Download A Dissertation entitled Quantum Theory of Ion

A Dissertation
entitled
Quantum Theory of Ion-Atom Interactions
by
Ming Li
Submitted to the Graduate Faculty as partial fulfillment of the requirements for the
Doctor of Philosophy Degree in Physics
Dr. Bo Gao, Committee Chair
Dr. Song Cheng, Committee Member
Dr. Steven R. Federman, Committee Member
Dr. Thomas J. Kvale, Committee Member
Dr. Biao Ou, Committee Member
Dr. Patricia R. Komuniecki, Dean
College of Graduate Studies
The University of Toledo
August 2014
Copyright 2014, Ming Li
This document is copyrighted material. Under copyright law, no parts of this
document may be reproduced without the expressed permission of the author.
An Abstract of
Quantum Theory of Ion-Atom Interactions
by
Ming Li
Submitted to the Graduate Faculty as partial fulfillment of the requirements for the
Doctor of Philosophy Degree in Physics
The University of Toledo
August 2014
This thesis consists of a series of theoretical efforts aimed at reformulating the
quantum theory of ion-atom interactions using quantum-defect theory that is based
on the analytic solutions for the long-range, −1/R4 , polarization potential.
Ion-atom interactions, especially at cold temperatures of a few kelvin or lower, are
complicated by the rapid energy variations induced by the long-range polarization
potential, by the generally large number of contributing partial waves, and by the
sensitive dependence of the interactions on the short-range potential. The standard
numerical method is not only inefficient in addressing these issues, but can also miss
important physics such as extremely narrow resonances. Ion-atom interaction at
cold temperatures is further complicated by what is normally considered as “weak”
interactions, such as the hyperfine interaction. While they may not be important at
high temperatures, they become exceedingly important at 1 K or lower temperatures.
The hyperfine effects, and the related effects of identical nuclei, have not been properly
treated in existing theories.
This thesis contains works that establish the quantum-defect theory for ion-atom
interactions, including both its single-channel version, and its multichannel version.
Through detailed comparison with numerical calculations, carried out for Na+ +Na
and proton-hydrogen systems, we show how quantum-defect theories provide a systematic and an efficient understanding of ion-atom interactions. Such an efficient
iii
description is not only important for two-body systems, but also the key to a systematic understanding of quantum few-body systems, chemical reactions, and many-body
systems involving ions. Proper treatments of hyperfine structure and identical nuclei
are also developed as a part of these studies.
iv
Acknowledgments
I am deeply indebted to my advisor, Prof. Bo Gao, for his continuous guidance and
support. He led me into the world of atomic physics and guided me with wisdom, intelligent insights, rich knowledge, and a deep understanding of physics. He has always
been encouraging and patient to me for which I am very grateful. His enthusiasm
and rigorous attitude towards physics have been a great influence on me.
Constantinos Makrides has always been a great friend and colleague to me, with
whom I enjoyed many discussions, collaborations, and more. Many helpful discussions
are due to the participants of the AMO seminar, among which are Prof. Robert
Deck, Prof. Steven Federman, and Prof. Thomas Kvale. I am also indebted to Prof.
Federman for his careful editing of this thesis. Special thanks also go to Prof. Song
Cheng and Prof. Scott Lee for their guidance on my role as a teaching assistant. I
am also especially thankful to my friends and collegues Thomas Allen, Brad Hubartt,
Sam Ibdah, and Carl Starkey.
I would like to show great gratitude to Prof. Li You for his guidance and support. I
would also like to express my appreciation to my colleagues from Tsinghua University,
with special thanks to Prof. Mengkhoon Tey and Dr. Hao Duan.
This work would have been impossible without the invaluable support from my
parents, for which I am deeply grateful. My thanks also go to Lia Van Dril, who has
given me much support and encouragement. Last but not least, I sincerely thank all
my friends, who were not mentioned above due to page limitation, for making my
time in Toledo enjoyable and delightful.
v
Contents
Abstract
iii
Acknowledgments
v
Contents
vi
List of Tables
ix
List of Figures
x
List of Abbreviations
xiv
1 Introduction
1
2 Theory background
5
2.1
General consideration for two-body interaction . . . . . . . . . . . . .
5
2.2
Born-Oppenheimer approximation . . . . . . . . . . . . . . . . . . . .
8
2.3
Channel definitions and frame transformation . . . . . . . . . . . . .
10
2.4
Physical boundary conditions . . . . . . . . . . . . . . . . . . . . . .
14
2.5
Brief introduction to the quantum-defect theory . . . . . . . . . . . .
16
3 Quantum-defect thetory for resonant charge exchange
20
3.1
Background and introduction . . . . . . . . . . . . . . . . . . . . . .
20
3.2
General theory for 1 S+2 S type of systems . . . . . . . . . . . . . . . .
22
3.2.1
22
Elastic approximation . . . . . . . . . . . . . . . . . . . . . .
vi
3.2.2
Single channel quantum-defect theory . . . . . . . . . . . . . .
24
3.3
Three-parameter QDT implementation . . . . . . . . . . . . . . . . .
25
3.4
The example of Na+ +Na . . . . . . . . . . . . . . . . . . . . . . . . .
27
3.4.1
Comparison of QDT results with previous results . . . . . . .
28
3.4.2
Comparison of QDT results with current numerical results . .
31
3.4.2.1
Potential energy curves adopted . . . . . . . . . . . .
31
3.4.2.2
Comparison of results . . . . . . . . . . . . . . . . .
33
3.4.3
Comparison of results of different potentials . . . . . . . . . .
37
3.4.4
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
3.5
4 Multichannel quantum theory for ion-atom interactions
43
4.1
Background and introduction . . . . . . . . . . . . . . . . . . . . . .
43
4.2
Theoretical framework . . . . . . . . . . . . . . . . . . . . . . . . . .
45
4.2.1
Channel structure and frame transformation . . . . . . . . . .
47
4.2.2
Scattering amplitude and cross sections . . . . . . . . . . . . .
48
4.2.3
Potential energy curves and numerical method . . . . . . . . .
50
4.2.4
Multichannel quantum-defect theory . . . . . . . . . . . . . .
52
4.2.4.1
General formulation . . . . . . . . . . . . . . . . . .
52
4.2.4.2
K c matrix and short-range parametrization . . . . .
53
4.2.4.3
Resonance structure . . . . . . . . . . . . . . . . . .
54
The example of Na+ +Na with hyperfine interaction . . . . . . . . . .
57
4.3.1
Baseline results from the simplest MQDT parametrization . .
57
4.3.2
Total scattering cross sections . . . . . . . . . . . . . . . . . .
65
4.3.3
Resonance structures . . . . . . . . . . . . . . . . . . . . . . .
69
Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
4.3
4.4
vii
5 Slow proton-hydrogen collision
72
5.1
Background and introduction . . . . . . . . . . . . . . . . . . . . . .
72
5.2
General considerations and potential energy curves . . . . . . . . . .
73
5.3
QDT parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
5.4
Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . .
77
5.4.1
Comparison with elastic approximation . . . . . . . . . . . . .
77
5.4.2
Total scattering cross sections . . . . . . . . . . . . . . . . . .
78
5.4.3
Threshold behavior of de-excitation rate . . . . . . . . . . . .
83
Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
5.5
6 Conclusions and outlook
87
References
90
A Quantum-defect theory functions for polarization potentials
viii
100
List of Tables
4.1
Channel structure for ion-atom interactions of the type 2 S +1 S with identical nuclei of spin I2 = I1 .
. . . . . . . . . . . . . . . . . . . . . . . . .
47
4.2
Positions, widths, and classifications of the 7 resonances labeled in Fig. 4-6. 70
5.1
Zero energy QDT parameters for proton-hydrogen interaction . . . . . .
ix
76
List of Figures
3-1 Comparison of the total and the partial “molecular” cross sections for
the gerade state of Na+
2 from the QDT calculation using parameters from
Ref. [21] (3-1a) and from Ref. [21] (3-1b). . . . . . . . . . . . . . . . . . .
29
3-2 Comparison of the total and the partial “molecular” cross sections for the
ungerade state of Na+
2 from the QDT calculation using parameters from
Ref. [21] (3-2a) and from Ref. [21] (3-2b). . . . . . . . . . . . . . . . . . .
30
3-3 Comparison of the BO potential energy curves adopted in this work (33a) and from Ref. [21] (3-3b) for gerade (solid lines) and ungerade (dashed
lines) states of Na+
2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
3-4 Charge exchange cross sections of Na+ +Na obtained from a three-parameter
QDT description (dashed line) and from numerical calculations (solid line). 34
3-5 Total cross sections of Na+ +Na obtained from a three-parameter QDT
description (dashed line) and from numerical calculations (solid line).
.
35
3-6 Charge exchange cross sections of Na+ +Na obtained from three-parameter
QDT descriptions using parameters corresponding to the potential of Ref. [21]
(solid line) and using parameters corresponding to our potential (dashed
line).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
3-7 Total cross sections of Na+ +Na obtained from three-parameter QDT descriptions using parameters corresponding to the potential of Ref. [21]
(solid line) and using parameters corresponding to our potential (dashed
line).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
38
4-1 Baseline MQDT results (solid line) and numerical results (dashed line) for
the total hyperfine de-excitation cross sections of Na+ +Na from channel
{F1 = 2, F2 = 3/2} to channel {F1 = 1, F2 = 3/2}. The vertical dotted
line identifies the upper hyperfine threshold at 0.08502 K.
. . . . . . . .
58
4-2 Baseline MQDT (solid line) results and numerical results (dashed line) for
the total elastic cross sections of Na+ +Na in channel {F1 = 1, F2 = 3/2}.
The vertical dotted line identifies the upper hyperfine threshold at 0.08502
K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
4-3 Baseline MQDT (solid line) results and numerical results (dashed line) for
the partial elastic cross sections of Na+ +Na for l = 5 and F = 5/2 in
channel {F1 = 1, F2 = 3/2}. The vertical dotted line identifies the upper
hyperfine threshold at 0.08502 K. . . . . . . . . . . . . . . . . . . . . . .
60
4-4 Total hyperfine de-excitation cross sections from channel {F1 = 2, F2 =
3/2} to channel {F1 = 1, F2 = 3/2} from MQDT (solid line) and numerical method (dashed line). The vertical dotted line identifies the upper
hyperfine threshold at 0.08502 K. . . . . . . . . . . . . . . . . . . . . . .
62
4-5 Total elastic cross sections in the lower channel {F1 = 1, F2 = 3/2} from
MQDT (solid line) and numerical method (dashed line). The vertical
dotted line identifies the upper hyperfine threshold at 0.08502 K. . . . .
63
4-6 Partial wave contribution to the elastic cross section of Fig. 4-5 from l = 5
and F = 5/2. There are seven resonances within the hyperfine splitting that are labelled with numbers 1 through 7. The vertical dotted
line identifies the upper hyperfine threshold at 0.08502 K. Their detailed
characteristics are tabulated in Table. 4.2. A magnified version of this
figure focusing on the resonances within the region between the hyperfine
thresholds is presented in Fig. 4-9. . . . . . . . . . . . . . . . . . . . . .
xi
64
4-7 Total hyperfine excitation cross sections from channel {F1 = 1, F2 = 3/2}
to channel {F1 = 2, F2 = 3/2} from MQDT (solid line) and numerical
method (dashed line). The vertical dotted line identifies the upper hyperfine threshold at 0.08502 K. . . . . . . . . . . . . . . . . . . . . . . . . .
67
4-8 Total elastic cross sections in the higher channel {F1 = 2, F2 = 3/2} from
MQDT (solid line) and numerical method (dashed line). The vertical
dotted line identifies the upper hyperfine threshold at 0.08502 K. . . . .
68
4-9 Magnified version of Fig. 4-6 focussing on the energy region below the
hyperfine threshold (vertical dotted line). Labelled 1 through 7 are seven
resonances whose detailed characteristics are tabulated in Table. 4.2. . .
69
5-1 BO potential energy curves of the gerade (solid line) and the ungerade
(dashed line) states constructed in our work for proton-hydrogen collision
74
5-2 Total hyperfine de-excitation cross section of the proton-hydrogen collision with the present numerical calculation (solid line) and the spinexchange cross section from Ref. [67] multiplied by the proper coefficient,
1/4 that accounts for nuclear statistics, and offset by the center-of-gravity
0.0511265K (stars).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
5-3 Total hyperfine de-excitation cross sections from channel {F1 = 1, F2 =
1/2} to channel {F1 = 0, F2 = 1/2} from MQDT (solid line) and numerical method (dashed line). The vertical dotted line identifies the upper
hyperfine threshold located at 2 /kB ≈ 0.0682 K. . . . . . . . . . . . . .
79
5-4 Total hyperfine excitation cross sections from channel {F1 = 0, F2 = 1/2}
to channel {F1 = 1, F2 = 1/2} from MQDT (solid line) and numerical
method (dashed line). The vertical dotted line identifies the upper hyperfine threshold located at 2 /kB ≈ 0.0682 K.
xii
. . . . . . . . . . . . . . . .
80
5-5 Total elastic cross sections in the lower channel {F1 = 0, F2 = 1/2} from
MQDT (solid line) and numerical method (dashed line). The vertical
dotted line identifies the upper hyperfine threshold at 0.0682 K. . . . . .
81
5-6 Total elastic cross sections in the higher channel {F1 = 1, F2 = 1/2} from
MQDT (solid line) and numerical method (dashed line). The vertical
dotted line identifies the upper hyperfine threshold at 0.0682 K. . . . . .
82
5-7 Threshold behavior of the hyperfine de-excitation rate Wde just above the
upper threshold E2 . The x-axis represents the temperature equivalence of
the initial kinetic energy ( − E2 )/kB . The results are produced using our
numerical method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiii
84
List of Abbreviations
a.u. . . . . . . . . . . . . . . . . . . . . . . .
BO . . . . . . . . . . . . . . . . . . . . . . .
CC . . . . . . . . . . . . . . . . . . . . . . .
MQDT . . . . . . . . . . . . . . . . . . .
PES . . . . . . . . . . . . . . . . . . . . . .
QDT . . . . . . . . . . . . . . . . . . . . . .
Atomic unit
Born-Oppenheimer
Coupled-channel
Multichannel quantum-defect theory
Potential energy surface
Quantum-defect theory
xiv
Chapter 1
Introduction
One of the fundamental contributions of cold-atom physics has been its revelation
of universal behaviors in quantum many-body [11, 47, 119] and few-body systems
[15, 51, 107]. Excluding scaling, ultracold atomic systems behave the same with
their only differences being characterized by a few parameters such as the scattering
length. At a more fundamental level, such universal behaviors have their origin in
the universal ultracold two-body interaction as described by the effective range theory
[113, 10, 9, 101]. Since this theory quickly breaks down at slightly higher energies and
at shorter distances, it is natural to ask the question of whether universal behaviors
exist beyond the ultracold energy regime and at higher densities, and whether they
exist for systems of mixed species of, e.g., atoms, ions, and electrons. These are
important questions in physics, the answers to which will determine the degree we
can understand the world around us, including phenomena as diverse as reactive
processes in atomic collisions, chemical reactions, and high-Tc superconductivity.
One specific question being asked is whether a two-body theory for the interaction between an ion and an atom can be developed to capture the universal behavior
beyond the ultracold regime and to be incorporated into few-body and many body
theories. Such a theory would have to be efficient and simple enough to be incorporated, but at the same time capable of addressing rapid energy dependence and
1
generally the large number of contributing partial waves accurately, especially to describe complex resonance structures, all of which are attributes of ion-atom interaction
in cold temperatures where quantum effects are important [21, 88, 45, 89].
Many recent experimental and theoretical efforts have been devoted or greatly
related to the development of such a theory. On the experimental side, these efforts
include the study of trapped ions interacting with atomic gas in the milli-Kelvin
regime [52, 111, 130, 129, 53, 116, 87, 57], ultracold plasmas [78, 22, 20, 76, 77, 108],
ultracold chemistry [118, 18, 106, 104, 55], and dissociation spectroscopy of molecular
ions [58, 80, 61, 115]. On the theoretical side, before this work, developments have
been made with numerical calculation [70, 21, 28, 92, 81, 12, 127, 83, 103, 128, 120, 84,
8, 110], semiclassical theories [33, 32], and quantum-defect theory (QDT) [68, 42, 69].
However, theories of the desired characteristics mentioned earlier have not been fully
established, especially for systems with fine or hyperfine structures that exhibit nontrivial multichannel characteristics.
In the present thesis, we aim to establish a theoretical framework developed around
QDT for ion-atom interactions that captures their universal behaviors beyond the
ultracold regime. To serve such an intent, we organise the thesis as described in the
following outline.
Outline of the thesis
Chapter 2 briefly introduces the theoretical framework upon which the present
work is developed. We start with the general consideration of the fully quantum mechanical description of a two-body interaction including the total Hamiltonian, conserved observables, time-independent Schrödinger equation [112, 85], and expansion
of the stationary wavefunction. The Born-Oppenheimer (BO) approximation [14]
is then introduced as an essential building block of our work. To effectively solve
2
the Schrödinger equation, symmetry properties of the system need to be fully incorporated, and are reflected in the choice of different sets of channel functions used to
expand the wavefunction [35]. They are briefly introduced, as well as the frame transformation between them. Next, we briefly go over the physical boundary condition
along with the means of extracting physical observables for scattering problems from
the solution of the Schrödinger equation [98]. Last, we look at the general concepts
behind QDT, which are shared by QDT for the −1/R4 potential [42, 45] used in our
work and the already established Coulombic QDT [72].
In Chapter 3, we present the first installment of QDT for the −1/R4 potential [42, 45], to study the resonant charge exchange process of systems of the 1 S+2 S
type as a prototypical system of ion-atom interaction. With the elastic approximation [23], such a problem can be simplified to effective single-channel problems which
presents an ideal testing ground for single-channel QDT. We briefly go over the concept of elastic approximation before elaborating on the formulation of single-channel
QDT and the three-parameter implementation of QDT for resonant charge exchange.
To demonstrate the predictive power of this QDT implementation, we present the
comparison of partial and total cross section results from QDT and numerical calculations for the
23
Na+ +23 Na resonant charge exchange process. We then further
compare the results from slightly different potential energy curves to show the dependence of the scattering results on the short range interaction. The content in this
chapter is based on our work in Ref. [88].
In Chapter 4, we look at the more complete picture of ion-atom interaction at low
temperatures that includes hyperfine structures and the effect of indistinguishable
nuclei. To fully resolve the complication raised by these factors, we take a close
look at the channel structures of the interaction as well as the physical boundary
conditions. Two different sets of channels representing different symmetry properties
at the short-range and long-range of internuclear separations are defined and the
3
frame transformation between them is presented. The scattering amplitude and cross
sections for the case of resonant charge exchange that take proper account of the
effect of identical nuclei are given. With these building blocks in place, we apply
the multichannel quantum-defect theory (MQDT) [46] on ion-atom interactions for
the first time. We also present the analytical characterization of the resonances
through the MQDT formulation, especially within hyperfine splitting, including their
positions, widths, and categorization.
We demonstrate the predictive power of MQDT by comparing two different implementations to numerical calculations for
23
Na+ +23 Na, with hyperfine structure
included this time. The first implementation takes the same number of parameters as
our single-channel QDT calculation, and the second slightly more advanced MQDT
implementation includes two more short-range parameters characterizing the small
partial wave dependences of the short-range interaction. The results of the comparison are presented and discussed. Resonances in a particular partial wave are also
analysed to demonstrate the capability of MQDT to analytically characterize them.
The content in Chapter 4 is based on our work in Ref. [89].
Chapter 5 presents another application of the theory in Chapter 4, this time to the
proton-hydrogen collision at low temperatures. We first demonstrate the construction
of the potential energy curves as well as the extraction of the short-range parameters
for the MQDT implementation for this system. Then we present the cross section
results of fully multichannel calculation from zero to five kelvin. MQDT results and
numerical results are again compared. Threshold behavior of the de-excitation cross
sections is investigated.
Chapter 6 summarizes the theory and results from the previous chapters. The
prospects of further application of the theory are also briefly discussed.
4
Chapter 2
Theory background
In this chapter, we briefly overview the general framework of the quantum theory
for two-body interactions, especially for low energy collisions with spin-orbit and/or
hyperfine interaction involved. The framework is the foundation of our work, and
contains important physics that helps the development of our theories and their applications. We focus on outlining a relatively self-contained picture of how the theory
works for the systems we are interested in this work, without getting too much into
the technical details. If further information is needed, please refer to the references
cited.
2.1
General consideration for two-body interaction
We consider the interaction between two atoms A and B (we use the term “atom”
here in a broader sense that can also refer to an ion, as in a charged atom) in free space.
The energy and dynamics of the system are characterized by its total Hamiltonian
H, and the wavefunction of the system is governed by the Schrödinger equation [112].
Since the two atoms are in free space, H is time-independent, thus the stationary
wavefunction of the system, which is an eigenfunction of the Hamiltonian that satisfies
certain boundary conditions, is what we are interested in.
5
The total Hamiltonian H, including all the relativistic effects, can be written as
~2 2
H = − ∇ + HBO + Hf + Hhf ,
2µ
(2.1)
where µ is the reduced mass of the two particles. HBO is the adiabatic BO Hamiltonian, which is used in the determination of the BO molecular states and electronic
potential energy surfaces (PESs) within the BO approximation [14]. This will be
briefly overviewed in Section 2.2. Hf is the Hamiltonian describing the spin-orbit
interactions, and Hhf describes the hyperfine interactions.
The total Hamiltonian H is rotationally invariant, thus commuting with the total
angular momentum T of the system. This guarantees the conservation of the total
angular momentum through Noether’s theorem [49], and that the eigenfunctions of T
are also eigenfunctions of H. The same conclusion stands for the magnetic quantum
number MT of T . H is also invariant under coordinate-inversion, thus the total parity
PT is a conserved quantum number and any eigenfunction of H bears a fixed parity.
The total stationary wavefunction can therefore be identified by the quantum numbers mentioned above and is denoted as ψ T MT PT . The time-independent Schrödinger
equation that governs the stationary wavefunction takes the form
Hψ T MT PT = Eψ T MT PT .
(2.2)
The total wavefunction can be expanded in terms of adiabatic channel functions
(basis functions) as
ψ T MT PT =
X
ΦTa MT PT (R)GTa MT PT (R)/R .
(2.3)
a
R is the vector of internuclear separation, where a (and b that will be used later in this
work) denotes a particular set of channel functions ΦTa MT PT (R), and the summation is
6
over the complete set of these channel functions. The channel functions contain both
the electronic wave functions and the angular part of the motion of the two centers
of mass of the ion and atom, relative to the center of mass of the whole system. The
construction of channel functions and the transformation between different sets of
them are presented in Section 2.3.
Substituting Eq. (2.3) into the Schrödinger equation (2.2) with the Hamiltonian
from Eq. (2.1), and making use of the orthogonality properties of the channel functions, we obtain a set of coupled-channel (CC) equations:
2 2
la (la + 1)~2
~ d
+
− E GaT MT PT (R)
−
2
2
2µ dR
2µR
X
[VabBO (R) + Vabf (R) + Vabhf (R)]GTb MT PT (R) = 0 ,
+
(2.4)
b
where
VabBO (R) ≡ hΦTa MT PT |HBO |ΦbT MT PT i ,
(2.5a)
Vabf (R) ≡ hΦaT MT PT |Hf |ΦbT MT PT i ,
(2.5b)
Vabhf (R) ≡ hΦaT MT PT |Hhf |ΦbT MT PT i .
(2.5c)
The term la (la + 1)~2 /2µR2 , usually referred to as the centrifugal barrier, arises from
the spherical harmonics describing the angular part of the motion in channel a acting on the total Hamiltonian, and la is the corresponding orbital angular momentum
quantum number. Notice that the channel functions have to comply with the symmetry properties of the Hamiltonian and conserve the three quantum numbers: T ,
MT , and PT .
The boundary conditions for the CC equations and a brief discussion on how to
extract physical information from the wavefunctions for scattering problems will be
presented in Section 2.4.
7
2.2
Born-Oppenheimer approximation
The BO Hamiltonian from Eq. (2.1) is the complete non-relativistic Hamiltonian
of the entire system except for the kinetic term for the nuclei, which includes the
kinetic terms for electrons and Coulomb potential terms between all charged particles
(electrons and nuclei). It can be written in atomic units as
HBO = −
X1
ie
2
∇2ie −
X Zi
X 1
X Zi Zj
n
n
n
+
+
,
r
r
R
i
i
i
j
i
j
e
n
e
e
n
n
i ,i
i >j
i >j
e n
e
e
n
(2.6)
n
where ie and je are labellings for electrons, and in and jn are labellings for nuclei.
Zin or Zjn is the charge carried by nucleus in or jn respectively, rie je is the distance
between electrons ie and je , and Rin jn is the separation between nuclei in and jn .
With the internuclear separations Rin jn fixed, in other words, with a specific nuclear configuration {Rin jn }, we can obtain the eigenvalues and eigenstates of HBO .
The eigenvalues are points on the PESs at that specific nuclear configuration {Rin jn },
and the eigenstates are adiabatic BO electronic wavefunctions at {Rin jn }. The electronic wavefunctions at a nuclear configuration can be uniquely identified by a set
of quantum numbers, and the ones that share the same set of quantum numbers (or
channel) at different internuclear separations form the adiabatic BO molecular states
(shortened to BO states for convenience in the following text).
For the case of a diatomic system, the BO states are the electronic wavefunctions
calculated with the direction and magnitude of the internuclear axis fixed. Thus the
BO states conserve ML (R̂) which is the projection of the total electronic orbital angular momentum along the internuclear axis R̂. Also conserved are the total electronic
spin S and its projection onto the internuclear axis MS (R̂) when relativistic effects
such as spin-orbit couplings are not accounted for. Therefore, the eigenvalue equation
8
for the BO Hamiltonian can be written as
HBO |ML (R̂)SMS (R̂)Γ; Ri = εML SΓ (R)|ML (R̂)SMS (R̂)Γ; Ri ,
(2.7)
where Γ is the rest of the quantum numbers that can be used to characterize the BO
state (i.e. certain symmetry properties for specific systems and the energy ordering.
Please refer to Section 3.2.1 for an example).
The Wigner-Witmer rule [124] states that the molecular BO states are correlated
with the electronic states of individual atoms when the two atoms are far apart.
Thus the BO states can be expanded asymptotically with individual atomic states
of the same symmetry with internuclear orientation carefully taken care of. This is
an essential step in our work because the channel functions can be related to the
BO states through the asymptotic expansions at large R. The technical details not
presented here can be found in Ref. [35].
The BO approximation [14] ignores all the couplings (which we call nonadiabatic
couplings) that arise from the BO states acting on the rest of the total Hamiltonian,
namely the kinetic term of the nuclei, the spin-orbit coupling term, and the hyperfine interaction term. Besides neglecting all relativistic terms, the approximation
assumes that the electronic motion can be decoupled from the nuclear motion. This
is usually physically realistic because the mass of an electron is three orders of magnitude smaller than the mass of a nucleus which makes the electron move much faster
than the nucleus in most circumstances. However, there are situations where these
nonadiabatic couplings become strong enough that they need to be treated properly.
The nonadiabatic couplings due to spin-orbit and/or hyperfine interaction, which
are important when the interaction energy is comparable to or smaller than the relativistic effect, can be incorporated fairly easily with careful construction of channel
functions. Some details will be discussed in Section 2.3. The nonadiabatic cou9
pling that arises from the kinetic term of the nuclei can become significant within
the proximity of avoided crossings between PESs, where the two BO states become
near-degenerate and strongly coupled. It is especially important when the avoided
crossings are away from the inner region of the PESs where the nuclei are not likely to
be. However, this usually happens when electronically excited BO states are involved,
which often requires high collision energy, or involves an excited atom or molecule.
These are situations beyond the scope of the present work. Another effect that comes
from the kinetic term of the nuclei is the isotope effect for similar atoms (nuclei carrying the same charge but having different masses), which gives a different asymptotic
threshold due to the mass difference that is not included in the BO Hamiltonian.
This effect can also be incorporated with careful construction of the channel functions. Overall, we will be able to cope with the nonadiabatic couplings except for the
unlikely avoided crossings, and the BO approximation will be the building block that
our theory is built with.
2.3
Channel definitions and frame transformation
Channel functions are used to expand the total stationary wavefunction of the
system that reflects certain symmetry properties of the Hamiltonians (other than the
shared ones: T , MT , and PT ) of different systems or different regions of one system via
the evaluation of the potential terms in Eq. (2.5). The structure of the total potential
matrix under a specific basis generally varies with internuclear separation, sometimes
diagonalized or block-diagonalized, sometimes with large off-diagonal terms (otherwise it will be an effective single-channel problem, see Section 3). Employing the
channels that can most reflect the symmetry in a certain region can dramatically
reduce the complexity of solving the CC equations, and more importantly is the essential base for developing analytic theories, in this case the QDT. Note that the
10
BO potential matrix is always diagonalized at infinite internuclear separation in the
circumstance considered here regardless of the channels chosen, since the difference
between different BO energy curves vanishes when they go to the same threshold
(without the splittings due to relativistic effects).
The channel functions are defined by the angular momentum coupling scheme
which dictates in what order the total angular momentum T is constructed from
fundamental angular momenta (including spins). These different schemes, which are
the embodiment of corresponding symmetry properties, links the channel functions
to the BO states, as well as other channel bases. Practically this can be done through
comparing the asymptotic forms of the channel functions (or BO states as mentioned
in Section 2.2) constructed from individual atomic states plus other ingredients such
as rotational wavefunction and possibly nuclear spin state. To be more specific, we
will introduce three angular momentum coupling schemes that define three sets of
channel functions.
Before going into the detailed discussion, it is necessary to introduce the general
notations of different angular momenta used here. We use Ls as the electronic orbital
angular momenta, Ss as the spins of electrons, Is as the nuclear spins, and ls as
the relative orbital angular momenta of the two centers of mass of the two atoms.
The bold letters represent the angular momentum vectors, the normal font letters
represent the quantum numbers of these angular momentum vectors, and M s with
these letters as subscripts represent the corresponding magnetic quantum numbers
along a space-fixed axis (lower case ml in the case of l). We also define J = L + S
and F = I + J .
The first kind of channels to be introduced is called the fragmentation channels,
the F F channels, or the F F coupled basis. It reflects the symmetry when two atoms
are far apart, where the electrons and nuclei mainly interact within individual atoms,
and the coupling between the internal angular momenta of the two atoms is only
11
through the two total angular momenta F1 and F2 mediated by the relative orbital
angular momentum l. Thus the spin-orbit and hyperfine potential are diagonalized
in this basis when the two atoms are far apart. Adding to the diagonalization of the
BO potential in the long-range, the total long-range interaction is diagonalized.
The angular momentum coupling scheme of the F F basis can be specified by the
quantum numbers as
(α1 L1 S1 J1 I1 F1 )A (α2 L2 S2 J2 I2 F2 )B F lT MT ,
(2.8)
where F = F1 + F2 , and αs are the rest of the quantum numbers that characterize the atomic states. This coupling scheme starts with the two atomic states,
|α1 L1 S1 J1 I1 F1 iA and |α2 L2 S2 J2 I2 F2 iB , with full coupling within individual atoms including spin-orbit and hyperfine interactions. It then couples the two states through
F = F1 + F2 , and finishes with adding the relative orbital angular momentum to it,
as T = F + l. To show as an example of how this works, the F F channel function,
for the case of the two nuclei carrying different charges (ZA 6= ZB ), can be written
asymptotically as
R→∞
ΦT(αM1 LT1PST1 J1 I1 F1 )(α2 L2 S2 J2 I2 F2 )F l −−−→
X
hF1 M1 , F2 M2 |F MF i
M1 M2 MF ml
× hF MF , lml |T MT i|α1 L1 S1 J1 I1 F1 iA |α2 L2 S2 J2 I2 F2 iB Ylml (R̂) . (2.9)
As the equation demonstrates, the channel function is expanded asymptotically with
the atomic wavefunctions as well as the rotational motion of the nuclei, expressed as
spherical harmonics Ylml (R̂), through angular momentum coupling. The expansion
coefficients here are the Clebsch-Gordan coefficients. For the case of similar atoms
(ZA = ZB ) with different nuclei (different isotopes) and the case of identical nuclei,
extra symmetrization treatment of the channel functions has to be applied. Since the
12
fragmentation channel functions can be seen as the combination of complete atomic
internal wavefunctions with the external motion broken down by the partial wave
expansion (the spherical harmonics), the scattering boundary conditions are most
easily imposed in this channel.
The second kind of channels to be introduced is called the LS channels or the LS
coupled basis. The channel representation can be specified by the quantum numbers
as
(α1 L1 S1 I1 )A (α2 L2 S2 I2 )B LlN SKIT MT ,
(2.10)
where L = L1 + L2 , S = S1 + S2 , I = I1 + I2 , N = L + l, K = N + S, and finally
T = K + I. This coupling scheme starts with the uncoupled angular momenta from
individual atoms, which are the same as the BO states since the BO states ignore
the couplings due to relativistic effects. Therefore the LS coupled basis is more
closely related to the molecular BO states, and as a result, the BO Hamiltonian is
block-diagonalized in this basis. However, the spin-orbit and hyperfine interaction
couplings are not explicitly reflected in the symmetry of this basis; therefore these
two Hamiltonians are not diagonalized at large internuclear separation.
The third kind of channels is called the JJ channels or the JJ coupled basis. The
channel representation can be expressed by the quantum numbers as
(α1 L1 S1 J1 I1 )A (α2 L2 S2 J2 I2 )B JlKIT MT ,
(2.11)
where J = J1 + J2 and K = J + l. This coupling scheme starts with atomic basis
with spin-orbit couplings while the hyperfine coupling is done in the molecular level.
It falls in between the F F coupling where the atomic angular momenta are completely
coupled and the LS coupling where they are completely uncoupled within individual
atoms. This coupling scheme is especially useful when spin-orbit coupling is strong
in the inner region of the internuclear separation.
13
The LS channels and the JJ channels are called the condensation channels where
the short-range interaction is (approximately) block-diagonalized. They should be
used in the short-range in order to simplify the CC equations. At long-range, when the
BO potential diminishes, the spin-orbit and hyperfine interaction become important
and the fragmentation channels should be used. To use two different channel bases,
a frame transformation is needed. This can be done by comparing the expressions of
the channel functions as R goes to infinity, like the one shown in Eq. (2.9). Note that
like the case described for the F F channels, symmetry properties of the nuclei have
to be taken into consideration while constructing the asymptotic channel functions.
2.4
Physical boundary conditions
To obtain physically viable solutions from the CC equations and to extract useful
physical information from them, we need to enforce physical boundary conditions,
namely the total wavefunction has to be finite everywhere.
There are 2N independent solutions for the CC equations, assuming there are N
coupled channels. When R approaches zero, there is a practically infinite wall at a
finite R for the overall potential in the CC equations due to the exchange interaction of
electrons that results from the exclusion principle and the Coulomb repulsion of nuclei.
The solutions and their derivatives well inside the wall can be obtained by solving the
CC equations with the infinite potential wall at an approximate R. To prevent the
solutions from diverging in this classically forbidden region, half of the solutions that
contain an exponentially growing term with decreasing R have to be removed, which
leaves only N linearly independent solutions. These N linearly independent solutions
and their derivatives can then serve as initial conditions for the global solutions.
At large internuclear separation, the potential matrix in the F F coupled basis
becomes diagonalized, and the thresholds depend on the fine and/or hyperfine struc14
tures of the asymptotic atoms when R goes to infinity. If the total energy E is lower
than the asymptotic threshold of a certain channel, that channel is called a closed
channel. Otherwise, it is called an open channel. If all the channels are closed, it is a
bound state problem. If there is at least one open channel, it is a scattering problem.
Regardless of how many channels are open or closed, the total wavefunction satisfies the physical boundary conditions
R→∞
ψiT MT PT −−−→
X
(T )
ΦTj MT PT [fj δji − gj Kji ]/R ,
(2.12)
j
where i and j denote F F channels, fj and gj are asymptotic reference functions
(T )
determined by the asymptotic behavior of the potential matrix, Kji is an element
in the physical K matrix, and the summation is over all channels. For interactions
between two neutral atoms or between an atom and an ion (or an electron), the
potential behaves asymptotically as
Vijtot (R) = (Ei − Cni /Rni )δij ,
(2.13)
where Ei is the threshold energy associated with a fragmentation channel i and ni >
2. The reference functions are spherical Bessel functions for open channels and are
modified spherical Bessel functions for closed channels. For scattering problems, the
total wavefunction is normalized to unit energy as
hψiT MT PT (Ef )|ψjT
0M 0P 0
T
T
(Ef 0 )i = δT T 0 δMT MT 0 δPT PT 0 δij δ(Ef − Ef 0 ) ,
(2.14)
and for bound state problems, the total wavefunction is normalized to unity.
The physical K matrix K (T ) is defined for all channels including open ones and
closed ones. It contains the physical information needed to determine scattering
properties which is similar to the concept of phase shift in single-channel problems
15
(can be understood as a phase shift between incoming and outgoing waves due to the
interaction). With proper ordering of the channels, the K matrix can be written in
the blocked submatrices form


Koo Koc 
K=
,
Kco Kcc
(2.15)
where Koo , Koc , Kco , and Kcc are open-open, open-closed, closed-open, and closedclosed submatrices of K (T ) respectively. The S matrix used in the standard scattering
theory [98] is given in terms of Koo by
S = (1 + iKoo )−1 (1 − iKoo ) ,
(2.16)
where 1 is the unit matrix with the same dimension as Koo . The scattering amplitude can be given in terms of the S matrix in a standard fashion [98] with extra
consideration of the nuclear symmetry, and all kinds of cross sections can be further
derived.
2.5
Brief introduction to the quantum-defect theory
The name “quantum defect” first appeared to denote the parameter, µl , in the
famous equation for atomic energy levels Enl of a Rydberg series of the excited electron
in a hydrogen-like atom [109],
Enl = −
Ry
,
(n − µl )2
16
(2.17)
which we will call the Rydberg formula, where n is the principle quantum number
of the electron, l is the quantum number of the orbital angular momentum of the
electron, and Ry is the Rydberg constant. The originally empirical parametrization
using the quantum defect in the Rydberg formula contains important physics that
would lead to the development of the Coulombic QDT [72] and later the QDTs for
−1/Rn type of potentials with n > 2 [42, 69, 45, 36, 37, 46, 40, 96, 16]. In the rest of
this section, we focus on introducing the physical concepts that connect the original
empirical parametrization and the later QDTs. For a full acount of the history of the
development of QDT, especially the QDT for Coulombic interactions, please refer to
Ref. [114] and [105].
The quantum defect is the difference between the Rydberg formula and its counterpart for a hydrogen atom, the Balmer formula [6]. It arises from the deviation of
the interaction between the excited electron and the ionic core from a pure Coulombic
interaction, as the electron-proton interaction in a hydrogen atom described by the
Balmer formula. A closer examination of the physical picture in a hydrogen-like atom
reveals that the ionic core mostly occupies a small volume centered on the nucleus,
while the excited electron roams much more freely, spending most of its time outside
the ionic core. When the excited electron is far away from the ionic core, the interaction between the two is practically Coulombic, with the interaction potential given in
the form of 1/r. As the distance between the electron and the ionic core decreases, the
non-Coulombic interactions contributing more and more significantly, such as other
electrostatic interactions from higher multipole moments of the ionic core, second or
higher order purterbation terms, and the exchange interaction between electrons [74].
The Rydberg formula succeeds in combining the Balmer formula that describes the
bound state structure of the pure Coulombic interaction in the long range with the
quantum defect that encapsulates the effect of the non-Coulombic part of the interaction in the short range. This idea of separating the total interaction into different
17
zones that dominated by different effects leads to the eventual development of the
Coulombic QDT.
To examine the structure of the Coulombic QDT in more detail, we can look
at the radial Schrödinger equation describing the combined system of the excited
electron and the ionic core in a hydrogen-like atom, which takes a similar form to
Eq. (2.4). Applying the idea from above, the radial Schrödinger equation is treated in
the short range and in the long range respectively, and then combined. The potential
energy term in the long range should take the form of a Coulombic potential, i.e. 1/r.
The radial Schrödinger equation can then be solved analytically, and the solution is
a combination of Coulomb functions [114, 72]. The short-range interaction is much
more complicated to describe using ab initio method and the accurate wavefunction is
difficult to obtain even with advanced numerical techniques. However, the effect from
the short-range interaction on the long-range wavefunction can be viewed as an initial
condition which only determines the coefficients for the Coulomb functions. Thus,
physical properties such as bound state energies and scattering cross sections that
only require the phase of the wavefunction at infinity1 can be characterized by the
corresponding analytic form derived from the analytic long-range wavefunction, such
as the Balmer formula for energy levels, in combination with short-range parameters
that encapsulate the short-range interaction, such as the quantum defect. The same
idea also leads to the QDTs for −1/Rn type of potentials with n > 2, including
the case when n = 4 developed in this thesis. The main difference here is that
the long-range interactions take different functional forms, which results in different
mathematical properties of the long-range wavefunctions. This gives rise to some
interesting characteristics unique to the QDTs for −1/Rn type of potentials with
1
Calculations of physical processes that require detailed knowledge of wavefunctions, such as
transitions, ionizations, detachments, and dissociations, can also make use of QDT in many situations
when long-range wavefunctions dominate the integral of the matrix element that corresponds to that
physical process.
18
n > 2, such as the concept of quantum reflection which does not exist in the Coulombic
QDT [40].
Technically, the quantum defect, or any equivalent short-range parameters, depend on the specific quantum state and energy. However, insights into the shortrange physics can help greatly simplify the parametrization and reduce the number
of short-range parameters required, which is crucial to the development of any QDT
for practical applications. One example of such physical insights in the Coulombic
QDT is the insensitivity of the energy dependence of the quantum defect around the
ionization threshold, which allows us the use the quantum defect extracted from the
Rydberg states, of which the energies are negative (taking the ionization threshold as
zero energy), to predict the scattering properties of an electron with the ionic core, of
which the energies are positive. We will show in the following chapters that similar
patterns of short-range physics exist and are very important for the simplification
and optimization of the QDT for ion-atom interactions.
19
Chapter 3
Quantum-defect thetory for
resonant charge exchange
3.1
Background and introduction
Despite being one of the simplest reactive processes that has been a subject of
study for a long time [98, 21, 12, 127], quantitative understanding of resonant charge
exchange, such as
Na+ + Na −→ Na + Na+ ,
remains difficult, especially at cold temperatures. This difficulty stems from the
sensitive dependence on the PESs when solving the radial Schrödinger equations. It is
a common difficulty shared by all heavy particle interactions at cold temperatures (see,
e.g., [35]), including not only ion-atom interactions, but also atom-atom interactions
[19], and chemical reactions (whenever the Langevin assumption breaks down [41,
43]).
In the case of atom-atom interactions, this difficulty has only been overcome by
incorporating a substantial amount of spectroscopic data, especially data close to the
dissociation limit, to fine tune the PES (see, e.g., Ref. [31, 79]). Without such fine
tuning, no ab initio PES for alkali-metal systems has been sufficiently accurate to
20
predict the scattering length and other scattering characteristics around the threshold. The availability of such data, however, is limited mostly to alkali-metal atoms
and a few other species that can be cooled. For ion-atom systems, with a few exceptions that came close [59], no such data are yet available, though recent efforts on the
trapping and cooling of molecular ions (see, e.g., Refs. [64, 100, 99, 126]) show considerable promise. This status on the ion-atom PES is such that at the moment, with
the possible exception of H+ +H and its isotopic variations [12, 70, 28], no other predictions for cold or ultracold ion-atom processes, including resonant charge exchange,
can yet be trusted before experimental verification.
We present here a QDT, not only as a general approach to ion-atom interactions,
but also as one specific method of dealing with this difficulty of sensitive dependence
on PES. It is an initial application of the QDT for a −1/R4 potential [121, 29, 42, 45],
as formulated in Ref. [42], to the resonant charge exchange process. We show that
by taking advantage of the partial-wave-insensitive nature of the QDT formulation
[37, 40, 42], resonant charge exchange of the type of 1 S+2 S, applicable to Group IA
(alkali), Group IIA (alkaline earth), and helium atoms in their ground states, can
be accurately described over a wide range of energies using only three parameters
even at energies where many partial waves contribute to the cross sections. The
theory further relates ultracold ion-atom interactions to interactions at much higher
temperatures.
We adopt the widely used elastic approximation (see, e.g., Refs. [23, 21, 127]),
which ignores the hyperfine and isotope effects. The radial Schrödinger equations are
effectively single-channel which allows us to examine the range of applicability of QDT
in the simplest and purest setting. The more complicated multichannel formulation,
which would not be possible without the work in this chapter, will be presented in
later chapters.
21
3.2
3.2.1
General theory for 1S+2S type of systems
Elastic approximation
Consider the system of an atom and its ion, one in a 1 S state, one in a 2 S state.
This type covers resonant charge exchange of both Group IA (alkali), Group IIA
(alkaline earth), and helium atoms in their ground states. Such a system correlates
to two BO molecular curves, characterized by 2 Σ+
g,u . The other, energetically higher,
BO states can be ignored for collision energies much smaller than the first electronic
excitation energy [98]. Here the molecular term symbol 2 Σ+
g,u means ML = 0 and
S = 1/2 with MS = ±1/2 degenerate. g and u representing gerade and ungerade
(meaning even and odd respectively in German) are quantum numbers characterizing
the parity symmetry of the total electronic wavefunction with respect to the geometric
center of the two nuclei. This symmetry only exists for similar atoms (ZA = ZB ). The
plus sign characterizes the mirror-reflecting symmetry of the electronic wavefunction
with respect to the plane containing the internuclear axis, which only exists for Σ
states where ML = 0.
The elastic approximation [23] ignores hyperfine structures and isotope shifts.
Then the interaction energy for each channel can be treated approximately as the
same. Due to this approximation, the channels all behave the same asymptotically,
and the short-range interaction is determined by the two BO curves. Therefore, there
are two coupled radial Schrödinger equations in the fragmentation channels, and they
can be decoupled in the BO basis (LS coupling with no nuclear spin involved). Since
the asymptotic energies in the fragmentation channels are degenerate, the BO basis
is diagonal in the long-range, too. Thus, in the BO basis, the two radial Schrödinger
equations are completely decoupled at all R, which reduces the understanding of
resonant charge exchange to two single-channel equations for the gerade, g, and the
22
ungerade, u, states, respectively, as [98, 21, 127]
~2 d2
~2 l(l + 1)
−
+
+ Vg,u (R) − ug,u
l (R) = 0 .
2
2
2µ dR
2µR
(3.1)
Here is the energy in the center-of-mass frame, Vg,u (R) represent the two BO potential energy curves for the gerade and the ungerade states, and ug,u
l (R) are the
corresponding radial wave functions for the lth partial wave.
In terms of the two phase shifts, δlg,u , for the gerade and ungerade states in partial
wave l, as determined from the solutions of Eq. (3.1), the total charge exchange cross
section σex can be derived from standard scattering theory, expressed as [98, 21, 127]
∞
π X
(2l + 1) sin2 (δlg − δlu ) .
σex () = 2
k l=0
(3.2)
It contains the physical concept that resonant charge exchange is due to the phase
difference between the gerade and the ungerade molecular states. For elastic and
total cross sections, it is convenient to first define two single-channel “molecular”
cross sections
σ
g,u
∞
4π X
(2l + 1) sin2 (δlg,u ) ,
= 2
k l=0
(3.3)
in terms of which the total cross section is given by σtot = (σ g + σ u )/2, and the elastic
cross section is given by σel = σtot − σex [98, 21, 127].
All cross sections for resonant charge exchange can be written explicitly in terms
of tan δlg and tan δlu for the gerade and the ungerade states.
σex () =
∞
π X
(tan δlg − tan δlu )2
(2l
+
1)
,
k 2 l=0
(1 + tan2 δlg )(1 + tan2 δlu )
(3.4)
and
σ g,u =
∞
4π X
tan2 (δlg,u )
(2l
+
1)
.
k 2 l=0
1 + tan2 (δlg,u )
23
(3.5)
For sufficiently large l and away from a resonance (a shape resonance in this case due
to quasi-bound states within the potential well created by the centrifugal barrier),
tan δl is independent of the short-range parameter and is given, for both g and u
states, by the Born approximation (see, e.g., Ref. [85])
tan δl ∼
π
s ,
(2l + 3)(2l + 1)(2l − 1)
(3.6)
where s is the scaled energy that will be defined in the next subsection. This 1/l3
type of behavior for large l ensures convergence in summations over l in all total cross
section calculations.
3.2.2
Single channel quantum-defect theory
For a 1 S+2 S type of system with either a 1 S or a 2 S atom in its ground state,
the potentials Vg,u (r) in Eq. (3.1) have the same leading term −C4 /R4 at long range,
where C4 > 0 is given in atomic units by C4 = α1 /2 with α1 being the static dipole
polarizability of the atom. Application of the single-channel QDT for a −1/R4 type
of potential [42, 45] gives an efficient characterization of the phase shifts δlg,u , leading
to an efficient characterization of the resonant charge exchange process.
Specifically, for a potential with a long-range behavior of V ∼ −Cn /Rn , the
tangent of the phase shift is given in QDT by [40]
c
tan δl = Zgc
K c − Zfc c
c
Kc
Zfc s − Zgs
−1
.
(3.7)
c
Here K c (, l) is a dimensionless short-range K c matrix [40]. Zxy
(s , l) are universal
QDT functions for −1/Rn types of potentials. They are functions of the angular
momentum l and a scaled energy s = /sE , where sE = (~2 /2µ)(1/βn )2 is the
energy scale and βn = (2µCn /~2 )1/(n−2) is the length scale for the −Cn /Rn potential.
24
c
Explicit expressions for Zxy
, applicable to the polarization potential, are given in the
Appendix A. As explained in Ref. [40], Eq. (3.7) includes not only the effect of longrange phase shift, but also effects of quantum reflection and tunneling, which are the
key differences between long-range potentials with n > 2 and those with n < 2.
The application of QDT allows the description of resonant charge exchange in
terms of the C4 coefficient, equivalently the atomic polarizability α1 , and two shortrange K matrices, Kgc (, l) for the gerade state and Kuc (, l) for the ungerade state.
Such a description is exact if the energy and the partial wave dependences of the K c s
are fully accounted for. More importantly, QDT allows for efficient parametrizations
of resonant charge exchange by taking advantage of the fact that the short-range K c
matrices depend not only weakly on energy, but also weakly on the partial wave l
for both atom-atom and ion-atom interactions [37, 40, 42]. Through an example of
Na+ +Na, we show that even the simplest parametrization, corresponding to ignoring
the and l dependences of the K c s completely, provides an accurate description of
resonant charge exchange over a wide range of energies, including energies at which
tens of partial waves contribute.
3.3
Three-parameter QDT implementation
A three-parameter parametrization of resonant charge exchange for a 1 S+2 S system results from ignoring both the energy and the partial wave dependences of
Kgc (, l) and Kuc (, l). Specifically, it corresponds to the approximation of Kgc (, l) ≈
Kgc ( = 0, l = 0) and Kuc (, l) ≈ Kuc ( = 0, l = 0). Using Kgc and Kuc as the shorthand notation for the resulting constant K c s, we have one of the three-parameter
parametrizations for resonant charge exchange, with two short-range parameters Kgc
and Kuc , characterizing the short-range ion-atom interaction, and one long-range parameter, C4 or the atomic polarizability α1 , characterizing the strength of the long25
range interaction.
A mathematically equivalent three-parameter parametrization is in terms of two s
wave scattering lengths, agl=0 and aul=0 , for the gerade and the ungerade state, respectively, and the atomic polarizability α1 . This is derived from the first parametrization
by noting that the K c ( = 0, l = 0), for both g and u states, are related rigorously to
the corresponding s wave scattering lengths by [38, 39]
c
K (0, 0) + tan(πb/2)
2b Γ(1 − b)
,
al=0 /βn = b
Γ(1 + b) K c (0, 0) − tan(πb/2)
(3.8)
where b = 1/(n − 2). It reduces to, for n = 4,
al=0 /β4 =
K c (0, 0) + 1
.
K c (0, 0) − 1
(3.9)
We note that in the context of the effective range theory [113, 10, 9, 101], the
three parameters, agl=0 , aul=0 , and α1 , can only be expected to describe ion-atom
interactions in the ultracold regime as characterized by sE , in which only the
s wave makes a significant contribution. The QDT for ion-atom interactions asserts
that the very same set of parameters can in fact describe an ion-atom interaction over
a much wider range of energies, of the order of 105 sE , as tested in the next section
for
23
Na and expected to be qualitatively the same for all alkali-metal atoms.
The two equivalent parametrizations are complementary in terms of the physical
understanding that they provide. The parametrization using Kgc , Kuc , and α1 gives
more direct insight as to why it works over a wide range of energies. It is because Kgc
and Kuc are short-range parameters that are both insensitive to and l [37, 40, 42].
The parametrization using agl=0 , aul=0 , and α1 enforces the concept that the understanding of ultracold interactions immediately provides understanding of interactions
over a much wider range of energies through QDT. This is because embedded in the
knowledge of the scattering lengths, agl=0 and aul=0 , are the knowledge of the Kgc and
26
Kuc parameters, through Eq. (3.8).
3.4
The example of Na++Na
Low energy Na+ +Na charge exchange for 23 Na has been studied in detail by Côté
and Dalgarno in Ref. [21], within the elastic approximation. It serves as a prototypical
system to test the QDT formulation for resonant charge exchange, in particular the
range of validity of the three-parameter description.
In Section 3.4.1, we make a preliminary evaluation of the QDT description by
showing, visually, that a three-parameter QDT parametrization, using parameters
as given in Ref. [21], reproduces the cross sections of Ref. [21], including all the
resonance structures, without any knowledge of the short-range potential. A more
detailed comparison is not possible since Ref. [21] made use of unpublished potential
energy results by Magnier et al. in the short-range that are unavailable to us.
For a more detailed comparison between fully quantum numerical calculations and
2 +
three-parameter QDT results, we construct in Section 3.4.2 the 2 Σ+
g and Σu potential
curves for 23 Na+
2 using the same procedure as prescribed in Ref. [21], except by using
the later published results of Magnier et al. [91] in the short range. These potentials
are meant to be as close to those of Ref. [21] as possible. Fully quantum numerical
calculations of cross sections are carried out with these potentials and compared to
the results of corresponding three-parameter QDT descriptions. The comparison of
the QDT results with parameters from our potential and from the earlier results of
Côté and Dalgarno [21] are presented in Section 3.4.3 to illustrate the sensitivity of
the phase shift calculations to the PES.
In both sets of calculations, we take the sodium static dipole polarizability to be
α1 = 162.7 atomic unit (a.u.) [26], the same as that adopted by Côté and Dalgarno
2 1/2
[21]. The corresponding length scale for 23 Na+
= 1846 a0 , where
2 is β4 = (2µC4 /~ )
27
a0 is the Bohr radius, and the corresponding energy scale is sE /kB = 2.21 µK or
sE /h = 46.05 kHz.
3.4.1
Comparison of QDT results with previous results
2 +
Numerical results of the “molecular” cross sections for 2 Σ+
g and Σu states, as
defined by Eq. (3.3), are given for energies ranging from 10−16 a.u. to 1 a.u. in
Ref. [21]. The reference also gives the zero energy s wave scattering lengths for 2 Σ+
g
and 2 Σ+
u states,
agl=0 = 763.3a0 ,
(3.10a)
aul=0 = 7721.4a0 .
(3.10b)
These s wave scattering lengths, plus the Na polarizability of α1 = 162.7 a.u. [26],
give us all the parameters required for a three-parameter QDT description of resonant
charge exchange, from which all relevant cross sections can be calculated, without
detailed knowledge of the potentials.
Specifically, from the s wave scattering lengths of Eq. (3.10), we first calculate,
using Eq. (3.9), the short range K c parameters Kgc and Kuc , and obtain
Kgc = −2.4095 ,
(3.11a)
Kuc = 1.6286 .
(3.11b)
In the three-parameter QDT description, they are taken as constants applicable at all
energies and for all partial waves. These parameters, together with the QDT equation
for the phase shift, Eq. (3.7), give us all phase shifts and all cross sections. The results
2 +
for the total and partial “molecular” cross sections for the 2 Σ+
g and the Σu states
are illustrated in Figs. 3-1 and 3-2, respectively. To compare them with the results
28
8 .0
7
M o le c u la r c r o s s s e c tio n ( u n its o f 1 0 a
0
2
)
2 .0
7 .0
to ta l
l= 0
l= 1
l= 2
l= 3
l= 4
l= 5
6 .0
5 .0
4 .0
1 .0
3 .0
0 .0
1 0
-1 6
1 0
-1 2
1 0
-6
1 0
-8
1 0
-4
2 .0
1 .0
0 .0
1 0
-1 6
1 0
-1 4
1 0
-1 2
1 0
-1 0
1 0
-8
1 0
-4
1 0
-2
1 0
0
E n e rg y (a .u .)
(a) Results from QDT calculation.
(b) Results from Ref. [21].
Figure 3-1: Comparison of the total and the partial “molecular” cross sections for
the gerade state of Na+
2 from the QDT calculation using parameters from Ref. [21]
(3-1a) and from Ref. [21] (3-1b).
29
)
8 .0
M o le c u la r c r o s s s e c tio n ( u n its o f 1 0 8 a
0
2
0 .1 0
7 .0
6 .0
0 .0 5
5 .0
4 .0
to ta l
l= 0
l= 1
l= 2
l= 3
l= 4
l= 5
3 .0
2 .0
1 .0
0 .0
1 0
-1 6
1 0
-1 4
0 .0 0
1 0
1 0
-1 2
1 0
-1 0
-1 6
1 0
1 0
-8
-1 2
1 0
1 0
-6
1 0
-4
-8
1 0
1 0
-2
-4
1 0
0
E n e rg y (a .u .)
(a) Results from QDT calculation.
(b) Results from Ref. [21].
Figure 3-2: Comparison of the total and the partial “molecular” cross sections for
the ungerade state of Na+
2 from the QDT calculation using parameters from Ref. [21]
(3-2a) and from Ref. [21] (3-2b).
30
from Ref. [21], the corresponding results shown in Figs. 2 and 3 of Ref. [21] are also
included in the figures, which are visually nearly identical to our results. All detailed
features of the cross sections that are visible on the figures are found to be at the
right places judged by visual examination. The results show, at least tentatively, that
the three-parameter QDT description can provide an accurate account of resonant
charge exchange over a wide range of energies, including all the complex structures
which in this case are shape resonances from a wide range of partial waves.
3.4.2
Comparison of QDT results with current numerical results
For a more detailed comparison between fully quantum numerical results and
2 +
three-parameter QDT calculations, we construct here a version of the 2 Σ+
g and Σu
potential curves for Na+
2 . Numerical results are calculated using these potentials and
compared to the QDT results corresponding to the same potentials.
3.4.2.1
Potential energy curves adopted
+
2 +
For both 2 Σ+
g and Σu states of Na2 , we use the ab initio data of Magnier et
al. [91] ranging from 5.0 a0 to 20.0 a0 . Outside of this region, the potentials are
extended using the same procedure as prescribed in Ref. [21], in the hope of getting
potentials as close to those of Ref. [21] as possible. Specifically, for distances larger
than 22.0 a0 , we extended the potential by the asymptotic form of [21]
Vg,u (R) = Vdisp (R) ∓ Vexch (R) ,
31
(3.12)
1 0 .0
0 .3
E n e r g y ( u n its o f 1 0
-2
a .u .)
8 .0
6 .0
0 .0
4 .0
-0 .3
2 .0
1 4
1 6
1 8
2 0
2 2
2 4
2 6
2 8
0 .0
-2 .0
-4 .0
0
4
8
1 2
1 6
2 0
2 4
2 8
3 2
2
Σg
2
Σu
3 6
+
+
4 0
D is ta n c e R ( a 0 )
(a) Potential energy curves adopted in this work.
(b) Potential energy curves from Ref. [21].
Figure 3-3: Comparison of the BO potential energy curves adopted in this work (3-3a)
and from Ref. [21] (3-3b) for gerade (solid lines) and ungerade (dashed lines) states
of Na+
2.
32
2 +
with ∓ for 2 Σ+
g and Σu , respectively. The dispersion term and exchange term are
given by [21]
C4
C6
C8
− 6− 8 ,
4
R
R R B
1
ARa e−bR 1 +
.
Vexch (R) =
2
R
Vdisp (R) = −
(3.13)
(3.14)
Except for different notations 1 , all coefficients are taken to be the same as in Ref. [21],
which, in atomic units, are given by C4 = α1 /2 = 81.35, C6 = 936.5, C8 = 27069.5,
A = 0.111, a = 2.254, b = 0.615, and B = 0.494. Potential energies between 20.0a0
and 22.0a0 are interpolated using a cubic spline [102] to make a smooth connection
between the ab initio data and the long-range behavior. The same cubic spline is also
used to interpolate data points within the range of ab initio data. At short distances
(R < 5.0a0 ), we extended the potential with an exponential wall as in Ref. [21]
V (R) = W exp(−wR) ,
(3.15)
∂ ln V (R) .
W = V (R) exp(wR)|5.0a0 , w = −
∂R 5.0a0
(3.16)
with
+
2 +
The resulting 2 Σ+
g and Σu potentials for Na2 , thus constructed, are illustrated in
Fig. 3-3, along with the potential used in Ref. [21] from Fig. 1 of the paper. The two
sets of potentials have the same long-range behavior and differ only slightly in the
short range due to slightly different ab initio data adopted.
3.4.2.2
Comparison of results
For the QDT calculations, we first calculate the parameters Kgc and Kuc , more
specifically the Kgc ( = 0, l = 0) and Kuc ( = 0, l = 0) from the potentials. The radial
1
Our C4 , C6 , and C8 are denoted as C4 /2, C6 /2, and C8 /2 in Ref. [21]
33
8
1 0
7
1 0
6
1 0
5
C h a r g e e x c h a n g e c r o s s s e c tio n ( u n its o f a
0
2
)
1 0
N u m e ric a l re s u lt
Q D T re s u lt
1 0
4
1 0
-1 0
1 0
-9
1 0
-8
1 0
-7
1 0
-6
1 0
-5
E n e rg y /k
B
1 0
-4
1 0
-3
1 0
-2
1 0
-1
1 0
0
(K )
Figure 3-4: Charge exchange cross sections of Na+ +Na obtained from a threeparameter QDT description (dashed line) and from numerical calculations (solid line).
34
8
1 0
7
1 0
6
1 0
5
T o ta l c r o s s s e c tio n ( u n its o f a
0
2
)
1 0
N u m e ric a l re s u lt
Q D T re s u lt
1 0
4
1 0
-1 0
1 0
-9
1 0
-8
1 0
-7
1 0
-6
1 0
-5
E n e rg y /k
B
1 0
-4
1 0
-3
1 0
-2
1 0
-1
1 0
0
(K )
Figure 3-5: Total cross sections of Na+ +Na obtained from a three-parameter QDT
description (dashed line) and from numerical calculations (solid line).
35
wave function is matched to
ul (r) = Al [fcs l (rs ) − K c (, l)gcs l (rs )] ,
(3.17)
at progressively larger R until the resulting K c converges to a constant to the desired
accuracy [37, 38, 40]. Here f c and g c are the zero-energy QDT reference functions for
the −1/R4 potential [39, 40]. We obtain
Kgc = −1.5953 ,
(3.18a)
Kuc = 0.25416 .
(3.18b)
From the K c s, the s wave scattering lengths can be obtained by substitution into
Eq. (3.9). We obtain
agl=0 = 423.51a0 ,
(3.19a)
aul=0 = −3104.8a0 .
(3.19b)
We note in passing that this method of calculating the scattering length converges
at much smaller R and provides more accurate results than by matching to the freeparticle solutions or by matching the phase shift to the effective range expansion
[101, 21], especially for cases with al=0 β4 .
Our numerical calculations of the phase shifts and cross sections are carried out
using a log-derivative method [71, 94]. Figure 3-4 shows the comparison of the charge
exchange cross sections obtained from numerical calculations and from the threeparameter QDT description. Figure 3-5 shows a similar comparison of the total cross
sections. They both show that the QDT description and the numerical results are
in excellent agreement for energies below 0.2 mK. For energies between 0.2 mK and
0.1 K, the QDT prediction still works well with the only discernible differences being
36
9
1 0
8
1 0
7
1 0
6
1 0
5
1 0
4
C h a r g e e x c h a n g e c r o s s s e c tio n ( u n its o f a
0
2
)
1 0
P o te n tia l I
P o te n tia l II
1 0
-1 0
1 0
-9
1 0
-8
1 0
-7
1 0
-6
1 0
E n e rg y /k
B
-5
1 0
-4
1 0
-3
1 0
-2
1 0
-1
(K )
Figure 3-6: Charge exchange cross sections of Na+ +Na obtained from three-parameter
QDT descriptions using parameters corresponding to the potential of Ref. [21] (solid
line) and using parameters corresponding to our potential (dashed line).
due to shape resonances in high partial waves. A better QDT description of such
resonances is possible, and will be discussed in later chapters. Overall, the QDT
prediction is satisfactory below 0.1 K, or about 105 sE . For energy higher than 0.1 K,
the discrepancy between the two results grows larger, mainly due to the energy and
partial wave dependences of the short-range parameters.
3.4.3
Comparison of results of different potentials
The considerable differences between the scattering lengths for our potentials, as
given by Eq. (3.19), and for the potentials of Ref. [21], as given in Eq. (3.10), are
37
)
2
0
T o ta l c r o s s s e c tio n ( u n its o f a
1 0
9
1 0
8
1 0
7
1 0
6
1 0
5
1 0
4
P o te n tia l I
P o te n tia l II
1 0
-1 0
1 0
-9
1 0
-8
1 0
-7
1 0
-6
1 0
-5
E n e rg y /k
B
1 0
-4
1 0
-3
1 0
-2
1 0
-1
1 0
0
(K )
Figure 3-7: Total cross sections of Na+ +Na obtained from three-parameter QDT
descriptions using parameters corresponding to the potential of Ref. [21] (solid line)
and using parameters corresponding to our potential (dashed line).
38
illustrations of the sensitive dependence of cold or ultracold ion-atom interactions
on the short-range potential. The two sets of potentials have the same long-range
behaviors as characterized by Eqs. (3.12)-(3.14), and differ only slightly in the short
range, to a degree that is visually almost indistinguishable on the scale of Fig. 33. Figures 3-6 and 3-7, which compare QDT results for the two sets of potentials,
give a more complete picture of this dependence. They show that the interactions
in the ultracold regime are the most sensitive to the short-range potential. Beyond
the ultracold regime of ∼< sE , the shape resonance positions remain sensitive to
the potential and the QDT parameters over a considerable wider range of energies, of
the order of 1 mK or about 1000sE . While the sensitive dependence of the ion-atom
interaction on the potential gradually diminishes at higher energies for the total cross
section, it remains to a considerable degree for the charge exchange cross section.
Fortunately, the sensitive dependence of the ion-atom interaction on the PESs is
fully encapsulated in a few (two) QDT parameters, as shown in Figs. 3-4 and 3-5.
Instead of from the PESs, this small number of parameters can be determined from
a few experimental data points, such as two resonance positions for Na+ +Na, or two
binding energy measurements of highly vibrationally excited Na+
2 . Such a determination, in a similar manner as illustrated earlier for atom-atom interactions [36, 37],
is further facilitated by the concept of a universal spectrum including the concept of
a universal resonance spectrum introduced in Ref. [42] for the −1/R4 potential.
3.4.4
Discussion
The example of Na+ +Na charge exchange shows excellent agreement between the
QDT parametrization and numerical results from 0 K all the way through 0.1 K,
including all resonances within this range. To put this energy range into perspective,
we note that /kB = 0.1 K corresponds roughly to s = /sE ∼ 105 . At this energy,
√ 1/4
one can estimate that there are at least 2s ∼ 25 partial waves contributing to
39
the cross sections. Such a simple parametrization over such a wide range of energies
is made possible in QDT not only by the energy insensitivity of the short-range
parameters, Kgc and Kuc , but also by their partial wave insensitivity [37, 40, 42]. The
energy insensitivity is ensured here by the large length scale separation as is typical
for ion-atom interactions. More quantitatively, it is reflected in β6 /β4 ≈ 6.56 × 10−3
for Na, where β6 = (2µC6 /~2 )1/4 is the length scale associated with the −C6 /R6 term
of the potential in Eq. (3.13). This value gives an order-of-magnitude measure of the
length scale separation that is representative of all alkali-metal atoms. The partial
wave insensitivity is ensured by the combination of length scale separation and the
smallness of the electron-to-nucleus mass ratio [37, 39].
Also illustrated in this example is the sensitive dependence of cold or ultracold
ion-atom interactions on the PESs. While one can always construct the potentials for
ion-atom systems, their accuracies are generally far from sufficient in predicting cold
collisions, and the related highly vibrationally excited molecular-ion spectrum [42].
The QDT deals with this difficulty by encapsulating this sensitive dependence into a
few short-range parameters that can be determined experimentally. The simplicity
of the resulting description has the following implications. (a) Since there are only
a few parameters, they can be determined from very few experimental data points.
(b) Since the parametrization works over a wide range of energies, it allows the determination of the parameters from measurements of structures (either resonance or
binding energy) away from the threshold, where they are much further separated [42]
and can be resolved with much less stringent requirements on either the energy resolution or the temperature [59]. This can be an important consideration for ion-atom
interactions, where going below millikelvin has proven to be difficult experimentally.
(c) Such a parametrization offers a systematic understanding of a class of systems.
For example, the parametrization for Na+ +Na works the same for all resonant charge
exchange processes of the type of 1 S+2 S. Different systems differ only in scaling as
40
determined by the atomic polarizability α1 , and the two short-range parameters. (d)
The parameters that are used to characterize the interaction in the absence of any
external field also characterize the interaction in the presence of external fields, thus
relating field-free interactions to interactions within a field [46, 54].
The QDT results in this example represent only the simplest QDT description for
resonant charge exchange, with a goal of establishing key qualitative features that
form the conceptual foundation for further theoretical development. Rigorous treatments of hyperfine effects, more accurate treatment of the partial wave dependence of
the short range parameters, especially for high partial waves, which give rise to only
deviations of any significance in Figs. 3-4 and 3-5, will be presented in later chapters. We point out that the existing semiclassical approximation [21], while helpful
qualitatively, is not accurate quantitatively.
3.5
Chapter summary
In conclusion, we have presented the application of the single-channel QDT for the
−1/R4 polarization potential to the resonant charge exchange problem with the elastic approximation. We have shown that resonant charge exchange of the type of 1 S+2 S
can be accurately described over a wide range of energies with only three parameters, which can either be two short-range K c matrices, Kgc and Kuc , and the atomic
polarizability α1 , or two s wave scattering lengths, agl=0 and aul=0 , and α1 . Since
the polarizability is well known for most atoms, this is effectively a two-parameter
description. Everything else is described by analytic QDT functions for the −1/R4
polarization potential (see, Ref. [42, 45] and Appendix A).
Comparing to the conventional method of numerical calculation using PESs, the
QDT formulation reflects more of the underlying physics that both the energy dependence [36] and the partial wave dependence [37] of the ion-atom interaction around
41
a threshold are due to the long-range potential. This helps overcome the sensitive
dependence of cold atomic interactions on short-range PESs shown in the example of
the resonant charge exchange of Na+ +Na.
The single-channel QDT will serve as the foundation of the development of the
multichannel QDT in latter chapters.
42
Chapter 4
Multichannel quantum theory for
ion-atom interactions
4.1
Background and introduction
The QDT for atom-atom [96, 16, 36, 37, 46, 40] and ion-atom interactions [68,
42, 69, 88, 45], as we have demonstrated in the previous chapter, has served to establish the existence of universal behaviors far beyond the ultracold regime in atomic
interactions. They imply universal behaviors, over a wide range of temperatures, in
processes such as molecule formulation involving ions
A+ + A + A → A+ + A2 ,
→ A + A+
2 .
(4.1a)
(4.1b)
For the hydrogen atom, these 3-body reactions constitute mechanisms for H2 formation that are not yet fully understood, and can potentially change the prevailing
belief that H2 in the interstellar medium is formed mainly on grain surfaces [122]. For
the Rb atom, process (4.1a) for the formation of Rb2 has been observed in a recent
cold-ion experiment [56].
43
While the prospects of few-body theories built upon better understanding of pairwise interactions would seem clear and straightforward [30, 90, 4, 86], their actual
implementation and success depend critically not only on the accuracy of the underlying two-body theories, but also on their efficiency and simplicity, especially in their
descriptions of resonances. Two-body theories of such characteristics have not been
fully established beyond the ultracold regime for fundamental atomic interactions.
They are highly non-trivial for systems with fine or hyperfine structure because of
their multichannel characteristics, and especially so for ion-atom interaction because
of its rapid energy dependence and generally large number of contributing partial
waves [21, 88, 45] as we have witnessed in Chapter 3.
In this chapter, we present a MQDT for ion-atom interactions, which we hope
to establish as the ion-atom component of future few-body theories involving ions.
We demonstrate the application of MQDT to resonant charge exchange of group
I, II, and helium atoms. Following the work of the previous chapter, we look at
the Na+ +Na charge exchange problem again, but with hyperfine structure included
instead of applying the elastic approximation. Via the example of resonant charge
exchange, we illustrate the complexity of cold ion-atom interactions and show how
such complexity can be described efficiently and quantitatively using MQDT. For
the seemingly simple process of resonant charge exchange, earlier theories using the
elastic approximation [21, 13, 127] have not fully accounted for the effects of hyperfine
structure, limiting their range of applicability to approximately 1 K and above. We
show that a proper treatment of hyperfine structure leads to qualitatively different
behavior for cold ion-atom interactions including a change of threshold behavior for
hyperfine-changing collisions. We further show that the small energy scale associated
with ion-atom interactions [42, 88, 45] is such that there exists a large number of
resonances: a collection of shape, Feshbach, and diffraction resonances [42, 45], even
within the small energy interval of hyperfine splitting (∼ 0.1 K). We show how such
44
complexity is fully characterized using MQDT with a small number of parameters.
4.2
Theoretical framework
Consider the interaction of an atom (group I , II, or He) of nuclear spin I1 in its
ground electronic state with an ion of identical nucleus (I2 = I1 ) also in the ground
electronic state. At low energies, the relevant processes, including elastic, m-changing,
and hyperfine-changing processes, can be described, for group I atoms, by
A(F1i , M1i ) + A+ (F2 , M2i ) → A(F1j , M1j ) + A+ (F2 , M2j ) ,
(4.2)
and, for group II atoms or He, by
A+ (F1i , M1i ) + A(F2 , M2i ) → A+ (F1j , M1j ) + A(F2 , M2j ) .
(4.3)
Here F1 = I1 ±1/2 is the total angular momentum corresponding to the 2 S atomic (or
ionic) electronic state, and F2 = I2 = I1 is the total angular momentum corresponding
to the 1 S state. The M s are the corresponding magnetic quantum numbers. The subindices i and j refer to the internal states before and after the collision. Notice that
this process can alter the magnitude of the asymptotic total angular momentum of
the atom, as well as the corresponding magnetic quantum numbers of the atom and
the ion. The asymptotic total quantum angular momentum of the ion F2 will not
change.
The total Hamiltonian describing the system of interest can be written as
H=−
~2 2
O + HBO + Hhf ,
2µ R
(4.4)
which does not have the spin-orbit term compared to the generic one in Eq. (2.1)
45
because the orbital angular momenta are zero for both colliding parties (S electrons).
In our discussion of a slow collision that only involves the S electron, there are two
relevant BO curves characterized by 2 Σ+
g,u (see Section 3.2.1 for detailed explanation).
How to treat the hyperfine term effectively and efficiently depends on the importance of the hyperfine interaction compared to the BO Hamiltonian and the kinetic
energy term in Eq. (4.4). When the two nuclei are well separated, the BO potential
energy is small and only gives one asymptotic threshold regardless of the final spin
states of the atom. When the kinetic energy is comparable to or smaller than the
hyperfine interaction energy, this physical picture is not complete and the hyperfine
term needs to be incorporated to address the correct asymptotic thresholds. On the
other hand, when the two nuclei are close, the BO term becomes dominant over the
hyperfine interaction which can be neglected. Since the hyperfine interaction is only
important when the two nuclei are well separated, it can be approximately replaced
with the asymptotic atomic hyperfine interaction.
The following subsections demonstrate some of the details of the theory. Section 4.2.1 describes the channel structure of the system with two different angular
momentum coupling schemes. The frame transformation between the two coupling
basis functions is also given here. Section 4.2.2 gives the scattering amplitude in terms
of the S matrix, with proper symmetry consideration of identical nuclei, and various
cross sections that can be derived from it. To obtain the S matrix, we present two
methods here: numerical calculation and MQDT. Section 4.2.3 shows the procedure
to construct the proper potential energy curves for numerical calculations and the
numerical method used in this work. MQDT is then introduced and explained in
detail in Section 4.2.4.
46
Table 4.1: Channel structure for ion-atom interactions of the type 2 S +1 S with
identical nuclei of spin I2 = I1 .
4.2.1
F
F F coupling {F1 , F2 }
JI coupling {J, I}
1/2 ≤ F ≤ 2I1 − 1/2
{I1 − 1/2, I1 }
{I1 + 1/2, I1 }
{1/2, F − 1/2}
{1/2, F + 1/2}
F = 2I1 + 1/2
{I1 + 1/2, I1 }
{1/2, 2I1 }
Channel structure and frame transformation
Since both the atom and the ion have zero electronic orbital angular momentum,
in other words S electrons, the total “spin” angular momentum F , and the relative
orbital angular momentum l of the nuclei, are decoupled due to restrictions on angular
momentum coupling and parity consideration. Therefore F and l are independently
conserved. For each l, the total number of states is 2(2I1 + 1)2 , and the total number
of curves is 4I1 + 1. These curves are separated into two uncoupled groups for the
different F s. For 1/2 ≤ F ≤ 2I1 − 1/2, there are two coupled channels, and both
elastic and inelastic including excitation and de-excitation scatterings can happen.
For F = 2I1 + 1/2, there is only one channel and only elastic scattering can happen.
To reflect proper symmetry properties in different regions of the internuclear separation, we adopt two angular momentum coupling schemes to define two sets of
channels, following the discussion in Chapter 2 and the theory of Ref. [35, 46]. For
the fragmentation channels that diagonalize the Hamiltonian in the long-range, we
use the F F coupling scheme where F = F1 + F2 . The scattering boundary conditions, hence the S matrix, are defined in the F F channels. Since there is no spin-orbit
coupling term in the Hamiltonian and the electronic orbital angular momenta of the
molecule as well as individual atoms are zero, LS coupling and JJ coupling are the
same for the sake of defining the condensation channel. Since the decomposition of F
in this scheme follows F = J + I, we will denote it the JI coupling. The JI coupling
47
basis forms the condensation channels that diagonalize the adiabatic BO Hamiltonian, hence approximately diagonalizes the total Hamiltonian in the short-range due
to the insignificance of the hyperfine interaction compared to the BO energies in that
region. The JI channels are directly associated with the BO curves 2 Σ+
g,u , and serve
as the channels in which the short-range K c matrix is defined to be used in the MQDT
formulation. The detailed channel structure is illustrated in Table 4.1.
For 1/2 ≤ F ≤ 2I1 − 1/2, the asymptotic thresholds E1 and E2 of the two coupled
channels in the F F coupling basis are separated by the atomic hyperfine splitting
E2 − E1 = ∆E hf which is usually well known from experiments. Following the theory
of Ref. [35], the F F and the JI coupling basis functions are related by a frame
transformation given by a two by two orthogonal matrix


p
p
2I1 + F + 3/2 
(−1)2F +1  − 2I1 − F + 1/2
UF = p
 p
 .
p
2(2I1 + 1)
2I1 + F + 3/2
2I1 − F + 1/2
(4.5)
The ordering of the channels here as well as in the matrices later in this paper follow
the ordering in Table 4.1.
4.2.2
Scattering amplitude and cross sections
The scattering amplitude for processes of Eqs. (4.2)-(4.3) that satisfies the scattering boundary condition in the F F channels are given by [35]
f ({F1i M1i , F2 M2i }ki → {F1j M1j , F2 M2j }kj )
X
2πi
Y ∗ (k̂ )Y (k̂ )
=−
1/2 lm i lm j
(k
k
)
i j
lmF M
× hF1j M1j , F2 M2j |F MF i S F l (E) − 1 ji
× hF MF |F1i M1i , F2 M2i i ,
48
(4.6)
where 1 is the unit matrix and S F l is the S matrix defined in the F F channels. ~ki,j
are the initial and the final relative momenta in the center-of-mass frame.
Notice that the subscript labelling of 1 or 2 in the scattering amplitude equation
refers to the individual quantum states, but not the nuclei, because the nuclei are
identical. Therefore, if we use f (i → j, kj ) as a short-hand notation for the amplitude
of Eq. (4.6), due to the indistinguishability of the nuclei, the differential cross section
of the process of i → j at angle k̂j has both contributions from f (i → j, kj ), for the
particle with state (F2 , M2j ) to be detected at that direction, and f (i → j, −kj ), for
the particle with state (F1j , M1j ) to be detected in the same direction. Thus, the
differential cross section that carefully takes account of the symmetry property of
identical nuclei is given by [35, 44]
dσ
({F1i M1i , F2 M2i }ki → {F1j M1j , F2 M2j }kj )
dΩj
kj 1
=
|f (i → j, kj )|2 + |f (i → j, −kj )|2 . (4.7)
ki 2
Notice that the differential cross section is the same for direction k̂j and −k̂j .
Many different cross sections can be derived from the differential cross section
in Eq. 4.7. In particular, the total cross sections for elastic (include m-changing)
collisions, and the hyperfine excitation or de-excitation processes, after averaging
over initial states and summing over final states, are given by
σ({F1i , F2 } → {F1j , F2 }) =
π
(2F1i + 1)(2F2 + 1)ki2
X
Fl
×
(2l + 1)(2F + 1)|Sji
− δji |2 .
(4.8)
Fl
At energies much higher than the hyperfine splitting, the hyperfine interaction
can be neglected and the cross sections above can be simplified, by taking ∆E hf to
zero, to cross sections given by the elastic approximation [23]. Specifically, the total
49
hyperfine de-excitation cross section simplifies to
∆E hf
σde −−−−−→
I1
π X
(2l + 1) sin2 (δlu − δlg ) ,
2
2I1 + 1 k l
(4.9)
where σde is short for σ({I1 + 1/2, I1 } → {I1 − 1/2, I1 }). The coefficient I1 /(2I1 + 1)
takes into account nuclear statistics. When comparing to the multichannel result,
the zero energy should be offset by the center-of-gravity, which is given by (I1 +
1)/(2I1 + 1) · ∆E hf . The corresponding hyperfine excitation cross section is related
to the de-excitation cross section by a detailed balance relation that is guaranteed by
time-reversal symmetry [85]. From Eq. (4.8), the relation is given by
I1 + 1 − ∆E hf ∆E hf I1 + 1
σex
=
·
−−−−−→
,
σde
I1
I1
(4.10)
where σex is short for σ({I1 − 1/2, I1 } → {I1 + 1/2, I1 }).
4.2.3
Potential energy curves and numerical method
The traditional and more common way to calculate the S matrix is by numerical
integration. To apply such a method, we need detailed BO potential energy curves
for the gerade and ungerade states and to construct the potential energy terms in
Eqs. (2.5a) and (2.5c) in the F F channels out of the BO potential energy curves.
For 1/2 ≤ F ≤ 2I1 − 1/2, there are two coupled channels for each F . The
hyperfine term can be approximated as diagonal and constant in the F F channels,
which is given by


0 
 0
V hf = 
 .
hf
0 ∆E
(4.11)
The BO potential energy matrix in the F F channels is given in terms of the matrix
50
in the JI channels with a frame transformation, as
V BO = U F † V BO(JI) U F .
(4.12)
The BO potential energy matrix in the JI channels is diagonal and can be written as


V BO(JI) = 

(Vg +Vu )+e2 (Vg −Vu )
2
0
0
(Vg +Vu )−e2 (Vg −Vu )
2

 ,
(4.13)
in which Vg,u are the two BO potential energy curves for the 2 Σ+
g,u molecular states
respectively, and e2 = (−1)F +l−1/2 .
For F = 2I1 + 1/2, there is only one channel that only opens when the collision
energy is above the upper hyperfine threshold. The hyperfine term V hf = ∆E hf ,
and the BO term is either given by Vg or Vu depending on whether e1 = (−1)2I1 +l is
positive or negative, which can be written as
V BO =
1
[(Vg + Vu ) + e1 (Vg − Vu )] .
2
(4.14)
In the multichannel numerical calculations employed in this work, the CC equations are integrated numerically using a hybrid propagator [93, 3] constructed similarly to the one used in the Hibridon scattering code [1]. It employs a modified
version of the log-derivative method of Johnson [71] by Manolopoulos [93] at short
range, and a modified version of the potential-following method of Gordon [50] by
Alexander and Manolopoulos [3] at long range. Convergence can be tested on the
resulting S matrix after being converted from the log-derivative matrix following the
method of Johnson [71].
51
4.2.4
Multichannel quantum-defect theory
The MQDT for ion-atom interactions consists of the formulation of Ref. [46] in
combination with the QDT functions for the −1/R4 -type potential as detailed in
Ref. [45]. It takes full advantage of the physics that both the energy dependence [36]
and the partial wave dependence [37] of the atomic interaction around a threshold
are dominated by effects of the long-range potential, which are encapsulated in the
universal QDT functions. The short-range contribution is isolated to a short-range
K c matrix that is insensitive to both the energy and the partial wave.
4.2.4.1
General formulation
For an N -channel problem and at energies where all channels are open, the physical
K matrix, in this case denoted as K F l , is given by MQDT as [46]
c
c
K F l = −(Zfc c − Zgc
K c )(Zfc s − Zgs
K c )−1 ,
(4.15)
c
c
c
where Zxy
s are N × N diagonal matrices with elements Zxy
(si , l) being the Zxy
functions ([45], Appendix A) evaluated at scaled energy si = ( − i )/sE relative to
the respective channel threshold i . i can take the value 1 to N corresponding to
the lowest to the highest channel. Here sE = (~2 /2µ)(1/β4 )2 and β4 = (µαA /~2 )1/2
are the characteristic energy and the length scales, respectively, associated with the
polarization potential, −αA /2R4 , defined in the same fashion as in Chapter 3. αA is
the static dipole polarizability of the atom.
At energies where No channels are open, and Nc = N − No channels are closed, it
gives [46]
c
c −1
c
c
) ,
Keff
)(Zfc s − Zgs
Keff
K F l = −(Zfc c − Zgc
52
(4.16)
where
c
c
c
c −1 c
Keff
= Koo
+ Koc
(χc − Kcc
) Kco ,
(4.17)
in which χc is a Nc × Nc diagonal matrix with elements χcl (si , l) ([45], Appendix A),
c
c
c
c
and Koo
, Koc
, Kco
, and Kcc
, are submatrices of K c corresponding to open-open, open-
closed, closed-open, and closed-closed channels, respectively, as defined in Eq. (2.15).
Equation (4.16) is formally the same as Eq. (4.15), except that the K c matrix is
c
replaced by Keff
that accounts for the effects of closed channels.
The conversion from K matrix to S matrix is given by Eqs. (2.16) and (2.15).
4.2.4.2
K c matrix and short-range parametrization
The short-range K c matrix has similar structure as the potential energy matrix
constructed previously for the numerical calculation. Only two slowly varying funcc
(, l), are needed. They are
tions of energy and l, the single-channel K c matrices Kg,u
directly related to their corresponding quantum defects by [45]
c
Kg,u
(, l) = tan[πµcg,u (, l) + π/4] .
(4.18)
For 1/2 ≤ F ≤ 2I1 − 1/2 where there are two coupled channels, the K c matrix in
the F F channels can be obtained from the one in the JI channels through a frame
transformation, given by
K c = U F † K c(JI) U F ,
where


K c(JI) = 
(Kgc +Kuc )+e2 (Kgc −Kuc )
2
(4.19)

0
(Kgc +Kuc )−e2 (Kgc −Kuc )
0
2
53

 .
(4.20)
For F = 2I1 + 1/2 where there is only one channel, K c is given by
Kc =
1 c
(Kg + Kuc ) + e1 (Kgc − Kuc ) .
2
(4.21)
In the simplest MQDT implementation, similar to the single-channel QDT implementation in Chapter 3, instead of two full potential curves used in the numerical
calculation, only three constant parameters are needed besides the hyperfine splitting
and the reduced mass. The static dipole polarizability of the neutral atom characterizes the long range part of the potential. The two QDT parameters, the zero energy
c
(0, 0), characterize the short range part
zero angular momentum single-channel Kg,u
of the potential due to the energy and partial wave insensitive nature of the short
range interaction. They are related to the corresponding s wave scattering lengths
by Eq. (3.9) [38, 39].
More accurate results over a greater range of energies can be obtained by incorporating the energy dependence, and especially, for the range of energies under
consideration, the partial wave dependence of the short-range parameters. These
weak dependences are well described by expansions
µcg,u (, l) ≈ µcg,u (0, 0) + bµg,u + cµg,u [l(l + 1)] ,
(4.22)
in which the parameters bµg,u and cµg,u characterize the energy and the partial wave
dependences of the quantum defects for the gerade and ungerade states, respectively.
They can be determined easily through single-channel calculations at a few energies
and for a few partial waves.
4.2.4.3
Resonance structure
Ion-atom interactions usually have many resonances due to the relatively small
energy scale. They correspond to singularities in the physical K matrix. From
54
Eqs. (4.16), the resonance locations when all channels are open satisfy
χ
ecl − K c = 0 ,
(4.23)
c −1 c
Zf s [45]. For resonance locations where there are channels closed,
where χ
ecl = Zgs
they satisfy
c
χ
ecl − Keff
=0,
(4.24)
where χ
ecl is defined in the same way in the open channels.
Specific to our two-channel charge exchange problem, we are able to further characterize the resonances within the hyperfine splitting. The cross sections at these
resonance locations can reach the unitarity limit where |S F l − I| = 2. The locations
of the resonances are determined by Eq. (4.24) which we denote as sl . Near these
positions, the K matrix can be characterized by
Fl
K F l (s ) = Kbg
(s ) −
1 Γsl
,
2 s − sl
(4.25)
Fl
where Kbg
(s ) is the background and Γsl is the scaled width which is related to the
width Γl by Γsl ≡ Γl /sE . They are similar in form to the single-channel ones defined
c(F F )
c
in Ref. [?], but have Keff
in place of K c . If we treat KF l
to be independent of
energy, which is almost exact within the hyperfine splitting, the scaled width can be
given explicitly by
2
Γsl = −
c ( , l) 2
Zgs
sl
de
χcl ds +
s1l
,
c
c Kc
Koc
co dχl c )2 d (χcl −Kcc
s
(4.26)
s2l
where s1l and s2l are the scaled resonance positions relative to the lower and the
upper thresholds, respectively. A positive scaled width corresponds to a time delay
and enhanced density-of-state [45], which can be viewed, in the time-dependent pic55
ture, as the two scattering particles spend more time close together. On the other
hand, a negative scaled width corresponds to a time advance and reduced densityof-state [42, 45]. The magnitude of the width is inversely proportional to the time
change.
We classify the resonances within the hyperfine splittings into three categories:
Feshbach, shape, and diffraction. Resonances of negative widths are diffraction resonances that correspond to reductions of density-of-states [42, 45] and tend to be
broad. Feshbach and shape resonances have positive widths and tend to be narrow.
They may play a significant role in the three-body interaction rate by producing a
relatively long time delay for the two-body scattering process, thereby enhancing the
chance of three bodies to come together. The difference between Feshbach and shape
resonances lies at the origin of the resonances. Feshbach resonances are the ones
where the continuum is coupled to the would-be bound states of the closed channel.
The locations of the would-be bound states, which would correspond to bound states
if the closed channel is not coupled to the open channel, are called the bare (unshifted)
locations of Feshbach resonances. They are determined by the solutions of
c
χcl − Kcc
=0.
(4.27)
The same equation also gives the bound spectrum of true bound states, at energies
where all channels are closed [?]. Shape resonances are the ones where the continuum
is coupled to the semi-bound states within the centrifugal barriers in the open channel.
Thus, their locations are solutions of Eq. (4.24) that do not have a nearby solution
of Eq. (4.27).
Equations (4.23), (4.24), (4.25) and (4.26) are part of the multichannel generalizations of the concept of a resonance spectrum and the corresponding width function
[45]. They describe the pole structures of the physical K matrix that has to be
56
understood effectively in applications beyond two-body physics [90, 4, 86].
4.3
The example of Na++Na with hyperfine interaction
We illustrate our theory with the
nuclear spin of
23
23
Na+ +
23
Na charge exchange process. The
Na is I1 = 3/2. Therefore, the total number of channels per partial
wave is 4I1 + 1 = 7, corresponding to two channels each for F = 1/2, 3/2, and 5/2,
and one channel for maximum F = 7/2. For 1/2 ≤ F ≤ 2I1 − 1/2, the two channels
in the F F coupled basis are separated by the atomic hyperfine splitting between
the F1 = 1 and 2 states of the neutral atom, given by ∆E hf /h ≈ 1771.6 MHz [5]
(∆E hf /kB ≈ 0.08502 K). The asymptotic long range potential is characterized by the
static atomic dipole polarizability αA = 162.7 a.u. [27].
We compare the results from MQDT with the ones from numerical calculations
from zero energy to 3 K. The BO potential energy curves used to construct the
potential matrix are the same as our single-channel QDT work given in Section 3.4.2.1.
4.3.1
Baseline results from the simplest MQDT parametrization
In the simplest MQDT parametrization, we ignore both the energy and the partial
wave dependences of the short-range parameters, corresponding to the approximation
c
c
of Kg,u
(, l) ≈ Kg,u
( = 0, l = 0), or equivalently, µcg,u (, l) ≈ µcg,u ( = 0, l = 0), where
c
Kg,u
(0, 0) and µcg,u (0, 0) are related to the s wave scattering lengths agl=0 for the g
state and aul=0 for the u state [45]. We call results from this simplest parametrization
the baseline MQDT results.
For Na+ +Na, the baseline MQDT results are obtained using Kgc (0, 0) = −1.5953
57
E n e rg y /h (M H z )
1 0
6
1 0
5
1 0
4
3
1 0
4
{2 , 3 /2 } to {1 , 3 /2 }
2
T o ta l C r o s s - s e c tio n ( a 0 )
1 0
M Q D T
N u m e r ic a l
1 0
3
1 0
-1
1 0
E n e rg y /k
B
0
(K )
Figure 4-1: Baseline MQDT results (solid line) and numerical results (dashed line)
for the total hyperfine de-excitation cross sections of Na+ +Na from channel {F1 =
2, F2 = 3/2} to channel {F1 = 1, F2 = 3/2}. The vertical dotted line identifies the
upper hyperfine threshold at 0.08502 K.
58
E n e rg y /h (M H z )
1 0
1 0
-1
1 0
0
1 0
1
1 0
2
1 0
3
1 0
4
8
1 0
{1 , 3 /2 } to {1 , 3 /2 }
2
T o ta l C r o s s - s e c tio n ( a 0 )
-2
1 0
7
1 0
6
M Q D T
N u m e r ic a l
1 0
5
1 0
-6
1 0
-5
1 0
-4
1 0
-3
E n e rg y /k
B
1 0
-2
1 0
-1
1 0
0
(K )
Figure 4-2: Baseline MQDT (solid line) results and numerical results (dashed line)
for the total elastic cross sections of Na+ +Na in channel {F1 = 1, F2 = 3/2}. The
vertical dotted line identifies the upper hyperfine threshold at 0.08502 K.
59
E n e rg y /h (M H z )
1 0
1 0
1
1 0
2
1 0
3
1 0
4
{1 , 3 /2 } to {1 , 3 /2 }
l = 5 ; F = 5 /2
6
1 0
2
P a r tia l C r o s s - s e c tio n ( a 0 )
0
1 0
5
1 0
4
1 0
3
U n
ita
r it
y
L i
m
it
M Q D T
N u m e r ic a l
1 0
2
1 0
-5
1 0
-4
1 0
-3
-2
1 0
E n e rg y /k
B
1 0
-1
1 0
0
(K )
Figure 4-3: Baseline MQDT (solid line) results and numerical results (dashed line)
for the partial elastic cross sections of Na+ +Na for l = 5 and F = 5/2 in channel
{F1 = 1, F2 = 3/2}. The vertical dotted line identifies the upper hyperfine threshold
at 0.08502 K.
60
and Kuc (0, 0) = 0.25416 from Eq. (3.18) from the previous chapter calculated using
our potential energy curves. They correspond to µcg (0, 0) = 0.42823 and µcu (0, 0) =
0.82922, or agl=0 = 423.51 a.u. and aul=0 = −3104.8 a.u., for the g and the u state,
respectively. Figure 4-1 depicts the baseline MQDT results for the total hyperfine deexcitation cross section and its comparison with numerical result. Figure 4-2 depicts
a similar comparison for the total elastic cross section in the lower channel. Figure 4-3
depicts the baseline partial elastic cross section for l = 5 and F = 5/2 in the lower
channel {F1 = 1, F2 = 3/2}, and its comparison with numerical results. They show
that the simplest MQDT parametrization gives good descriptions of the Na+ +Na interaction up to around 0.4 K, except for resonances from high partial waves. Beyond
0.4 K, the l dependence begins to have more substantial effects. At higher energies,
such as 10 K and beyond, the energy dependence of the short-range parameter, induced by interactions of shorter range than the polarization potential, will also come
into play. More detailed discussion on the cross sections will be presented in the next
subsection.
To put the baseline MQDT results into perspective, we note that the theory at
this level uses exactly the same number of parameter as one would have used in
an effective range theory for ion-atom interactions [101, 21]. For
23
Na+ +23 Na, the
effective range theory is applicable only for sE ≈ 2.21 µK. At 0.4 K, there
√
are 2(/sE )1/4 ∼ 29 number of partial waves above the barrier making significant
contributions [88]. The MQDT has expanded the energy range described by the same
parameters by more-than 5 orders of magnitude, including the descriptions of all tens
of partial waves contributing in this energy range. This is achieved purely through a
better theoretical formulation and a better understanding of the QDT functions.
61
E n e rg y /h (M H z )
1 0
T o t a l C r o s s - s e c t i o n ( a 20 )
1 0
6
3
1 0
4
1 0
5
{2 , 3 /2 } to {1 , 3 /2 }
1 0
5
1 0
4
M Q D T
N u m e r ic a l
1 0
3
1 0
-1
1 0
E n e rg y /k
B
0
(K )
Figure 4-4: Total hyperfine de-excitation cross sections from channel {F1 = 2, F2 =
3/2} to channel {F1 = 1, F2 = 3/2} from MQDT (solid line) and numerical method
(dashed line). The vertical dotted line identifies the upper hyperfine threshold at
0.08502 K.
62
E n e rg y /h (M H z )
1 0
1 0
-1
1 0
0
1 0
1
1 0
2
1 0
3
1 0
4
1 0
5
8
1 0
T o t a l C r o s s - s e c t i o n ( a 20 )
-2
{1 , 3 /2 } to {1 , 3 /2 }
1 0
7
1 0
6
M Q D T
N u m e r ic a l
1 0
5
1 0
-6
1 0
-5
1 0
-4
1 0
-3
E n e rg y /k
B
1 0
-2
1 0
-1
1 0
0
(K )
Figure 4-5: Total elastic cross sections in the lower channel {F1 = 1, F2 = 3/2} from
MQDT (solid line) and numerical method (dashed line). The vertical dotted line
identifies the upper hyperfine threshold at 0.08502 K.
63
E n e rg y /h (M H z )
1 0
0
1 0
1
1 0
2
3
1 0
1 0
4
1
U n
ita
6
r it
2
P a r tia l C r o s s - s e c tio n ( a 0 )
1 0
{1 , 3 /2 } to {1 , 3 /2 }
l = 5 ; F = 5 /2
y
L i
m
it
5
7
2
1 0
6
3
1 0
4
1 0
3
4
5
M Q D T
N u m e r ic a l
1 0
2
1 0
-5
1 0
-4
1 0
-3
-2
1 0
E n e rg y /k
B
1 0
-1
1 0
0
(K )
Figure 4-6: Partial wave contribution to the elastic cross section of Fig. 4-5 from
l = 5 and F = 5/2. There are seven resonances within the hyperfine splitting
that are labelled with numbers 1 through 7. The vertical dotted line identifies the
upper hyperfine threshold at 0.08502 K. Their detailed characteristics are tabulated
in Table. 4.2. A magnified version of this figure focusing on the resonances within the
region between the hyperfine thresholds is presented in Fig. 4-9.
64
4.3.2
Total scattering cross sections
To achieve more accurate results over a greater range of energies, we try to incorporate the energy dependence and the partial wave dependence of the short-range
parameters following the expansion in Eq. (4.22). Through single-channel calculations
at a few energies, we found that the energy variation of µc is negligible for energies
up to a few Kelvin, namely bµg,u ≈ 0. To determine the partial wave dependences cµg,u ,
we perform the single-channel calculations at zero energy with a few partial waves.
We can propagate the radial wavefunctions at zero energy and match them to the
proper boundary conditions, which is given by Eq. (3.17), at progressively larger R
c
until the resulting Kg,u
converges to a constant to a desired accuracy. Then the rec
( = 0, l) are converted into µcg,u ( = 0, l) and fitted into Eq. (4.22) for
sulting Kg,u
various partial waves. We then obtained the following results: cµg = 5.265 × 10−4 ;
cµu = 1.030 × 10−3 .
With the expansion of the short-range parameters, we are able to perform more
accurate MQDT calculations. Figures 4-4, 4-5, and 4-6 depict the total hyperfine
de-excitation, a total elastic, and a sample partial elastic cross sections in which the
MQDT results are evaluated with this l-dependent µc , to compare with the baseline results Fig. 4-1, 4-2, and 4-3. They show that with the addition of two more
parameters (one per channel) that characterize the partial wave dependences of the
short-range parameters, MQDT provides quantitatively accurate results that are in
full agreement with numerical results and cover the entire energy range of 0 to 3 K
in which hyperfine and quantum effects are the most important.
More specifically, Fig. 4-4 shows the total cross section for hyperfine de-excitation
in which the Na atom is de-excited from its F1 = 2 hyperfine level to its F1 = 1
hyperfine level. In earlier studies of resonant charge exchange [21, 13, 127] using
the elastic approximation [23], the de-excitation cross section goes to a constant at
65
the threshold [21, 48, 34]. Figure 4-4 shows the altered threshold behavior for the
de-excitation cross section with the proper treatment of hyperfine structure, which
behaves as (E − E2 )−1/2 above the upper threshold, implying a constant rate in the
zero temperature limit, as opposed to a zero rate.
Figure 4-5 depicts the total cross sections for elastic scattering in the lower channel
{F1 = 1, F2 = 3/2}, in which the atom stays in the lower hyperfine level (but its M1
may change). It shows the complexity of the ion-atom interaction as a result of the
rapid energy variation induced by the long-range polarization potential. Even within
a small energy interval of a hyperfine splitting (∼ 0.085 K), the small energy scale
associated with the long-range potential, sE ≈ 2.21 µK, is such that there are many
√
contributing partial waves (more than 2(/sE )1/4 ∼ 20 [88] at the upper threshold).
Each partial wave contribution contains a variety of resonances, which, in the energy
region below the second threshold, include Feshbach resonances in addition to shape
and diffraction resonances [42, 45].
Figure 4-6 depicts the partial wave contribution from l = 5 and F = 5/2, for which
we illustrate the analysis of resonances using MQDT. There are seven resonances
between the two hyperfine thresholds that reaches the unitarity limit, and will be
displayed in more detail and discussed further in the next subsection.
In addition, we also obtained the hyperfine excitation cross sections and elastic
cross sections in the upper hyperfine channel. Figure 4-7 shows the total cross section
for hyperfine excitation in which the the Na atom is excited from its F1 = 1 hyperfine
state to its F1 = 2 hyperfine state. It behaves as (−E2 )1/2 above the upper threshold.
The excitation cross section is related to the de-excitation cross section by Eq. (4.10)
which is guaranteed by time-reversal symmetry [85]. Figure 4-8 depicts the total cross
sections for elastic scattering in the higher channel {F1 = 2, F2 = 3/2}, in which the
atom stays in the same hyperfine level while its M1 may or may not change. Just like
the de-excitation cross section, it diverges as (E − E2 )−1/2 above the upper threshold,
66
E n e rg y /h (M H z )
1 0
T o t a l C r o s s - s e c t i o n ( a 20 )
1 0
5
3
1 0
4
1 0
5
{1 , 3 /2 } to {2 , 3 /2 }
1 0
4
M Q D T
N u m e r ic a l
1 0
-1
1 0
E n e rg y /k
B
0
(K )
Figure 4-7: Total hyperfine excitation cross sections from channel {F1 = 1, F2 = 3/2}
to channel {F1 = 2, F2 = 3/2} from MQDT (solid line) and numerical method (dashed
line). The vertical dotted line identifies the upper hyperfine threshold at 0.08502 K.
67
E n e rg y /h (M H z )
1 0
T o t a l C r o s s - s e c t i o n ( a 20 )
1 0
7
3
1 0
4
1 0
5
{2 , 3 /2 } to {2 , 3 /2 }
1 0
6
1 0
5
M Q D T
N u m e r ic a l
1 0
-1
1 0
E n e rg y /k
B
0
(K )
Figure 4-8: Total elastic cross sections in the higher channel {F1 = 2, F2 = 3/2}
from MQDT (solid line) and numerical method (dashed line). The vertical dotted
line identifies the upper hyperfine threshold at 0.08502 K.
68
E n e rg y /h (M H z )
1 0
1
1 0
2
3
1 0
1 0
6
1 0
5
{1 , 3 /2 } to {1 , 3 /2 }
l = 5 ; F = 5 /2
U n
ita r
ity
L im
it
2
P a r tia l C r o s s - s e c tio n ( a 0 )
1
7
2
6
3
1 0
4
1 0
3
1 0
2
4
5
M Q D T
N u m e r ic a l
1 0
-4
1 0
-3
1 0
E n e rg y /k
B
-2
1 0
-1
(K )
Figure 4-9: Magnified version of Fig. 4-6 focussing on the energy region below the
hyperfine threshold (vertical dotted line). Labelled 1 through 7 are seven resonances
whose detailed characteristics are tabulated in Table. 4.2.
implying a constant rate in the zero temperature limit.
4.3.3
Resonance structures
Figure 4-9 is a magnified version of Fig. 4-6 focusing on the energy region within
the hyperfine splitting. For the elastic partial cross-sections for l = 5 and total F =
5/2 in channel {F1 = 1, F2 = 3/2}, seven resonances within the hyperfine splitting
are identified by number 1 through 7, with three Feshbach resonances, one shape
resonance, and three diffraction resonances. The resonance positions, widths, and
69
Table 4.2: Positions, widths, and classifications of the 7 resonances labeled in Fig. 4-6.
Resonance
Energy/kB (K)
Width/h (MHz)
Type
1
2
3
4
5
6
7
1.801 × 10−4
8.795 × 10−3
2.600 × 10−2
3.970 × 10−2
5.271 × 10−2
7.085 × 10−2
8.370 × 10−2
1.562 × 10−2
−1.573 × 102
−3.235 × 102
1.030 × 102
−5.669 × 102
3.946 × 101
1.647 × 101
Shape
Diffraction
Diffraction
Feshbach
Diffraction
Feshbach
Feshbach
classification are calculated as illustrated in Section 4.2.4.3 and tabulated in Table 4.2.
As discussed in Section 4.2.4.3, Feshbach and shape resonances are relatively narrower
than diffraction resonances. This example fully demonstrates the complex resonance
structure of the ion-atom interaction and the capability of MQDT to predict these
complex resonances. We note that while the resonances in a single partial wave
tend to be smeared in the total cross section (Fig. 4-5) due to the summation over
a large number of contributing partial waves, they are in principle observable in
photodissociation of a molecular ion [59] from an excited rovibrational state.
4.4
Chapter summary
In conclusion, we have presented a fully quantum-mechanical multichannel theoretical frame work that employs frame transformation and MQDT to efficiently
characterize ion-atom interactions and illustrated its application to resonant charge
exchange. The theory provides a systematic and basically an analytic description of
ion-atom systems in its most complex energy regime where quantum effects are important. Other than well-known atomic properties such as the atomic mass, hyperfine
splitting, and the atomic polarizability, different group I, II, and He atoms differ pri70
marily only in two parameters such as the gerade and ungerade scattering lengths or
quantum defects, and secondarily (when interest is over a greater range of energies)
in two more parameters, cµg,u , that characterize their partial wave dependences.
Theoretically, MQDT gives a complete understanding and characterization of
threshold behaviors and complex resonance structures, and helps to overcome the
sensitive dependence of cold atomic interactions on short-range potentials, just as
what we were able to do with the single-channel QDT in Chapter 3. Computationally, MQDT is much more efficient than numerical calculations even when the QDT
functions are calculated on the fly. Since the QDT functions are universal mathematical functions that are the same for all applications, and can be computed to arbitrary
precision with efficient algorithms [45], their computation can be further accelerated
to be as efficient as most other mathematical special functions.
We believe that the systematic and efficient understanding of ion-atom interactions
that our theory provides, especially in the cold temperature regime where quantum
effects are important, will be the key to systematic understanding of quantum fewbody systems, chemical reactions, and many-body systems involving ions.
71
Chapter 5
Slow proton-hydrogen collision
5.1
Background and introduction
Scattering properties of proton-hydrogen collisions have been investigated theoretically throughout the last six decades with increasing accuracy and expanding energy
range [24, 7, 117, 67, 62, 82, 13]. Such data is important for understanding of physics
in planetary atmospheres [62, 65] and the interstellar medium [95], especially for the
interpretation of the brightness of the 21cm transition, which depends essentially on
the spin temperature of atomic hydrogen [48, 34, 65, 95].
While the past studies cover most of the temperature regime that is of astrophysical interest, one uncharted territory is the regime of cold and ultracold temperatures.
The lowest energy reached by previous studies is 10−4 eV [67, 82], which is equivalent
to 1.16 K, using the elastic approximation [23]. However, the elastic approximation
is expected to start to break down below several kelvin since the hyperfine splitting is
approximately 0.1 K. This has been showed for hydrogen-hydrogen scattering problem
by a previous study using multichannel theory that rigorously incorporated the hyperfine structure [131]. Furthermore, the multichannel nature of the collision results
in a diverging threshold behavior of the hyperfine de-excitation cross section [125],
which was not predicted in earlier works by simply using the elastic approximation
72
and the effective range theory [48, 34, 13].
With the multichannel theory described in Chapter 4 that effectively incorporates
the hyperfine interaction, we are able to investigate the proton-hydrogen collision
problem in the cold and ultracold temperature regime, which also provide another test
ground for our MQDT for ion-atom interactions introduced in Chapter 4. We briefly
go over the channel structure, the potential energy curves adopted for numerical
calculation, and the QDT parameters used for our MQDT implementation before
presenting the cross section results. We compare the results of MQDT with numerical
calculation to demonstrate, again, in addition to the example presented in Chapter 4,
the capability of MQDT to characterize scattering properties of ion-atom interactions
from ultracold temperature through several kelvin with just a handful of parameters.
Threshold behavior of the de-excitation cross section is also presented and discussed.
5.2
General considerations and potential energy
curves
The proton-hydrogen system is of the 1 S+2 S type with nuclear spin I1 = 1/2
that can be described by the theory from Chapter 4. Following the theory, the total
number of channels per partial wave is 4I1 + 1 = 3, corresponding to two channels
for F = 1/2 and one channel for F = 3/2. For F = 1/2, the two channels in the
F F coupled basis are separated by the atomic hyperfine splitting between the F1 = 0
and 1 states of hydrogen, which results in the famous 21 cm line. It is given by
∆E hf /h ≈ 1420.405751768 MHz [60] (∆E hf /kB ≈ 0.068168729 K). The asymptotic
long range potential is characterized by the static dipole polarizability of hydrogen
αA = 9.0/2.0 a.u. [25].
To perform the numerical calculation, we need the BO potential energy curves
Vg,u for the construction of the potential matrix. For the proton-hydrogen interaction,
73
2 4 .0
0 .1 0
0 .0 5
1 6 .0
0 .0 0
E n e r g y ( u n its o f 1 0
-2
a .u .)
2 0 .0
1 2 .0
8 .0
-0 .0 5
4 .0
-0 .1 0
9
1 0
1 1
1 2
1 3
1 4
1 5
0 .0
-4 .0
2
Σg
+
-8 .0
2
Σu
+
-1 2 .0
0
4
8
1 2
1 6
2 0
D is ta n c e R ( a 0 )
Figure 5-1: BO potential energy curves of the gerade (solid line) and the ungerade
(dashed line) states constructed in our work for proton-hydrogen collision
74
the potential curves can be calculated analytically in prolate spheroidal coordinates
following the method of Ref. [66]. However, to simplify the calculation, we opt for
an easier way to construct these potential curves. For internuclear separation R from
0.4a.u. to 10.0a.u., we use a cubic spline [102] to interpolate the data points given in
Ref. [67] which are calculated using the exact method of Ref. [66]. For R larger than
10.0a.u., we use the asymptotic expansion from Ref. [25]. More specifically,
1
Vg,u = V0 (R) ∓ ∆V (R) ,
2
(5.1)
2 +
with ∓ for 2 Σ+
g and Σu , respectively. Here,
V0 (R) = −
9
15
213
7755
1773
−
−
−
−
,
4R4 2R6 4R7 64R8
2R9
(5.2)
and
−R−1
∆V (R) = 4Re
1
25
131
3923
145399
−
−
−
−
2R 8R2 48R3 384R4 3840R5
509102915 37749539911
521989
−
−
.
−
46080R6
645120R7
10321920R8
1+
(5.3)
For R smaller than 0.4a.u., we use fitted functions for the inner wall, given by
−1.0597
R
Vg (R) = 0.835
+ 0.0012
− 0.993 ,
R0
−1.031
R
Vu (R) = 0.932
− 0.00009
+ 0.0896 ,
R0
(5.4)
(5.5)
in which R0 = 1.000544628a.u.. All the potential equations are in atomic units. The
two potential energy curves that we constructed are shown in Fig. 5-1. Notice that
the ungerade state has a very shallow well with a depth around 0.0001a.u.. Despite
the shallow well depth and light reduced mass, the ungerade potential supports two
75
Table 5.1: Zero energy QDT parameters for proton-hydrogen interaction
g or u
gerade
ungerade
K c (0, 0)
µc (0, 0)
-0.35829 0.64049
1.1723 0.025194
al=0 (a.u.)
cµ
-30.371
810.52
0.0116
0.0214
molecular bound states [17, 73], one of which is extremely weakly bounded with the
bound state energy around 10−9 a.u. [17, 73]. The extremely-near-threshold bound
state of the gerade potential suggests there may be a resonance very near the threshold, which makes the evaluation of the s-wave scattering length very sensitive to the
potential in the inner region [45]. This may explain the relatively large discrepancy
of the s-wave scattering length for the ungerade state reported by previous studies
despite the availability of an analytical potential [17, 48, 13, 73].
5.3
QDT parameters
The simplest MQDT implementation works well for the first handful of partial
waves for hydrogen but starts to show noticeable deviation from the numerical calculation when l becomes large; thus it breaks down at higher energies where high partial
waves contribute significantly. To parametrize the short-range interaction more accurately for energies ranging from zero to more than ten hyperfine splittings, we need
to use the expansion of the quantum defects as in Eq. (4.22).
The zero energy zero partial wave short-range parameters as well as parameters
bµg,u and cµg,u can be determined easily through solving single-channel radial equations
with Vg,u as the potential terms at a few energies and for a few partial waves, as
elaborated in Chapter 4. For the energy range considered here, the energy variation
of µc is found to be negligible, namely bµg,u ≈ 0. The partial wave dependences cµg,u , as
c
well as the zero energy zero partial wave parameters Kg,u
(0, 0) and their equivalents
76
E n e rg y /h (M H z )
1 0
5
T o t a l C r o s s - s e c t i o n ( a 20 )
{1 , 1 /2 } to {0 , 1 /2 }
3
1 0
P a p e r I
P re s e n t
1 0
2
1
1 0
E n e rg y /k
B
(K )
Figure 5-2: Total hyperfine de-excitation cross section of the proton-hydrogen collision
with the present numerical calculation (solid line) and the spin-exchange cross section
from Ref. [67] multiplied by the proper coefficient, 1/4 that accounts for nuclear
statistics, and offset by the center-of-gravity 0.0511265K (stars).
µcg,u (0, 0) and ag,ul=0 , are listed in Table 5.1. The scattering lengths are in reasonable
agreement with numbers given in previous work [17, 48, 13, 73].
5.4
5.4.1
Results and discussion
Comparison with elastic approximation
First, to illustrate the validity of the construction of our potential energy curves,
we compare the present numerical results with the results from Ref. [67]. The spin77
exchange cross section used in Ref. [67] is identical to the approximated de-excitation
cross section through the elastic approximation in Eq. 4.9 except that the coefficient
in front is unity instead of 1/4. Therefore, for energies calculated in Ref. [67] which
are magnitudes larger than the hyperfine splitting, we expect our de-excitation cross
section to be approximately equal to one fourth of the spin-exchange cross section
in Ref. [67] offset by the center-of-gravity (I1 + 1)/(2I1 + 1) · ∆E hf = 0.0511265K.
This comparison is presented in Fig. 5-2 which shows almost exact agreement between the two works for energies between 5 and 30 kelvin. The first two data points
from Ref. [67] below 5 kelvin are slightly different from our calculation which is
to be expected, since those energies are the closest to the upper hyperfine threshold
where there might be slight differences between exact and approximated results. This
comparison demonstrates the validity of our potential energy curves and numerical
method, and confirms that our theory, which accounts for hyperfine structure, corresponds to the correct single-channel theory at high energies where hyperfine splitting
is negligible. It also suggests that for the proton-hydrogen interaction the elastic approximation is valid for energies above 5 kelvin and the multichannel treatment should
be employed for energies below 5K, which is similar to the conclusion in Ref. [131]
for the hydrogen-hydrogen hyperfine interaction.
5.4.2
Total scattering cross sections
For collision energies lower than five kelvin, we present the total cross sections
(Eq. 4.8) calculated using both the multichannel numerical method and MQDT follwing the theory in Chapter 4.
Figure 5-3 shows the total cross section for hyperfine de-excitation, in which the
Hydrogen atom is de-excited from its F1 = 1 hyperfine level to its F1 = 0 hyperfine
level, for energies from zero to five kelvin. MQDT results are almost exactly on top
of the numerical calculation below one kelvin, and, although there is slight deviation,
78
E n e rg y /h (M H z )
3
1 0
4
1 0
5
{1 , 1 /2 } to {0 , 1 /2 }
2
T o ta l C r o s s - s e c tio n ( a 0 )
1 0
1 0
4
1 0
3
M Q D T
N u m e r ic a l
1 0
2
1 0
-1
1 0
E n e rg y /k
B
0
(K )
Figure 5-3: Total hyperfine de-excitation cross sections from channel {F1 = 1, F2 =
1/2} to channel {F1 = 0, F2 = 1/2} from MQDT (solid line) and numerical method
(dashed line). The vertical dotted line identifies the upper hyperfine threshold located
at 2 /kB ≈ 0.0682 K.
79
E n e rg y /h (M H z )
1 0
3
1 0
4
1 0
5
4
1 0
2
T o ta l C r o s s - s e c tio n ( a 0 )
{0 , 1 /2 } to {1 , 1 /2 }
M Q D T
N u m e r ia l
1 0
3
1 0
-1
1 0
E n e rg y /k
B
0
(K )
Figure 5-4: Total hyperfine excitation cross sections from channel {F1 = 0, F2 = 1/2}
to channel {F1 = 1, F2 = 1/2} from MQDT (solid line) and numerical method (dashed
line). The vertical dotted line identifies the upper hyperfine threshold located at
2 /kB ≈ 0.0682 K.
the two methods still agree relatively well from one to five kelvin. The discrepancy is
in general within one percent which can be attributed to the slight energy dependence
of the short-range parameters. The cross section diverges as ( − E2 )−1/2 right above
the upper hyperfine threshold, which will be looked at in detail in the next subsection.
Figure 5-4 shows the total cross section for hyperfine excitation in which the
Hydrogen atom is excited from its F1 = 0 hyperfine state to its F1 = 1 hyperfine
state. It behaves as ( − E2 )1/2 above the upper threshold. The excitation cross
section is related to the de-excitation cross section by Eq. (4.10) which is guaranteed
80
E n e rg y /h (M H z )
0
1 0
1
1 0
2
1 0
3
1 0
4
1 0
5
6
1 0
5
1 0
4
1 0
3
{0 , 1 /2 } to {0 , 1 /2 }
1 0
2
T o ta l C r o s s - s e c tio n ( a 0 )
1 0
M Q D T
N u m e r ic a l
1 0
-5
1 0
-4
1 0
-3
1 0
E n e rg y /k
-2
B
1 0
-1
1 0
0
(K )
Figure 5-5: Total elastic cross sections in the lower channel {F1 = 0, F2 = 1/2} from
MQDT (solid line) and numerical method (dashed line). The vertical dotted line
identifies the upper hyperfine threshold at 0.0682 K.
by time-reversal symmetry [85].
Figure 5-5 and 5-6 depict the total cross sections for elastic scattering in the lower
channel {F1 = 0, F2 = 1/2} and the higher channel {F1 = 1, F2 = 1/2} respectively,
in which the atom stays in the same hyperfine level while its MF s may or may not
change. The elastic cross-section in the higher hyperfine channel, just like the deexcitation cross section, diverges as (E − E2 )−1/2 above the upper threshold, implying
a constant rate in the zero temperature limit.
Compared to the rich resonance structures for the resonant charge exchange prob81
E n e rg y /h (M H z )
1 0
1 0
4
1 0
5
6
1 0
{1 , 1 /2 } to {1 , 1 /2 }
2
T o ta l C r o s s - s e c tio n ( a 0 )
3
1 0
5
1 0
4
1 0
M Q D T
N u m e r ic a l
3
1 0
-1
1 0
E n e rg y /k
B
0
(K )
Figure 5-6: Total elastic cross sections in the higher channel {F1 = 1, F2 = 1/2}
from MQDT (solid line) and numerical method (dashed line). The vertical dotted
line identifies the upper hyperfine threshold at 0.0682 K.
82
lem of
23
Na+ +23 Na from Chapter 4, there is not at all any significant resonance
structure within the hyperfine splitting in the current results. This can be attributed
to the small atomic mass and polarizability of hydrogen that result in an unusually
large energy scale, sE = 0.0416K (compared to 2.21 µK for the sodium system) for
the proton-hydrogen system. With the large energy scale, much fewer partial waves
contribute within the hyperfine splitting, which results in much less chance of finding
resonances.
Figures 5-4, 5-5, and 5-6 are similar to Fig. 5-3 in showing the agreement between MQDT and the numerical calculation, and the conclusion we drew for the
de-excitation cross section also stands for the others. This again demonstrates the
capability of MQDT to accurately characterize multichannel ion-atom interactions
from zero energy up to several kelvin with only a handful of parameters (five in
this case), as we have demonstrated in Chapter 4 for the example of sodium charge
exchange. Also examined and verified by these comparisons is the physical picture
behind our MQDT formulation, that in this energy range the energy and partial
wave dependences are primarily due to the long-range interaction which can be accurately characterized by the analytical solution of the long-range potentials, and the
short-range interaction is energy and partial wave insensitive [46, 45].
5.4.3
Threshold behavior of de-excitation rate
For a two-level system, the hyperfine de-excitation cross section follows Wigner’s
threshold law [125], which diverges as (−E2 )−1/2 above the upper hyperfine threshold.
Therefore, the hyperfine de-excitation rate without thermal averaging,
s
Wde ≡ vσde =
83
2( − E2 )
σde ,
µ
(5.6)
9 .1 0
D e - e x c ita tio n R a te ( 1 0
-1 0
c m
3
⋅s
-1
)
9 .1 5
9 .0 5
9 .0 0
8 .9 5
8 .9 0
1 0
-1 2
1 0
-1 1
1 0
-1 0
1 0
-9
(ε- E
2
1 0
)/k
B
-8
1 0
-7
1 0
-6
1 0
-5
(K )
Figure 5-7: Threshold behavior of the hyperfine de-excitation rate Wde just above the
upper threshold E2 . The x-axis represents the temperature equivalence of the initial
kinetic energy ( − E2 )/kB . The results are produced using our numerical method.
84
will reach a constant when the relative velocity v approaches zero. As illustrated in
Fig. 5-7 with results from our numerical calculation, the de-excitation rate rises while
initial energy decreases, until eventually reaching a plateau with a constant rate of
9.098 × 10−10 cm3 · s−1 approximately. This was not expected in previous attempts to
extend the elastic approximation to lower energy with effective range theory, which
anticipated a constant cross-section with the corresponding rate approaching zero
with decreasing energy [48, 34, 13]. The same threshold behavior should also be
present for electron-hydrogen and hydrogen-hydrogen collisions.
5.5
Chapter summary
We have carried out fully quantum calculations for low temperature protonhydrogen collisions with the theory elaborated in Chapter 4. We compared our
hyperfine de-excitation cross section with previous results by the elastic approximation [67] and concluded that the results from the elastic approximation deviate
gradually from our results from multichannel treatment for temperatures below five
kelvin. For energies from zero to five kelvin, we presented the total hyperfine deexcitation and excitation, and elastic cross sections, which are available for the first
time. Further, The de-excitation cross section diverges at the upper hyperfine threshold as predicted by Wigner’s threshold law [125]. This contradicts the prediction of
a constant cross section at the threshold made with the elastic approximation and
effective range theory.
We compare the results from MQDT and the numerical method for temperatures
below five kelvin, where the discrepancies are in general within one percent. This
demonstrates again, in addition to the results presented in Chapter 4, that MQDT
gives a complete understanding and characterization of the universal behavior of
ion-atom interactions, especially at cold temperatures where quantum effects are im85
portant, which can be the key to systematic understanding of quantum few-body
systems, chemical reactions, and many-body systems involving ions.
86
Chapter 6
Conclusions and outlook
In this thesis, a fully quantum-mechanical theoretical framework built around
QDT is presented for ion-atom interactions.
We started with single-channel QDT in Chapter 3, applying it on the resonant
charge exchange problem with the elastic approximation for group I, II, and He atoms.
A three parameter implementation of the single-channel QDT, using the gerade and
ungerade scattering lengths and the atomic polarizability, or equivalent parameters,
is tested on the
23
Na+ +23 Na resonant charge exchange process. We found excellent
agreement between QDT results and numerical results from below 0.1 nK through
0.1 K including all resonances within this range. The comparison provides a strong
testament of the validity of the physical picture behind QDT, which attributes the
energy and partial wave dependence of the ion-atom interaction at low temperature
mainly to the long-range interaction that can be analytically characterized. We further showed the sensitive dependence of the scattering results on the short-range
interaction. This lends favor on QDT over numerical methods, because our QDT implementation relies on only two short-range parameters which encapsulate the entire
short-range interaction compared to the two potential energy curves used in numerical
calculations.
We then moved onto presenting a multichannel theory in Chapter 4 that combines
87
the scheme of frame transformation and MQDT to address the complexity introduced
by hyperfine structures. Analytical descriptions of the resonances are also presented
as a feature of MQDT. We demonstrated the theory again on the
23
Na+ +23 Na sys-
tem, first with the same three-parameter implementation as in the single-channel case,
and then with two more secondary parameters to characterize the slight partial wave
dependence. The results from the first implementation agree well with numerical calculations up to 0.4 K, which proves the universal behavior in ion-atom interactions
can be captured by the QDT formulation. The results from the second implementation of MQDT overlap almost exactly with numerical results from zero up to three
kelvin, which include the entire complex resonance structure. We also investigated
the resonances in a particular partial wave to demonstrate their characterizations
using MQDT.
In Chapter 5, we further presented multichannel quantum calculation results on
the proton-hydrogen collision from zero to five kelvin, at which temperature the elastic
approximation results start to deviate from multichannel results. The multichannel
nature of our theory guarantees a diverging threshold behavior of the de-excitation
cross section above the upper hyperfine threshold which contradicts previous predictions with the elastic approximation. MQDT results and numerical results were again
compared, resulting in excellent agreement throughout the entire temperature range
of calculation.
Overall our theory that developed around QDT for the 1/R4 potential provides
a basically analytical description of ion-atom systems in its most complex energy
regime where quantum effects are important. Further, connections among ion-atom
interaction properties in the ultracold regime all the way up to over one kelvin can be
established with our theory through a few energy and partial wave insensitive parameters in a systematic way. Compared to our theory, the standard numerical method is
not only inefficient and heavily dependent on the accuracy of PESs, but also missing
88
important physics such as the prediction and characterization of narrow resonances,
which is especially important for ion-atom interactions due to their generally small
energy scale. We believe that the systematic and efficient understanding of ion-atom
interactions that our theoretical framework provides with only a few parameters will
be the key to systematic understanding of quantum few-body systems, chemical reactions, and many-body systems involving ions, especially in the cold temperature
regime where quantum effects are important.
89
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Appendix A
Quantum-defect theory functions
for polarization potentials
In this appendix, we present the formulas for QDT functions for a −1/R4 type of
c
polarization potential, namely, the four Zxy
functions for open channels and χcl for
closed channels. The entire appendix is based on Ref. [45] with permission from the
author.
c
and χcl functions are well defined for any −1/Rn type of potential with
The Zxy
n > 2 [40]. For the polarization potential corresponding to n = 4, they are given by
Zfc s (s , l) =
Zfc c (s , l) =
c
Zgs
(s , l) =
cos[π(ν − ν0 )/2]
2Ms l cos(πν/2)
× 1 − (−1)l M2s l tan[π(ν − ν0 )/2] ,
(A.1)
cos[π(ν − ν0 )/2]
2Ms l cos(πν/2)
× tan[π(ν − ν0 )/2] − (−1)l M2s l ,
(A.2)
cos[π(ν − ν0 )/2]
2Ms l sin(πν/2)
× 1 + (−1)l M2s l tan[π(ν − ν0 )/2] ,
(A.3)
100
cos[π(ν − ν0 )/2]
2Ms l sin(πν/2)
× tan[π(ν − ν0 )/2] + (−1)l M2s l .
c
Zgc
(s , l) =
(A.4)
And
χcl = tan(πν/2)
1 + M2s l
.
1 − M2s l
(A.5)
In these formulas, ν0 = l + 1/2.
Ms l is given by
Γ[1 − (ν + ν0 )/2]
Ms l (ν) = 2 |s |
Γ[1 + (ν + ν0 )/2]
Γ[1 − (ν − ν0 )/2]
Cs l (−ν)
×
,
Γ[1 + (ν − ν0 )/2]
Cs l (+ν)
−2ν
ν/2
(A.6)
where
Cs l (ν) =
∞
Y
Q(ν + 2j) ,
(A.7)
j=0
and Q(ν) is given by a continued fraction
Q(ν) =
1
1−
s
Q(ν
[(ν+2)2 −ν02 ][(ν+4)2 −ν02 ]
+ 2)
.
(A.8)
ν is the characteristic exponent for the −1/R4 potential (corresponding to the
modified Mathieu equation [2, 63, 75, 45]). It is known that the characteristic exponent for the Mathieu class of functions can be determined using two different methods
[97]. One is as the root of a Hill determinant. The other is as the root of a characteristic function. In the Hill determinant formulation, it is a solution of
DlH (ν; s ) = 0 ,
101
(A.9)
where DlH (ν; s ) is the Hill determinant










H
Dl ≡ det 








...
..
.
..
.
..
.
..
.
..
.

...
1
h−2
0
0
0
...
...
h−1
1
h−1
0
0
...
...
0
h0
1
h0
0
...
...
0
0
h1
1
h1 . . .
...
0
..
.
0
..
.
0
..
.
h2
..
.
1
..
.
...
..
.









 .








(A.10)
hm is defined by
hm = s1/2 /[(ν + 2m)2 − ν02 ] .
(A.11)
It corresponds to the three-term recurrence relation
hm bm+1 + bm + hm bm−1 = 0
(A.12)
for the Mathieu class functions [63, 45] in the Laurent expansions [63, 45]
M+ν (x) =
M−ν (x) =
∞
X
m=−∞
∞
X
bm xν+2m ,
(A.13)
b−m x−ν+2m .
(A.14)
m=−∞
In the characteristic function method, ν is a solution of
Λl (ν; s ) = 0 ,
(A.15)
Λl (ν; s ) ≡ (ν 2 − ν02 ) − s [Q̄(ν) + Q̄(−ν)] ,
(A.16)
where
102
is the characteristic function, with Q̄(ν) defined in terms of the Q(ν) function by
Q̄(ν) =
1
Q(ν) .
(ν + 2)2 − ν02
(A.17)
It can be shown that the Hill determinant and the characteristic function are
related by [45]
DlH (ν, s ) =
1
(ν 2
−
ν02 )Cs l (ν)Cs l (−ν)
Λ(ν; s ) .
(A.18)
This relationship not only makes it immediately clear that the two approaches to ν
are equivalent, but also provides an efficient method for the evaluation of the Hill
determinant and therefore the characteristic exponent.
Due to the special characteristics of a Hill determinant [123], the solution of
Eq. (A.9) can be found through the evaluation of DlH (ν; s ) at a single ν such as
ν = 0. Defining Hl (s ) ≡ DlH (ν = 0; s ), we have from Eq. (A.18)
Hl (s ) =
1
ν02 [Cs l (0)]2
ν02
2s
Q(0) .
+
4 − ν02
(A.19)
From Hl , the ν, as a function of the scaled energy, can be found as the solution
of [63, 42]
sin2 (πν/2) = Hl (s )/2 .
(A.20)
For example, for Hl < 0 or Hl > 2, ν = νr + iνi is complex, with its imaginary part
νi given by
1
cosh−1 (|1 − Hl |) ,
π
i
p
1 h
=
ln |1 − Hl | + (1 − Hl )2 − 1 .
π
νi =
103
(A.21)
(A.22)
Its real part is given by
νr =


 l,
l = even
,
(A.23)
,
(A.24)

 l + 1 , l = odd
for Hl < 0, and by
νr =


 l + 1 , l = even

 l,
l = odd
for Hl > 2. The real part of ν is defined within a range of 2. All ν + 2j, where j is
an integer, are equivalent.
104