Invited article in DST-SERC-School publication (Narosa, November 2011); collection of articles based on
lecture course given at the DST-SERC School at the Birla Institute of Technology, Pilani, January 9-28, 2011.
Symmetry in Electron-Atom Collisions
and Photoionization Process
1*
2
Pranawa C. Deshmukh , Dilip Angom , and Alak Banik
3
1
Indian Institute of Technology Madras, Chennai;
2
Physical Research Laboratory, Ahmadabad;
3
Space Applications Centre, Ahmadabad.
*pcd@physics.iitm.ac.in
ABSTRACT
The final state of an electron-ion collision process, and that of photoionization of
an atom, both consist of a ‘free’ electron and an ion in the ground/excited state.
The initial ingredients of the two reactions are of course different; they are an
electron and an ion in the former case, and a photon and a neutral atom in the
latter. It is the quantum mechanical discrete time-reversal symmetry which
connects solutions of the electron-ion collision process to those of atomic
photoionization. An introduction to these connections is attempted in the present
article. A related topic where time-reversal symmetry plays a crucial role in
atomic physics, violation of the symmetry, is in the detection of permanent
electric dipole moment.
I.
INTRODUCTION
Physical processes are governed by conservation laws, which in turn are
connected intimately to symmetry processes. For example, many conservation
laws are obtained from physical laws, but often the physical laws themselves are
obtainable from symmetry considerations that govern them [1,2]. The
connections between symmetry and conservation laws have far reaching
implications in physics that impact our understanding of the laws of nature. This
subject is both vast and deep, but the purport of the present article is limited; it is
only aimed at providing an introduction to mathematical connections based on
the quantum mechanical time reversal symmetry which relate solutions of the
quantum collision problem for an electron impinging on an atomic ion to those
of atomic photoionization process in which an atom absorbs electromagnetic
radiation resulting in an atomic electron get knocked out of the atom and escape
as a ‘free’ electron. This is illustrated in Fig.1.
2
Theoretical Techniques in Atomic and Molecular Collision Physics
(a)
(b)
Fig.1. The final state of two reactions whose initial state ingredients are quite
different is the same for the two processes: (a) atomic photoionization and (b)
electron-ion collision.
It is thus natural to expect that the quantum mechanical description of the two
processes, (a) atomic photoionization and (b) electron-ion collision, must be
related. Indeed it is, but the relationship is subtle. The difficulty in relating the
two processes comes from the fact that even if the final state of the two
processes is the same, the initial states are different: even the ingredients of the
initial states are different for the two processes! By simply running the process
backward in time, one cannot recover the original ingredients because of the
multiplicity of channels in which the central complex can decay! The temporal
evolution of the process does not regenerate the history of the process simply by
letting the time t go to t in the equation of motion! In this respect, it is
necessary to understand the difference in the role time-reversal symmetry plays
in quantum mechanics as opposed to classical mechanics. In classical mechanics,
the equation of motion contains either the second order differential operator with
2
2
d
d r
respect to time, namely the operator
, in Newton’s equation F  m
, or
2
2
dt
dt
d
two first order differential operators in time,
, as in Hamilton’s equations
dt
q 
H
p
and p  - H . The classical equations of motion are symmetric with
q
respect to the transformation t  t . This result is of course independent of the
formulation, whether Newtonian, Lagrangian, or Hamiltonian as long as there
are no unspecified degrees of freedom that lead to dissipation.
Atoms and molecules are the building blocks of all matter around us. These are,
however, bound states of the elementary particles electrons and quarks. There is
a large gap, several orders of magnitude, in the energy scales relevant to the
physics of atoms and physics of elementary particles. The energy scales in atoms
are at the most few eV, the excitation energies, whereas the energy range of
interest in particle physics is TeV (1012). Despite the large difference in energy
scales, it is possible to probe phenomena in particle physics through high
precision spectroscopy of atoms. One remarkable example is the permanent
Symmetry in Electron- Atom Collisions……. 3
electric dipole moment (EDM). It is the observable signature of simultaneous
violations of parity and time-reversal symmetries. Among the two, a proper
understanding of time-reversal violation is of paramount importance to resolve
the preponderance of matter in the Universe. Another observable of equal
importance arises from the parity violation, which modifies the selection rules of
radiative transitions in atoms.
II.
COLLISIONS:DESCRIPTION IN TERMS OF OUTGOING
WAVE BOUNDARY CONDITION
In the context of the relationship between (i) the solution to the electron-ion
collision process and (ii) photoionization of a neutral atom, shown in Fig.1, we
shall first briefly review the well-known solution to the Schrodinger equation
formulation of the scattering problem given by [3]:
ˆ ) ikr 
f (

 (r ; r  )  A eik r 
e ,
ki

r
(1)

ˆ ) is the well-known scattering amplitude.
in which f (
The total wavefunction is then given by:
l
 Total (r ) 
r    cl i (2l  1) Pl (cos  )
l
in which  l is the phase-shift caused by the
sin(kr 
l
2
kr
 l )
(2)
scattering potential, and the
‘normalization’ constant cl must be chosen ‘appropriately’ i.e., as per the
‘boundary conditions’. We restrict ourselves to central field potentials for which
the current formulation is applicable.
→z axis
Fig.2: Pictorial depiction of an electron-ion collision process
In the collision problem described pictorially in Fig.2, a mono-energetic beam of
electrons is incident from the left along the Z-axis of a Cartesian coordinate
system on a scattering central field atomic potential. The electron flux scattered
4
Theoretical Techniques in Atomic and Molecular Collision Physics
by the target cannot have any spherical ingoing wave, and this requirement fixes
 i
the coefficient cl  e l , which gives the following scattered wave solution:
 i 2l 
e
1

 scattered (r ; r  ) 
 (2l  1) Pl (cos  ) 
 2ik  .
r l


e
ikr
(3)
The scattering amplitude of Eq.1 is then given by:
2i
e l 1
ˆ
f( )
(2l 1) Pl (cos )
,
2ik
l
(4)
and the differential scattering cross-section is given by:
2
d
f (ˆ)
d
We note that the time-dependence of the wavefunction is given by:
i E t
it
e
e
(5)
(6)
and accordingly the time-dependent solution to the scattering problem is given
by:
 2il 
i ( kr t )
e
1
i ( kz t ) e


 T (r , t ) 

 (2l  1) Pl (cos  ) 
r   e
 2ik  (7)
r
l


We observe from the solution given in Eq.7 that the two terms on the right hand
side provide the asymptotic description of the incident plane wave and a
scattered outgoing spherical wave shown in Fig.2. To highlight the fact that the
solution is based on the outgoing spherical wave in the final total solution, a
superscript ‘  ’ is placed on the symbol for the wavefunction on the left hand
side of Eq.7. The boundary condition that has been used is based on cancellation
of all spherical ingoing waves in the scattered solution and is referred to as
OUTGOING WAVE BOUNDARY CONDITION.
We now raise the question: what kind of boundary conditions should be
employed to describe an atomic photoionization event, as opposed to electronion scattering? As shown in Fig.1, the main issue here is that the photoelectron
that escapes the reaction zone as a result of photoionization did not really exist as
a free electron in the initial state. It was an integral part of the neutral atom in the
‘nucleus + electron(s)’ bound system. Following the description in Reference
[4,5] which is both the inspiration and the primary source for this article, we
shall first discuss the one-dimensional analogue that would relate the collision
dynamics to photoionization.
We consider a collision process in which the electron is incident from the left
and impinges on a one-dimensional scattering center as shown in Fig.3. The
Symmetry in Electron- Atom Collisions……. 5
reflection and transmission coefficients can be determined readily by employing
the equation of continuity for the conservation of electron charge density flux.
The experiment we envisage has an electron incident on the reaction zone from
the left, along the X-axis. It is thus clear that the boundary condition is set by
requiring G  0 (see Fig.3).This boundary
Region I
Region II
Region III
Fig.3: Relationship between one-dimensional electron-ion scattering/collision process
and photoionization. Collision of an electron incident from the left is described by the
boundary condition G=0, while photoionization resulting in the photoelectron escaping
to the left is described by F=0.
condition determines the collision experiment in which the entrance channel is
unique; it has an electron incident from the left. In a (one-dimensional)
photoionization experiment in which we envisage the photoelectron to escape to
the left, the boundary condition is then represented by the choice F  0 (Fig.3).
It is the exit channel which is unique in this case, represented by the
photoelectron flux escaping to the left as a result of photoionization. We observe
that the relationship of collision to photoionization is thus in some sense one of
‘motion reversal’, except that this is not merely the time-reversal of classical
mechanics in which the equations of motion are symmetric under the
transformation t  t .
III.
PHOTOIONIZATION: DESCRIPTION IN TERMS OF
INGOING WAVE BOUNDARY CONDITION
To understand the relationship between solutions to the collision problem with
those of photoionization, we re-write the traveling wave solutions of the onedimensional problem in terms of a new base pair: u ( x), u ( x) , defined as
follows:
x  a (region I): u ( x)  N E cos(kx    );
u ( x)   N E cos(kx    )
(8)
and
x  a (region III); u ( x)  N E cos(kx    );
u ( x)  N E cos(kx    )
(9)
In terms of the new base pair, the traveling wave solutions shown in Fig.3
become:
6
Theoretical Techniques in Atomic and Molecular Collision Physics
u
u
(I )
( x) 
( III )
( x) 
i
i
i
i
ikx
ikx 
N E (c e   ce  )e  (c e   ce  ) e

 (10)

2
1
i
i
ikx
i
i
ikx 
N E (c e   ce  )e  (c e   ce  ) e

 (11)
2
1
The ‘collision’ boundary condition G  0 is then expressed as:
(c e

 i 
 c e
 i 
)  0;
i . e.
c
 e
 i (    )
c
(12)
Likewise, the ‘photoionization’ boundary condition F  0 is expressed as:
c e
i 
 c e
i 
 0;
i.e.
We observe the complex-conjugation of
c
c
 e
 i (    )
(13)
c
c in the description of the
photoionization boundary condition (Eq.13) in relation to the collision boundary
condition (Eq.12). This complex conjugation is characteristic of ‘motion
reversal’ in quantum mechanics, usually referred to as ‘time reversal’. Using a
quantum mechanical operator  for ‘time/motion reversal’, one can depict, as
in Fig.4, the photoionization process as time/motion reversed electron-ion
collision. Photoionization is referred to in the literature often as ‘half-scattering’
on account of this relation. The term ‘motion reversal’ was preferred by Wigner,
since the relationship involves complex-conjugation of the wavefunction in
addition to t  t under the operator  which nevertheless is most often
referred to as the ‘Time-reversal Operator’. Time-reversal is a discrete
symmetry, just like parity and charge-conjugation [1]. It is effected through an
operator generally denoted by  which is an anti-unitary operator. The (antiunitary) time-reversal operator is certainly not the inverse of the (unitary) timeevolution operator in quantum mechanics; it has the following commutation/anticommutation properties with respect to the position, momentum and angular
momentum operators:
[r , ]-  0 : commute
[ p, ]  0 : anticommute
(14)
[ J , ]  0 : anticommute
It is important to understand the difference between the implications of timereversal in quantum mechanics as opposed to what it is in classical mechanics
[7]. As mentioned above, the classical equations of motion are symmetric under
time-reversal. From the equation of motion, one can thus predict the future, and
also determine the past of the mechanical state of the system. The meaning in
quantum mechanics is however different. Suppose it is known that the system at
Symmetry in Electron- Atom Collisions……. 7
time t is in the state
state
 (t ) . Then, the system is said to be in the time-reversed
 R (t ) under the time-reversal transformation, if the transformation
R (t ) is equal to
ensures that the probability of finding the system in a state
finding it at time –t in the state
 (t )
We recall that Eq.7 gave us the scattering/collision solution subject to the socalled ‘OUTGOING WAVE BOUNDARY CONDITION’. We must now write
the total wavefunction given in Eq.2 with a different set of boundary conditions
as would be appropriate for the photoionization process.
Now,
l
sin( kr 
 l )
l
2
 Tot ( r ) 

c
i
(2
l

1)
P
(cos

)

r 
l
kr
l l
 Tot ( r ) 
r  

1
 c (2l  1)  Pl (cos  )e
2ikr l l

i   kr  
i (  kr )

l
 Pl (  cos  ) e  l


(15)
and the incident wavefunction is:
1
ikr
l ikr  .
 inc (r ) 
 (2l  1) Pl (cos  ) e  ( 1) e


2ikr l
Thus,
(16)
 Scattered ( r )   Tot ( r )   incident ( r )
 Scattered ( r ) 

 i
 ikr
 (2l  1) Pl (cos  )  cl e l  1  e
2ikr l


1

1

 (2l  1) Pl (  cos  )  cl e
2ikr l

i
l  1  e ikr


(17)
It is clear that the choice cl  e cancels the outgoing spherical waves in the
scattered solution as must happen in the photoionization event whose exit
channel is unique which represents the escaping photoelectron plane wave,
receding away from the reaction zone. This choice of the coefficient is, not
surprisingly, the complex conjugate of that employed to describe the collision
process.
The total wavefunction, now written with a superscript ‘-’, with the new
 i
choice cl  e l , and inclusive of the time-dependence then is:
 i l
8
Theoretical Techniques in Atomic and Molecular Collision Physics
 T (r , t ; r  )  ei ( kz t )

e  i ( kr t )
r
 e 2i l  1 
(2l  1) Pl (  cos  ) 

 2ik 

l
(18a)
On complex-conjugation, it becomes:
i (kz t ) e
 T (r , t; r  )*  e

i (kr t )
r
 2il 
e
1

 (2l  1) Pl ( cos  ) 
 2ik 
l


(18b)
→x axis
Fig.4a
Collision
Photoionization
Fig.4b
→z axis
Fig.4a and Fig.4b: Shown in the above two panels respectively is the onedimensional and three-dimensional pictorial depiction of the relationship
between atomic photoionization and electron-ion collision process effected by
the ‘time/motion reversal operator.
Further, by letting t  t , we get:
i ( kz t ) e
 T ( r , t ; r  )*  e

i ( kr t )
r
 2il 
e
1
.
 (2l  1) Pl (  cos  ) 
 2ik 
l


(18c)
Symmetry in Electron- Atom Collisions……. 9
Note that the surface of constant phase of the wave represented by the first term
must have:
d  kz t   0 i.e. kdz   dt  0,
dz

which gives:
dt   k
dz
The fact that dt is intrinsically negative implies that the surface of constant
phase this term represents is a plane wave moving toward z   from the
reaction zone. It represents the plane wave moving toward the left in Fig.4b
showing the photoelectron’s escape along the unit exit channel.
Likewise, the surface of constant phase of the wave represented by the second
term must have:
d ( kr  t )  0;
i.e. kdr  dt  0
which gives
dr

 
dt
k
dr
dt being intrinsically negative, it corresponds to a spherical wavefront of
diminishing radius representing the spherical ingoing wave shown in Fig.4b.
 i
We see that the choice cl  e l has provided us the correct boundary condition
on the total wavefunction appropriate for the description of the photoionization
process.
The scattered solution for the photoionization process is thus given by Eq.17,
 i
with cl  e l , which is referred to as INGOING WAVE BOUNDARY
CONDITION since this choice cancels the outgoing spherical waves in the
scattered solution. Hence,
 i 2l 
ikr
e
e
1
,
(19)
 Scattered ( r )  
 (2l  1) Pl (  cos  ) 
 2ik 
r l


and the final state of the photoelectron then becomes:
ikr

ikz e
| f   e 

r l
 i 2l 
e
1 
.
(2l  1) Pl (  cos  ) 
 2ik 


(20)
10
Theoretical Techniques in Atomic and Molecular Collision Physics
Fig.5: Geometry showing the reversal of motion corresponding to Fig.4
To determine the photoionization transition matrix element  f | T |  i  for
transition from an initial state  i
to a final continuum state  f , one must
therefore employ the final state with INGOING WAVE BOUNDARY
CONDITION expressed in Eq.20. The form given in Eq.20, not the one given in
Eq.3, must therefore be used to determine the angular distribution of the
photoelectrons, as for example in the famous Cooper-Zare formula [8]. The two
forms differ in respect of the outgoing/ingoing wave boundary conditions.
IV
DISCRETE SYMMETRY VIOLATIONS
We have, so far, examined the importance of symmetry in the collision
calculations and in particular, we have utilized the invariance of dynamics under
time reversal. The other equally important facet of the same symmetry is the
time reversal violation. The related phenomena and associated observables can
probe fundamental problems in physics. One of the unresolved riddles of nature
is the dearth of antimatter in the Universe. All observable signatures points to
Big Bang as the event which created the Universe [9]. One important outcome is
then the creation of matter and antimatter in equal amounts. However, till date
the astronomical observations up to the edge of the Universe have detected only
matter. This begs an explanation of how and where have all the antimatter
vanished?
Fig.6: Cosmic microwave background anisotropy from the WMAP data, it is
represents the matter distribution at the epoch when matter and radiation decoupled.
(Image from WMAP website of NASA).
Symmetry in Electron- Atom Collisions……. 11
Although, the question pertains to the scales equal in magnitude to the size of the
Universe, the answer lies in the physics of the smallest constituents of the
Universe; the elementary particles. A simple resolution of the puzzle is, there
must be physical process or processes which convert antimatter into matter. The
necessary condition for this is the violation of time reversal symmetry. This,
however, leads to another question: how to detect time reversal violation? The
answer, it turns out, is to detect an observable which could arise from time
reversal violation. Following symmetry conditions, the all familiar EDM is the
observable. EDM, the ones we are familiar with, are induced by an external field
or arise from degeneracy of opposite parity states. However, a permanent EDM
in a non-degenerate quantum system is a signature of time reversal and parity
violation. Of the two the former is of overriding importance as the latter, which we
shall dwell upon subsequently, is well established in weak interactions.
To prove permanent EDM violates parity and time reversal; consider a particle
or a composite non-degenerate quantum system has a permanent EDM d.
1
Under parity transformation, PdP  d .
that is, d is odd under parity transformation as it is a vector observable.
Following projection theorem in quantum mechanics, the experimentally
observable EDM of a system is the component along an internal vector quantity.
It is the spin s for an elementary particle like electron or the total angular
momentum J for a composite quantum system like atoms, then we can
write d  cJ , where, c is a constant. Since the angular momentum is r  p , there
is time dependence through the momentum p, under time reversal transformation
J is transformed to –J. The EDM transform under time reversal as
d 1  cJ 1  cJ .
Again, like in parity transformation; the EDM is odd under time reversal
transformation. These transformations are schematically represented in Fig.7.
Fig.7: Schematic diagram of parity and time reversal transformation of EDM. The
arrow represents the direction of the angular momentum and EDM.
12
Theoretical Techniques in Atomic and Molecular Collision Physics
It turns out that the standard model (SM) of particle physics, the most successful
and well accepted theory of elementary particle physics, does predict a non-zero
38
EDM of electron. And, the value is d e  2.9  10 e cm.
This is an extremely small value. Perhaps this is surprising as one tends to think
of electrons as point particles and not associated with a charge distribution. This
is not a precise description, in the proper quantum description of an electron; a
cloud of virtual particles surrounds it (vacuum polarization). An asymmetry in
the distribution of the virtual charges is the origin of electron EDM. To measure
de one must apply an external electric field E and observe the energy shift arising
from the interaction de  E . This interaction, like Larmor precession, causes
precession about E. But, it is an impossible task as the electron accelerates away
in presence of E. It is, however, possible to measure the EDM of neutral particle
like neutron. EDM of neutron, surprising, isn’t it? Not really, it is a bound state
of quarks (two down quarks and one up quark) which are charged elementary
particles. Experiments with neutrons are very challenging and the best bound is
26
[10] d n  2.9  10 e cm.
Even better candidates are atoms and molecules. These are charge neutral and
one can apply large external electric fields. An atom or molecule can have nonzero EDM due to the EDM of electron de [11]. From detailed theoretical studies,
it is now well established that there is an enhancement of EDM in atoms and
molecules due to relativistic corrections [12]. That is, for an atom, the EDM,
d a   de .
Where, 
EDM
1 , is the enhancement factor. Determining or extracting the electron
from
experimentally
measured
d a requires
accurate
theoretical
calculations to obtain  . This is where reliable atomic many-body theories like
coupled-cluster are extensively used. Though, we have discussed about atomic
EDM arising from de , atoms and molecules are also sensitive to EDMs arising
from: EDMs of neutrons and protons, and; parity and time reversal violating
interactions within the atom or molecule. In particular, the open-shell or
paramagnetic atoms are sensitive to de and closed-shell or diamagnetic atoms are
sensitive to EDMs arising involving the nuclear sector. The implication of a
precise determination of de is, there are extensions to SM of particle physics
which predict much larger de. So, an unambiguous detection of de different from
the SM prediction is a signature of physics beyond the standard model (BSM).
Parity violation, often referred to as parity non-conservation is another discrete
symmetry violation in atoms and molecules which has important implications to
Symmetry in Electron- Atom Collisions……. 13
elementary particle physics. Within the SM of particle physics, parity is
maximally violated in weak interactions. It is the fundamental interaction
associated with phenomena like beta decay. However, there are BSM which
predicts larger degree of parity violation. In atoms and molecules, one
consequence of parity violation is modification to selection rules of radiative
transitions. As example, consider the 1s and 2s states of Hydrogen atom. The
states are of same parity and electric dipole (E1) transition between the states is
forbidden.
2p
2s
E1
M1
E1
1s
Fig 8: Electric dipole (E1) transition between the 1s and 2s states of Hydrogen is
forbidden. However, in presence of parity violation 2s (solid line) acquires a mixture of
opposite parity (dashed line). The odd parity state 2p dominates the opposite parity
mixing and is denoted by the doubled sided arrow. The E1 transition between 1s and
parity mixed 2s is then allowed.
This is schematically shown in Fig 8. However, in the presence of a parity
violating interaction HPNC, the 2s state acquires a mixture from the opposite
parity states. From time-independent perturbation theory, the parity mixed 2s
state is
I H PNC 2s
2~s   I
,
 2s   I
I
Where I are the odd parity intermediate states and  i are the energies of the
states. Similarly, the 1s state also acquire an odd parity admixture. The E1
transition amplitude between the parity mixed states is then non-zero
 2s d I I H PNC 1s
2s H PNC I I d 1s 
E1PNC  2 ~s d 1~s   

  0,
 1s   I
 2s   I
I 

where, E1PNC is the HPNC induced electric dipole transition amplitude. In atomic
experiments, E1PNC is measured using very sensitive interference techniques.
Besides probing the physics of elementary particles, parity violation may be
cause for handedness of organic molecules. That is, during any organic chemical
reaction right and left handed molecules are produced in equal amounts.
However, in nature, most of the organic molecules are right handed. To
determine E1PNC, like in EDM, one has to use accurate atomic theory
calculations to extract the parameters related to particle physics.
14
Theoretical Techniques in Atomic and Molecular Collision Physics
In terms of the atomic theory calculations, the EDM and E1PNC calculations are
very similar. The former is an expectation value and the later is transition
amplitude. So, one may use the same method. In general, the E1PNC calculations
are more complicated as it involves two different states and at least one is an
excited state. Although, we have considered Hydrogen atom as a case study to
show the role of parity violation in altering the selection rules, it is preferable to
use heavier atoms. The reason is, the observables scales as Z3. An important
advantage of EDM or E1PNC as probes of particle physics is, the atomic
experiments are table top experiments and far cheaper than accelerator based
experiments. In addition, these probe parameter space complementary to the
accelerators. So, it helps to constrain the parameter space.
IV.
CONCLUSIONS
The initial state ingredients, an electron and an ion, in an electron-ion scattering
process are quite different from the ingredients (a photon and a neutral atom) of
an atomic photoionization process. Nevertheless, their end-states both contain a
‘free’ electron and an ion. The quantum mechanical description of collision and
photoionization is intimately related through the (discrete) time-reversal
symmetry. This involves complex conjugation of the wavefunction in addition
to t  t . While outgoing wave boundary conditions are employed to describe
the quantum collision process, it is the ingoing wave boundary condition that
must be employed to describe the photo-ionization process.
Atoms are suitable systems to probe the observable signatures of discrete
symmetry violations. The results from the precision atomic and molecular
experiments, when combined with theoretical results, provide stringent bounds
on parameters in elementary particle physics.
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