2
Correlations in the initial states characterise in general the properties of the matter
and can have relevant effects in the reaction process. After the reaction has occurred,
the reaction products interact among themselves, due to the long range of Coulomb
force, and in some cases the effects due to the indistinguibility of some particle can
occur. These effects are refereed as final state correlations. Finally, when a reaction
proceeds in two (or more) steps and an intermediate excited state(s) is created, EC
can occur in such state(s), causing intermediate state correlations.
In order to measure effects due to EC, high efficiency or high resolution conditions
are often required. These conditions have been seldom achieved in the past years. The
recent experimental developments permitted to overcome part of these difficulties,
and the study of EC is actually a very active field of research. In particular the third
generation Synchrotron sources gave availability of high brilliance- , high polarised-
light, tunable with high resolution over a wide energy range (namely form infra-red
up to hard X-ray). The improvement of electron/ion spectroscopy techniques as well
as the new electronic standard for signal processing and data acquisition, increased
the overall resolution and efficiency performances of the experimental apparatus.
In this work we shall discuss two processes which are particularly suited for the study
of EC: the direct double photoionisation (DPI) of He and the Auger cascade decay of
the Ne 1s
-1
3p resonantly excited state. Both of them can be thought as different
aspects of the same physical process, namely the double photoemission, i.e. the
emission of two electrons following the absorption of one photon.
When the photon absorption is treated within the dipole approximation, the double
photoionisation process is entirely due to EC, so DPI measurements are supposed to
be particularly useful to understand EC. The DPI is particularly sensitive to final state
EC for photon energy exceeding threshold for few electronvolts. Conversely, at
higher energies the initial state EC are supposed to become relevant. The question of
the energy range where initial state correlations can be investigated, and the
feasibility of experiments at such energies is still debated.
3
The importance of the study of the Auger cascade decay has been clearly illustrated
both theoretically [I.1] and experimentally [I.2, I.3, I.4]. It has been shown that by
combining time correlated and non-correlated measurements it is possible in principle
to determine all the dynamical parameters characterising the process. Thus a
quantum-mechanical �complete� or �perfect� experiment can be performed. The
simplest closed-shell systems where the direct DPI and the Auger cascade can occur
are Ne and He respectively.
The kinematics of double photoemission is fully determined by measuring correlated
in time the momenta, i.e. kinetic energy and directions, of two out of three final
products of the reaction. This implies to measure in coincidence two electrons, or one
electron and the ion, selecting their kinetic energy and direction. It is well known that
the coincidence spectroscopy has the disadvantage of the reduced detection efficiency
respect to the measurements not-correlated in time. This is particularly dramatic
when the cross section of the process under exam is low, as it is the case of double
photoemission in the systems we studied. In our measurements, a typical coincidence
counting rate of few tens of mHz was achieved, which was 3-4 order of magnitude
lower than the counting rates of the two electrons independently detected.
Up till now, time correlated measurements of DPI in He in equal energy sharing
condition have been performed from DPI threshold up to 40 eV excess energy.
Within this range, the process appears to be dominated by final state EC. We shall
present in this work the measurements of direct DPI in He in equal energy sharing
conditions performed at 20, 40 and 80 eV above DPI threshold. The 20 eV
measurements were devoted to reproduce the results of the experiments performed by
different authors [I.5, I.6, I.7] at that energy, while the ones at 40 eV was devoted to
extend the published results [I.8] to cases with different geometry. The purpose of
extending the investigation to higher energy was to find the limit beyond which the
initial state correlations start to be dominant for the DPI process. The results we
obtain agree, up to 40 eV, with the published ones, and show that the DPI process at
80 eV above threshold can still be described in terms of final state EC only.
4
Time correlated measurements of the Auger cascade process have never been
performed in Ne in the past. We observed the Auger cascade decay of the resonantly
excited Ne 1s
-1
3p state along five different decay path. Two electrons emitted in the
process have been detected both separately and in coincidence, selecting their kinetic
energy and direction. For each decay path, a number of dynamical parameters have
been obtained, which are directly related to the matrix elements of the Auger decay.
The investigation of the cascade processes has been complemented by a series of
spectroscopic measurements of the Auger decay of the Ne 1s
-1
mp (m=3,4,5) excited
states. These measurements provide original angle- and energy-resolved
spectroscopic information of the Ne
+
satellite states at moderate resolution, and they
will be presented here.
All the experiments were performed at the GasPhase Photoemission beamline [I.9] at
Elettra storage ring (Trieste-Italy). The multicoincidence end station there operating
holds 10 hemispherical electrostatic analysers for angle- and energy-resolved electron
spectroscopy, which can be positioned at different angles respect to the photon beam.
The original solution of a multi-detector array permits to cover a larger solid angle,
without degrading the angle-resolution. The electronic data acquisition system
permits the simultaneous detection of up to 21 pairs of electrons in coincidence.
This work is organized as it follows. In the first chapter we shall discuss in detail the
processes we have considered. The different aspects of single and double
photoemission will be treated showing how the latter can help to understand the
problem of EC. The experimental apparatus we used will be described in details in
Chap.2. Also the main features of Synchrotron radiation will be discussed, which
make it a valid tool for our goals, and ELETTRA Storage Ring and Gasphase
Photoemission Beamline will be described. In Chap.3 an overview of the procedures
of calibration of the apparatus will be given. The techniques we adopted to extract
coincidence signals from time spectra and the normalization procedures will be
summarized. Finally, the experimental results will be presented and discussed in
Chapter 4.
5
[I.1] N.M. Kabachnik et al., J. Phys. B. 32, 1769 (1999)
[I.2] K. Ueda et al., AIP Conference proceedings 506, 222 (1999)
[I.3] K. Ueda et al., Phys. Rev. Lett. 83, 5463 (1999)
[I.4] R. Wehlitz et al., Phys. Rev. A 59, 421 (1999)
[I.5] O. Schwarzkopf et al., Phys. Rev. Lett. 70, 3008 (1993)
[I.6] H. Br�uning et al., J. Phys. B 31, 5149 (1998)
[I.7] R. D�rner et al., Phys. Rev. A 57,1074 (1998)
[I.8] S. Cvejanovich et al. , J. Phys. B 33, 265 (2000)
[I.9] K. Prince et al., J. Synchrotron Rad. 5, 565 (1998)
6
CHAPTER 1 : THE PHOTOEMISSION PROCESS IN ATOMS
When the absorption of electromagnetic radiation by matter results in the emission of
one or more electrons, we talk about Photoemission process. Depending on the
number of the emitted electrons (usually referred as photoemitted electrons) we talk
about single photoemission, double photoemission etc. The photon absorption can
also result in the emission of radiation. This is the Fluorescence decay which is in the
general case a competitive process with electron emission.
Actually there are different mechanisms which can cause photoemission. In general
we can distinguish between direct processes, as the photoionisation, and indirect
processes, as the resonant Auger decay. The experimental features and the details of
these processes can be very different, so it is not surprising that they are often thought
as different physical processes.
In fact a more careful analysis shows that the differences are more apparent than real.
When photoemission is considered, all the possible mechanisms leading to the same
final state have to be considered. This can cause interference effects which can not be
assigned to a specific process. Also, in some particular cases, apparently different
processes appear to be very similar, as it occurs for example with photoemission near
threshold.
These considerations are still more clear if it is considered that the photoemission
process does not only involve the electron(s) which take part to the decay, but it
affects the atom as a whole. In fact, the removal of an electron modifies the overall
Coulomb potential felt by the all the other electrons. The atom must move from an
eigenstate of N-body system to an eigenstate of N-1 (or N-m) body. These processes
are called Relaxation of the system and can strongly modify the photoemission
process.
Finally, in case of photoemission of two or more electrons, due to the long range of
Coulomb force, the electrons continue to interact with the ionised atom and with each
other also after having left the atom. Sometimes this interaction is generically called
7
PCI (Post Collisional Interaction). More often PCI is used to indicate the particular
case in which one electron is firstly emitted and the other one is emitted in a second
time at a higher energy. In this case the second electron �overcomes� the first one,
and the reciprocal screening of the nuclear potential by the two electrons changes
with time. The latter definition will be assumed in this work.
Both relaxation and PCI can contribute to make the different mechanisms of
photoemission indistinguishable as we shall discuss in the following. In the next
paragraphs we shall present the main processes which can cause electron emission
after photon absorption, as they are classified in the literature for historical reasons.
We shall show how the limit between them is often somehow artificial and how they
are all better interpreted as different aspect of the photoemission process.
1.1 Photoionisation process
The photoionisation process :
A + hν ! A
+
+ e
-
was first studied by A. Einstein in 1905 who investigated the photoelectric process,
which is considered as one of the main step in the development of quantum
mechanics.
The kinetic energy of the emitted electron, usually called photoelectron (PE) is given
by :
KE(e) = E(hν) - ( BE(A
+
) � BE(A) ) (1.1)
where BE(A) and BE(A
+
) are the binding energies of the A atom before and after the
photoionisation respectively. The neutral atom is usually in the ground state, while
the final ion can be left in different excited states. If the relaxation process is much
slower than the photoionisation (frozen orbitals hypothesis), the kinetic energy of the
PE is simply the difference between the photon energy and the binding energy of the
atomic shell the electron was removed from. The energy spectrum of the PE will
8
show in this case as many peaks as the energy levels of the neutral atom with binding
energy lower than the photon energy. These peaks are often called �main peaks�. The
situation is more complex if relaxation occurs fast , and different hypothesis can be
introduced. In the adiabatic limit, the ion reaches a complete relaxation before the
electron is removed and is left in its ground state. The PE spectrum will still show
one peak per each level of the neutral atom, but their kinetic energy will be shifted
with respect to the peak of the non-relaxed ion. On the contrary, in the sudden
approximation, only part of the relaxation occurs and the interaction between the PE
and the hole is ignored. In this case, the different possible excitations of the electrons
of the ion, which are called shake-up/shake-down processes depending on the energy
balance, are accounted. The shake-up and shake-down processes generate different
discrete peaks at kinetic energy lower and higher respectively than the associated
main peak. Such peaks are called �satellite peaks� or simply �satellites�, and can be
more intense than the main peak. Also the states of the final ion are called satellite
states or simply satellites. When the photoionisation occurs just above the double
photoionisation threshold, the shake-up process can be so large that the shaken
electron is emitted in the continuum. This process is called shake-off .
The description of all the features of the photoionisation process is given by the
Double Differential Cross Section (DDCS)
dE d
d
2
Ω
σ
which expresses the probability of
emission of one PE within the solid angle dΩ at the energy dE, per atom and per
photon flux unit. Because the energy spectrum shows discrete peaks, the Differential
Cross Section (DCS), defined as the energy-integrated DDCS, is often used. The
DCS can be easily calculated in a semi-classical approximation, neglecting
relativistic effects. Within this picture, the monochromatic photon is described by
mean of the classical vector potential :
A(r,t) = A
0
(
t)(i
e
ω-r . k
+ c.c.)
A
0
= -iA
0
P (1.2)
E =
t∂
∂ A
-
9
where E and P are the electric field and the polarisation vector respectively and k is
the wavenumber vector. The Hamiltonian function for the free electron in an
electromagnetic field can be written as [1.1] :
H =
m2
1
(p � eA)
2
(1.3)
where p is the linear momentum operator, and the coupling between the electron and
the electromagnetic field is expressed by a term proportional to [1.1]:
H�
int
(t) =
∑
j
A(r
j
,t) . p
j
=
∑
j
(A
0
t)(i
e
ω-r . k
j
P . p
j
+ C.C.) (1.4)
where the sum index j runs over all the electrons. Writing formulas (1.3) and (1.4) the
so-called Coulomb gauge [1.1] : 0 =•∇ A , Φ = 0 (being Φ the classical scalar
potential) has been assumed .
Applying the time-dependent perturbation theory, the Fermi Golden Rule yields for
the transition rate w :
w ∝
2
int
iHf
H
int
∝ A
0
∑
j
)(i
e
j
r . k
P . p
j
(1.5)
where i and f are the eigenfunction of the stationary atomic Hamiltonian.
Expanding the exponential function in series of power e
i k . r
= 1 + k . r + �..and
discarding all the terms but the first, leads to :
e
i k . r
≅ 1 (1.6)
which is known as dipole approximation. For a given atomic shell nls, the associated
eigenfunction has significant amplitude within a certain mean radius a
nls
, which is of
the order of the atom radius a = 10
-10
m , so the effect of H
int
will be negligible for
r >> a
nls
. The condition k . r < k a
nls
= 2π/λ a
nls
<<1 can be written as
a
nls
<< λ (1.7)
i.e. the wavelength of the ionising photon is much bigger than the mean radius of the
bound electron. So the dipole approximation is expected to hold for low photon
10
energy. This condition is in general satisfied for soft x-ray, λ (hν =1KeV) ≅ 10
-9
m,
and the dipole approximation has been long used to describe photoionisation.
However very recently significant deviations from the dipolar limit have been
observed also at kinetic energies below 1 KeV [1.2, 1.3].
The dipole approximation introduces some constrains on the quantum states which
can be populated after photoionisation. If L
i
, M
Li
, S
i
, M
Si
, J
i
, M
Ji
are the quantum
number of the initial atom and L
f
, M
Lf
, S
f
, M
Sf
, J
f
, M
Jf
are the ones of the system in
the final state, i.e. the combination of the ion and the photoelectron, the conditions :
L
f
� L
i
= -1, 0, +1 (but L
f
= L
i
= 0 is forbidden) M
Lf
� M
Li
= 0
S
f
� S
i
=0 M
Sf
� M
Si
= 0 (1.8)
J
f
� J
i
= -1, 0, +1 (but J
f
= J
i
= 0 is forbidden) M
Jf
� M
Ji
= 0
must be satisfied to have a non-vanishing transition probability. The relations (1.8)
are known as dipole selection rules for photoionisation. A further selection rule is
imposed by the parity relation :
Π
i
* Π
f
= - 1 (1.9)
Note that (1.8) and (1.9) reduce to the conservation of angular momenta and parity if
we assign to the photon :
L
hν
= 1 S
hν
= 0 J
hν
= 1 Π
hν
= - 1 (1.10)
and we consider the initial system as the combination of the neutral atom and the
photon. Within this picture, the photon �carries� one unit of angular momentum, no
spin, and has a negative parity. To write relations (1.8-1.10) we assumed a pure LS-J
coupling. This statement is supposed to be valid for the simple atomic systems we
shall describe in this work (i.e. He and Ne) and in the following we shall assume a
LS-J coupling if not elsewhere specified.
In case of linearly polarised light, choosing the direction of quantisation z-axis along
the polarisation vector P, the DCS can be written as :
11
4π
σ
dΩ
dσ
t
=
[ 1 + β P
2
(cosθ ) ] (1.11)
where θ is the angle of the photoelectron respect to the polarisation vector and
P
2
(cosθ ) = � (3cos
2
θ -1) is the second order Legendre polynome. σ
t
is the total cross
section, while β affects the angular distribution of the photoelectrons and is called
asymmetry (or anisotropy) parameter .The values of σ
t
and β depend on the radial
part of the matrix elements (1.5) only, and β can assume values from �1 to +2 : the
distribution is isotropic for β =0. The angular distribution of the photoelectron does
not depend on the φ angle respect to the photon propagation direction; thus it is
axially symmetric around the polarisation axis. Note that for θ
M
= 1/3 cosar = 54.7
deg, the DCS does not depend on the β value and is proportional to the total cross
section : θ
M
is known as magic angle. In Fig.1.1, the DCS is plotted according to
formula (1.11) for different values of β .
Fig. 1.1: Angular distribution of Photoionisation in polar co-ordinates according to (1.11) for
different values of β. The polarisation vector P is reported as a black bold arrow. Magic angle has
been indicated by a black point.
0
30
60
90
120
150
180
210
240
270
300
330
β
= 2
β
= 1
β
= 0
β
= -0.5
β
= -1
P
12
The numerical values of σ
t
and β depend on the system and on the photon energy. A
particular case occurs for the photoionisation of an s-shell of a closed-shell atom: in
this case β becomes a constant value :
β (ns) = 2 (1.12)
This is for example the case of photoionisation of He, that we have used to calibrate
the efficiency of the experimental apparatus in the 10 � 40 eV kinetic energy range,
as we shall report in details in Chap.3.
The selection rules limit the possible partial waves of the photoelectron. In case of
photoionisation of the s-shell, only the p-wave is accessible to the photoelectron, and
this process is described by one radial matrix element only R
ε p,ns
. We shall indicate
with
lε
the wavefunction of a free electron with ε kinetic energy and in the l-state,
R
ε l
its radial part and the matrix element with R
ε l�,nl
=
nl'l
RrR
ε
. The
photoionisation of the p-shell instead produces electron in the continuum in s and d
states (the p state is permitted by the angular momentum but is forbidden for parity
selection rules) : R
ε s,np
and R
ε d,np
. In case of Neon, the expressions reported in
Tab.1.1 can be obtained (from [1.1],pag.52)
Atomic
Shell
Partial waves
of the
photoelectron
σ
t
β
1s p
1/2,3/2
C R
2
ε p,1s
2
2s p
1/2,3/2
C R
2
ε p,2s
2
2p s
1/2
d
3/2,5/2
C (R
2
ε s,2p
+ 2 R
2
ε d,2p
)
2
p2,d
2
p2,s
p2,sp2,d
2
p2,d
RR
cosRR4R2
εε
εεε
+
∆−
Tab.1.1: total cross section and asymmetry parameters for photoionisation in Neon, from [1.1],
pag.52.
13
where C =
3
8
2
π
α E(hν ) , α = 1/137 is the fine-structure constant and ∆ is the phase
difference between
dε
and
sε
. The experimental numerical values of σ
t
and β as
reported in literature are reported in Fig.1.2 (from [1.4], pag.146) .
Fig. 1.2: experimental numerical values of σ
t
and β in Neon, at different photon energy. Figure is
taken from [1.4], pag.146.
From an experimental point of view, neglecting spin, the anisotropy parameters β and
the total cross section σ
t
are the only quantities which can be obtained by the
measurement of the DCS. Furthermore, the later one can be measured only if the
absolute efficiency of the experimental apparatus is known. From Tab.1.1, it is clear
that, also in a simple case as Ne photoionisation, it is not possible to extract the
values of the matrix elements R
ε l�,nl
and their phase shifts once β and σ
t
are known.
The values of the matrix elements are important, because they are related to the
wavefunctions of the electron in the atom. Their reproduction is a stringent test for
any atomic theory. So, the energy- and angle-resolved photoelectron spectroscopy is
14
a very powerful tool for the study of atomic levels, but fails to obtain a full
information on the dynamics of the photoionisation process.
1.2 Auger decay
The ionisation process may leave the atom in an unstable state. The de-excitation
may result in the emission of electromagnetic radiation (fluorescence decay) or in the
emission of one electron. The latter is the Auger process and the emitted electron is
often referred as the Auger electron. In general the two decay path are in competition,
but for low Z-elements the Auger process dominates.
A + hν ! A
+
+ e
P
-
! A
++
+ e
P
-
+ e
A
-
In the Auger process three atomic shells are involved, namely the shell where the
hole is created, the shell from which one electrons is removed to refill the hole and
the shell from which one electron is emitted in the continuum. The two final holes
can be in the same shell. The conventional way to indicate the Auger process is to
report the three involved shells starting with the hollowed shell; the x-ray notation is
usually preferred to spectroscopic notation. For example, the Auger decay where an
1s shell is refilled by a 2s electron and one electron is emitted in the continuum from
the 2p shell is indicated as KL
1
L
23
. Note that the role of the two outer shells can be
interchanged. The process in which the electron in the continuum is emitted from the
n�l� shell and the electron refilling the hole is removed from the n��l�� shell can not be
distinguished from the one in which the electron is emitted in the continuum from the
n��l�� shell and the hole is refilled from the n�l� shell. For this reason the notations
KL
1
L
23
and KL
23
L
1
are equivalent. If one of the involved electrons is removed from a
shell with the same principal quantum number of the hollowed shell, the decay is
particularly fast, due to the large overlap of the orbitals. These kind of decays are
called Coster-Kronig Auger decays: if all the three involved levels are within a
principal shell, then we talk about super Coster-Kronig Auger decays.
15
The kinetic energy of the Auger electron is :
KE(e) = BE(A
+
) � BE(A
++
) (1.3)
where BE(A
+
) and BE(A
++
) are the binding energies of the singly ionised and doubly
ionised atom respectively. The energy of the Auger electron does not depend on the
energy of the photon which created the hole. This implies that the photon bandwidth
does not affect the linewidth of the Auger electron.
The same consideration on the atom relaxation we made for the photoionisation
process are also valid for the Auger process, so satellite Auger peaks can appear in
the energy spectrum together with the Auger main peak. Within the hypothesis of
frozen orbitals in the A
+
ion, the kinetic energy of the Auger electron is given by the
BE(nl)-BE(n�l�)-BE(n��l��) where BE are the binding energies of the nl, n�l�, n��l��
shell of the A
+
ion being nl the shell hollowed by the photon. Auger spectroscopy can
thus be used to measure the binding energies of ionised systems.
Two different models have been introduced to study the theoretical formulation of the
Auger process. In the first model, the photoionisation and the Auger process are
supposed to be independent from each other. The Auger decay is supposed to start
from a fully relaxed ion state, and its transition rate does not depend on the history of
the initial hole. This picture is known as two-step model. The transition rate for the
photoionisation and subsequent Auger decay :
A + hν ! A
+
+ e
P
! A
++
+ e
P
+ e
A
is simply proportional to the product of the matrix elements of the two processes
separately calculated, so :
P ∝
∗
∑∑∑
0
2
f
2
0DiiCf
i
(1.14)
where f ,i ,0 are the eigenstates of the initial, intermediate and final state
respectively. The summations on the 0 and f index accounts for possible unresolved
initial and final states. D and C are the operator causing the photoionisation and the
Auger process respectively. The operator D takes the dipole form described in the
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