a r X i v : p h y s i c s / 0 2 0 3 0 6 5 v 2 [ p h y s i c s . a t o m  p h ] 8 A p r 2 0 0 2
Angular correlation theory for double photoionization in arare gas atom : ionization by polarized photons
Chiranjib Sur and Dipankar Chattarji
Department of Physics, VisvaBharati, Santiniketan 731 235, INDIA
This is a sequel to an earlier article on the theory of angular correlation for double photoionization. Here we consider thetwostep double photoionization of a rare gas atom under the inﬂuence of a polarized photon beam described by appropriateStokes parameters. Cylindrical mirror analyzers (CMA) are used to detect the outgoing electrons. Theoretical values of thecorrelation function are obtained for linearly polarized light. Two diﬀerent situations are handled. Once, the value of thecorrelation function is obtained keeping the photoelectron in a ﬁxed direction. In the other case the direction of the Augerelectron is kept ﬁxed. Comparison with experiments on xenon shows excellent agreement for the case of 4
d
5
/
2
photoionizationfollowed by a subsequent
N
5
−
O
23
O
231
S
0
Auger decay for a linearly polarized incident photon of energy 94
.
5
eV
[J. Phys.B,
26
, 1141 (1993)].PACS: 32.80, 32.80.H, 32.80.F, 03.65.T, 79.20.F
I. INTRODUCTION
In an earlier paper [1] we considered the double photoionization (DPI) of a rare gas atom under the inﬂuence of a unpolarizedphoton. The atom was taken to be in a randomly oriented
1
S
0
state. We considered the angular correlation between the twosuccessively emitted electrons, their emissions being adequately separated in time [2]. Using a statistical theory we obtainedgood agreement with the experimental results of K¨
a
mmerling and Schmidt [3].In the present paper we take the incident photon beam to be polarized. The rare gas atoms receiving the photon beam nolonger remain randomly oriented, but become aligned. If a photon of adequate energy is absorbed by an atom, a photoelectronis emitted from one of its inner shells, leaving the atom singly ionized. This ion subsequently deexcites by emitting an Augerelectron [4] from one of its outer shells. We are left with a doubly ionized atom and two electrons in the continuum.The doublephotoionization process described above therefore amounts to
hν
+ A
−→
A
+
+
e
−
1
−→
A
++
+
e
−
1
+
e
−
2
.
(1)As in reference [1] we denote the initial state (photon+atom) by the set of quantum numbers (
J
a
M
a
α
a
), or by virtual quantumnumbers (
J
′
a
M
′
a
α
′
a
), keeping in mind possible interaction with other atoms and electrons. (
J
a
,M
a
) or (
J
′
a
M
′
a
) are angularmomentum quantum numbers, and
α
a
,
α
′
a
stand for the set of remaining quantum numbers. Similarly for the intermediate andﬁnal states. The polarization properties of the photon beam are described by appropriate Stokes parameters
S
1
,S
2
and
S
3
[5].
II. DPI BY POLARIZED PHOTONS :
We proceed by calculating the density matrix [6] and the angular correlation function, which is the expectation value of theeﬃciency operator for the detection of electrons. The density matrix of the initial state equals the product of the density matrixof the intermediate singly ionized atom and the density matrix of the photoelectron. Similarly, the density matrix of the singlyionized atom can be written as the product of the density matrices of the doubly ionized atom and the Auger electron.Using the WignerEckart theorem, the matrix element of the density operator for the initial atomic state can be expressedas [1]
J
a
M
a
α
a

ρ
J
′
a
M
′
a
α
′
a
=
k
a
κ
a
(
−
1)
J
′
a
−
M
′
a
C
J
a
J
′
a
k
a
M
a
M
′
a
κ
a
ρ
k
a
κ
a
(
J
a
α
a
,J
′
a
α
′
a
)
.
(2)Here the statistical tensor
ρ
k
a
κ
a
is an irreducible tensor of rank
k
a
, which transforms according to the (2
k
a
+ 1) dimensionalirreducible representation
D
k
a
of the rotation group. In Eq.(2)
C
J
a
J
′
a
k
a
M
a
M
a
κ
a
is a ClebschGordan coeﬃcient satisfying the trianglerule
k
a
=
J
a
+
J
′
a
and
κ
a
is the projection of
k
a
. Using the unitarity property of ClebschGordan coeﬃcients we get
1
ρ
k
a
κ
a
(
J
a
α
a
,J
′
a
α
′
a
) =
M
a
M
′
a
(
−
1)
M
a
−
M
′
a
C
J
a
J
′
a
k
a
M
a
M
a
κ
a
J
a
M
a
α
a

ρ
J
′
a
M
′
a
α
′
a
.
(3)We assume that the initial state is formed after the randomly oriented rare gas atom absorbs a photon. Then the densitymatrix of the initial state becomes
ρ
k
a
κ
a
(
J
a
α
a
,J
′
a
α
′
a
) = 3(2
J
0
+ 1)
k
0
κ
0
k
γ
κ
γ
√
2
k
0
+ 1
2
k
γ
+ 1
C
k
0
k
γ
k
a
κ
0
κ
γ
κ
a
×
J
0
J
′
0
k
0
1 1
k
γ
J
a
J
a
k
a
ρ
k
0
κ
0
(
J
0
,J
0
)
ρ
γ k
γ
κ
γ
(1
,
1)
.
(4)This equation satisﬁes the triangle rule
k
0
=
J
0
+
J
′
0
, where
J
0
and
J
′
0
are the angular momentum quantum numbers of therandomly oriented atom before absorption of the photon and its virtual counterpart respectively. Here
ρ
k
0
κ
0
(
J
0
,J
0
) representsthe density matrix of the randomly oriented atom and can be expressed as
ρ
k
0
κ
0
(
J
0
,J
0
) = 1
√
2
J
0
+ 1
δ
k
0
0
δ
κ
0
0
J
b
j
1
J
a
J
b
j
′
1
J
a
⋆
J
c
j
2
J
b
J
c
j
′
2
J
b
⋆
.
(5)Here the symbol
stands for a reduced matrix element.In Eq. (4) the expression
ρ
γ k
γ
κ
γ
(1
,
1) represents the density matrix of the photon with its polarization properties. Its elementsare
ρ
γ
00
=
1
√
3
ρ
γ
10
=
S
3
√
3
ρ
γ
20
=
1
√
6
ρ
γ
1
±
1
= 0
ρ
γ
2
±
1
= 0
ρ
γ
2
±
2
=
−
12
(
S
1
∓
iS
2
)
.
(6)
S
1
,S
2
and
S
3
are the Stokes parameters [5] describing the polarization of the photon.Then Eq.(4) yields
ρ
k
a
κ
a
(
J
a
J
′
a
) = 3(
−
1)
J
a
+
k
a
+1
J
a
1 01
J
a
k
a
ρ
γ k
a
κ
a
(1
,
1)
×
J
b
j
1
J
a
J
b
j
′
1
J
a
⋆
J
c
j
2
J
b
J
c
j
′
2
J
b
⋆
=
3(
−
1)
J a
√
2
J
a
+1
√
2
J
′
a
+1
ρ
γ k
a
κ
a
(1
,
1)
J
b
j
1
J
a
J
b
j
′
1
J
a
⋆
J
c
j
2
J
b
J
c
j
′
2
J
b
⋆
.
(7)We deﬁne the angular correlation function as the expectation value of the eﬃciency operator [1]. Following the same notationas in reference [1] we can write it as
ε
=
J
a
J
′
a
α
a
α
′
a
k
a
κ
a
ρ
k
a
κ
a
(
J
a
α
a
,J
′
a
α
′
a
)
ε
⋆k
a
κ
a
(
J
a
α
a
,J
′
a
α
′
a
)
.
(8)Some simpliﬁcation gives
ε
=
ρ
k
a
κ
a
(
J
a
,J
′
a
)
ε
⋆k
c
κ
c
(
J
c
,J
′
c
)
ε
⋆k
1
κ
1
(
J
1
,J
′
1
)
ε
⋆k
2
κ
2
(
J
2
,J
′
2
)
×
C
k
b
k
1
k
a
κ
b
κ
1
κ
a
C
k
c
k
2
k
b
κ
c
κ
2
κ
b
√
2
J
a
+ 1
√
2
J
′
a
+ 1
√
2
k
b
+ 1
√
2
k
1
+ 1
×√
2
J
b
+ 1
2
J
′
b
+ 1
√
2
k
c
+ 1
√
2
k
2
+ 1
×
J
c
j
2
J
b
J
′
c
j
′
2
J
′
b
k
c
k
2
k
b
J
b
j
1
J
a
J
′
b
j
′
1
J
′
a
k
b
k
1
k
a
,
(9)where the summation extends over
J
a
,J
′
a
,J
b
,J
′
b
,J
c
,J
′
c
,j
1
,j
′
1
,j
2
,j
′
2
,k
a
,κ
a
,k
c
,κ
c
,k
1
,κ
1
,k
2
and
κ
2
.In Eq.(9)
ε
⋆k
i
κ
i
(
j
i
,j
′
i
) is the eﬃciency tensor component for detection of the
i
th electron. Here
i
= 1 corresponds to thephotoelectron, and
i
= 2 to the Auger electron. In DPI experiments the detectors usually used are cylindrical mirror analyzers(CMA) [7] which have cylindrical symmetry with respect to the axis of the detector. Details of the choice of detectors are givenin reference[8]. The eﬃciency tensor component now becomes
ε
⋆k
i
κ
i
(
j
i
j
′
i
) =
κ
′
i
z
k
i
(
i
)
c
k
i
κ
′
i
(
j
i
j
′
i
)
D
k
i
κ
′
i
κ
i
(
ℜ
i
)
.
(10)Since the residual doubly ionized state is unobserved, the corresponding quantum numbers are averaged over. This gives
ε
⋆k
c
κ
c
(
J
c
J
′
c
) =
√
2
J
c
+ 1
δ
k
c
0
δ
κ
c
0
δ
J
c
J
′
c
.
(11)Then Eq.(9) becomes
2
ε
=
ρ
k
a
κ
a
(
J
a
,J
′
a
)
√
2
J
c
+ 1
C
k
b
k
1
k
a
κ
b
κ
1
κ
a
C
0
k
2
k
b
0
κ
2
κ
b
√
2
J
a
+ 1
√
2
J
′
a
+ 1
×√
2
k
b
+ 1
√
2
k
1
+ 1
√
2
J
b
+ 1
2
J
′
b
+ 1
√
2
k
c
+ 1
√
2
k
2
+ 1
×
J
c
j
2
J
b
J
′
c
j
′
2
J
′
b
0
k
2
k
b
J
b
j
1
J
a
J
′
b
j
′
1
J
′
a
k
b
k
1
k
a
×
z
k
1
(1)
c
k
1
κ
′
1
(
j
1
j
′
1
)
z
k
2
(2)
c
k
2
κ
′
2
(
j
2
j
′
2
)
D
k
1
κ
′
1
κ
1
(
ℜ
1
)
D
k
2
κ
′
2
κ
2
(
ℜ
2
)
.
(12)Here we have used the relation
D
k
1
κ
′
1
κ
1
(
ℜ
1
)
D
k
2
κ
′
2
κ
2
(
ℜ
2
) =
k
C
k
1
k
2
kκ
1
κ
2
κ
C
k
1
k
2
kκ
′
1
κ
′
2
κ
′
D
kκκ
′
(
ℜ
) (13)to get the actual angular dependence of the angular correlation function. In Eq.(13) the Euler rotation
ℜ
= (
β
1
θβ
2
) [8]. Thisgeometrical dependence of the tensor matrix element is separated out from the dynamics by using the WignerEckart theorem.As a result, the dynamics of the DPI process resides in the reduced matrix elements and the geometric dependence is containedin the angular part.We deﬁne
ζ
=
√
2
J
c
+ 1
2
k
1
+ 1
√
2
J
b
+ 1
2
J
′
b
+ 1
2
k
2
+ 1 (14)and
ξ
=
J
b
j
1
J
a
J
b
j
′
1
J
a
⋆
J
c
j
2
J
b
J
c
j
′
2
J
b
⋆
.
(15)Then the expectation value of the eﬃciency operator in Eq.(12) becomes
ε
∼
(
−
1)
J
a
ζξ
J
c
j
2
J
b
J
′
c
j
′
2
J
′
b
0
k
2
k
2
J
b
j
1
J
a
J
′
b
j
′
1
J
′
a
k
2
k
1
k
×
C
k
2
k
1
kκ
2
κ
1
κ
z
k
1
(1)
z
k
2
(2)
ρ
γ kκ
(1
,
1)
c
k
1
κ
′
1
(
j
1
j
′
1
)
×
c
k
2
κ
′
2
(
j
2
j
′
2
)
C
k
1
k
2
kκ
1
κ
2
κ
C
k
1
k
2
kκ
′
1
κ
′
2
κ
′
D
kκκ
′
(
ℜ
)
.
(16)
A. Attenuation corresponding to polarization sensitivity of a detector
The electron detector may or may not be sensitive to the spin state of the incoming electron. The attenuation of the signaldue to the detector will depend on this sensitivity. The factor
c
k
i
κ
i
(
j
i
j
′
i
), (
i
= 1
,
2) describes this property [8]. We shall nowconsider two diﬀerent cases.
1. CaseI : Detectors insensitive to electron polarization
If the detectors(CMAs) are insensitive to the spin polarization of electrons then the projection
κ
i
of the
k
i
th component of the angular momentum is eﬀectively zero, i.e. the electrons are emitted symmetrically with respect to the axis of the detector.Hence the attenuation factor can be written as
c
k
i
0
(
j
i
j
′
i
) =
√
2
j
i
+ 1
2
j
′
i
+ 14
π
(
−
1)
j
i
−
12
+
k
i
C
j
i
j
′
i
k
i
12
−
12
0
.
(17)
2. CaseII : Detectors sensitive to electron polarization
In reference [1] we deﬁned
c
k
i
κ
i
(
j
i
j
′
i
) as the attenuation factor due to the change in the state of polarization of an electroncaused by the detector. When the detectors are insensitive to electron polarization, one takes the average over the electronspin and its projection. Now consider the case where the detectors are sensitive to electron polarization. In this case the spinsensitivity of the detectors is described by a tensor of the form
c
k
si
κ
si
(
s
i
s
i
). The attenuation factor then turns out to be
c
k
i
κ
i
(
j
i
j
′
i
) =
c
k
li
0
(
l
i
l
′
i
)
c
k
si
κ
si
(
s
i
s
i
)
2
k
l
i
+ 1
2
k
s
i
+ 1
×√
2
j
i
+ 1
2
j
′
i
+ 1
l
i
l
′
i
k
l
i
s
i
s
i
k
s
i
j
i
j
′
i
k
i
C
k
li
k
si
k
i
0
κ
i
κ
i
,
(18)
3
where
c
k
li
0
(
l
i
l
′
i
) =
√
2
l
i
+ 1
2
l
′
i
+ 14
π
(
−
)
l
′
i
C
l
i
l
′
i
k
li
000
,
(19)and
c
k
si
κ
si
(
s
i
s
i
) can be expressed in terms of the Stokes parameters describing the spin polarization of the electron to bedetected[3]. The factor
c
k
si
κ
si
(
s
i
s
i
) picks out electrons with a particular spin projection and may be called a
SternGerlach operator
. Its components are
c
00
=
1
√
2
c
10
=
S
ez
√
2
c
11
=
−
(
S
ex
−
iS
ey
)
c
1
−
1
=
−
(
S
ex
+
iS
ey
)
.
(20)Here
S
ex
,S
ey
and
S
ez
are Stokes parameters describing the polarization of the electron. For polarization insensitive detectors onehas
S
ex
=
S
ey
=
S
ez
= 0, and the attenuation factor reduces to Eq.(17).The lifetime of the singly ionized state is very small. Depending on the photon energy there may be a situation where it isimpossible to diﬀerentiate between the photo and Auger electrons simply by energy analysis. Then, to distinguish between thetwo electrons it is necessary to measure the electron spin, i.e. their polarization. For spin analysis of the electrons we have touse a SternGerlach type experimental setup. Here the factor
c
k
si
κ
si
(
s
i
s
i
) serves exactly that purpose, i.e. picks out electronswith a particular spin projection. This type of experiment is known as ‘energy and angleresolved coincidence experiment’ andis being done by Schmidt and his coworkers [9].In general, for DPI of atoms using polarized photon of suﬃcient energy, one can distinguish the photo and the Auger electronsby diﬀerential energy analysis. In that case determination of electron spin is meaningless. Then, if the spin is unobserved, onecan take the average over the spin projection. In that case the projection
κ
i
of the
k
i
th component of the angular momentumis zero and the attenuation factor
c
k
i
κ
i
(
j
i
j
′
i
) turns out to be
c
k
i
0
(
j
i
j
′
i
).
III. CALCULATION AND RESULTS
In reference [1] we treated DPI in the xenon atom due to unpolarized light. In this paper we are concerned with the samexenon atom with the diﬀerence that DPI occurs due to a polarized light source. A randomly oriented xenon atom is irradiatedwith a polarized photon beam of energy 94
.
5
eV
. As a result, the xenon atom no longer remains randomly oriented but acquiresthe polarization of the photon beam. This leads to photoionization in the 4
d
5
/
2
shell followed by a subsequent
N
5
−
O
23
O
231
S
0
Auger decay. We use the dipole approximation, the letters
e,f
and
g
for the three possible photoionization channels [1]. Theseare characterised by
e
)4
d
5
/
2
−→
ε
p
f
7
/
2
,
f
)4
d
5
/
2
−→
ε
p
f
5
/
2
and
g
)4
d
5
/
2
−→
ε
p
p
3
/
2
respectively. And the Auger transition ischaracterised by the wave
ε
A
d
5
/
2
. The same selection rules for photoionization and Auger transitions hold good as in the caseof unpolarized light.In experiments for measuring angular correlation one usually chooses detectors which are insensitive to the spin polarizationof electrons. In such a case
κ
1
=
κ
2
=
κ
′
1
=
κ
′
2
=
κ
=
κ
′
= 0, and
D
k
00
(
β
1
θβ
2
) =
P
k
(cos
θ
). Then Eq. (16) becomes
ε
∼
(
−
1)
J
a
ζξ
J
c
j
2
J
b
J
′
c
j
′
2
J
′
b
0
k
2
k
2
J
b
j
1
J
a
J
′
b
j
′
1
J
′
a
k
2
k
1
k
z
k
1
(1)
z
k
2
(2)
ρ
γ kκ
(1
,
1)
×
C
k
2
k
1
k
000
c
k
1
0
(
j
1
j
′
1
)
c
k
2
0
(
j
2
j
′
2
)
P
k
(cos
θ
)
.
(21)The summation extends over
k
1
,k
2
and
k
.In the limiting case of unpolarized photons Eq.(21) reduces to a simple form. Using Eq. (17) and some properties of 9
−
j
symbols and Racah coeﬃcients [10], we get
ε
=
k
z
k
(1)
z
k
(2)(
−
1)
j
1
+
j
2
c
k
0
(
j
1
j
′
1
)
c
⋆k
0
(
j
2
j
′
2
)
×
J
c
j
1
J
b
J
c
j
′
1
J
b
⋆
J
b
j
2
J
a
J
b
j
′
2
J
a
⋆
×
w
(
J
b
J
′
b
j
1
j
′
1
;
kJ
a
)
w
(
J
b
J
′
b
j
2
j
′
2
;
kJ
c
)
P
k
(cos
θ
)
.
(22)Note that this is identical with Eq.(25) of reference[1], as it should be.Experiments on the xenon atom were carried out by Schmidt and his coworkers using 94
.
5
eV
synchrotron radiation [3].They used a perpendicular plane geometry to describe the process. The collision frame
x,y,z
is attached to the target where the
z
axis coincides with the direction of the photon beam. The arbitrary polarization of the incident beam from the synchrotronis described by the Stokes parameters
S
1
,S
2
and
S
3
. Both
S
1
and
S
2
refer to the same quantity, but with diﬀerently orientedaxes. One can make
S
2
= 0 by choosing the
x
axis of the collision frame to coincide with the direction of maximum linearpolarization, i.e. the major axis of the polarization ellipse. To compare our results with experimental values we use the samepolar and azimuthal angles in the perpendicular plane geometry. Fig.1 shows the perpendicular plane geometry describedabove,
e
1
and
e
2
being the directions of emission of the photo and the Auger electron respectively.
θ
is the angle between theirdirections of emission. We have calculated the theoretical value of the angular correlation function for two diﬀerent cases.
4
(i) The photoelectron is observed in a ﬁxed direction and the Auger electron spectrometer is turned around to get the angulardistribution of the Auger electrons with respect to the photoelectron. Here the maximum allowed value of
k
is 2
j
2
.(ii)The second one is the complementary case, i.e. the Auger electron is observed in a ﬁxed direction and the photoelectronspectrometer is turned around to get the angular distribution of the photoelectrons with respect to the Auger electron. Herethe maximum allowed value of
k
is 2
j
1
,
max
,
j
1
,
max
is the maximum value of
j
1
for the possible photoionization channels.The value of
k
gives the highest order of the Legendre polynomials occurring in the correlation function. Interchannelinteraction of the diﬀerent photoelectron channels contributes to the angular correlation pattern by introducing the diﬀerentterms, however, the total intensity remain unchanged. This inter channel interaction is treated as it was in reference [1].As in reference [1] we have deﬁned the angular correlation function to be the angular part of the expectation value of theeﬃciency operator. Solid lines represents the theoretically calculated plot and the dots represents the experimental plot [3].For a linearly polarized incident photon beam the angular correlation function for our case turns out to bei) Case 1:
S
1
= 1
,S
2
= 0
,S
3
unknown. The photoelectron is observed in a ﬁxed direction and the Auger electronspectrometer is turned around to get the angular distribution of the Auger electron with respect to the photoelectron.
W
(
θ
)
∼
1 + 1
.
314
P
2
(cos
θ
) + 1
.
100
P
4
(cos
θ
)
.
(23)ii) Case 2:
S
1
= 1
,S
2
= 0
,S
3
unknown. The Auger electron is observed in a ﬁxed direction and the photoelectronspectrometer is turned around to get the angular distribution of the photoelectron with respect to the Auger electron.
W
(
θ
)
∼
1 + 0
.
817
P
2
(cos
θ
) + 0
.
602
P
4
(cos
θ
) + 0
.
570
P
6
(cos
θ
)
.
(24)In both the cases one of the electron spectrometers is kept ﬁxed along the direction of the electric ﬁeld vector (xaxis).The index
k
in the general theoretical expression for
ε
depends on the angular momenta of the emitted electrons. Hence, thestructure of the angular correlation pattern depends on this index. If higher order angular momenta are involved the angularcorrelation pattern has more structure. This is clear from the ﬁgures 2 and 3. Since the distribution of the photoelectron withrespect to the ﬁxed Auger electron direction involves higher order angular momenta, the angular correlation patterns has morestructure.
yz
θ
e1ephoton2xtarget
Figure 1 : Perpendicular plane geometry used in the experiments by Schmidt and his coworkers
5