron is detected then the DDCS which describes its energy and angular distribution can be obtained by integration of Eq. 45 over the Auger electron energy , d2 dEd Naturally, the DDCS, d the standard form, d dEd where d = dE d d2 d dE 49
2 2

FIG. 1. Color online Two-dimensional plots of the amplitude squared A , E 2 in atomic units for sequential double photoionization of Ar 2 p by a short pulse. The excess photon energy above 2 p ionization threshold is 2 eV. The pulse durations are a 3.3 fs, b 1.1 fs, c 540 as, and d 270 as in correspondence with the spectral width of the pulse indicated at the top of the panels.

=

d

d3 . d dEd

47

Since the integral over the Auger electron energy cannot be presented as a product of two factors as Eq. 42 , not only the cross section d / dE but also the averaged asymmetry parameter ¯ E can depend on the duration and spectral distribution of the pulse. We note that the asymmetry parameter is changing across the photoelectron line. What is usually premeasured with the sented as an experimental value of quasimonochromatic light very long pulse is an average value of ¯ E over the photoelectron line profile.

/ dEd , can be also presented in
E. Computational details

=

1d 1+ ¯ E P2 cos 4 dE

,

48

In the particular calculations discussed below we have chosen the inverse hyperbolic-cosine shape of the pulse, t = 1/cosh t/ , 50

describes the photoelectron line profile averaged over the emission angles, while ¯ E , the asymmetry parameter averaged over the Auger electron line, characterizes the variation of the photoelectron line profile with the emission angle.

where characterizes the duration of the pulse. The single ^ electron potentials vi r in the Hamiltonians hi r see Eq. 16 have been calculated in Hartree-Slater approximation 46 separately for the initial atom and the final ion. In this approximation the potential is the same for different . The choice of the potential was determined by its simplicity. Besides, for particular examples which we have considered see

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FIG. 2. Color online A two-dimensional plot of the function S ,E 2= S , E 2 in atomic units for sequential double photoionization of Ar 2 p see text for details . The excess energy is 2 eV.

next section , the results of calculations within the HartreeSlater and the Hartree-Fock approximations are very close 36 . Note that our calculation scheme is quite flexible, and it allows one to easily incorporate any other local or even nonlocal and -dependent potential, if necessary. The numerical solution of the time-dependent equations 14 and 15 have been obtained using the Crank-Nicolson integration scheme 47,48 . All details of the mathematical procedure have been presented and discussed previously 10 . An important ingredient of our algorithm is the substitution of the variables r = x2 49 . The nonstationary ShrÆdinger equation 14 has been discretized on the mesh uniform in the x variable, with the step x = 0.007 a.u. and the total number of steps N = 8000. The corresponding com-

FIG. 4. a The photoelectron line profile F E for the 2 p photoionization of Ar at the excess energy of 2 eV for various pulse durations: 1 ­ 3.3 fs, 2 ­ 1.1 fs, 3 ­ 540 as, and 4 ­ 270 as. b The corresponding asymmetry parameter ¯ E .

FIG. 3. Color online A two-dimensional plot of the asymmetry parameter , E for sequential double photoionization of Ar 2 p see text for details . The excess energy is 2 eV.

putation box reaches rmax 3100 a.u., but starting from rgobb 2000 a.u. an artificial absorbing potential appears. Thus the wave packets are correctly computed only for r rgobb. At this maximal value of r the mesh step is r =2x x 0.7 a.u. that allows us to consider very accurately the waves with the wave length 10 r 7 a.u. corresponding to the electron energy E about 10 eV. During the time of computation dt = 0.02 a.u., Tmax 2000 a.u. the wave packet d r , t does not essentially leave the region of effective computation, decaying almost completely before reaching the region where the adsorbing potential becomes operative. Therefore the conditions of implementation of the computational scheme are quite comfortable and the accuracy of more than first three digits in the cross sections is obtained. The computational scheme works stably and quite fast, therefore the computations can be performed on a PC. Time of computation of the entire set of the outcomes for a fixed pulse and one excess energy is within the 2 ­ 3-h range for PC Athlon 1300 MHz . Luckily, the shorter the pulse, ,E the better we can compute the functions S , E and in a quite broad range of energies. Therefore the computation time is mainly determined by the decay rate of the Auger state and is not limited by the pulse duration. Only one computation, for any shape of the pulse, allows one to obtain these functions and, consequently, compute the outcome for any pulse shape.

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PHYSICAL REVIEW A 73, 062712 2006

FIG. 6. Color online A two-dimensional plot of the function S ,E 2= S , E 2 in atomic units for sequential double photoionization of Kr 3d see text for details . The excess energy is 2 eV.

FIG. 5. Color online Two-dimensional plots of the amplitude squared A , E 2 in atomic units for sequential double photoionization of Kr 3d by a short pulse. The excess photon energy above 3d ionization threshold is 2 eV. The pulse durations are a 1.1 fs and b 270 as in correspondence with the spectral width of the pulse indicated at the top of the panels. III. Ar,,2p... AND Kr,,3d... PHOTOIONIZATION. RESULTS AND DISCUSSION

An application of the above described approach to a particular atomic system is illustrated below by calculations of near-threshold photoionization of Ar 2 p and Kr 3d with the following Auger decay LMM and MNN, respectively. The 2 p vacancy in Ar has the width of 120 meV 50 which corresponds to the lifetime of 5.4 fs. The width of the 3d vacancy in Kr is smaller, = 90 meV 50 . The corresponding lifetime is 7.2 fs. In Fig. 1 the results of calculations for Ar 2 p sequential double photoionization are shown for four different values of the pulse duration and the excess energy

of 2 eV. The values A , E 2 which are proportional to the DDCS see Eq. 41 are presented as gray color -scale two-dimensional plots, as functions of the photoelectron energy and the Auger electron energy the latter is counted from the position of the diagram Auger line . Both energies are in eV. Note different scales on the axes. In the case of the longest pulse = 3.3 fs the pattern has a typical narrow, almost linear, shape concentrated along the straight line + E = 0 - 0 = Eexc which reflects the conservation of energy. Long pulse is close to the quasimonochromatic light with the energy 0. The maximum of the cross section is shifted to the positive values of which is typical for the PCI effect. We remind that the position of the diagram line corresponds to =0. One sees that the spectral distribution of the cross section along the line + E = Eexc has an asymmetric shape with a sharp onset at small and a long descent at larger , what is also typical for a PCI-distorted line profile. Energy distributions of this kind for emitted Auger and photoelectrons has been recently discussed by Smirnova et al. 51 . However, they have used a phenomenological model for the Auger decay and the PCI effects were not included. This led to a symmetric distribution along the line + E = const. For the shorter pulses, the shape of the DDCS becomes broader and asymmetric see Figs. 1 b ­1 d . It becomes oriented more along the vertical axis axis of photoelectron energies . All these properties of the distributions can be easily explained by comparing them with the universal function S , E 2 presented in Fig. 2. The DDCS is a S ,E 2= product of this function and the spectral distribution of the incident pulse see Eq. 42 . When the pulse is long the distribution is narrow. It cuts a narrow stripe from the function S , E 2 along the line + E = Eexc. For decreasing pulse duration, the spectral distribution of the pulse becomes broader, covering a larger area of the S , E 2 distribution, which leads to the cross section patterns shown in Figs. 1 b ­1 d .

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PHYSICAL REVIEW A 73, 062712 2006

FIG. 7. Color online A two-dimensional plot of the asymmetry parameter , E for sequential double photoionization of Kr 3d see text for details . The excess energy is 2 eV.

The two-dimensional distribution of the asymmetry pa, E is shown in Fig. 3. It is rather smooth. At any rameter decreases from a maximum at value of the value of E = 0. This distribution is universal, independent of the pulse duration. The nonzero asymmetry parameter leads to the fact that the pattern of the TDCS measured in coincidence experiments depends on the observation angle. It is of interest to consider also a noncoincident experiment, when only the photoelectron is detected. In such a case, the angular dependence of the cross section is determined by Eq. 48 with the total cross section and ¯ parameter shown in Fig. 4. Both these values depend on the duration of the pulse. However, for the asymmetry parameter ¯ this dependence is rather weak. In Fig. 4 a we show the photoelectron profile function F E which is determined as F E =K
-1

FIG. 8. a The photoelectron line profile F E for the 3d photoionization of Kr at the excess energy 2 eV for various pulse durations: 1 ­ 1.1 fs, 2 ­ 540 as, and 3 ­ 270 as. b The corresponding asymmetry parameter ¯ E .

d

A

,E

2

.

51

The function F E is proportional to the cross section d / dE see Eqs. 49 and 41 . For our choice of the pulse profile, Eq. 50 , and sufficiently large T, the value K =2 . One sees from Fig. 4 a that the photoelectron line profile has a typical asymmetric shape which is determined by the PCI effects 12,13 and it strongly depends on the pulse duration as it was discussed previously 10 . In contrast, the asymmetry parameter ¯ E shown in Fig. 4 b depends only slightly on the pulse duration especially in the region of the maximum of the spectral line about E =2 eV . For comparison we note that the value for the monochromatic light with the excess energy 2 eV calculated using the same Hartree-Slater potential 36 is 1.05 which is rather close to the values shown in Fig. 4 b . Figures 5­7 show the results of similar calculations for the Kr 3d photoionization by ultrashort pulses with the following MNN Auger decay. The character of the DDCS

shown in Fig. 5 for two values of the pulse duration is rather similar to what we obtained for Ar compare with Fig. 1 . However, the functions S , E 2 Fig. 6 and especially , E Fig. 7 look rather different. The value of is increasing from the threshold for any value of . This is in agreement with the calculations in Ref. 36 . The line profile F E and the corresponding asymmetry parameter ¯ E for three values of the pulse duration are shown in Fig. 8. As in the Ar case, the line profile strongly depends on the pulse duration. The asymmetry parameter varies with the pulse length only at the wings of the line. In the central part, near the maximum of the yield, ¯ E is practically insensitive to the pulse duration. The asymmetry parameter calculated for the monochromatic light with the excess energy 2 eV is 0.722 36 which is close to the value of ¯ E at the yield maximum.

IV. CONCLUSIONS

We have developed a time-dependent approach for calculation of the triple differential cross section of sequential double photoionization of atoms by femto- and attosecond pulses. The method is based on the numerical solving of the system of coupled nonstationary SchrÆdinger equations

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PHYSICAL REVIEW A 73, 062712 2006

which describe the inner-shell photoionization and the following Auger decay. The developed formalism is applicable to a multichannel problem with several partial electronic waves involved. It allows one to consider the shape of the spectra as well as the angular distributions of photoelectrons in the near-threshold region for the case of photoelectronAuger electron coincidence experiments as well as for noncoincident measurements. As examples, the cross sections for the Ar 2 p and Kr 3d photoionization with the following Auger decay are calculated and discussed. It is demonstrated that the cross sections strongly depend on the duration of the ionizing pulse. On the other hand, the asymmetry parameter which characterizes the

angular anisotropy of the photoelectron emission is independent of the pulse properties in the case of two-electron coincidence experiments and rather weakly depends on those properties in the case of noncoincident measurements.
ACKNOWLEDGMENTS

We are grateful to Bielefeld University for hospitality and financial support. A.K.K. acknowledges with gratitude the financial support from Deutsche Forshungsgemeindshaft DFG . We also acknowledge the financial support from Russian Foundation for Fundamental Researches via Grant No. 05-02-16216 and 06-02-16289.

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