Applications of the Classical Two-Coulomb-Center Systems to Atomic/Molecular Physics
by
Nikolay Kryukov
A dissertation submitted to the Graduate Faculty of
Auburn University
in partial fulfillment of the
requirements for the Degree of
Doctor of Philosophy
Auburn, Alabama
December 14, 2013
Keywords: two-Coulomb-center systems, diatomic Rydberg quasimolecules, crossings of energy
terms, effects of external fields and plasma screening, continuum lowering, Runge-Lenz vector
Copyright 2013 by Nikolay Kryukov
Approved by
Eugene Oks, Chair, Professor of Physics
Michael S. Pindzola, Professor of Physics
Joe D. Perez, Professor of Physics
Abstract
We advanced the classical studies of the two-Coulomb-center (TCC) systems consisting
of two nuclei of charges Z and Z' and an electron in a circular or a helical state centered on the
internuclear axis. These systems represent diatomic Rydberg quasimolecules encountered, e.g., in
plasmas containing more than one kind of multicharged ions. Diatomic Rydberg quasimolecules
are one of the most fundamental theoretical playgrounds for studying charge exchange. Charge
exchange and crossings of corresponding energy levels that enhance charge exchange are
strongly connected with problems of energy losses and of diagnostics in high temperature
plasmas; besides, charge exchange is one of the most effective mechanisms for population
inversion in the soft x-ray and VUV ranges. The classical approach is well-suited for Rydberg
quasimolecules. First, we considered diatomic Rydberg quasimolecules subjected to a static
electric field parallel to the internuclear axis. We found the appearance of an additional (fourth)
term, which was absent at the zero field, and which had a V-type crossing with the lowest term.
We also found X-type crossings (absent at the zero field) which significantly enhance charge
exchange. Second, we studied effects of the electron screening in plasmas on diatomic Rydberg
quasimolecules. We found that the screening stabilizes the nuclear motion for Z = 1 and
destabilizes it for Z > 1. We also found that a so-called continuum lowering in plasmas was
impeded by the screening, creating the effect similar to that of the magnetic field and opposite to
that of the electric field. The continuum lowering plays a key role in calculations of the equation
ii
of state, partition function, bound-free opacities, and other collisional atomic transitions in
plasmas. Third, we considered diatomic Rydberg quasimolecules in a laser field. For the situation
where the laser field is linearly-polarized along the internuclear axis, we found an analytical
solution for the stable helical motion of the electron valid for wide ranges of the laser field
strength and frequency. We also found resonances, corresponding to a laser-induced unstable
motion of the electron, that result in the destruction of the helical states. For the case of a
circularly-polarized field, polarization plane being perpendicular to the internuclear axis, we
found an analytical solution for circular Rydberg states valid for wide ranges of the laser field
strength and frequency. For this case we demonstrated also that there is a red shift of the primary
spectral component. We showed that both under the linearly-polarized laser field and under the
circularly-polarized laser field, in the electron radiation spectrum in the addition to the primary
spectral component at (or near) the unperturbed revolution frequency of the electron, there
appear satellites. Under a laser field of a known strength, in the case of the linear polarization the
observation of the satellites would be the confirmation of the helical electronic motion in the
Rydberg quasimolecule, while in the case of the circular polarization the observation of the red
shift of the primary spectral component would be the confirmation of the specific type of the
phase modulation of the electronic motion. Conversely, if the laser field strength is unknown,
both the relative intensities of the satellites and the red shift of the primary spectral component
could be used for measuring the laser field strength. Fourth, we considered TCC systems
consisting of a proton, muon and an electron. We found that a muonic hydrogen atom can attach
an electron, with the muon and electron being in circular states. The technique of the separation
of rapid and slow subsystems was used, where the muon represented the rapid subsystem and the
iii
electron the slow subsystem. We showed that the spectral lines emitted by the muon experience a
red shift compared to the corresponding spectral lines in a muonic hydrogen atom. Observing
this red shift should be one of the ways to detect the formation of such muonic-electronic
negative hydrogen ions. Studies of muonic atoms and molecules, where one of the electrons is
substituted by the heavier lepton ??, have several applications, such as muon-catalyzed fusion, a
laser-control of nuclear processes, and a search for strongly interacting massive particles
proposed as dark matter candidates and as candidates for the lightest supersymmetric particle.
Fifth, we studied fundamental algebraic symmetry of the TCC systems leading to an additional
conserved quantity: the projection of a super-generalized Runge-Lenz vector on the internuclear
axis. We derived the correct super-generalized Runge-Lenz vector, whose projection on the
internuclear axis is conserved, and showed that the corresponding expressions by other authors
did not correspond to a conserved quantity and thus were incorrect. The correct super-
generalized Runge-Lenz vector for the TCC systems that we derived should be of a general
theoretical interest since the TCC systems represent one of the most fundamental problems in
physics. It can also have practical applications: e.g., it can be used as a necessary tool while
applying to the TCC systems the robust perturbation theory for degenerate states based on the
integrals of the motion.
iv
Dedicated to my beloved grandmother,
Nina Stepanovna Lipatova
v
Acknowledgments
First, I would like to thank my academic advisor, Prof. Dr. Eugene A. Oks, for directing
my steps towards completing this work. His outstanding experience and knowledge guided me
throughout the whole process of thinking, finding out, deriving, calculating and writing. I am
also obliged to him for leading me into and supporting my graduate study.
Accomplishing this work would not have been possible without necessary preparation in
graduate-level physics. Besides Dr. Oks, my student's gratitude goes to Profs. J. D. Perez, M. S.
Pindzola, A.-B. Chen, J. D. Hanson, S. Hinata, J. J. Dong, who taught me the graduate physics
courses. Unfortunately, Dr. Chen is no longer with us, and, besides his help with my application
for the Ph.D. program, I will forever remember the interest he inspired in me in his class, and
that one homework I did for twenty-two hours nonstop. Since mathematical preparation is
indispensable for physics research, it was also a pleasure to attend the classes of Profs. M. Liao
and D. Glotov, which I took at the mathematics department.
I owe my admission to physics graduate school to the education I received in Russia. Russian
traditional physics education and teaching is considered to be one of the best in the world, and I
was certainly privileged, perhaps not always aware of this, to take the courses at the University
of Nizhny Novgorod and the Institute of Applied Physics of the Russian Academy of Sciences
led by Profs. M. I. Bakunov, S. B. Biragov, B. V. Lisin, O. N. Repin, A. M. Feygin, A. V.
Abrosimov, G. M. Zhislin, A. G. Demekhov, I. A. Shereshevsky and other professors and
vi
teachers. Profs. Bakunov and Biragov, in addition to teaching university students, organized
preparatory physics courses and physics olympiads for secondary-school students, which not
only stimulated the spirit of healthy competition, but also eased the transition from school to
university. Here, I also acknowledge the support of my tutor Dr. A. S. Garevsky, who prepared
me for the entrance exams to the University of Nizhny Novgorod.
Going back to the time when physics was shown to me as a tool to satisfy my curiosity about
the world around me, I have to mention the master who introduced me to it: my secondary-
school physics teacher V. Yu. Kovalev. His ability to ignite curiosity, encourage analysis and
discussion, together with his sense of humor, planted the seed of interest in me, then a seventh-
grader, which budded and, after nineteen years, fed by many, troubled at times, but always
moving upwards, has grown into the present work.
Finally, returning to here and now, I would like to thank Dr. F. Robicheaux for an important
discussion which resulted into a notable contribution to this dissertation. The discrepancy, which
was noticed while preparing Chapter 6, helped discover errors in earlier publications of other
authors and obtain new, correct results.
vii
Table of Contents
Abstract.........................................................................................................................................ii
Acknowledgments........................................................................................................................vi
List of Figures..............................................................................................................................ix
List of Abbreviations...................................................................................................................xii
Chapter 1. Introduction: Two-Coulomb Center Systems Representing Diatomic Rydberg
Quasimolecules .................................................................................................................1
Chapter 2. Enhancement of Charge Exchange and of Ionization by a Static Electric Field .....10
Chapter 3. Effect of Plasma Screening on Circular States of Diatomic Rydberg Quasimolecules
and their Application to Continuum Lowering in Plasmas ............................................30
Chapter 4. Helical and Circular States of Diatomic Rydberg Quasimolecules in a Laser Field 65
Chapter 5. Attachment of an Electron by Muonic Hydrogen Atoms: Circular States ................84
Chapter 6. Super-generalized Runge-Lenz Vector ...................................................................109
Chapter 7. Conclusions ............................................................................................................117
References ...............................................................................................................................121
viii
List of Figures
Figure 1.1 ......................................................................................................................................4
Figure 1.2 ......................................................................................................................................7
Figure 2.1 ....................................................................................................................................15
Figure 2.2 ....................................................................................................................................16
Figure 2.3 ....................................................................................................................................18
Figure 2.4 ....................................................................................................................................19
Figure 2.5 ....................................................................................................................................20
Figure 2.6 ....................................................................................................................................20
Figure 2.7 ....................................................................................................................................21
Figure 2.8 ....................................................................................................................................23
Figure 2.9 ....................................................................................................................................23
Figure 3.1 ....................................................................................................................................35
Figure 3.2 ....................................................................................................................................36
Figure 3.3 ....................................................................................................................................37
Figure 3.4 ....................................................................................................................................38
Figure 3.5 ....................................................................................................................................38
Figure 3.6 ....................................................................................................................................39
Figure 3.7 ....................................................................................................................................39
ix
Figure 3.8 ....................................................................................................................................40
Figure 3.9 ....................................................................................................................................40
Figure 3.10 ..................................................................................................................................41
Figure 3.11 ..................................................................................................................................43
Figure 3.12 ..................................................................................................................................44
Figure 3.13 ..................................................................................................................................44
Figure 3.14 ..................................................................................................................................45
Figure 3.15 ..................................................................................................................................46
Figure 3.16 ..................................................................................................................................47
Figure 3.17 ..................................................................................................................................47
Figure 3.18 ..................................................................................................................................49
Figure 3.19 ..................................................................................................................................49
Figure 3.20 ..................................................................................................................................50
Figure 3.21 ..................................................................................................................................51
Figure 3.22 ..................................................................................................................................52
Figure 3.23 ..................................................................................................................................53
Figure 3.24 ..................................................................................................................................53
Figure 3.25 ..................................................................................................................................53
Figure 3.26 ..................................................................................................................................54
Figure 3.27 ..................................................................................................................................55
Figure 3.B.1 ................................................................................................................................60
Figure 3.B.2 ................................................................................................................................60
x
Figure 3.B.3 ................................................................................................................................61
Figure 3.C.1 ................................................................................................................................62
Figure 3.C.2 ................................................................................................................................63
Figure 3.C.3 ................................................................................................................................63
Figure 3.C.4 ................................................................................................................................64
Figure 3.C.5 ................................................................................................................................64
Figure 4.1 ....................................................................................................................................71
Figure 4.2 ....................................................................................................................................73
Figure 4.3 ....................................................................................................................................73
Figure 4.4 ....................................................................................................................................75
Figure 4.5 ....................................................................................................................................78
Figure 4.6 ....................................................................................................................................81
Figure 4.7 ....................................................................................................................................82
Figure 5.1 ....................................................................................................................................87
Figure 5.2 ....................................................................................................................................91
Figure 5.3 ....................................................................................................................................93
Figure 5.4 ....................................................................................................................................98
Figure 5.5 ..................................................................................................................................101
Figure 5.6 ..................................................................................................................................102
Figure 5.7 ..................................................................................................................................103
Figure 5.A.1 ..............................................................................................................................108
xi
List of Abbreviations
CL Continuum lowering
CRS Circular Rydberg states
OCC One Coulomb center
TCC Two Coulomb centers
TCP Two-centers problem
xii
Chapter 1. Introduction: Two-Coulomb Center Systems Representing Diatomic Rydberg
Quasimolecules
Charge exchange and crossings of corresponding energy levels that enhance charge exchange are
strongly connected with problems of energy losses and of diagnostics in high temperature
plasmas ? see, e.g., [1.1, 1.2] and references therein. Besides, charge exchange was proposed as
one of the most effective mechanisms for population inversion in the soft x-ray and VUV ranges
[1.3 ? 1.6]. One of the most fundamental theoretical playgrounds for studying charge exchange is
the problem of electron terms in the field of two stationary Coulomb centers (TCC) of charges Z
and Z' separated by a distance R. It presents fascinating atomic physics: the terms can have
crossings and quasicrossings.
The crossings are due to the fact that that the well-known Neumann-Wigner general theorem
on the impossibility of crossing of terms of the same symmetry [1.7] is not valid for the TCC
problem of Z' ? Z ? as shown in paper [1.8]. Physically it is here a consequence of the fact that
the TCC problem allows a separation of variables in the elliptic coordinates [1.8]. As for the
quasicrossings, they occur when two wells, corresponding to separated Z- and Z'-centers, have
states ? and ?', characterized by the same energies E = E', by the same magnetic quantum
numbers m = m', and by the same radial elliptical quantum numbers k = k' [1.9 ? 1.11]. In this
situation, the electron has a much larger probability of tunneling from one well to the other (i.e.,
of charge exchange) as compared to the absence of such degeneracy.
These rich features of the TCC problem also manifest in a different area of physics such as
1
plasma spectroscopy as follows. A quasicrossing of the TCC terms, by enhancing charge
exchange, can result in unusual structures (dips) in the spectral line profile emitted by a Z-ion
from a plasma consisting of both Z- and Z'-ions, as was shown theoretically and experimentally
[1.12 ? 1.17]. From the experimental width of these dips it is possible to determine rate
coefficients of charge exchange between multicharged ions, which is a fundamental reference
data virtually inaccessible by other experimental methods [1.17].
Before year 2000, the paradigm was that the above sophisticated features of the TCC problem
and its flourishing applications were inherently quantum phenomena. But then in year 2000,
papers [1.18, 1.19] were published presenting a purely classical description of both the crossings
of energy levels in the TCC problem and the dips in the corresponding spectral line profiles
caused by the crossing (via enhanced charge exchange). These classical results were obtained
analytically based on first principles without using any model assumptions.
In the classical studies the TCC systems represent diatomic Rydberg quasimolecules
encountered, e.g., in plasmas containing more than one kind of multicharged ions. Naturally, the
classical approach is well-suited for Rydberg quasimolecules.
Later applications of the results from [1.18, 1.19] included a magnetic stabilization of Rydberg
quasi-molecules [1.20], a problem of continuum lowering in plasmas (which plays a key role in
calculations of the equation of state, partition function, bound-free opacities, and other
collisional atomic transitions in plasmas) [1.21], and the study of the classical Stark effect for
Rydberg quasi-molecules ([1.22] and Chapter 2).
In these studies a particular attention was given to circular Rydberg states. Circular states of
atomic and molecular systems are an important subject in its own right. They have been
2
extensively studied both theoretically and experimentally for several reasons (see, e.g., [1.18 ?
1.20, 1.24 ? 1.37] and references therein): (a) they have long radiative lifetimes and highly
anisotropic collision cross sections, thereby enabling experiments on inhibited spontaneous
emission and cold Rydberg gases, (b) these classical states correspond to quantal coherent states,
objects of fundamental importance, (c) a classical description of these states is the primary term
in the quantal method based on the 1/n-expansion, and (d) they can be used in developing atom
chips.
As examples of experimental studies of Rydberg states, we refer to paper [1.23] where such
studies made in the last three decades have been enumerated [1.23]. In particular, Day and Ebel
[1.38] predicted theoretically in 1979 that probability of a wake electron being captured by fast-
moving ions traversing a solid to a state with large principal (n) and angular momentum (l)
quantum numbers is quite high and much of the time the electron is captured into circular
Rydberg states (l = n ? 1) distributed over a narrow band near nmax. Note here that l = n ? 1
defines circular orbits, whereas the full qualification of circular Rydberg states (CRS) requires
| ml | = l = n ? 1. Day and Ebel proposed existence of an "optical window" in ion velocity as a
possible explanation for non-observability of the CRS in beam-foil spectroscopy work. Also, the
CRS are both long lived with respect to radiative transitions and short lived with respect to
collisions, hence their observation requires a wide aperture and very good vacuum. Pegg et al. in
1977 [1.39] observed a strong cascade tails in the decay curves of Cu18+ in a beam-foil interaction
and attributed it to the successive decay of long-lived CRS or "yrast states". Note that CRS can
radiate only to the next lower state, which leads to a chain of successive yrast transitions till they
reach to the ground state. Recently, from the study of the time-resolved beam-foil X-ray spectra
3
of projectile or projectile-like ions of 2p,2s ? 1s transitions in H-like Fe, Ni, Cu, and Zn at
different delay times (in the range 250?1600 ps). Nandi identified, in each case, a single circular
Rydberg and/or an elliptic Rydberg state cascading successively to the 2p or 2s level.
Coming back to the ground-breaking theoretical papers [1.18, 1.19], it should be emphasized
that the analysis there was not confined to circular orbits of the electron. Paper [1.19] presented a
detailed study of helical Rydberg states of these diatomic Rydberg quasimolecules. For the stable
motion, the electron trajectory was found to be a helix on the surface of a cone, with the axis
coinciding with the internuclear axis. In this helical state, the electron, while spiraling on the
surface of the cone, oscillates between two end-circles which result from cutting the cone by two
parallel planes perpendicular to its axis (Fig. 1.1).
Fig. 1.1. Sketch of the helical motion of the electron in the ZeZ'-system at the absence of the magnetic
field. We stretched the trajectory along the internuclear axis to make its details better visible.
Let us now reiterate how classical energy terms of diatomic Rydberg quasimolecules were
obtained in paper [1.19]. (The meaning of ?classical energy terms? is clarified below.) They were
obtained by considering the Hamiltonian of the particle in a circular state in the cylindrical
4
coordinates:
H=12 ? p?2? p?
2
?2 ?pz
2?? Z
?z2??2?
Z'
??R?z?2??2
(1.1)
where (?, ?, z) are the cylindrical coordinates with the z-axis being the internuclear axis and
(p?, p?, pz) are the corresponding canonical momenta, Z and Z' are the charges at z = 0 and z = R.
Since ? is cyclic, p? = const = L and in the circular state pz = p? = 0, so (1.1), which is the particle
energy, can be written as
E= L
2
2?2 ?
Z
?z2??2?
Z'
??R?z?2??2 (1.2)
which, using the scaled quantities defined below,
w= zR, v= ?R, ?= L?ZR, b=Z'Z , ?=?ERZ (1.3)
takes the form
?= 1?w2?v2? b?
?1?w?2?v2
? ?
2
2v2 (1.4)
In equilibrium, the derivatives of ? by both scaled coordinates (w, v) must vanish, so by taking
the partial derivatives of (1.4) by each coordinate we obtain two more equations. The partial
derivative of (1.4) by w set equal to zero yields
5
v2=w
2/3?1?w?4/3?b2/3w2
b2/3?w2/3?1?w??2/3 (1.5)
and the partial derivative of (1.4) by v set equal to zero yields
?2=v4? 1?w2?v2?3/2 ? b??1?w?2?v2?3/2? (1.6)
Equation (1.5) determines the points (w, v) where the equilibrium is located. For b > 1, the
equilibrium value of v exists for 0 < w < w1 and for w3 < w < 1; for b < 1, it exists for 0 < w < w3
and w1 < w < 1. Here we introduced the quantities w1 = 1/(1 + b1/2) and w3 = b/(1 + b). For
definiteness, we shall consider the cases of b > 1 (or Z' > Z). There are no equilibrium points for
w < 0 or w > 1 (i.e., for z < 0 or z > R): the z-component of the total Coulomb attraction force of
the two centers has no balancing force at those points.
Solving (1.5) for v and substituting it into (1.6) and (1.4) and then solving (1.6) for ? and
substituting it into (1.4), we obtain ?(w, b) ? the scaled energy depending on the scaled
internuclear coordinate w for a given ratio of the charges of the nuclei b.
If we scale the internuclear distance R as r = (Z/L2)R, and given ? = ?ER/Z from (1.3), the
energy of the electron can be represented in the form E = ? (Z2/L2)?1, where we define ?1 = ?/r.
The scaling of E to ?1 includes no more R and includes L, just like the scaling of R to r. Next,
from the scaled quantities (1.3) we have ?2 = L2/(ZR) and from earlier in this paragraph,
r = ZR/L2, therefore, r = 1/?2; with ?2 taken from (1.6) and with v substituted from (1.5) we obtain
r(w, b). Thus, for any L > 0 and any b > 0, the dependence ?1(r), which represents the classical
energy terms for this system, can be presented in a parametric form ?(w, b), r(w, b) via the
6
parameter w for a given b. Here we give the explicit form of this dependence:
?1=p2? 1?w2?p?3/2 ? b??1?w?2?p?3/2 ?? w
2?p/2
?w2?p?3/2 ?
b??1?w?2?p/ 2?
??1?w?2?p?3/2 ? (1.7)
r=p?2? 1?w2?p?3/2? b??1?w?2?p?3/2?
?1
(1.8)
where we have defined the quantity p
p=v2=w
2/3?1?w?4/3?b2/3w2
b2/3?w2/3?1?w??2/3 (1.9)
The plot of the energy terms given by (1.7) and (1.8) for b = 3 is given below. The same plot,
with different ranges on both axes, is given in paper [1.19].
Fig. 1.2. Classical energy terms: the dependence of the scaled classical energy ?1 on the scaled
internuclear distance r given by eq. (1.7) and (1.8) for b = 3.
7
At this point it might be useful to clarify the relation between the classical energy terms ??1(r)
and the energy E. The former is a scaled quantity related to the energy as specified above:
E = ? (Z/L)2 ?1 . The projection L of the angular momentum on the internuclear axis is a
continuous variable. The energy E depends on both ?1 and L. Therefore, while the scaled quantity
?1 takes a discrete set of values, the energy E takes a continuous set of values (as it should be in
classical physics).
It turns out that the form of the parametric dependence can be significantly simplified and
some of the properties of it can be found analytically by introducing a new parameter
?=? 1w?1?
1/3
(1.10)
In this case, w = 0 will correspond to ? = +? and w = 1 will correspond to ? = 0, thus ? > 0 in the
allowed regions. The points w1 = 1/(1 + b1/2) and w3 = b/(1 + b) defining the allowed regions
0 < w < w1, w3 < w < 1 (here we assume b > 1) will correspond to ?1 = b1/6 and ?3 = 1/b1/3 (notice
that 0 < w < w1 corresponds to +? > ? > ?1 and w3 < w < 1 corresponds to ?3 > ? > 0). The energy
terms ?1(r) will take the following parametric form:
?1??,b?=?b
2/3??4?2????3?2??b2/3?2?3?1??
2??3?1?2??6?1? (1.11)
r??,b?=?b
2/3 ?2?1??6?1?3/2
??b2/3??4?2 (1.12)
8
The parametric plot of (1.11) and (1.12) with the parameter ? varied from 0 to 1/b1/3 and from b1/6
to +? for b = 3 will yield the same graph as in Fig. 1.2.
The crossing of the top two terms corresponds to the point where r(?, b) has a minimum or
?1(?, b) has a maximum for a given b. Thus, taking the derivative of either function by ? and
setting it equal to zero will yield a solution for the ? on the interval ? > 1 corresponding to the
crossing. The equation for ? obtained from differentiating r(?) is a 6th-power polynomial and
cannot be solved analytically; however, the equation for ? obtained from differentiating ?1(?) can
be solved analytically for ?. Below is the critical value ?0 corresponding to the crossing.
?0=?b1/3??b?1?
1/3
b1/6 ???b?1?
1/3???b?1?1/3? (1.13)
Substituting (1.13) into (1.12), we can obtain analytically the value of r corresponding to the
crossing.
In the following chapters we consider the same system subjected to external potentials of
various nature, as well as other configurations, such as a quasimolecule consisting of a proton, an
electron and a muon, and examine the integrals of motion in the TCC system. Atomic units
(? = e = me = 1) are used throughout the whole study.
9
Chapter 2. Enhancement of Charge Exchange and of Ionization by a Static Electric Field
2.1. Introduction
Charge exchange is one of the most important atomic processes in plasmas. Charge exchange
and crossings of corresponding energy levels that enhance charge exchange are strongly
connected with problems of energy losses and of diagnostics in high temperature plasmas ? see,
e.g., [1.1, 1.2] and references therein. Charge exchange was proposed as an effective mechanism
for population inversion in the soft x-ray and VUV ranges [1.3 ? 1.6]. One of the most
fundamental theoretical domains for studying charge exchange is the problem of electron terms
in the field of two stationary Coulomb centers (TCC) of charges Z and Z' separated by a distance
R. It presents an intriguing atomic physics: the terms can have crossings and quasicrossings.
The crossings are due to the fact that that the well-known Neumann-Wigner general theorem
on the impossibility of crossing of terms of the same symmetry [1.7] is not valid for the TCC
problem of Z' ? Z ? as shown in paper [1.8]. Physically it is here a consequence of the fact that
the TCC problem allows a separation of variables in the elliptic coordinates [1.8]. As for the
quasicrossings, they occur when two wells, corresponding to separated Z- and Z'-centers, have
states ? and ?', characterized by the same energies E = E', by the same magnetic quantum
numbers m = m', and by the same radial elliptical quantum numbers k = k' [1.9 ? 1.11]. In this
situation, the electron has a much larger probability of tunneling from one well to the other (i.e.,
of charge exchange) as compared to the absence of such degeneracy.
These intrinsic features of the TCC problem also manifest in a different area of physics such as
10
plasma spectroscopy as follows. A quasicrossing of the TCC terms, by enhancing charge
exchange, can result in unusual structures (dips) in the spectral line profile emitted by a Z-ion
from a plasma consisting of both Z- and Z'-ions, as was shown theoretically and experimentally
[1.12 ? 1.17].
Before year 2000, the paradigm was that the above sophisticated features of the TCC problem
and its flourishing applications were inherently quantum phenomena. But then in year 2000
papers [1.18, 1.19] were published presenting a purely classical description of both the crossings
of energy levels in the TCC problem and the dips in the corresponding spectral line profiles
caused by the crossing (via enhanced charge exchange). These classical results were obtained
analytically based on first principles without using any model assumptions. Later applications of
these results included a magnetic stabilization of Rydberg quasi-molecules [1.20] and a problem
of continuum lowering in plasmas [1.21].
In papers [1.18, 1.20, 1.21] the study was focused at Circular Rydberg States (CRS) of the TCC
system (the analysis in paper [1.19] went beyond CRS). CRS of atomic and molecular systems,
with only one electron, correspond to |m| = (n ? 1) >>1, where n and m are the principal and
magnetic electronic quantum numbers, respectively. They have been extensively studied [2.1 ?
2.4] both theoretically and experimentally for several reasons: (a) CRS have long radiative
lifetimes and highly anisotropic collision cross sections, thereby enabling experiments on
inhibited spontaneous emission and cold Rydberg gases [1.36, 2.5], (b) classical CRS correspond
to quantal coherent states, objects of fundamental importance, and (c) a classical description of
CRS is the primary term in the quantal method based on the 1/n-expansion (see, e.g. [2.6] and
references therein).
11
While the authors of paper [1.20] studied analytically the effect of a magnetic field (along the
internuclear axis) on CRS of the TCC system, in the present chapter we study the effect of an
electric field (along the internuclear axis) on CRS of the TCC system. We provide analytical
results for strong fields, as well as numerical results for moderate fields. We show that the
electric field leads to the following consequences.
First, it leads to the appearance of an extra energy term: the fourth classical energy term ? in
addition to the three classical energy terms at zero field. Second, but more importantly, the
electric field creates additional crossings of these energy terms. We show that some of these
crossings enhance charge exchange while other crossings enhance the ionization of the Rydberg
quasi-molecule.
2.2. Calculations of the classical Stark effect
for a Rydberg quasi-molecule in a circular state
We consider a TCC system, where the charge Z is at the origin and the Oz axis is directed to the
charge Z', which is at z = R. A uniform electric field F is applied along the internuclear axis ? in
the negative direction of Oz axis. We study CRS where the electron moves around a circle in the
plane perpendicular to the internuclear axis, the circle being centered at this axis.
Two quantities, the energy E and the projection L of the angular momentum on the internuclear
axis are conserved in this configuration. We use cylindrical coordinates to write the equations for
both.
12
E=12 ? ??2??2 ??2??z2??Zr ?Z'r' ?F z (2.1)
L=?2 ?? (2.2)
where ? is the distance of the electron from the internuclear axis, ? is its azimuthal angle, z is the
projection of the radius-vector of the electron on the internuclear axis, r and r' are the distances
of the electron from the particle to Z and Z', respectively.
The circular motion implies that d?/dt = 0; as the motion occurs in the plane perpendicular to
the z-axis, dz/dt = 0. Further, expressing r and r' through ? and z, and taking d?/dt from (2.2), we
have:
E= L
2
2?2 ?
Z
??2?z2?
Z'
??2??R?z?2 ?Fz
(2.3)
With the scaled quantities
w= zR, v= ?R, b=Z'Z , ?=?ERZ , ?= L?Z R, f =F R
2
Z , r=
Z R
L2
(2.4)
our energy equation takes the form below:
?= 1?w2?v2? b?
?1?w?2?v2
? f w? ?
2
2v2
(2.5)
We can seek the equilibrium points by finding partial derivatives of ? by the scaled coordinates
w, v and setting them equal to zero. This will give the following two equations.
f ? b?1?w???1?w?2?v2?3/2= w?w2?v2?3/2 (2.6)
13
?2=v4? 1?w2?v2?3/2 ? b??1?w?2?v2?3/2? (2.7)
From the definitions of the scaled quantities (4), ?2 = 1/r and E = ? (Z/R) ?. Since r = ZR/L2,
E = ? (Z/L)2 ?/r, where r = 1/?2 can be obtained by solving (2.7) for ?. Substituting ? into the
energy equation, we get the three master equations for this configuration.
?1=p2? 1?w2?p?3/2 ? b??1?w?2?p?3/2 ?? w
2?p/2
?w2?p?3/2 ?
b??1?w?2?p/2?
??1?w?2?p?3/2 ? f w? (2.8)
r=p?2? 1?w2?p?3/2 ? b??1?w?2?p?3/2 ?
?1
(2.9)
f ? b?1?w???1?w?2?p?3/2= w?w2?p?3/2 (2.10)
where E = ? (Z/L)2 ?1 and p = v2. Thus, ?1 is the ?true? scaled energy, whose equation for E does
not include R. The scaled energy ?1 and internuclear distance r in (2.8) and (2.9) now explicitly
depend only on the coordinates w and p (besides the constants b and f). Therefore, if we solve
(2.10) for p and substitute it into (2.8) and (2.9), we will have the parametric solution ?1(r) with
the parameter w.
Our focus is at crossings of energy terms of the same symmetry. In the quantum TCC problem,
?terms of the same symmetry? means terms of the same magnetic quantum number m [1.8 ?
1.11]. Therefore, in our classical TCC problem, we fixed the angular momentum projection L
14
and study the behavior of the classical energy at L = const ? 0 (the results for L and ?L are
physically the same).
Equation (2.10) does not allow an exact analytical solution for p. Therefore, we will use an
approximate analytical method.
Figure 2.1 shows a contour plot of (2.10) for a relatively weak field f = 0.3 at b = 3, with w on
the horizontal axis and p on the vertical. The plot has two branches. The left branch spans from
w = 0 to w = w1. The right one actually has a small two-valued region between some w = w3 and
1 (w3 < 1). Indeed, at w = 1, there are two values of p: p = 0 and p = f ?2/3 ? 1. Thus, the two-
valued region exists only for f < 1.
Fig. 2.1. Contour plot of (2.10) for a relatively weak field f = FR2/Z = 0.3 at b = Z'/Z = 3.
15
The right branch touches the abscissa at w = 1 and at some w = w2. Analytical expressions for w1
and w2 are given in Appendix A. The quantity w3 is a solution of the equation
f 2/5?2w3?1?3/5=w32/5?b2/5?1?w3?2/5 (2.11)
The method, by which w3 was found from (2.11), is presented in Appendix B.
Figure 2 shows a contour plot of (2.10) for a relatively strong field f = 20 at b = 3. It is seen
that there is no two-valued region.
Fig. 2.2. The same as in Fig. 2.1, but for a relatively strong field f = FR2/Z = 20.
From now on we consider the situation where the radius of the electronic orbit is relatively
small, meaning that p << 1. Physically this corresponds to strong fields f > fmin ~ 10.
16
Solving (2.10) in the small-p approximation, we obtain
p=? w
f ? b?1?w?2 ?
2/3
?w2 (2.12)
for the left branch (0 < w < w1) and
p=?b?1?w?1
w2? f ?
2/3
??1?w?2 (2.13)
for the right branch (1 < w < w2). Substituting these results into (2.8) and (2.9), we get
approximate solutions for energy terms ??1(r) in both regions in a parametric form, w being the
parameter. Now we plot classical energy terms ??1(r) by varying the parameter w over both
regions, using the appropriate formula for each one.
Figure 2.3 shows classical energy terms at b = 3 for f = 20. Figure 2.4 presents classical energy
terms at b = 3 for f = 5.
17
Fig. 2.3. Classical energy terms at b = 3 for f = 20.
18
Fig. 2.4. Classical energy terms at b = 3 for f = 5.
We also solved the same problem numerically. By comparison we found that the approximate
analytical solution is accurate for fields f = 5 and above.
Figures 2.5 and 2.6 show the numerically obtained classical energy terms at b = 3 for f = 2 and
f = 0.1, respectively. For comparison, Fig. 2.7 shows the classical energy terms at b = 3 at the
absence of the electric field (it had been previously presented in papers [1.18, 1.19] and is the
same as Fig. 1.2).
19
Fig. 2.5. Classical energy terms at b = 3 for f = 2.
Fig. 2.6. Classical energy terms at b = 3 for f = 0.1.
20
Fig. 2.7. Classical energy terms at b = 3 at the absence of the electric field.
The electric field causes several important new features compared to the zero-field case. While
at f = 0 there are three classical energy terms, the electric field brings up the fourth classical
energy term. Indeed, let us take as an example the case of f = 5 at b = 3 presented in Fig. 2.4.
There are four energy terms that we label as follows:
#1 ? the lowest term;
#2 ? the next term up (which has a V-type crossing with term 1);
#3 ? the next term up;
#4 ? the highest term (which has a V-type crossing with term 3).
We will use this labeling also while discussing all other plots (except the plot in Fig. 2.7 for
f = 0): terms 1 and 2 will be those having the V-type crossing at the lower energy, terms 3 and 4
21
will be those having the V-type crossing at the higher energy,
At f = 0 term 2 is absent, but it appears at any non-zero value of f ? no matter how small.
Actually, as f approaches zero, this term behaves like ?f/r, which is why it disappears at f = 0.
The existence of this additional term can be explained physically as follows. When f = 0,
equilibrium of the orbital plane to the right of Z' (i. e., for w > 1) does not exist, so that the values
of w1 and w3 reduce to the ones presented in papers [1.18, 1.19] and the right branch of p(w)
asymptotically goes to infinity when w goes down to w3. When an infinitesimal field f appears,
the right branch flips over positive infinity and ends up on the abscissa at w2 ? ?, thus enabling
the whole region w > 1 for equilibrium. As the field grows, w2 decreases. Physically, the force
from the electric field at w > 1 balances out the Coulomb attraction of the Z?Z' system on the left
? the situation not possible for f = 0. This term is obtained by varying the parameter w from 1 to
w2.
We emphasize that the above examples presented for Z'/Z = 3 represent a typical situation. In
fact, for any pair of Z and Z' ? Z, at the presence of the electric field, there are four classical
energy terms of the same symmetry for CRS.
Another important new feature caused by the electric field is X-type crossings of the classical
energy terms. This kind of crossings and their physical consequences are discussed in the next
section.
2.3. X-type crossings of classical energy terms and their physical consequences
Figure 2.8 shows a magnified version of the energy terms 2, 3, and 4 at b = 3 for f = 2. Figure 2.9
22
shows a further magnified version of the energy terms 2 and 4 at b = 3 for f = 2. Compared to
Fig. 2.5 for the same b and f, in Figs. 2.8 and 2.9 we decreased the exhibited energy range, but
increased the exhibited range of the internuclear distances r.
Fig. 2.8. Magnified plot of the classical energy terms 2, 3, and 4 at b = 3 for f = 2.
Fig. 2.9. Further magnified plot of the classical energy terms 2 and 4 at b = 3 for f = 2.
23
It is seen that term 2 has the X-type crossing with term 3 at r = 7.8 and the X-type crossing
with term 4 at r = 32. The situation where there are two X-type crossings exists in a limited range
of the electric fields. For example, for b = 3:
- two X-type crossings exist at 1.31 < f < 2.4;
- there are no X-type crossings at f < 1.31;
- there is one X-type crossing at f > 2.4 (the crossing of terms 2 and 3).
To reveal physical consequences of the X-type crossings, let us first discuss the origin of all
four classical energy terms for arbitrary Z'/Z ? 1. At r ? ?, term 3 corresponds to the energy of a
hydrogenlike ion of the nuclear charge Zmin = min(Z', Z), slightly perturbed by the charge
Zmax = max(Z', Z), as shown in [1.18, 1.19].
At r ? ?, term 4 corresponds to a near-zero-energy state (where the electron is almost free), as
shown in [1.18, 1.19]. If the ratio Z'/Z is of the order of (but not equal to) unity, this term at
r ? ? can be described only in the terminology of elliptical coordinates (rather than parabolic or
spherical coordinates), meaning that even at r ? ? the electron is shared between the Z- and Z'-
centers. However, in the case of Z' >> Z, this term can be asymptotically considered as the Z'-
term, as shown in [1.18, 1.19]. It has the V-type crossing with term 3, which asymptotically is the
Z-term (since Zmin = Z for Z' > Z). Likewise, in the case of Z' << Z, term 4 can be asymptotically
considered as the Z-term, as shown in [1.18, 1.19]. It has the V-type crossing with term 3, which
asymptotically is the Z'-term (since Zmin = Z' for Z' < Z).
At r ? ?, term 1 corresponds to the energy of a hydrogenlike ion of the nuclear charge Zmax
slightly perturbed by the charge Zmin , as shown in [1.18, 1.19]. As for the term 2, at r ? ? it has
properties similar to term 4, but with the interchange of Zmax and Zmin . In particular, in the case of
24
Z' >> Z, this term can be asymptotically considered as the Z-term, having the V-type crossing
with term 1, which asymptotically is the Z'-term (since Zmax = Z' for Z' > Z). In the case of
Z' << Z, term 2 can be asymptotically considered as the Z'-term, having the V-type crossing with
term 1, which asymptotically is the Z-term (since Zmax = Z for Z' < Z).
Thus, when Z and Z' differ significantly from each other, the V-type crossings occur between
two classical energy terms that can be asymptotically labeled as Z- and Z'-terms. This situation
classically depicts charge exchange, as explained in papers [1.18, 1.19]. Indeed, say, initially at
r ? ?, the electron was a part of the hydrogenlike ion of the nuclear charge Zmin . As the charges
Z and Z' come relatively close to each other, the two terms undergo a V-type crossing and the
electron is shared between the Z- and Z'-centers. Finally, as the charges Z and Z' go away from
each other, the electron ends up as a part of the hydrogenlike ion of the nuclear charge Zmax .
So, the first distinction caused by the electric field is an additional, second V-type crossing
leading to charge exchange ? compared to the zero-field case where there was only one such
crossing. However, the second V-type crossing (the crossing of terms 1 and 2) occurs at the
internuclear distance rV2 << rV1, where rV1 is the internuclear distance of the first V-type crossing
(the crossing of terms 3 and 4). Therefore the cross-section of the charge exchange
corresponding to the second V-type crossing is much smaller than the corresponding cross-
section for the first V-type crossing.
Now let us discuss the X-type crossing from the same point of view. When Z and Z' differ
significantly from each other, the X-type crossing of terms 2 and 4 is the crossing of terms that
can be asymptotically labeled as Z- and Z'-terms. Thus, this situation again classically depicts
charge exchange. The most important is that this crossing occurs at the internuclear distance
25
rX1 >> rV1 >> rV2 . Therefore, the cross-section of charge exchange due to this X-type crossing is
much greater than the corresponding cross-sections of for the V-type crossings. This is the most
fundamental physical consequence caused by the electric field: a significant enhancement of
charge exchange.
When Z and Z' differ significantly from each other, the X-type crossing of terms 2 and 3 is the
crossing of terms having the same asymptotic labeling: either both of them are Z-terms or both of
them are Z'-terms. Therefore this second X-type crossing (at r = rX2) does not correspond to
charge exchange ? rather it represents an additional ionization channel. Indeed, say, initially at
r ? ?, the electron resided on term 3 of the hydrogenlike ion of the nuclear charge Z. As the
distance between the charges Z and Z' decreases to r = rX2, the electron can switch to term 2,
which asymptotically corresponds to a near-zero-energy state (of the same hydrogenlike ion of
the nuclear charge Z) where the electron would be almost free. So, as the charges Z and Z' go
away from each other, the system undergoes the ionization. Thus, another physical consequence
caused by the electric field is the appearance of the additional ionization channel. This should
have been expected since the electric field promotes the ionization of atomic and molecular
systems.
2.4. Conclusions
We studied the effect of an electric field (along the internuclear axis) on circular Rydberg states
of the two Coulomb centers system. We provided analytical results for strong fields, as well as
numerical results for moderate fields. We showed that the electric field had the following effects.
26
The first effect is the appearance of an extra energy term: the fourth classical energy term ? in
addition to the three classical energy terms at zero field. This term exhibits a V-type crossing
with the lowest energy term. The two highest energy terms continue having a V-type crossing
like at the zero field. In the situation where the charges Z' differ significantly from each other,
both V-type crossings correspond to charge exchange.
The second effect is the appearance of a new type of crossings ? X-type crossings. One of the
X-type crossings (existing in a limited range of the electric field strength) corresponds to charge
exchange at a much larger internuclear distance compared to the V-type crossings. Therefore the
cross-section of charge exchange due to this X-type crossing is much greater than the cross-
section of charge exchange due to V-type crossings. Thus, the electric field can significantly
enhance charge exchange. We believe that this is the most important result of the present chapter.
The other X-type crossing does not correspond to charge exchange. Instead, it represents an
additional ionization channel.
Appendix A. The limits w1 and w2 on the graph of p(w) in Eq. (2.10).
The analytical results for the quantities w1 and w2, obtained using the software Mathematica,
have the following form.
For w1:
27
)6)1(4)1(
3)1(2)1(
)1(36
3)1(2)1(3(63
1
1
2
1
1
2
1
1
2
1
+?+???+?
+?+?+?+
++
++?+?+?+?=
f
fb
f
a
fa
fb
f
fb
f
a
fa
fbf
b
f
fb
f
a
fa
fbw
where
3 3233
1 ))7(1(3)1()1(336)1(45 +++?+?++?++= ffbffbbfbfbfba
For w2:
)2)1(4)1(
1)1(2)1(
)1(36
1)1(2)1(3(63
2
2
2
2
2
2
2
2
2
2
+++?+??
+++++?
?+
++++++?+=
f
b
f
a
fa
fb
f
b
f
a
fa
fbf
b
f
b
f
a
fa
fbw
where
3 323323
2 )1(3))7(1(3)1(36))61(1(3)1(3)1( fbfbbfbfbffbbfbfa ?++?+?++++???+?=
Appendix B. Finding the lower limit w3 of the two-valued region
on the graph of p(w) in Eq. (2.10).
Defining a function
F?p,w?= f ? b?1?w???1?w?2?p?3/2 ? w?w2?p?3/2 , (2.B.1)
28
we can rewrite (2.10) as F(p, w) = 0. From the graph it is seen that at w3, dw/dp = 0. Since
F(p, w) = 0, dF/dp = 0 as well. On the other hand, F(w, p) = F(w(p), p) = 0 and
dF
dp=
?F
?w
dw
dp ?
?F
? p =0 (2.B.2)
from where we get
dw
dp =?
?F/? p
?F/?w (2.B.3)
Setting the right side of (2.B.1) and (2.B.3) to zero, we obtain the system of two equations,
solving which for w will give us the point on the contour plot of F(p, w) = 0 where the derivative
dw/dp vanishes, i. e., the desired point. Excluding p from the system, we reduce the equation to
f 2/5?2w3?1?3/5=w32/5?b2/5?1?w3?2/5 (2.B.4)
where w was renamed to w3 for clarity. This is (2.11) of this chapter.
29
Chapter 3. Effect of Plasma Screening on Circular States of Diatomic Rydberg
Quasimolecules and their Application to Continuum Lowering in Plasmas
3.1. Introduction
Circular Rydberg States (CRS) of atomic and molecular systems containing only one electron
have been extensively studied both theoretically and experimentally for several reasons (see, e.g.,
[1.18 ? 1.20, 1.24 ? 1.37, 3.1] and references therein). First, they have long radiative lifetimes
and highly anisotropic collision cross sections, thereby enabling experiments on inhibited
spontaneous emission and cold Rydberg gases. Second, these classical states correspond to
quantal coherent states, i.e., objects of fundamental importance. Third, a classical description of
these states is the primary term in the quantal method based on the 1/n-expansion (n is the
principal quantum number). Fourth, they can be used in developing atom chips.
In the previous chapters we studied analytically CRS of the two-Coulomb-center system, the
system (denoted as ZeZ') consisting of two nuclei of charges Z and Z', separated by a distance R,
and one electron ? see also [1.18 ? 1.20, 1.24, 1.25, 1.34, 3.1]. Energy terms of these Rydberg
quasimolecules were obtained for a field-free case, as well as under an electric field or under a
magnetic field.
The Rydberg quasimolecules of this type naturally occur in high density plasmas of several
types of ions, where a fully-stripped ion of the charge Z' is in the proximity of a hydrogenlike ion
of the nuclear charge Z (or where a fully-stripped ion of the charge Z is in the proximity of a
30
hydrogenlike ion of the nuclear charge Z'). Therefore in the present chapter we study the effects
of plasma screening on CRS of these Rydberg quasimolecules ? the effects not taken into
account in the previous works. We provide analytical results for weak screening, as well as
numerical results for moderate and strong screening. We show that the screening leads to the
following consequences.
The screening causes an additional energy term to appear ? compared to the absence of the
screening. This new term has a V-type crossing with the lowest energy term. The internuclear
potential is also affected by the screening, destabilizing the nuclear motion for Z > 1 and
stabilizing it for Z = 1.
We also study the effect of plasma screening on continuum lowering (CL) in the ionization
channel. CL was studied for over 50 years ? see, e.g., books/reviews [3.2 ? 3.6] and references
therein. Calculations of CL evolved from ion sphere models to dicenter models of the plasma
state [3.4, 3.7 ? 3.12]. One of such theories ? a percolation theory [3.4, 3.9] ? calculated CL
defined as an absolute value of energy at which an electron becomes bound to a macroscopic
portion of plasma ions (a quasi-ionization). In 2001 the value of CL in the ionization channel was
derived analytically, which was disregarded in the percolation theory: a quasimolecule,
consisting of the two ion centers plus an electron, can get ionized in a true sense of this word
before the electron would be shared by more than two ions [1.21]. It was also shown in [1.21]
that, whether the electron is bound primarily by the smaller or by the larger out of two positive
charges Z and Z', makes a dramatic qualitative and quantitative difference for this ionization
channel. The results in [1.21] were obtained for circular states of the corresponding Rydberg
quasimolecules.
31
In the present chapter we show that the screening decreases CL in the ionization channel,
making CL vanish as the screening factor increases.
3.2. Calculation of the effect of plasma screening and classical energy terms for a Rydberg
quasimolecule in a circular state
Plasma screening of a test charge is a well-known phenomenon. For a hydrogen atom or a
hydrogen-like ion (an H-atom, for short), it is effected by replacing the pure Coulomb potential
by a screened Coulomb potential which contains a physical parameter ? the screening length a.
For example, the Debye-H?ckel (or Debye) interaction of an electron with the electronic shielded
field of an ion of charge Z is U(R) = ?(Ze2/R)exp(?R/a), where
a = (kT / (4?e2Ne))1/2 ? 1.304 ? 104(1010/Ne)1/2T1/2a0 , where Ne (cm?3) and T (K) are the electron
density and temperature, respectively.
We study a two-Coulomb center (TCC) system with the charge Z placed at the origin, and the
Oz axis is directed at the charge Z', which is at z = R, the system being submerged in a plasma of
a screening length a. We consider the circular orbits of the electron which are perpendicular to
the internuclear axis and centered on the axis.
Two quantities, the energy E and the projection L of the angular momentum on the internuclear
axis are conserved in this configuration. We use cylindrical coordinates to write the equations for
both:
E=12 ? ??2??2 ??2??z2??Zr e?r/a?Z'r' e?r'/a (3.1)
32
L=?2 ?? (3.2)
where r and r' are distances from the electron to Z and Z'. The circular motion implies that
d?/dt = 0; as the motion occurs in the plane perpendicular to the z-axis, dz/dt = 0. Further,
expressing r and r' through ? and z, and taking d?/dt from (3.2), we have:
E= L
2
2?2?
Z
??2?z2 e
???2?z2/a? Z'
??2??R?z?2 e
???2??R?z?2/a (3.3)
With the scaled quantities
w= zR, p=??R?
2
, b=Z'Z , ?=?ERZ , ?= L?ZR , ?=Ra , r=ZRL2 (3.4)
our energy equation takes the form below:
?=e
???w2? p
?w2?p?
be????1?w?2?p
??1?w?2?p?
?2
2 p
(3.5)
We can seek the equilibrium points by finding partial derivatives of ? by the scaled coordinates
w, p and setting them equal to zero. This will give the following two equations.
we???w2?p
w2?p ?
1
?w2?p???=
b?1?w?e????1?w?2?p
?1?w?2?p ?
1
??1?w?2?p???
(3.6)
?2
p2=
e???w2?p
w2?p ?
1
?w2?p????
be????1?w?2?p
?1?w?2?p ?
1
??1?w?2?p???
(3.7)
From the definitions of the scaled quantities (3.4), ?2 = 1/r and E = ? (Z/R) ?. Since r = ZR/L2,
33
E = ? (Z/L)2 ?/r, where r = 1/?2 can be obtained by solving (3.7) for ?. Thus, the scaled energy
without explicit dependence on R is ?/r, which we shall denote ?1. Using this, equations (3.5),
(3.6) and (3.7) can be transformed into the following three master equations for this
configuration.
?1=? p?1???w
2?p?e???w2? p
?1?w??w2?p?3/2 ?
2??1?w??w2?p?
1???w2?p
? w??1?w?
2?p?
1????1?w?2?p
? p2 ? (3.8)
r=?1?w??w
2?p?3/2e??w2? p
p2?1???w2?p?
(3.9)
w?1???w2?p?e???w2?p
?w2?p?3/2 =
b?1?w??1????1?w?2?p?e????1?w?2?p
??1?w?2?p?3/2
(3.10)
The quantities ?1 and r now depend only on the coordinates w and p (besides the constant ?).
Therefore, if we resolve (3.10) for p and substitute it to (3.8) and (3.9), we obtain the parametric
solution for the energy terms ?1(r) with the parameter w for the given b and ?.
Equation (3.10) does not allow an exact analytical solution for p. Therefore, we will use an
approximate analytical method.
Below is the contour plot of this equation for b = 3 and ? = 0.1.
34
Fig. 3.1. Contour plot of equation (3.10) for b = 3 and ? = 0.1.
As in [1.19] and Chapter 2, the plot has two branches, the left one spanning from w = 0 to
w = w1, and the right one from the asymptote w = w3 to w = 1. w1 is a solution of the equation
?1?w1?2 ?1??w1?e??1?2w1?=bw12?1???1?2w1?? (3.11)
in the interval 0 < w1 < 1, and w3 does not depend on ? and equals b/(1 + b) ? the same as in
[1.19] for ? = 0 ? the "default" case described in Chapter 1. As ? increases, w1 and the p-
coordinate of the maximum of the left branch increase, but the general shape of both curves is
preserved. Below is the plot for a relatively strong ? = 2.
35
Fig. 3.2. Contour plot of equation (3.10) for b = 3 and ? = 2.
An approximation was made for small values of ?. Approximating (3.10) in the first power of
?, we obtain the expression involving only the second and higher powers of ?. Therefore, an
attempt was made using the value of p(w) for ? = 0 presented in [1.19], which we shall denote as
p0; it is the same as the quantity in (1.5). Further, taking the higher powers of ? into account, we
obtained the next-order approximation for p(w):
p?w?=p0??
2
6 ?1?2w??1??1?2w??
w2/3?b2/3?1?w?2/3
w2/3?b2/3?1?w?2/3 ?
2?
(3.12)
where
p0=w
2/3?1?w?4/3?b2/3w2
b2/3?w2/3?1?w??2/3 (3.13)
36
? the zero-? value as in (1.5).
Equation (3.11) can be approximated by substituting 1 + ?(1 ? 2w1) in place of exp(?(1 ? 2w1)),
which will render it a 4th-degree polynomial in w1. The analytical expression for it is given in
Appendix A.
Substituting (3.12) into (3.8) and (3.9), we obtain the approximate parametric solution for the
energy terms ? ?1(r) by running the parameter w on 0 < w < w1 and w3 < w < 1. Empirically, by
comparison with the numerical results, it was found that using the value of p from (3.13) on the
0 < w < w1 range and from (3.12) on the w3 < w < 1 range gives the best approximate results.
Below are the approximate terms for b = 3 and different values of ?.
Fig. 3.3. Approximate classical energy terms for b = 3 at ? = 0.1.
37
Fig. 3.4. Approximate classical energy terms for b = 3 at ? = 0.2.
Fig. 3.5. Approximate classical energy terms for b = 3 at ? = 0.3.
38
A numerical solution has also been made. It confirmed that the analytical solution was a good
approximation for ? < 0.3. Below are the terms plotted for selected arbitrary values of ?.
Fig. 3.6. Numerical classical energy terms for b = 3 at ? = 0.2.
Fig. 3.7. Numerical classical energy terms for b = 3 at ? = 0.5.
39
Fig. 3.8. Numerical classical energy terms for b = 3 at ? = 1.
Fig. 3.9. Numerical classical energy terms for b = 3 at ? = 2.
40
Fig. 3.10. Numerical classical energy terms for b = 3 at ? = 3.
The following reminder should be made. The above plots represent ?classical energy terms? of
the same symmetry. (In physics of diatomic molecules, the terminology "energy terms of the
same symmetry" means the energy terms of the same projection of the angular momentum on the
internuclear axis.) For a given R and L, the classical energy E takes only several discrete values.
However, as L varies over a continuous set of values, so does the classical energy E (as it should
be in classical physics).
3.3. Crossings of the energy terms
Several properties of these energy terms have been studied. We note that in case of small or
41
moderate ?, we observe four terms, both pairs of which have a V-type crossing. As an example,
we shall take the plot of the terms for ? = 0.2 and number the lowest term 1 and the highest term
2; the remaining terms will be numbered 3 and 4, from the lower one to the higher one.
Therefore, terms 1 and 2 and terms 3 and 4 undergo V-type crossings, to which we shall refer as
V12 and V34. Using a small-? approximation by choosing (3.13) as the p(w) solution for the
parametric energy terms (essentially, a zero-? approximation), we can substitute (3.13) into (3.9),
which will give it the form below.
r=
?1?2w?3/2 ?b2/3?? w1?w?
2/3
w3?b2/3?? 1?ww ?
4/3?2 (3.14)
For a given b, the terms 3 and 4 are produced by varying w between 0 and w1. The V34 crossing
occurs at the value of w where r(w) has a minimum [1.19]. Therefore, setting the derivative
dr/dw to zero, we obtain the equation whose solution for w in the range 0 < w < w1 gives us the
point on the parametric axis which produces the V34 crossing.
9w4/3?1?w?4/3?w4/3?b4/3?1?w?4/3?=b2/3?1?4w?22w2?36w3?18w4? (3.15)
This equation has no dependence on ? and is therefore equivalent to the Coulomb-potential case
described in Chapter 1. Therefore, an analytical solution exists for (3.15), which is shown in
(1.13). Going back to the w-parametrization, we obtain the analytical solution of (3.15):
42
wV34= 1
1??b1/3??b?1?
1/3
b1/6 ???b?1?
1/3???b?1?1/3??
3/2 (3.16)
Substituting it into (3.9) and using the numerical solution for p of (3.10), we obtain the semi-
analytical dependence rV34(?) for a given b. Below is the plot for b = 3.
Fig. 3.11. Semi-analytical plot of rV34(?) for b = 3.
Since the plot in Fig. 3.11 was obtained using a zero-? approximation for the point of the V34
crossing, we also graphed this dependence numerically point by point. The graph below for the
same b shows that in relation to terms 3 and 4, this approximation works well even for moderate
values of ?.
43
Fig. 3.12. Numerical plot of rV34(?) for b = 3.
The energy of the V34 crossing can be obtained semi-analytically by substituting the numerical
solution for p of (3.10) into (3.8), and by further substituting (3.16) into the resulting formula. It
could be seen that as ? grows, the energy of the crossing grows and at a relatively large ?
becomes positive. A numerical graph can also be made in a fashion similar to Fig. 3.12. A visual
comparison shows a good similarity between the two. Below, a numerical graph is given for
b = 3.
Fig. 3.13. Numerical plot of ??V34(?) for b = 3.
44
As we see, the energy of the V34 crossing becomes positive after ? = 2.96, has a maximum,
and later asymptotically approaches zero. For b = 4/3, the V34 crossing reaches zero energy at
? = 2.13.
The shape of the terms 3 and 4 is also affected by the screening. Term 3, whose energy
increases as r increases, becomes nearly horizontal at energy ?0.5 at a certain value of ?; at
further ?, its energy decreases with r. For b = 3, this value of ? is about 1.1; for b = 4/3, it is
about 0.7. The plots are shown below.
Fig. 3.14. Classical energy terms for b = 3 at ? = 1.1;
term 3 is nearly constant at energy ??1 = ?0.5.
45
Fig. 3.15. The same as Fig. 3.14 for b = 4/3 at ? = 0.7.
For V12 crossing, the small-? approximation is not applicable since this crossing is not
observed at ? = 0. Therefore, only numerical methods were used. A situation of particular interest
is the behavior of term 1 at very small r, because as r ? 0 it corresponds to the energy of the
hydrogenic ion of the nuclear charge Z + Z' [1.25]. The point with the smallest r is the V12
crossing. A comparison was made of the dependence of the electronic energy on the screening
parameter ? between [1.25] and the limiting case r ? 0 in our situation. Since in the paper
mentioned above, the calculation was performed for a single Coulomb center Z, we had to re-
scale the quantities to make a valid comparison. The electronic energies are related as
?1(TCC) = (1 + b)2?1(OCC), where OCC stands for "one Coulomb center". Since the scaling for the
screening parameter in the OCC case did not include R (the internuclear distance), the scaling
factor between the screening parameter includes r: ?(TCC) = r(1 + b)?(OCC). Taking this into
46
account, we can plot the energy dependence on ? for the limiting case r ? 0.
Fig. 3.16. Plot of the energy of the electron versus the scaled screening factor for b = 3 in the limit r ? 0.
Below is the dependence obtained in [1.25] for OCC:
Fig. 3.17. Plot of the energy of the electron in a OCC system versus the scaled screening factor.
47
3.4. The effect of the plasma screening on the internuclear potential
Another aspect of this problem worth studying is the internuclear potential. Previously its
properties were studied for the same system with ? = 0 and a magnetic field parallel to the
internuclear axis [1.20]. Particularly, the magnetic field created a deep minimum in the
internuclear potential, which stabilized the nuclear motion and transformed a Rydberg quasi-
molecule into a real molecule. Here we shall investigate the effect of the screening on the
internuclear potential. Its form in atomic units is
Uint=ZZ'R ?E (3.17)
where E is the electronic energy. Using the scaled quantities from (3.4), we have the scaled
internuclear potential
uint=bZr ??1 (3.18)
where Uint = (Z/L)2uint. By plotting its dependence on r, we found out that in cases of Z > 1 the
screening tends to flatten the minimum, producing the effect opposite to the one of the magnetic
field. Compare the plots of uint(r) in the case of Z = 2, b = 2 for ? = 0 and ? = 0.3.
48
Fig. 3.18. The plot of the scaled internuclear potential versus the scaled internuclear distance for Z = 2,
Z' = 4, ? = 0.
Fig. 3.19. The plot of the scaled internuclear potential versus the scaled internuclear distance for Z = 2,
Z' = 4, ? = 0.3.
49
The screening increases the potential of the point of intersection of the two branches; the upper
branch, which has a very shallow minimum at ? = 0, loses it as ? increases.
A completely different behavior was observed for Z = 1. A small ? creates a deep minimum in
the upper branch of the potential. For comparison, we present the plots of the potential in the
case of Z = 1, b = 2 for ? = 0 and ? = 0.3.
Fig. 3.20. The plot of the scaled internuclear potential versus the scaled internuclear distance for Z = 1,
Z' = 2, ? = 0.
50
Fig. 3.21. The plot of the scaled internuclear potential versus the scaled internuclear distance for Z = 1,
Z' = 2, ? = 0.3.
The figures above reveal the case of the screening stabilization of the nuclear motion for the
case of Z = 1 and destabilization for Z > 1.
3.5. The effect of the plasma screening on the continuum lowering
Our analysis of the stability of the electronic motion shows results similar to those obtained
previously in [1.18, 1.19]. Namely, term 3 corresponds to a stable motion while term 4 ? to an
unstable motion. So, the crossing point of terms 3 and 4 corresponds to the transition from the
stable motion to the unstable motion, leading the electron to the zero energy (i.e., to the free
motion) along term 4, which constitutes the ionization of the molecule.
51
Therefore, we arrive at the following. For the ionization of the hydrogenlike ion of the
nuclear charge Zmin perturbed by the charge Zmax, it is sufficient to reach the scaled energy
?c(b) = ?(wV34(b), b) < 0. At that point, the electron switches to the unstable motion and the
radius of its orbit increases without a limit. This constitutes CL by the amount of
Z<1/R> | ?(wV34(b), b) |, where <1/R> is the value of the inverse distance of the nearest
neighbor ion from the radiating ion averaged over the ensemble of perturbing ions.
Thus, obtaining CL in the ionization channel requires calculations of the scaled energy ? at the
crossing point wV34 of terms 3 and 4.
CL for the "default" (? = 0) TCC system was studied in [1.21]. Particularly, the scaled CL
energy ?c(b) = ?(wV34(b), b) = ?E/(Z<1/R>) was graphed on a double-logarithmic scale, where ?
is defined in (1.3) and wV34 is given by (3.16). The graph is given below; "lg x" stands for
"log10 x".
Fig. 3.22. The plot of the CL energy versus b on a double logarithmic scale for ? = 0.
We have made several plots of ?c(b) for several values of ?. A numerical value for wV34 was
taken to increase precision.
52
Fig. 3.23. The plot of the CL energy versus b on a double logarithmic scale for ? = 0.1.
Fig. 3.24. The plot of the CL energy versus b on a double logarithmic scale for ? = 0.5.
Fig. 3.25. The plot of the CL energy versus b on a double logarithmic scale for ? = 2.
53
From the figures above we can see that the plasma screening decreases the value of CL in the
ionization channel. Also, starting from about ? = 1.7, we observe the "cutoff" value of b > 1,
below which ?c becomes negative, i.e., the electron energy at wV34 becomes positive. This means
that there is no more CL in this ionization channel ? instead, the continuum becomes higher than
for the isolated hydrogen-like ion of the nuclear charge Z. This effect cannot be observed in the
logarithmic graphs above because the cutoff value of energy (zero) corresponds to lg ?c = ??.
Below we made the standard, non-logarithmic plots of ?c(b) for selected values of ? at which this
effect is observed.
Fig. 3.26. The plot of the CL energy versus b for ? = 2.
54
Fig. 3.27. The plot of the CL energy versus b for ? = 3.
In Fig. 3.27 we can see that there is no CL for b = 2 and b = 3 at ? = 3.
In Appendices B and C we present the effects of the electric and magnetic fields on CL. The
effect of the magnetic field was observed to decrease the value of CL, similar to the case above,
while the electric field increases the value of CL, promoting ionization.
3.6. Conclusions
We studied the effects of the plasma screening on the classical energy terms of the electron in the
field of two Coulomb centers. We provided analytical results for the small values of the
screening factor and numerical results for the medium values.
We found that the plasma screening leads to the appearance of the fourth energy term ? in
addition to the three classical energy terms with no screening. This term exhibits a V-type
crossing with the lowest energy term. The two highest energy terms continue having a V-type
55
crossing like at the zero field.
We studied the effect of the screening on the internuclear potential. We found that the nuclear
motion was stabilized by screening for Z = 1 and destabilized for Z > 1.
The effect of the screening on the continuum lowering was studied as well. The plasma
screening decreases the value of CL in the ionization channel, similar to the effect of the
magnetic field [1.20].
Appendix A. The analytical expression for the limit w1 in Eq. (3.11) in the small-?
approximation.
56
57
58
Appendix B. The effect of the electric field on continuum lowering
Using the value of the scaled energy of the electron in the TCC system given in (2.5) with the
substitution of the numeric or approximate solution for p from (2.10) into (2.5) and (2.7) and the
further substitution of ? from (2.7) into (2.5), we obtain the dependence of the scaled energy on
the scaled coordinate w in the situation considered in Chapter 2, where the electric field was
parallel to the internuclear axis. Then we numerically find the point on the w-axis corresponding
to the V34 crossing for a given value of the scaled electric field f and substitute it into the
59
formula for the scaled energy, obtaining the critical energy, which is the value of CL.
Below are a few logarithmic plots (lg ?c versus lg b) made for selected values of f.
Fig. 3.B.1. The plot of the CL energy versus b for f = 0.1.
Fig. 3.B.2. The plot of the CL energy versus b for f = 1.
60
Fig. 3.B.3. The plot of the CL energy versus b for f = 10.
CL increases as the electric field increases. This is expected because the electric field promotes
ionization.
Appendix C. The effect of the magnetic field on continuum lowering
In the case of the magnetic field B parallel to the internuclear axis, the default energy given in
(1.2) will acquire an additional term
?L??
2?2
2 (3.C.1)
where ? = B/(2c) is the Larmor frequency. We apply the same method as we used in the
beginning of each chapter to find the energy dependent on one spatial parameter. Substituting the
scaled quantities as defined in (3.4) and defining ? = ?L3/Z2, taking the derivatives by w and p
and setting them equal to zero, solving for p and ?, substituting them back into the formula for
61
energy, and further substituting the parameter ? as given in (1.10), we arrive at the following
expressions of the scaled energy ? = ?ER/Z: and the scaled internuclear distance r:
?=??
4?2??b2/3?2?3?1?????3?1??b2/3?2?1?
2 ???3?1?3/2 ?
? ?
2?b2/3??4?
??3?1?2?b2/3 ?2?1???????
2???
3?1?5/2?b2/3?2?1?3/2
?3??3?1?3/2 ?
(3.C.2)
r= ??
3?1?4?b2/3?2?1?2
?4?b2/3??4?2??2???
3?1?5/2 ?b2/3?2?1?3/2
?3??3?1?3/2 ?
(3.C.3)
To find the point of the V34 crossing, we take the derivative of r by ? and set it equal to zero.
The numerical solution for this equation determines the value of ? corresponding to the minimum
of r(?) for given b and ?, which corresponds to the crossing. Substituting it to the expression for
the energy in (3.C.2), we obtain ?c(b, ?) ? the dependence of CL on b for a given ?.
Below we present several double-logarithmic plots, similar to those in Section 3.5 and
Appendix B, for selected values of ?.
Fig. 3.C.1. The plot of the CL energy versus b on a double-logarithmic scale for ? = 0.5.
62
Fig. 3.C.2. The plot of the CL energy versus b on a double-logarithmic scale for ? = 1.
Fig. 3.C.3. The plot of the CL energy versus b on a double-logarithmic scale for ? = 2.
From the graphs in Figs. 3.C.1 ? 3.C.3 it is seen that the effect of the magnetic field on CL is
similar to the effect of the plasma screening ? it decreases CL. The "cutoff" values of b, below
which there is no CL for a given ?, are also observed as ? becomes large. For example, for
? = 2.8 CL at b = 2 vanishes, so the values of b corresponding to CL start at b > 2. At ? = 4.3,
CL starts at b > 3. (See Figs. 3.C.4 and 3.C.5 below.)
63
Fig. 3.C.4. The plot of the CL energy versus b for ? = 2.8.
Fig. 3.C.5. The plot of the CL energy versus b for ? = 4.3.
64
Chapter 4. Helical and Circular States of Diatomic Rydberg Quasimolecules in a Laser
Field
4.1. Introduction
Circular states of atomic and molecular systems in general, as well as circular Rydberg states
(CRS) in particular, have been extensively studied both theoretically and experimentally for
several reasons (see, e.g., [1.18 ? 1.20, 1.24 ? 1.37] and references therein). Namely: (a) they
have long radiative lifetimes and highly anisotropic collision cross sections, thereby enabling
experiments on inhibited spontaneous emission and cold Rydberg gases, (b) these classical states
correspond to quantal coherent states, objects of fundamental importance, (c) a classical
description of these states is the primary term in the quantal method based on the 1/n-expansion,
and (d) they can be used in developing atom chips.
In the previous works [1.18 ? 1.20, 1.24, 1.25, 1.28, 1.34] and the previous chapters, analytical
studies of circular Rydberg states of two-Coulomb-center systems consisting of two nuclei of
charges Z and Z', separated by a distance R, and one electron were carried out. Energy terms of
these Rydberg quasimolecules for a field-free case [1.18, 1.19] were obtained, as well as under a
static electric field ([1.24] and Chapter 2) or under a static magnetic field [1.20], and crossings of
the energy terms were studied ? the crossings that enhance charge exchange in these systems.
The analysis was not confined to circular orbits of the electron. For example, paper [1.19]
studied in detail helical Rydberg states of these Rydberg quasimolecules. In order to make those
results more transparent, we briefly outline here the scheme of that analysis. In cylindrical
65
coordinates (z, ?, ?) with the z-axis along the internuclear axis, using the axial symmetry of the
problem, the z- and ?-motions can be separated from the ?-motion. The ?-motion can be then
determined from the calculated ?-motion. Equilibrium points of the two-dimensional motion in
the z?-space were studied and a condition distinguishing between two physically different cases,
where the effective potential energy either has a two-dimensional minimum in the z?-space or
has a saddle point in the z?-space, was explicitly derived. In particular, it turned out that the
boundary between these two cases corresponds to the point of crossing of the upper and middle
energy terms (out of the three energy terms in this system). For the stable motion, the trajectory
was found to be a helix on the surface of a cone, with axis coinciding with the internuclear axis.
In this helical state, the electron, while spiraling on the surface of the cone, oscillates between
two end-circles which result from cutting the cone by two parallel planes perpendicular to its
axis (Fig. 1.1).
In the present chapter we study such Rydberg quasimolecules under a laser field. For the
situation where the laser field is linearly-polarized along the internuclear axis, we found an
analytical solution for the stable helical motion of the electron valid for wide ranges of the laser
field strength and frequency. We also found resonances, corresponding to a laser-induced
unstable motion of the electron, that result in the destruction of the helical states. For the
situation where such Rydberg quasimolecules are under a circularly-polarized field, polarization
plane being perpendicular to the internuclear axis, we found an analytical solution for circular
Rydberg states valid for wide ranges of the laser field strength and frequency. We showed that
both under the linearly-polarized laser field and under the circularly-polarized laser field, in the
electron radiation spectrum in the addition to the primary spectral component at (or near) the
66
unperturbed revolution frequency of the electron, there appear satellites. We found that for the
case of the linearly-polarized laser field, the intensities of the satellites are proportional to the
squares of the Bessel functions Jq2(s), (q = 1, 2, 3, ? ), where s is proportional to the laser field
strength. As for the case of the circularly-polarized field, we demonstrated that there is a red shift
of the primary spectral component ? the shift linearly proportional to the laser field strength.
4.2. Analytical solution for the case of a linearly-polarized laser field
We consider the case where the laser is polarized parallel to the internuclear axis and oscillates
sinusoidally with the frequency ?. The angular momentum L is conserved here due to ?-
symmetry. The corresponding Hamiltonian is
H= p?
2?p
?
2
2 ?
L2
2?2?
Z
??2?z2?
Z'
??2??R?z?2 ?zFcos ?t
(4.1)
Below we scale all frequencies using the factor (R3/Z)1/2 : for example, the scaled laser frequency
is ? = ?(R3/Z)1/2 . We also use scaled coordinates as in papers [1.18, 1.19] and the other chapters
w= zR, v= ?R (4.2)
where R is the internuclear distance. The origin is at the location of charge Z.
Without the electric field, in the vicinity of the equilibrium the motion in z?-space corresponds
to a two-dimensional harmonic oscillator [1.19]. Its scaled eigen-frequencies are
67
??= 1?w2?v2?3/4 ? 11?w? 3w?
?w2?v2???1?w?2?v2?
(4.3)
where the equilibrium value of v connected to w as follows ? as in (1.5) and in [1.18, 1.19]:
v?w,b?=?w
2/3?1?w?4/3?b2/3w2
b2/3?w2/3?1?w??2/3 (4.4)
The motion occurs on the axes (w', v'), which are the original axes (w, v) rotated by an angle ?
given in [1.19]. The dependence of the angle ? on the scaled coordinate w can be expressed in
the most compact form by introducing the notation as in (1.15):
?=? 1w?1?
1/3
(4.5)
In the ?-representation it has the form
?=12 arctg ??b
2/3 ?2?1???4?b2/3?
??b2/3??? (4.6)
The scaled eigen-frequencies ?? and ?+ are the scaled frequencies of small oscillations about the
equilibrium along the coordinates w', v' accordingly.
As we introduce the oscillating electric field, these oscillations become forced, with the forces
F cos ? cos ?t on w' and F sin ? cos ?t on v'. Therefore, the deviations from equilibrium on
(w', v') are (see, e.g., textbooks [4.1, 4.2])
68
?w'= f cos??
?
2??2 cos??, ?v'=
f sin?
??2??2 cos?? (4.7)
where ? = ?(R3/Z)1/2 and ? = t(Z/R3)1/2. Now we revert to the original coordinates (w, v) and
obtain the equations of motion in the linearly-polarized oscillatory electric field in the vicinity of
the equilibrium: the electron follows the circular path corresponding to the case with no electric
field with deviations from equilibrium depending on the scaled time ?:
?w=f ? cos
2?
??2??2?
sin2?
??2??2?cos??,
?v= f sin?cos?? 1?
?
2??2?
1
??2??2?cos ??
(4.8)
From (4.8) it is seen that the strength and frequency of the laser field affect the amplitudes of
the forced oscillations on w- and v-axes; in fact, these amplitudes are proportional to the field
strength f. The frequencies of the forced oscillations on the axes are equal to that of the laser
field, instead of ?? and ?+.
Since the Hamiltonian from (4.1) does not depend on ?, the corresponding momentum is
conserved:
p?=?2 d?dt ?L=const (4.9)
We can re-write (4.9) in the scaled notation as
d?
d?=
?
v2??? (4.10)
69
where ? = L/(ZR)1/2 is the scaled angular momentum. Substituting in (4.10) v(?) = v0 + ?v(?),
where v0(w) is the equilibrium value of the scaled radius v of the electron orbit from (4.4) and
?v(?) is given by (4.8), we obtain
d?
dt ?
?
v02 ?
2?
v03 ?v??? (4.11)
which after the integration with respect to time yields:
??t?? ?v
0
2 ??
2?
?v03 f sin?cos??
1
??2??2?
1
??2??2 ?sin?? (4.12)
From (4.12) it is seen that the ?-motion is a rotation about the internuclear axis with the scaled
frequency ?/v02, slightly modulated by oscillations of the scaled radius of the orbit v at the laser
frequency ? (i.e., at the laser frequency ? in the usual notation).
Thus, from (4.8) and (4.12) it is clear that the electron is bound to a conical surface which
incorporates the original circular orbit. In Fig. 4.1 below the three-dimensional actual trajectory
is plotted for b = 3, f = 1, ? = 1 at w = 0.2.
70
Fig. 4.1. The actual trajectory of the electron in the linearly-polarized laser
field for b = 3, f = 1, ? = 1 at w = 0.2. The z-axis is along the internuclear axis.
The expression for ?(?) from (4.12), i.e., ?(t(Z/R3)1/2), enters the following Fourier-transform
that determines the amplitude of the power spectrum of the electron radiation
Al???= 1??
0
?
dtcos??t???t? ZR3 ?? (4.13)
where ? is the radiation frequency measured, e.g., by a spectrometer. The sinusoidal modulation
of the phase ? is analogous to the situation where hydrogen spectral lines are modified by an
external monochromatic field at the frequency ?, the latter problem being solved analytically by
Blochinzew as early as in 1933 [4.3] (a further study can be found, e.g., in book [4.4]).
From Blochinzew?s results it follows for our case in the electron radiation spectrum, this
71
helical motion should manifest as follows. The most intense emission would be at the frequency
? = d?/dt of the rapid ?-motion. In addition, there will be satellites at the frequencies ? ? q?,
where q = 1, 2, 3, ? , whose relative intensities Iq are controlled by the Bessel functions Jq(s):
Iq=Jq2 ?s?, s= 2??v
0
3 f sin?cos??
1
??2??2?
1
??2??2 ? (4.14)
The oscillatory motion of the electron in the z?-space with the laser frequency ? should lead
also to the radiation at this frequency. However, since ? << ?, this spectral component would be
far away from the primary spectral line and its satellites.
From (4.8) it is also seen that there are resonances when the laser frequency is equal to one of
the eigen-frequencies of the motion in the z?-space, i.e., when either ? = ?+ or ? = ?? . It turns
out that these conditions yield three resonance points on the w-axis for the laser field frequency ?
below a certain critical value ?c, or five resonance points for ? > ?c ? see the figures below.
72
Fig. 4.2. Eigen-frequencies of the motion in the z?-space ?+ (solid curves) and ?? (dashed curves) versus
w, i.e., versus the scaled z-coordinate of original circular Rydberg state. The scaled laser frequency ? is
shown by the horizontal straight line. The plot is for b = 3 and ? = 4. Three resonant points are seen.
Fig. 4.3. Same as in Fig. 4.2, but for b = 3 and ? = 9. Five resonant points are seen.
73
For instance, in the case of b = 3, for ? = 8, we observe resonances at the following five values of
w: 0.02883, 0.1106, 0.2497, 0.9852, 0.9878. The critical value corresponds to the minimum of
?+ (w) for a given b in the interval 0 < w < w1 at the equilibrium point (the equilibrium scaled
coordinate v being expressed via w by (4.4)). Calculating the derivative of ?+ with respect to w
and setting it equal to zero, we find the point of the minimum. The value of ?+ at this point will
be equal to the critical value of the scaled laser frequency ?c. For example, for b = 3 at
w = 0.17642 (the minimum of ?+ in Figs. 4.2 and 4.3) this critical value is ?c = 7.5944. As the
ratio of nuclear charges b increases, so does also the critical value ?c of the scaled laser
frequency.
These resonances correspond to a laser-induced unstable motion of the electron that result in
the destruction of the helical states. This is illustrated in Fig. 4.4 showing the three-dimensional
actual trajectory of the electron (for various directions of its initial velocity) for a resonance case
where b = 3, f = 1, ? = 8, and w = 0.111 (w = 0.111 is one of the three values of w, at which the
laser frequency ? coincides with the eigen-frequency ?+). A striking difference is seen compared
to the stable helical motion depicted in Fig. 4.1: the resonance destroyed the helical state.
74
Fig. 4.4. The actual trajectory of the electron (for various directions of its initial velocity) in the linearly-
polarized laser field for a resonance case where b = 3, f = 1, ? = 8, and w = 0.111. The z-axis is along the
internuclear axis.
4.3. Analytical solution for the case of a circularly-polarized laser field
Now we consider the case of a circular polarization of the laser field, polarization plane being
perpendicular to the internuclear axis. The laser field varies as
75
F=F?ex cos ?t?ey sin ?t? (4.15)
where ex and ey are the unit vectors along the x- and y-axes, F is the amplitude and ? is the
frequency. The Hamiltonian for the electron in this configuration will take the following form.
H=12 ? p?2?pz2? p?
2
?2 ??
Z
??2?z2 ?
Z'
??2??R?z?2?F?cos????0?
(4.16)
where we introduced ?0 = ?t. As in [1.19], we consider ?-motion to be the rapid subsystem, i.e.
d?/dt is much greater than the laser frequency ? and the frequencies of z- and ?-motion. The
canonical equations for the ?-motion obtained from (4.16) are
d?
dt =
?H
? p?=
p?
?2
(4.17)
d p?
dt =?
?H
?? =F ?sin????0?
(4.18)
Combining (4.17) and (4.18), we get
d2?
dt2 =
F
? sin????0? (4.19)
After a substitution ? ? ?0 = ? + ?, (4.19) becomes
d2?
dt2 =?
F
? sin? (4.20)
76
which is the equation of motion of a mathematical pendulum of length ? in gravity F. Its two
possible modes are libration and rotation; since ? is the rapid coordinate, we have the case of
rotation. The solution for ?(t) is well-known and can be expressed in terms of Jacobi amplitude:
??t?=2am??t2 , 4F??2? (4.21)
Here we denoted d?/dt at t = 0 as ?. For rapid rotations, the change in the angular speed on ? is
insignificant compared to the initial speed and d?/dt ? ?.
The expression for ?(t) enters the following Fourier-transform that determines the amplitude of
the power spectrum of the electron radiation:
Ac??, 4F??2?= 1??
0
?
dt cos??t???t , 4F??2 ?? (4.22)
Figure 4.5 shows as an example the power spectrum of the electron radiation spectrum (i.e., Ac2)
versus the dimensionless radiation frequency ?/? for the case where 4F/(??) = 0.1. It is seen
that the most intense component in the spectrum is at the frequency ? approximately equal to, but
slightly less than ?. It is also seen that the laser modulation of the primary frequency of the
electron rotation results in a series of relatively small satellites of the primary spectral
component.
77
Fig. 4.5. The power spectrum of the electron radiation P (in arbitrary units) versus the dimensionless
radiation frequency ?/? for the case where 4F/(??) = 0.1. Here ? is the frequency of the electron
radiation at the absence of the laser field. A certain width is assigned to all spectral components to display
a continuous spectral line profile.
The red shift of the primary spectral component can be calculated analytically as follows. Since
?-motion is rapid, we can average the Hamiltonian in (4.16) with respect to time. Integrating
(4.20) with the initial condition d?/dt = ?, we get
?2??d?dt ?
2
=4F? sin2 ?2 (4.23)
By averaging this equation with respect to time, we obtain
78
?2???d?dt ?
2?
=2F? (4.24)
Thus, the ?-momentum term in the Hamiltonian (4.16) becomes
? p?2?2 ?=?2 ??d?dt ?2?=?2?2?1? 2F??2? (4.25)
The last term in the Hamiltonian from (4.16) vanishes after the time averaging so that the time-
averaged Hamiltonian depends only on ?- and z-coordinates and their corresponding momenta.
The result is the following quasi-stationary Hamiltonian with no explicit time dependence:
H=12 ? p?2?pz2?? Z?
?2?z2
? Z'?
?2??R?z?2
?12 ?2 ?2??F (4.26)
Introducing the scaled quantities
w= zR, v= ?R, f =FR
2
Z , ?=??
R3
Z
(4.27)
and using the Hamiltonian equations, we obtain the following two differential equations of
motion:
? ?w= w?w2?v2?3/2? b?1?w???1?w?2?v2?3/2 (4.28)
??v=v? 1?w2?v2?3/2? b??1?w?2?v2?3/2??2??f (4.29)
79
(the dot above the letter indicates the differentiation by the scaled time ? = t(Z/R3)1/2).
In this section we consider these Rydberg quasimolecules in circular (not helical) states, so that
the plane of the electron orbit has a stationary position on the internuclear axis. Therefore, the
right-hand side of (4.28) vanishes and the relationship between w and v becomes the same as
given by (4.4). This makes the scaled radius of the orbit v a constant as well.
Since the angular momentum is L = ??2 for a stationary circular orbit, the averaging of the ?-
momentum in (4.25) is equivalent to changing L for L(1 ? F?3/L2). Using scaled units and the
relationship L = ??2, we find out that the case of the circularly-polarized laser field is equivalent
to a field-free case, but with an effective frequency ? given by the substitution:
????1????? f ? (4.30)
where
????= ?
6??3?1?3/2??4?b2/3?3/2
??3?1?11/2?b2/3 ?2?1?3 (4.31)
The quantity ? (?)? f is the red shift of the primary spectral component. This result is valid as
long as the relative correction (?)? f to the unperturbed angular frequency ? of the electron
remains relatively small. Figures 4.6 and 4.7 illustrate the situation for the case where the ratio of
the nuclear charges is b = 2. On the horizontal axis is the scaled coordinate w, i.e., the scaled
coordinate along the internuclear axis of the Rydberg quasimolecule. The solid curve, having two
branches, shows the unperturbed angular frequency ? of the electron. The dashed curve shows
80
the correction ? (?)? f . It is seen that the correction remains relatively small for the entire left
branch of ? and for a significant part of the right branch of ?. (Figures 4.6 and 4.7 differ only by
the range of the vertical scale, so that Fig. 4.6 allows to see more clearly the region where the
solid and dashed curves intersect and the region of validity of the results for the right branch of
?.) Physically, the left branch corresponds to the situation where the electron is primarily bound
by the charge Z. The region of the right branch, where the correction is relatively small,
physically corresponds to the situation where the electron is primarily bound by the charge Z'.
Fig. 4.6. Dependence of the unperturbed angular frequency ? of the electron (solid curve, two branches)
and of the correction ? (?)? f for f = 1 (dashed curve) on the scaled coordinate w along the internuclear
axis of the Rydberg quasimolecule.
81
Fig. 4.7. The same as in Fig. 4.6, but with better visible details in the region of the right branch of ?(w).
4.4. Conclusions
While studying diatomic Rydberg quasimolecules under a laser field that is linearly-polarized
along the internuclear axis, we found an analytical solution for the stable helical motion of the
electron valid for wide ranges of the laser field strength and frequency. Namely, the linearly-
polarized laser field makes the motion in the z?-space to be forced oscillations at the frequency
of the laser field. We also found resonances, corresponding to a laser-induced unstable motion of
the electron, that result in the destruction of the helical states. For the situation where such
Rydberg quasimolecules are under a circularly-polarized field, polarization plane being
perpendicular to the internuclear axis, we found an analytical solution for circular Rydberg states
valid for wide ranges of the laser field strength and frequency.
We showed that both under the linearly-polarized laser field and under the circularly-polarized
laser field, in the electron radiation spectrum in the addition to the primary spectral component at
82
(or near) the unperturbed revolution frequency of the electron, there appear satellites. We found
that for the case of the linearly-polarized laser field, the intensities of the satellites are
proportional to the squares of the Bessel functions Jq2(s), (q = 1, 2, 3, ? ), where s is
proportional to the laser field strength. As for the case of the circularly-polarized field, we
demonstrated that there is a red shift of the primary spectral component ? the shift linearly
proportional to the laser field strength.
Under a laser field of a known strength, in the case of the linear polarization the observation of
the satellites would be the confirmation of the helical electronic motion in the Rydberg
quasimolecule, while in the case of the circular polarization the observation of the red shift of the
primary spectral component would be the confirmation of the specific type of the phase
modulation of the electronic motion described by (4.21). Conversely, if the laser field strength is
unknown, both the relative intensities of the satellites and the red shift of the primary spectral
component could be used for measuring the laser field strength.
83
Chapter 5. Attachment of an Electron by Muonic Hydrogen Atoms: Circular States
5.1. Introduction
Studies of muonic atoms and molecules, where one of the electrons is substituted by the heavier
lepton ??, have several applications. The first one is muon-catalyzed fusion (see, e.g., [5.1 ? 5.3]
and references therein). When a muon replaces the electron either in the dde-molecule (D2+),
which becomes the dd?-molecule, or in the dte-molecule, which becomes the dt?-molecule, the
equilibrium internuclear distance becomes about 200 times smaller. At such small internuclear
distances, the fusion can occur with a significant probability, which has been observed in dd? or
even with a higher rate in dt? [5.1 ? 5.3]. The second application is a laser-control of nuclear
processes. This has been discussed in the context of the interaction of muonic molecules with
superintense laser fields [5.4]. Another application is a search for strongly interacting massive
particles (SIMPs) proposed as dark matter candidates and as candidates for the lightest
supersymmetric particle (see, e.g., [5.5] and references therein). SIMPs could bind to the nuclei
of atoms, and would manifest themselves as anomalously heavy isotopes of known elements. By
greatly increasing the nuclear mass, the presence of a SIMP in the nucleus effectively eliminates
the well-known reduced mass correction in a hydrogenic atom. Muonic atoms are better
candidates (than electronic atoms) for observing this effect because the muon's much larger mass
(compared to the electron) amplifies the reduced mass correction [5.5]. This may be detectable in
astrophysical objects [5.5].
84
Another line of research is studies of the negative ion of hydrogen H?, which can be also
denoted as epe-system (electron-proton-electron), constitute an important line of research in
atomic physics and astrophysics. It has only one bound state ? the ground state having a
relatively small bound energy of approximately 0.75 eV. This epe-system exhibits rich physics.
Correlations between the two electrons are strong already in the ground state. With long-range
Coulomb interactions between all three pairs of particles, the dynamics is particularly subtle in a
range of energies 2 ? 3 eV on either side of the threshold for break-up into proton + electron +
electron at infinity [5.6]. There are strong correlations in energy, angle, and spin degrees of
freedom, so that perturbation theory and other similar methods fail [5.6]. Experimental studies of
H? provided a testing ground for the theory of correlated multielectron systems. Compared to the
helium atom, the structure of H? is even more strongly influenced by interelectron repulsion
because the nuclear attraction is smaller for this system [5.7]. In addition to the above
fundamental importance, the rich physics of H? is also important in studies of the ionosphere's D-
layer of the Earth atmosphere, the atmosphere of the Sun and other stars, and in development of
particle accelerators [5.6].
In this chapter we combine the above two lines of research: studies of muonic atoms/molecules
and studies of negative hydrogen ion. Namely, we consider whether a muonic hydrogen atom can
attach an electron and become a muonic negative hydrogen ion, i.e. ?pe-system. Specifically, we
study a possibility of circular states in such system. We show that the muonic motion can
represent a rapid subsystem, while the electronic motion can represent a slow subsystem ? the
result that might seem counterintuitive.
First, we find analytically classical energy terms for the rapid subsystem at the frozen slow
85
subsystem, i.e., for the quasimolecule where the muon rotates around the axis connecting the
immobile proton and the immobile electron. The meaning of classical energy terms is explained
below. We demonstrate that the muonic motion is stable. We also conduct the analytical
relativistic treatment of the muonic motion.
Then we unfreeze the slow subsystem and analyse a slow revolution of the axis connecting the
proton and electron. We derive the condition required for the validity of the separation into the
rapid and slow subsystems.
Finally we show that the spectral lines, emitted by the muon in the quasimolecule ?pe,
experience a red shift compared to the corresponding spectral lines that would have been emitted
by the muon in a muonic hydrogen atom (in the ?p-subsystem). Observing this red shift should
be one of the ways to detect the formation of such muonic negative hydrogen ions.
As for physical processes leading to the formation of muonic-electronic negative hydrogen
ions, one of the processes could be the following:
e + ?p ? ?pe
(which sometimes might be followed by the decay ?pe ? ? + pe). Such formation of the ?pe-
systems was discussed, e.g., in paper [5.8], where these systems were called resonances. The
theoretical approach based on the separation of rapid and slow subsystems requires in this case
the muon to be in a state of a high angular momentum. Luckily, the experimental methods to
create muonic hydrogen atoms ?p (necessary for the above reaction) lead to the muon being in a
highly-excited state (see, e.g., review [5.9] and paper [5.10]). We also mention paper [5.11]
where it has been shown, in particular, that the distribution of the muon principal quantum
number in muonic hydrogen atoms peaks at larger and larger values with the increase of the
86
energy of the muon incident on electronic hydrogen atoms.
5.2. Analytical solution for classical energy terms of the rapid subsystem
We consider a quasimolecule where a muon rotates in a circle perpendicular to and centered at
the axis connecting a proton and an electron ? see Fig. 5.1. As we show below, in this
configuration the muon may be considered the rapid subsystem while the proton and electron
will be the slow subsystem, which essentially reduces the problem to the two stationary Coulomb
center problem, where the effective stationary ?nuclei? will be the proton and electron. The
straight line connecting the proton and electron will be called here ?internuclear? axis. We use
the atomic units in this study.
Fig. 5.1. A muon rotating in a circle perpendicular to and centered at the axis connecting the proton and
the electron.
Because of the difference of muon and electron masses, the muon-proton separation is much
87
smaller than the electron-proton separation. Therefore, it should be expected that the spectral
lines, emitted by this system, would be relatively close to the spectral lines emitted by muonic
hydrogen atoms. In other words, the presence of the electron should result in a relatively small
shift of the spectral lines (compared to muonic hydrogen atoms); however, this shift would be an
important manifestation of the formation of the quasimolecule ?pe.
A detailed classical analytical solution of the two stationary Coulomb center problem, where an
electron revolves around nuclei of charges Z and Z', has been presented in papers [1.18, 1.19].
We base our results in part on the results obtained therein.
The Hamiltonian of the rotating muon is
H= 12m? pz2?p?2? p?
2
?2 ??
Z
?z2??2?
Z'
??R?z?2??2
(5.1)
where m is the mass of the muon (in atomic units m = 206.7682746), Z and Z' are the charges of
the effective nuclei (in our case, Z = 1 and Z' = ?1), R is the distance between the effective
nuclei, (?, ?, z) are the cylindrical coordinates, in which Z is at the origin and Z' is ar z = R, and
(p?, p?, pz) are the corresponding momenta of the muon.
Since ? is a cyclic coordinate, the corresponding momentum is conserved:
?p??=const=L (5.2)
With this substituted into (5.1), we obtain the Hamiltonian for the z- and ?-motions
Hz?= pz
2?p
?
2
2 ?Ueff ?z,??
(5.3)
where an effective potential energy is
88
Ueff ?z,??= L
2
2m?2?
Z
?z2??2?
Z'
??R?z?2??2
(5.4)
Because in a circular state pz = p? = 0, the total energy E(z, ?) = Ueff(z, ?).
With Z = 1, Z' = ?1 and the scaled quantities
w= zR, v= ?R, ?=?ER, ?= L?mR, r=mRL2 (5.5)
we obtain the scaled energy ? of the muon:
?= 1?w2?v2? 1?
?1?w?2?v2
? ?
2
2v2
(5.6)
The equilibrium condition with respect to the scaled coordinate w is ??/?w = 0; the result can be
brought to the form:
??1?w?2?v2?3/2
?w2?v2?3/2 =
w?1
w
(5.7)
Since the left-hand side of (5.7) is positive, the right-hand side must also be positive: (w ? 1)/w >
0. Consequently, the allowed ranges of w here are ?? < w < 0 and 1 < w < +?. This means that
equilibrium positions of the center of the muon orbit could exist (judging only by the equilibrium
with respect to w) either beyond the proton or beyond the electron, but there are no equilibrium
positions between the proton and electron.
Solving (5.7) for v2 and denoting v2 = p, we obtain:
p?w?=w2/3?w?1?2/3?w2/3??w?1?2/3? (5.8)
The equilibrium condition with respect to the scaled coordinate v is ??/?v = 0, which yields
89
?2=p2? 1?w2?p?3/2? 1??1?w?2?p?3/2? (5.9)
Since the left-hand side of (5.9) is positive, the right-hand side must be also positive. This entails
the relation w2 + p < (1 ? w)2 + p, which simplifies to 2w ? 1 < 0, which requires w < 1/2.
Thus, the equilibrium with respect to both w and v is possible only in the range ?? < w < 0,
while in the second range, 1 < w < +? (derived from the equilibrium with respect to w only)
there is no equilibrium with respect to v.
From the last two relations in (5.5), we find r = 1/?2; thus
r=p?2? 1?w2?p?3/2 ? 1??1?w?2?p?3/2 ?
?1
(5.10)
where p is given by (5.8). Therefore, the quantity r in (5.10) is the scaled "internuclear" distance
dependent on the scaled internuclear coordinate w.
Now we substitute the value of ? from (5.9), as well as the value of p from (5.8) into (5.6),
obtaining ?(w) ? the scaled energy of the muon dependent on the scaled internuclear coordinate
w. Since E = ? ?/R and R = rL2/m, then E = ? (m/L2)?1 where ?1 = ?/r. The parametric dependence
?1(r) will yield the energy terms.
The form of the parametric dependence ?1(r) can be significantly simplified by introducing a
new parameter ? = (1 ? 1/w)1/3, as was shown in Chapter 1 starting from (1.10). The region
? ? < w < 0 corresponds to 1 < ? < ?. The parametric dependence will then have the following
form:
?1???= ?1???
4?1??2?2
2?1????2?2?1??2??4?
(5.11)
90
r???=?1??
2??4?3/2
??1??2?2
(5.12)
Classical energy terms given by the parametric dependence of the scaled energy ?1 = (L2/m)E on
the scaled internuclear distance r = (m/L2)R are presented in Fig. 5.2.
Fig. 5.2. Classical energy terms: the scaled energy ??1 = (L2/m)E versus the
scaled internuclear distance r = (m/L2)R.
Fig. 5.2 actually contains two coinciding energy terms: there is a double degeneracy with
respect to the sign of the projection of the muon angular momentum on the internuclear axis. We
remind the readers that L is the absolute value of this projection ? in accordance to its definition
in (5.2).
The minimum value of R, corresponding to the point where the term starts, can be found from
(5.12). The term starts at w = ? ?, which corresponds to ? = 1; taking the value of (5.12) at this
point, we find
91
Rmin=3
3/2
4
L2
m
(5.13)
With the value of m = 206.7682746, (5.13) yields R = 0.00628258 L2.
The following note might be useful once again. The plot in Fig. 5.1 represents two degenerate
classical energy terms of "the same symmetry". (In physics of diatomic molecules, the
terminology "energy terms of the same symmetry" means the energy terms of the same
projection of the angular momentum on the internuclear axis.) For a given R and L, the classical
energy E takes only one discrete value. However, as L varies over a continuous set of values, so
does the classical energy E (as it should be in classical physics).
The revolution frequency of the muon ? is
?= Lm?2 = LmR2v2 = LmR2 p (5.14)
in accordance with the previously introduced notation p = v2 = (?/R)2. Since R = L2r/m (see
(5.5)), then (5.14) becomes ? = (m /L3)f, where f = 1/(pr2). Using (5.12) for r(?) and (5.8) for
p(w) with the substitution w = 1/(1 ? ?3), where ? > 1, we finally obtain:
?= mL3 f ???, f ???=?1??
2?3?1??3?2
?1??2??4?3
(5.15)
where f(?) is the scaled muon revolution frequency. Fig. 5.3 shows the scaled muon revolution
frequency f = (L3/m)? versus the scaled internuclear distance r = (m/L2)R.
92
Fig. 5.3. The scaled muon revolution frequency f = (L3/m)? versus
the scaled internuclear distance r = (m/L2)R.
It is seen that for almost all values of the scaled internuclear distance r = (m/L2)R, the scaled
muon revolution frequency f = (L3/m)? is very close to its maximum value fmax = 1,
corresponding to large values of R. (The quantity fmax can be easily found from (15) given that
large values of R correspond to ? >> 1 and that this limit yields fmax = 1.) In other words, for
almost all values of R, the muon revolution frequency ? is very close to its maximum value
?max=mL3 (5.16)
In Sect. 5.3, we will compare the muon revolution frequency with the corresponding frequency
of the electronic motion and derive the condition of validity of the separation into rapid and slow
subsystems.
To analyse the stability of the muon motion, corresponding to the degenerate classical energy
terms, while considering a classical circular motion of a charged particle (which was the electron
93
in [1.19]) in the field of two stationary Coulomb centers, using the same notation as in this
chapter, it was shown [1.19] that the frequencies of small oscillations of the scaled coordinates w
and v of the circular orbit around its equilibrium position are given by
??= 1?w2?p?3/4 ? 11?w?3wQ (5.17)
where
Q=??w2?p???1?w?2?p? (5.18)
These oscillations are in the directions (w', v') obtained by rotating the (w, v) coordinates by the
angle ?:
?w'=?wcos???vsin?, ?v'=??wsin???vcos ? (5.19)
where the "?" symbol stands for the small deviation from equilibrium. The angle ? is determined
by the following relation:
?=12 arctg ?1?2w??pw?1?w??p (5.20)
The quantity Q in (5.18) is always positive since it contains the squares of the coordinates. From
(5.17) it is seen that the condition for both frequencies to be real is
1
1?w?
3w
Q (5.21)
For the frequency ?? to be real, (5.17) requires Q ? 3w(1 ? w). For any w < 0 (which is the
allowed range of w), this inequality is satisfied: the left-hand side is always positive while the
right-hand side is always negative.
94
For the frequency ?+ to be real, the following function F(w) must be positive (in accordance
with (5.17) and (5.18)):
F?w?=?w2?p???1?w?2?p??9w2?1?w?2 (5.22)
After replacing w by ? = (1 ? 1/w)1/3, (5.22) becomes
F???=?
2??2?1?2?1?4?2??4?
??3?1?4
(5.23)
Since the allowed range of w < 0 corresponds to ? > 1, it is seen that F(?) is always positive.
Thus, the corresponding classical energy terms corresponds to the stable motion.
5.3. Electronic motion and the validity of the scenario
Now we unfreeze the slow subsystem and analyse a slow revolution of the axis connecting the
proton and electron, the electron executing a circular orbit. In accordance with the concept of
separating rapid and slow subsystems, the rapid subsystem (the revolving muon) follows the
adiabatic evolution of the slow subsystem. This means that the slow subsystem can be treated as
a modified ?rigid rotator? consisting of the electron, the proton, and the ring, over which the
muon charge is uniformly distributed, all distances within the system being fixed (see Fig. 5.1).
The potential energy of the electron in atomic units (with the angular-momentum term) is
Ee= M
2
2R2?
1
R?
1
??2??R?z?2
(5.24)
where M is the electronic angular momentum. Its derivative by R must vanish at equilibrium,
95
which yields
dEe
dR =?
M2
R3 ?
1
R2?
R?z
??2??R?z?2?3/2=0
(5.25)
which gives us the value of the scaled angular momentum
?e= M?R (5.26)
corresponding to the equilibrium:
?e2=1? 1?w??1?w?2?p?3/2 (5.27)
where the scaled quantities w, p of the muon coordinates are defined in (5.5). Using the muon
equilibrium condition from (5.7) with v2 denoted as p, we can represent (5.27) in the form
?e2=1? w?w2?p?3/2 (5.28)
After replacing w by ? = (1 ? 1/w)1/3, we obtain
?e???=?1??1???
2?1????2
?1????2?3/2
(5.29)
The electron revolution frequency is ? = M/R2 = ?e(?)/R3/2 given that M = ?e(?)R1/2 in
accordance with (5.26). Since R = L2r(?)/m (see (5.5)) with r(?) given by (5.12), then from
? = ?e(?)/R3/2 we obtain
?=m
3/2?
e???
L3r3/2???
(5.30)
96
From (5.15) and (5.30) we find the following ratio of the muon and electron revolution
frequencies:
?
?=
1
?m
f ???r3/2???
?e???
(5.31)
where f(?) is given in (5.15).
In addition to the above relation R = L2r(?)/m, the same quantity R can be expressed from
(5.26) as R = M2/?e2(?). Equating the right-hand sides of these two expressions, we obtain the
equality L2r(?)/m = M2/?e2(?), from which it follows:
L
M=
?m
?e????r???
(5.32)
The combination of (5.31) and (5.32) represents an analytical dependence of the ratio of the
muon and electron revolution frequencies ?/? versus the ratio of the muon and electron angular
momenta L/M via the parameter ? as the latter varies from 1 to ?. This dependence is presented
in Fig. 5.4.
97
Fig. 5.4. The ratio of the muon and electron revolution frequencies
?/? versus the ratio of the muon and electron angular momenta L/M.
For the separation into the rapid and slow subsystems to be valid, the ratio of frequencies ?/?
should be significantly greater than unity. From Fig. 5.4 it is seen that this requires the ratio of
angular momenta L/M to be noticeably greater than 20.
98
There is another validity condition to be checked for this scenario. Namely, the revolution
frequency ? of the muon must also be much greater than the inverse lifetime of the muon 1/Tlife,
where Tlife = 2.2 ?s = 0.91 ? 1011 a.u.: ?Tlife >> 1. Since for almost all values of R, the muon
revolution frequency ? is very close to its maximum value ?max = m/L3, as shown in Sect. 5.2,
then the second validity condition can be estimated as (m/L3)Tlife >> 1, from which it follows
L?Lmax=?mTlife?1/3=26600 (5.33)
(we remind that m = 206.7682746 in atomic units). So, the second validity condition is fulfilled
for any practically feasible value of the muon angular momentum L.
Thus, for the ratio of angular momenta L/M noticeably greater than 20, we deal here with a
muonic quasimolecule where the muon rapidly rotates about the axis connecting the proton and
electron following a relatively slow rotation of this axis.
5.4. Red shift of spectral lines compared to muonic hydrogen atoms
The muon, rotating in a circular orbit at the frequency ?(R), should emit a spectral line at this
frequency. The maximum value ?max = m/L3 corresponds to the frequency of spectral lines
emitted by the muonic hydrogen atom (by the ?p-subsystem). For the equilibrium value of the
proton-electron separation ? just as for almost all values of R ? the frequency ? is slightly
smaller than ?max . Therefore, the spectral lines, emitted by the muon in the quasimolecule ?pe,
experience a red shift compared to the corresponding spectral lines that would have been emitted
by the muon in a muonic hydrogen atom. The relative red shift ? is defined as follows
99
?=???0?
0
=?max??? (5.34)
where ? and ?0 are the wavelength of the spectral lines for the quasimolecule ?pe and the muonic
hydrogen atom, respectively. Using (5.15), the relative red shift can be represented in the form
????= 1f ????1 (5.35)
where f(?) is given in (5.15).
The combination of (5.35) and (5.32) represents an analytical dependence of the relative red
shift ? on the ratio of the muon and electron angular momenta L/M via the parameter ? as the
latter varies from 1 to ?. Figure 5.5 presents the dependence of ? on L/(m1/2M). In this form the
dependence is ?universal?, i.e., valid for different values of the mass m: for example, it is valid
also for the quasimolecule ?pe where there is a pion instead of the muon. Figure 5.6 presents the
dependence of ? on L/M specifically for the quasimolecule ?pe.
100
Fig. 5.5. Universal dependence of the relative red shift ? of the spectral lines of the quasimolecule ?pe (or
?pe) on L/(m1/2M), which is the ratio of the muon and electron angular momenta L/M divided by the
square root of the mass m of the muon or pion.
101
Fig. 5.6. Dependence of the relative red shift ? of the spectral lines of the
quasimolecule ?pe on the ratio of the muon and electron angular momenta L/M.
It is seen that is the relative red shift of the spectral lines is well within the spectral resolution
??res/? of available spectrometers: ??res/? ~ (10?4 ? 10?5) as long as the ratio of the muon and
electron angular momenta L/M < 80. Thus, this red shift can be observed and this would be one
of the ways to detect the formation of such muonic negative hydrogen ions.
Figure 5.7 presents the dependence of the relative red shift ? on the ratio of the muon and
electron revolution frequencies ?/?. It is seen that the relative red shift decreases as the ratio of
the muon and electron revolution frequencies increases, but it remains well within the spectral
102
resolution ??res/? of available spectrometers.
Fig. 5.7. Dependence of the relative red shift ? on the ratio of
the muon and electron revolution frequencies ?/?.
5.5. Conclusions
We studied whether a muonic hydrogen atom can attach an electron, the muon and the electron
being in circular states. We showed that it is indeed possible for a muonic hydrogen atom to
attach an electron and to become a muonic negative hydrogen ion. We demonstrated that in this
103
case, the muonic motion can represent a rapid subsystem while the electronic motion can
represent a slow subsystem ? the result that might seem counterintuitive. In other words, the
muon rapidly revolves in a circular orbit about the axis connecting the proton and electron while
this axis slowly rotates following a relatively slow electronic motion.
We used a classical analytical description to find the energy terms of such a system, i.e.,
dependence of the energy of the muon on the distance between the proton and electron. We
found that there is a double-degenerate energy term. We demonstrated that it corresponds to a
stable motion. We also conducted the analytical relativistic treatment of the muonic motion,
which is presented in Appendix. It was found that the relativistic corrections are relatively small.
Their relative value is ~ 1/(cL)2 ~ 0.5 ? 10?4/L2 (we remind the readers that here c = 137.036 is
the speed of light in atomic units).
Then we unfroze the slow subsystem and analysed a slow revolution of the axis connecting the
proton and electron. The slow subsystem can be treated as a modified ?rigid rotator? consisting
of the electron, the proton, and the ring, over which the muon charge is uniformly distributed, all
distances within the system being fixed. We derived the condition required for the validity of the
separation into the rapid and slow subsystems.
Finally we showed that the spectral lines, emitted by the muon in the quasimolecule ?pe,
experience a red shift compared to the corresponding spectral lines that would have been emitted
by the muon in a muonic hydrogen atom (in the ?p-subsystem). The relative values of this red
shift, which is a ?molecular? effect, are significantly greater than the resolution of available
spectrometers and thus can be observed. Observing this red shift should be one of the ways to
detect the formation of such muonic negative hydrogen ions.
104
Appendix. Relativistic treatment of the muonic motion
The Hamiltonian of the rotating muon is
H=c?m2c2?pz2?p?2? p?
2
?2 ?
Z
?z2??2?
Z'
??R?z?2??2?mc
2 (5.A.1)
Since ? is a cyclic coordinate, the corresponding momentum is conserved:
?p??=const=L (5.A.2)
With this substituted into (5.A.1) and taking into account that in a circular state, pz = p? = 0, we
obtain the energy of the muon in a circular state
E=c?m2c2?L
2
?2?
Z
?z2??2?
Z'
??R?z?2??2 ?mc
2 (5.A.3)
With Z = 1, Z' = ?1 and the scaled quantities
w= zR, v= ?R, ?=?ER, ?= LmcR , r=RL (5.A.4)
we obtain the scaled energy ? of the muon:
105
?= 1?
w2?v2
? 1?
?1?w?2?v2
?mc2 R?1??1??
2
v2 ? (5.A.5)
The equilibrium condition with respect to the scaled coordinate w is ??/?w = 0, which yields
p?w?=w2/3?w?1?2/3?w2/3??w?1?2/3? (5.A.6)
where p = v2.
The equilibrium condition with respect to the scaled coordinate v is ??/?v = 0, which yields:
?2= p
2
mc2 R ?1?
?2
p ?
1
?w2?p?3/2?
1
??1?w?2?p?3/2? (5.A.7)
From the relation before last in (5.A.4), we find R = L/(mc?). Substituting this in (5.A.7), we can
solve it for ? and obtain:
?=?c
2 L2
p4
1
? 1?w2?p?3/2? 1??1?w?2?p?3/2 ?2 ?
1
p?
?1/2
(5.A.8)
From the last two relations in (5.A.4), we find r = 1/(mc?); thus
r= 1mc ?c
2 L2
p4
1
? 1?w2?p?3/2? 1??1?w?2?p?3/2?2?
1
p (5.A.9)
where p is given by (5.A.6). Therefore, the quantity r in (5.A.9) is the scaled "internuclear"
distance dependent on the scaled internuclear coordinate w for a given absolute value of the
angular momentum projection on the internuclear axis L.
Now we substitute R = L/(mc?) and the value of ? from (5.A.8), as well as the value of p from
106
(5.A.6) into (5.A.5), obtaining ?(w, L) ? the scaled energy of the muon dependent on the scaled
internuclear coordinate w for a given value of the angular momentum L. Since E = ? ?/R and
R = rL, then E = ? ?1/L where ?1 = ?/r. The parametric dependence E(R), where E = ? ?1/L and
R = Lr will yield the energy terms for a given value of L.
After introducing the parameter ? = (1 ? 1/w)1/3, the parametric dependence takes the following
form:
E??,L?=?mc2?1? 1?
?6???
?
??1????2
??3?? (5.A.10)
R??,L?=L
2
m
??6??
??1??2 (5.A.11)
where quantities ? and ? are defined as follows:
?=?1??
2??4
1??2 , ?=?
1??3
cL ?
2
(5.A.12)
The revolution frequency of the muon is
?=mc
2
L
?
?6?? ?1?
?
?6?? (5.A.13)
Let us check the degree of the relativity of the muon motion. Figure 5.A.1 shows the ratio ? of
the muon velocity to the speed of light versus the "internuclear" distance R for L = 1 and L = 3. It
is seen that for all values of R ? n2 (n = 1, 2, 3, ...), this ratio is practically equal to some constant
value ?max. It is easy to find that ?max = 1/(cL) = 1/(137.036 L) ? since c = 137.036 in a.u.
107
Fig. 5.A.1. The ratio ? of the muon velocity to the speed of light versus the ?internuclear?
distance R (a.u.) for L = 1 (the upper curve) and L = 3 (the lower curve).
It is interesting to compare the above ?max with the corresponding average value of ?e for the
electron motion in hydrogen atoms: ?e = 1/(cn). So, ?max for the muon motion differs from ?e for
the electron motion in hydrogen atoms only by the substitution of the principal quantum number
n of the electron by the angular momentum quantum number L of the muon.
Thus, even for L = 1 (for which ?max is the highest), the muon motion is only weakly-
relativistic. The relativistic correction to the average frequency of the muon radiation is ~ 1/(cL)2
(a.u.), where c = 137.036. Thus, the relative correction is insignificant even for L ~ 1 and it
rapidly diminishes as L grows: for example, it is ~ 10?5 for L = 3 and ~ 10?7 for L = 15.
108
Chapter 6. Super-generalized Runge-Lenz Vector
6.1. Introduction
A so-called Two Centers Problem (TCP) has the following two mathematically equivalent, but
physically different embodiments. The first one is the motion of an electron in the field of two
stationary Coulomb centers of charges Z and Z' separated by a distance R, which is one of the
most fundamental problems in quantum mechanics (see, e.g., [1.10, 1.11]). The second one is the
motion of a planet in the gravitational field of two stationary stars of generally different masses,
which is one of the most fundamental problems in celestial mechanics (see, e.g., [6.1, 6.2]). The
geometrical symmetry of the TCP dictates the conservation of the energy and of the projection of
the angular momentum on the internuclear (or interstellar) axis. It is also well-known that the
TCP possesses a higher than geometrical symmetry and that there should be an additional
integral of the motion. The existence of the additional integral of the motion is intimately
connected with the fact that the TCP allows the separation of variables in the elliptical
coordinates ? the fact shown as early as in 1760 by Euler [6.3] (see also [6.4, 6.5]).
In the limit of large R, the problem of two Coulomb centers reduces to the problem of a
hydrogenic ion of the nuclear charge Z in the uniform electric field F = Z'/R2, which is another
fundamental problem in quantum mechanics (the corresponding celestial problem reduces to the
problem of the one-center Kepler system in the uniform gravitational field). This simpler quantal
problem also possesses higher than geometrical symmetry (connected with the fact that this
109
problem allows the separation of variables in the parabolic coordinates). The corresponding
integral of the motion is known as a projection of a generalized Runge-Lenz vector on the
internuclear axis. The generalized Runge-Lenz vector A for this simpler problem, introduced by
Redmond in 1964 [6.6], and its projection Az on the axis Oz||F can be represented in the forms,
respectively
A=A0??r?F??r2 , Az=A0z??x
2?y2?F
2
(6.1)
(atomic units are used throughout the chapter). Here A0 is the well-known Runge-Lenz vector for
one isolated Coulomb center*:
A0=p?L?Z rr (6.2)
We note that the corresponding expressions in Redmond?s paper [6.6] differed from (6.1), (6.2)
by factor ? 1/Z.
After Redmond introduced the generalized Runge-Lenz vector for the asymptotic case of the
TCP, the challenge was to find out whether a super-generalization of the Runge-Lenz vector is
possible for the general (non-asymptotic) TCP. At least, two groups of authors claimed that they
accomplished this task. Namely, Krivchenkov and Liberman in 1968 [6.11] and Gurarie in 1992
[6.12] presented expressions for a super-generalized Runge-Lenz vector and claimed that its
* Vector A0 is also called Laplace-Runge-Lenz vector. It is interesting to note that none of these three
scientists was the first to introduce this vector. Historically, it was introduced as early as in 1710 by
Hermann [6.7, 6.8] and Bernoulli [6.9] and therefore is also called Hermann-Bernoulli vector or
Ermanno-Bernoulli vector (different just by the spelling of the first author?s name, who is the same
person in both cases) ? more details on this history can be found in [6.10].
110
projection on the internuclear (or interstellar) axis is conserved.
In the present chapter, first, we show that their claims are incorrect: the projection of their
super-generalized Runge-Lenz vector is not conserved (despite the fact that in the limit of large
R, their expressions reduce to Redmond?s result). Second, we derive a correct super-generalized
Runge-Lenz vector whose projection on the internuclear/interstellar axis does conserve. Third,
since in the literature there are several expressions for the separation constant for the TCP ? the
expressions not having the form of a projection of any vector on the internuclear/interstellar axis
? we provide relations between those expressions and our result.
6.2. Super-generalized Runge-Lenz vector
Krivchenkov and Liberman [6.11] considered the TCP described by the Hamiltonian (or the
Hamilton function):
H= p
2
2 ?
Z1
r1 ?
Z2
r2 ?
Z2
R
(6.3)
The charge Z1 was placed at the origin and Z2 at z = R. They presented the following expression
for a super-generalized Runge-Lenz vector:
A?KL?=p?L?Z1 r1r
1
?Z2 r2r
2
?Z2 (6.4)
In the limit of large R, it reduces to Redmond?s result from (6.1).
We calculated the Poisson bracket [A(KL), H] of the projection of this vector on the internuclear
axis with the Hamiltonian from [6.11]. Surprisingly, the result was not zero:
111
[A?KL?,H]=?2Z1r
1
3 r?L?ez?
2Z2
r23 R?x px?y py?
(6.5)
Thus, the projection of Krivchenkov-Liberman?s vector on the internuclear axis does not
conserve ? contrary to their claim.
Gurarie [6.12] considered the TCP described by the Hamilton function:
H= p
2
2 ?
Z1
r1 ?
Z2
r2
(6.6)
where Z1 and Z2 are placed into z = a and z = ?a, respectively (atomic units are also used here).
He presented the following expression for a super-generalized Runge-Lenz vector:
A?G?=p?L?Z1 r?aez?r?ae
z?
?Z2 r?aez?r?ae
z?
(6.7)
In the limit of large R, it reduces to Redmond?s result from (6.1).
We calculated the Poisson bracket [A(G), H] of the projection of this vector on the interstellar
axis with the Hamiltonian from [6.12]. Surprisingly, again the result was not zero:
[A?G?,H]=?2a?x px?y py?? Z1?r?ae
z?
3 ?
Z2
?r?aez?3 ?
(6.8)
Thus, the projection of Gurarie?s vector on the interstellar axis does not conserve ? contrary to
his claim.
To derive a correct expression for the super-generalized Runge-Lenz vector, we started from
the Hamiltonian (or Hamilton function) given by (6.3) and followed the first few steps from
Krivchenkov-Liberman?s paper [6.11], arriving at the same expression in the elliptical
112
coordinates as they did, for the additional conserved quantity:
A= 1R 1w2?v2 ??w2?1??1?v2??pw2 ?pv2???w
2?1
1?v2 ?
1?v2
w2?1? p?
2??Z 1?wv
w?v ?
?Z' ?1?v??w?1?w?v
(6.9)
Then, using the relation between the elliptical coordinates and the Cartesian coordinates
x=R2 ??w2?1??1?v2?cos?, y=R2 ??w2?1??1?v2?sin?, z=R2 wv (6.10)
we obtained from (6.9) the following: the super-generalized Runge-Lenz vector, whose
projection on the internuclear axis is really conserved, has the form
A=p?L?L
2
R ez?Z
r
r?Z'
R?r
?R?r??Z'ez, ez=
R
R
(6.11)
The Poisson bracket of the projection of the vector from (6.11) on the internuclear axis
Az=p?L?ez?L
2
R ?Z
z
r ?Z'
R?z
?R?r??Z'
(6.12)
with the Hamilton function from (6.3) vanishes indeed.
In the limit of large R, the expression from (6.12) reduces to the following asymptotic form
?Az?as=p?L?ez?L
2
R ?Z
z
r?Z'
x2?y2
2R2
(6.13)
Here the direction of the z-axis is chosen from the charge Z to the charge Z' . Compared to
Redmond?s result from (6.1), there is an extra term in (6.13): ?L2/R. At the first glance, this might
seem puzzling. However, the Poisson bracket of ?L2/R with the Hamilton function from (6.3)
yields a term of the order of 1/R3. Thus, the Poisson bracket of Aas from (6.13) with the Hamilton
113
function from (6.3), being calculated up to (including) terms of the order of 1/R2, vanishes. This
resolves what might have seemed as the puzzle.
In the literature there are several expressions for the separation constant for the TCP ? the
expressions not having the form of a projection of any vector on the internuclear/interstellar axis.
Below we provide relations between those expressions and our result.
In 1949, Erikson and Hill [6.13] presented the following expression for the separation constant:
?=L??L??2a?Z1cos ?1?Z2 cos?2? (6.14)
where L' and L'' are the angular momenta of the electron with respect to Z1 and Z2, ?1 and ?2 are
the angles between the radii-vectors r1 and r2 (directed from the nuclei to the electron) and the
positive direction of the z-axis (from Z1 to Z2), and 2a is the internuclear distance.
We found that it is related to our result from (6.12) as follows:
?=R?Z'?Az? (6.15)
In Landau-Lifshitz?s book of 1960 [6.5], there is the following expression for the separation
constant:
?=?2? p?2? p?
2
?2 ??M
2?2m???
1 cos?1??2cos ?2?
(6.16)
where ?1 and ?2 are the nuclear charges with the sign opposite to ours, ? and ? are the cylindrical
coordinates corresponding to the z-axis which is the internuclear axis, p? and p? are the canonical
momenta corresponding to these coordinates, m is the mass of the particle (in our units m = 1), M
is the angular momentum of the electron with respect to the origin (which is at half the
internuclear distance), 2? is the internuclear distance, and ?1 and ?2 are the angles between the
114
radii-vectors r1 and r2 and the internuclear axis measured inside the triangle made by r1, r2 and
2?.
We found that it is related to our result from (6.12) as follows:
?=R?Az?Z'? (6.17)
In 1967, Coulson and Joseph [6.14] presented the following expression for the separation
constant:
B=L2?a2 pz2?2aq1zr
1
? 2aq2zr
2
(6.18)
where q1 and q2 are the nuclear charges with the sign opposite to ours, L is the angular
momentum of the particle with respect to the origin (which is at half the internuclear distance),
2a is the internuclear distance, r1 and r2 are the distances from the particle to the nuclei, z is the
coordinate along the internuclear axis and pz is its corresponding canonical momentum.
We found that it is related to our result from (6.12) as follows:
B=R?Z'?Az??R
2H
2
(6.19)
where H is the Hamiltonian.
In the book by Komarov, Ponomarev, and Slavyanov published in 1976 [1.11], the following
expression for the separation constant was presented:
?=L2?R
2 p
z
2
4 ?
RZ1 z
r1 ?
RZ2 z
r2 ?
R2H
2
(6.20)
We found that it is related to our result from (6.12) as follows:
115
?=R?Z'?Az? (6.21)
6.3. Conclusions
We showed that the expressions for a super-generalized Runge-Lenz vector presented for the
TCP by Krivchenkov and Liberman [6.11] and by Gurarie [6.12] are incorrect. Its projection on
the internuclear (or interstellar) axis is not conserved ? contrary to their claims.
We derived a correct super-generalized Runge-Lenz vector, whose projection on the
internuclear/interstellar axis does conserve. We also analysed the asymptotic form of the
projection in the limit of large R.
Finally, since in the literature there are several expressions for the separation constant for the
TCP ? the expressions not having the form of a projection of any vector on the
internuclear/interstellar axis ? we provided relations between those expressions and our result.
The correct super-generalized Runge-Lenz vector for the TCP that we derived should be of a
general theoretical interest since the TCP is one of the most fundamental problems in physics. It
can also have practical applications: for example, it can be used as a necessary tool while
applying to the TCP the robust perturbation theory for degenerate states (based on the integrals
of the motion) developed by Oks and Uzer [6.15].
116
Chapter 7. Conclusions
In this work we studied classically two-Coulomb-center systems consisting of two nuclei of
charges Z and Z' and an electron in the field of these nuclei in a circular or a helical state . These
systems represent diatomic Rydberg quasimolecules encountered, e.g., in plasmas containing
more than one kind of multicharged ions. Diatomic Rydberg quasimolecules are one of the most
fundamental theoretical playgrounds for studying charge exchange. Charge exchange and
crossings of corresponding energy levels that enhance charge exchange are strongly connected
with problems of energy losses and of diagnostics in high temperature plasmas; besides, charge
exchange is one of the most effective mechanisms for population inversion in the soft x-ray and
VUV ranges. The classical approach is well-suited for Rydberg quasimolecules.
Chapter 1 presented a history of classical studies of diatomic Rydberg quasimolecules. In these
studies one of the goals was to find the energy terms of this system ? the dependence of the
energy of the electron on the internuclear distance. There were three energy terms found, two of
them crossing; the crossing had a V-shape.
In Chapter 2, we considered diatomic Rydberg quasimolecules subjected to a static electric
field parallel to the internuclear axis. First, it led to the appearance of the fourth energy term,
which was absent at the zero field. This term had a V-crossing with the lowest energy term.
Second ? more importantly ? the electric field caused additional crossings and these crossings
had an X-shape. The X-crossings occurred at much larger internuclear distances than the V-
crossings, and one of them was found to significantly enhance charge exchange.
117
In Chapter 3, diatomic Rydberg quasimolecules were considered in a plasma of a given
screening length, which altered the original potential of either nucleus by an exponential factor
(see (3.1)). As in the previously mentioned case, the screening led to the appearance of the fourth
energy term having a V-type crossing with the lowest energy term. More importantly, it was
found that the screening stabilizes the nuclear motion for Z = 1 and destabilizes it for Z > 1. We
also found that a so-called continuum lowering in plasmas was impeded by the screening,
creating the effect similar to that of the magnetic field and opposite to that of the electric field.
The continuum lowering plays a key role in calculations of the equation of state, partition
function, bound-free opacities, and other collisional atomic transitions in plasmas.
In Chapter 4, diatomic Rydberg quasimolecules were studied in a laser field. For the situation
where the laser field is linearly-polarized along the internuclear axis, we found an analytical
solution for the stable helical motion of the electron valid for wide ranges of the laser field
strength and frequency. We also found resonances, corresponding to a laser-induced unstable
motion of the electron, that result in the destruction of the helical states. For the case of a
circularly-polarized field, polarization plane being perpendicular to the internuclear axis, we
found an analytical solution for circular Rydberg states valid for wide ranges of the laser field
strength and frequency. For this case we demonstrated also that there is a red shift of the primary
spectral component. We showed that both under the linearly-polarized laser field and under the
circularly-polarized laser field, in the electron radiation spectrum in the addition to the primary
spectral component at (or near) the unperturbed revolution frequency of the electron, there
appear satellites. Under a laser field of a known strength, in the case of the linear polarization the
observation of the satellites would be the confirmation of the helical electronic motion in the
118
Rydberg quasimolecule, while in the case of the circular polarization the observation of the red
shift of the primary spectral component would be the confirmation of the specific type of the
phase modulation of the electronic motion. Conversely, if the laser field strength is unknown,
both the relative intensities of the satellites and the red shift of the primary spectral component
could be used for measuring the laser field strength.
In Chapter 5, a system consisting of a proton, muon and an electron was studied. It was found
that a muonic hydrogen atom can attach an electron, with the muon and electron being in circular
states. The technique of the separation of rapid and slow subsystems was used, where the muon
represented the rapid subsystem and the electron the slow subsystem. The energy terms of the
rapid muon in the field of the two slow (quasi-static) proton and electron were found ? a double-
degenerate energy term, which corresponds to stable motion. By analysing the slow subsystem,
we derived the validity conditions for separation of rapid and slow subsystems. Finally we
showed that the spectral lines emitted by the muon experience a red shift compared to the
corresponding spectral lines in a muonic hydrogen atom. Observing this red shift should be one
of the ways to detect the formation of such muonic-electronic negative hydrogen ions. Studies of
muonic atoms and molecules, where one of the electrons is substituted by the heavier lepton ??,
have several applications, such as muon-catalyzed fusion, a laser-control of nuclear processes,
and a search for strongly interacting massive particles proposed as dark matter candidates and as
candidates for the lightest supersymmetric particle. Additionally, analytical relativistic treatment
of the muonic motion was conducted, which showed that the relativistic corrections are relatively
small.
Chapter 6 presented our study of fundamental algebraic symmetry of the TCC systems leading
119
to an additional conserved quantity: the projection of a super-generalized Runge-Lenz vector on
the internuclear axis. We derived the correct super-generalized Runge-Lenz vector, whose
projection on the internuclear axis is conserved, and showed that the corresponding expressions
by other authors did not correspond to a conserved quantity and thus were incorrect. The correct
super-generalized Runge-Lenz vector for the TCC systems that we derived should be of a
general theoretical interest since the TCC systems represent one of the most fundamental
problems in physics. It can also have practical applications: e.g., it can be used as a necessary
tool while applying to the TCC systems the robust perturbation theory for degenerate states
based on the integrals of the motion.
120
References
Chapter 1. Introduction: Two-Coulomb Center Systems Representing Diatomic Rydberg
Quasimolecules
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