Download Resonances in three-body systems S U L

S TOCKHOLM U NIVERISTY
L ICENTIAT T HESIS
Resonances in three-body systems
Author:
Muhammad Umair
Supervisor:
Svante Jonsell
Akademisk avhandling för avläggande av
licentiatexamen i fysik vid Stockholms Universitet
28 November 2014
c
Muhammad
Umair, Stockholm 2014
ISBN XXX-XX-XXXX-XXX-X
Distributor: Department of Physics, Stockholm University
Abstract
Three particles interacting via Coulomb forces represents a fundamental problem in quantum
mechanics whose approximate solution provides some insight into the more complex analysis
associated with few-body problems. We have investigated resonance states composed of three
particles interacting via Coulombic and more general potentials in non-relativistic quantum mechanics, using the complex scaling method. My calculations have been applied to two different
physical systems: (i) an investigation of the possibility of resonances in the peµ system, which
has been suggested as a possible reason for unexpected results from a recent measurement of the
proton radius in muonic hydrogen (ii) a calculation of resonances in positron-hydrogen scattering, which shows that we can represent this system with the accuracy needed for future scattering
calculations. The basis set used is built from Gaussians in Jacobi coordinates, thus automatically
including mass-polarisation effects which cannot be neglected in muonic systems.
My Family
List of Papers
The following papers, referred to in the text by their Roman numerals, are included in this thesis.
PAPER I: A search for resonances in pµe system
Umair M., Jonsell S.,
J. Phys. B. At. Mol. Opt. Phys., 47, 175003 (2014).
doi:10.1088/0953-4075/47/17/175003
PAPER II: Natural and Unnatural parity resonance states in positron-hydrogen scattering
Umair M., Jonsell S.,
J. Phys. B. At. Mol. Opt. Phys., 47, 225001 (2014).
doi:10.1088/0953-4075/47/22/225001
Author’s contribution
My contribution to the work reported in this thesis is substantial. In the following I will try to
summarize my individual contribution to the presented work:
Paper I. I have actively taken part in adapting the code, and adding the complex scaling part
as well as including the correction due to the vacuum polarization. I analyzed the data and wrote
the article with the close collaboration of my supervisor.
Paper II. In this paper, I did all the calculations and wrote the article.
Contents
Abstract
List of Papers
i
iii
Author’s contribution
v
Abbreviations
ix
1
Introduction
1
2
The Proton Radius Problem
2.1 Muonic hydrogen and the proton radius . . . . . . . . . . . . . . . . . . . . .
2.2 Why the proton radius is important . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Proton radius puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
3
3
4
3
Theory
3.1 Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Linear Variational Principle . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Hylleraas-Undheim-MacDonald Theorem and the Generalized Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Theory of Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Stabilization Method . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Complex Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Modelling of Quantum Three-Body Systems . . . . . . . . . . . . . . . . . . .
3.3.1 Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Mass Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 The Couple Rearrangement Channel method . . . . . . . . . . . . . .
3.4 Vacuum Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
5
5
6
6
7
8
10
10
11
14
21
4
Negative Hydrogen Ion
4.1 The pµe Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
27
5
Positron-Hydrogen Scattering (e+ H)
29
Summary and Outlook
References
Abbreviations
Ps
QED
RMS
S.E
µp
ep
CRC
pee
Positronium
Quantum Electrodynamics
Root mean square
Schrödinger equation
Muonic hydrogen
Electron proton scattering
Couple Rearrangement Channel
Negative hydrogen
1. Introduction
In this thesis we apply non-relativistic quantum mechanics to study three-body problems. We
use interaction potentials of Coulombic or Gaussian forms. However, the formal and numerical
approaches, used in this work, are in principle applicable to any three-body system. We are
interested in how the energies and widths of resonances scale with the masses of the particles. In
our calculations one particle is a proton, while the two other particles are negatively charged with
masses between the electron and the muon mass. The origin of the distinctive energy and size of
these so called exotic atoms lies in the energy difference between atomic and nuclear physics. It
provides an opportunity to undertake investigations of different basic physical phenomena and
quantities pertaining to atomic or nuclear physics e.g. the proton radius, the pion mass, strong
interaction scattering lengths and so on.
The primary motivation for our work is recent experiments on muonic hydrogen [1, 2].
Muonic hydrogen µ − p has been interest for a long time since a measurement of the Lamb shift
(2S-2P energy difference) could give a precise value for the root mean squared (RMS) charge
radius of the proton. An isolated (µ − p)2S is metastable with a life time mainly determined by
muon decay (about 2.2 µs). The 2S-2P splitting in muonic hydrogen arises mainly through
quantum electrodynamics (QED) effects such as vacuum polarization. It does however also
depend on a finite-size correction of the nuclear charge distribution. Recent experiments have
measured the correction and thereby determined a new value of the proton radius, 0.84184 fm
[1]. The proton radius is being recognized as a basic property of the simplest nucleus, the
proton, and treated in the published 2006-CODATA [4] adjustment as a fundamental physical
constant, but with a 5 standard deviation away from this recent (and presumably more accurate)
measurement. Various experiments have been performed to examine the formation of a longlived meta-stable state of µ p when muons are stopped in a low-pressure hydrogen target gas.
In combination with the recent advances in hydrogen spectroscopy [3], such an experiment will
enable assessment of bound state QED to a new level of precision.
We also calculate resonances in the positron-hydrogen system (or equivalently the charge
conjugate electron-antihydrogen system). These resonances could play an important role in
the production of anti-hydrogen atoms through rearrangement scattering of positronium (Ps) by
anti-protons [5]. Recently, attention has been drawn to the existence of chemical compounds
between anti-matter and matter. The bound or quasi-bound state (resonance) of a positron or
a Ps and atoms or molecules is a subject of intense studies [9–13]. Unfortunately, experiment
has not provided us with data about the positron-hydrogen scattering [6–8]. The excited Ps is
interesting since it probably gives a much higher formation cross-section. The process relating
to cross-sections
e+ + H → Ps + p
(1.1)
Ps + p̄ → e− + H̄
(1.2)
1
are of great importance since anti-hydrogen atom reflects an ideal system to study charge conjugation symmetries of physics. Other exotic atoms, for example, muonium, positronium and protonium have short lifetimes and are not so appropriate for high-precision spectroscopic research.
Nevertheless, there is a growing interest for studying various effects of positron-hydrogen scattering, especially for comparison with electron-hydrogen scattering (though we do not really
pursue this).
Outline
Chapter 2 gives an overview of the proton radius problem, its origin and why it is important.
In chapter 3, we describe the computational method used for the calculation in Paper I-II. In
chapter 4, negative hydrogen ion results are discussed and also scaling the mass of one of electrons to make this a pµe system. In chapter 5, positron-hydrogen system results are discussed.
Chapter 6 gives a summary and outlook of the work.
2
2. The Proton Radius Problem
Since the inception of quantum theory, the hydrogen atom has been an important system, which
can be solved exactly. Due to its relatively simple atomic structure, comprising only one electron
and a proton, the hydrogen atom has been important to the development of the theories of atomic
structure, quantum mechanics, and quantum electrodynamics (QED). For example, the small
observed deviation of the 2S-2P energy splitting from the prediction of the Dirac equation, called
the Lamb shift, established the development of QED [1, 15].
The ambiguity related to the proton radius was extracted from H-spectroscopy experiments,
which had been the most accurate determination of the hydrogen energy levels and accordingly
were limiting the comparison between theory and measurements. So, to further examine the
bound state QED representing the hydrogen energy levels it was necessary to have a more accurate verification of the proton radius. The most recent experiment to determine the proton
radius study the 2S → 2P transition in muonic hydrogen [1, 15]. The very accurate results from
these experiments do not agree with previous determination of the proton radius [17]. This has
motivated our investigation into possible exotic states involving 2S muonic H.
2.1
Muonic hydrogen and the proton radius
A muon is just like an electron in all regards with the exception of that it is heavier and has
a finite life time. Since the muon is heavier than the electron (about 200 times) and muonic
hydrogen is about 200 times smaller than ordinary hydrogen. Therefore, the proton is having
a diameter of about 60,000 times smaller than ordinary hydrogen, only 300 times smaller than
muonic hydrogen. As a consequence, the influence of the finite nuclear size on the energy levels
is significantly increased. That makes the details of muonic hydrogen more sensitive to the size
of the proton, and thus allows for a more precise measurement. Hence muonic atoms represent
a unique laboratory for the determination of RMS radii and other nuclear properties.
2.2
Why the proton radius is important
The small uncertainty in the new proton radius value (1 × 10−18 m) has opened the way for
checking bound-state QED calculations in hydrogen to an unprecedented level of accuracy (3 ×
10−7 ). This is very interesting, since bound state QED is challenging and both are part of
technical and fundamental aspects.
In summary, the new proton radius value will lead to:
• An order-of-magnitude more accurate analysis of the theory explaining hydrogen energy
levels.
3
• An order-of-magnitude correction to the Rydberg constant (to a relative level of 1 ×
10−12 ), which is a vital part of the constants adjustment [4].
• A criterion for lattice QCD calculation aiming to model the proton, starting from quarks
and their interactions [18].
• Confrontation with the electron-proton scattering domain, which is the historical way used
to determine the proton radius [19].
2.3
Proton radius puzzle
The proton radius puzzle is the disagreement between the proton charge radius determined
from muonic hydrogen and that determined from electron-proton scattering and hydrogen spectroscopy. Up until 2010, the accepted value for the proton radius was 0.8768(69) fm, determined
from atomic hydrogen measurements in the 2006 CODATA analysis [4]. The prime result obtained from ep scattering was 0.879(9) fm, from the analysis of Sick [20]. The first experiment
using muonic hydrogen by Pohl et al. [1], obtained a value of the proton radius of 0.84184(67)
fm. Recently, updated results [2] give a value of 0.84087(39) fm. This new result is consistent
with the first measurement [1] which virtually eliminates the possibility of experimental error.
But both measurements strongly disagree with the hydrogen spectroscopy and ep scattering results. The 5σ discrepancy compared to the earlier, less precise measurement has attracted much
attention. Presently, physicists all over the world are searching for the solution to this problem,
generally referred as a "Proton Radius Puzzle".
It is obvious from these numbers that the discrepancy is severe, and it is difficult to imagine an effect that could shift the resonance position by 5 standard deviations. Jentschura [15]
suggested that the presence of an electron could result in a shift of the resonance position if the
distance between the electron and the µ p(2S) atom was about one Bohr radius. He suggested
that the spectroscopy might have been carried out not on a µ p(2S) atom, but on the molecular
ion (pµe)− .
It is therefore important to search for (pµe)− resonances using theoretical calculations. One
such study was performed by Karr and Hilico [21], where they found a resonance in (pµ µ)− ,
but this resonance does not survive as the muon mass is scaled mµ → me . In our work, we
have used another numerical technique and discovered additional resonances in the (pex)− and
(pµx)− systems, where x is a muon with scaled mass.
4
3. Theory
3.1
Variational Principle
The variational principle [22, 23] is a numerical minimization technique used to find the approximate solutions of many-body problems by finding the best possible approximation to the true
ground state using trial wave functions of a certain form.
3.1.1
Linear Variational Principle
We deal with the stationary many-body Schrödinger equation HΨ = EΨ and represent the eigenfunctions and eigenvalues by Ψn and En , respectively. The well-established Rayleigh-Ritz variational principle states that the variational energy ε evaluated with an arbitrary trial-function Ψt
provides an upper bound to the exact ground state energy of the Hamiltonian H, i.e.
ε=
hΨt |H|Ψt i
hΨ|H|Ψi
≥ E0 =
,
t
t
hΨ |Ψ i
hΨ|Ψi
(3.1)
where E0 is the exact ground state energy. In order to find approximate variational solutions to
the full Schrödinger equation, we restrict the problem to a smaller space spanned by a set of K
basis functions ψk . Firstly, we expand the wave-function in this space, i.e.
K
Ψt =
∑ ck ψk
(3.2)
k=1
where ck are the expansion coefficients. These coefficients determine Ψt completely within the
space {ψk }.
How do we use the variational principle in practice to find the best Ψt , i.e. the values of ck which
gives the lowest ε?
To minimize ε we must have
∂ε
=0
∂ ck
f or all k = 1, 2.....K
(3.3)
From eq.(3.1) and eq.(3.2), we have
ε=
∑Ki=1 ∑Kj=1 ci c j Hi j
∑Ki=1 ∑Kj=1 ci c j Si j
(3.4)
5
where,
Hi j = hψi |H|ψ j i,
(3.5)
Si j = hψi |S|ψ j i
Hence,
K
K
K
K
ε ∑ ∑ ci c j Si j = ∑ ∑ ci c j Hi j
i=1 j=1
(3.6)
i=1 j=1
Taking the derivative w.r.t, cn
K
K
∂ε K K
c
c
S
+
ε
(c
S
+
S
c
)
=
∑ ∑ i j i j ∑ i in ni i ∑ (ci Hin + Hni ci )
∂ cn i=1
j=1
i=1
i=1
Since
∂ε
∂ cn
(3.7)
= 0 and Sin = Sni , Hin = Hni , one has
K
K
ε ∑ ci Sin = ∑ ci Hin
(3.8)
H̃ c̄ = ε S̃c̄
(3.9)
i=1
i=1
or
where, H̃ and S̃ are the matrix representations of the Hamiltonian and overlap operators and their
elements are defined in eq.(3.5). This is a generalized eigenvalue equation of size K, with K real
eigenvalues ε1 ≤ .... ≤ εK and corresponding eigenvectors c1 , ..., cK .
3.1.2
Hylleraas-Undheim-MacDonald Theorem and the Generalized Eigenvalue Problem
The Hylleraas-Undheim-MacDonald theorem states that for a trial wave function with linear
variational parameters such as (3.2), the higher eigenvalues εi > ε0 of eqn.(3.9) are upper bounds
to the excited states of the Hamiltonian. This result is based on a theorem showing that as the
number of basis functions is extended from N to N +1, the new eigenvalues interleave the former,
i.e. ε0N+1 ≤ ε0N ≤ ε1N+1 ≤ ε1N ≤ ...... Thus all eigenvalues, ordered according to increasing energy,
must decrease as the number of basis functions N is increased. As N → ∞ they will approach
the true eigenvalue of the Hamiltonian. It follows that for a finite N, εiN is an upper bound to Ei
as shown in figure (3.1).
3.2
Theory of Resonances
The identification of continuum and resonant states are two significant aspects in quantum
physics. Generally, a resonant state is explained as a long-lived state of the system which has
sufficient energy to divide up into two or more subsystems [35]. While conducting an experiment, a particle is scattered from the target. It can be an electron, atom or molecule and the
6
Figure 3.1: Approximate eigenvalues given by the Rayleigh-Ritz variational method with linear
functions. Each root εiN of the determinal equation (3.9) is an upper bound on the corresponding
exact eigenvalue Ei .
target can be a nucleus or any of the above. Three different processes can occur; one is elastic
scattering in which the energy of the particle is conserved. Another is inelastic scattering, in
which energy is exchanged between the particle and the target. The last is reactive scattering in
which the particle and the target collide with each other and form a different species. There are
numerous methods used to calculate the energy and the lifetime of a resonance, here we discuss
some of them.
3.2.1
Stabilization Method
The stabilization method provides an efficient approach to many problems in atomic and molecular physics [24, 31, 32]. This method deals with the real matrices and real basis functions,
which directly exploits the locality of the resonance in the interaction region. Applying this
stabilization technique, we introduce a real scaling parameter α by the transformations (for the
Coulombic potential):
T
V
(3.10)
,
V→ ,
2
α
α
Where T and V are the kinetic and potential energy matrix elements, respectively. When we
change the value of α, we get a stabilization graph as shown in figure (3.2) from which the resonance energies can be analysed. Eigenvalues E j (α) corresponding to bound states or resonances
are stable with respect to variation of α. The stabilization method diagonalizes the Hamiltonian
r → rα,
T→
7
Figure 3.2: stabilization graph showing resonance states of pee below the n = 2 threshold with J=0.
in a basis set of ever larger extension around the region where the wave function of the resonance
is localized. If the energies decrease for each region, these states are known to be the continuum
states, and do not possess any bound state properties. Also the wave-vector k scales as k → k/α,
meaning that the energy of a continuum state scales as Ek = h̄k2 /2m → (1/α 2 )Ek .
3.2.2
Complex Scaling
The Complex scaling method is a powerful tool in the numerical study of resonances in few
electron systems [33–40]. The Complex L2 method mathematically transform the Schrödinger
equation itself, avoiding the asymptotic boundary problems for resonances. This method enables
us to calculate the resonance energy and width directly, as real and imaginary parts of the complex eigenvalues. It helps us to investigate all the resonances that lie inside the energy region of
interest using the complex rotated Hamiltonian.
In such a method the radial coordinates r j are transformed into
r j → r j eiθ
(3.11)
∇ j → ∇ j e−iθ
(3.12)
and accordingly
where θ is the scaling angle which is restricted to 0 < θ < π/4.
The wave-function φ (r) transforms under the complex dilatation operator U(θ ) by definition as
U(θ )φ (r) = e3iθ /2 φ (eiθ r)
(3.13)
The factor e3iθ /2 gives the scaling of three dimensional volume element dV = (dxdydz). The
scaled Hamiltonian is defined as H(θ )
8
H(θ ) = U(θ )HU −1 (θ )
(3.14)
H(θ ) = e−2iθ T + e−iθ V
(3.15)
and the transformed Hamiltonian is
where e−i2θ , e−iθ are complex numbers which scale the kinetic and potential energy, H(θ ) has
complex eigenvalues with L2 eigenfunctions corresponding to the resonant states of the system.
When we apply this transformation to the Hamiltonian operator the S.E transforms to [35]
U(θ )HU −1 (θ ) (U(θ )φres ) = (Er − iΓ/2) (U(θ )φres )
Where Er and Γ represent the position and width of the resonance state, respectively, and φres
represents the diverging spherical out-going eigenfunction corresponding to the complex eigenvalues of the resonance.
• The bound state eigenvalues obtained from H(θ ) are independent of θ and similar to those
of H(θ = 0) for |θ | ≤ π/2.
• The continuous spectrum at each scattering threshold is rotated downward making an
angle of 2θ .
• H(θ ) may have isolated complex eigenvalues corresponding to the resonances energies
with L2 square integrable complex eigenfunctions.
To understand this technique we start with an example for short-range potential. In such
cases, the solutions for scattering states have asymptotic behaviour given by
φ scatt (r → ∞) = A(k)e−ikr + B(k)e+ikr
where
E=
(3.16)
(h̄k)2
2m
(3.17)
The complex-scaled scattering states, r → reiθ are given by
φ scatt (reiθ )r→∞ = A(k)e−ike
iθ r
+ B(k)e+ike
iθ r
(3.18)
Here we see in eq.(3.18), one of the exponentials diverges as r → ∞, violating the boundary
condition for a finite wave function. To preserve this asymptotic form we take k as complex values k → ke−iθ , then E = (h̄k)2 /2m → e−2iθ (h̄k)2 /2m for the allowed scattering eigenenergies.
The continuum is then rotated into the lower half of the complex energy plane by an angle of
2θ shown in fig. (3.3). The bound states are unaffected by the rotation and lie on the negative
real energy axis. This shows that the continuous spectrum of H(θ ) is different from that of the
unscaled Hamiltonian H which is the main purpose of r → reiθ transformation.
9
Dilatation Transformation
σ(H(θ))
σ(H)
{
{
Thresholds
Bound
states
Bound
states
Resonance
(Hidden)
2θ
Resonance
(Exposed)
Figure 3.3: Effect of dilatation transformation r → reiθ on a spectrum "σ " of many-body Hamiltonian. Bound states and thresholds are invariant. However, as the continua rotate, complex eigenvalues may be exposed. Such eigenvalues correspond to poles but are "hidden" if θ = 0, and will be
exposed if the cuts are appropriately moved.
3.3
3.3.1
Modelling of Quantum Three-Body Systems
Coordinate System
We know that a system with N particles has 3N degrees of freedom. If the particles are not
aligned on a straight line, the degrees of freedom may be reduced to 3N − 6 [22] by separating
out the center of mass motion and rotations about the center of mass. Thus, for a three-body
system three coordinates are needed to describe the degrees of freedom.
Various different coordinate systems are available when discussing three-body systems.
Some of them are Jacobi coordinates [24–26], hyper-spherical coordinates [24, 27, 28] and
Pekeris [29, 30] coordinates etc. All these coordinate systems are good when considering the
bound state calculation of the Schrödinger equation. Hyper-spherical coordinates consist of a
hyper-radius providing the size of the cluster and two hyper-angles describing the radial and
angular correlation of the three-body system respectively. This coordinate system is particularly interesting since the three dimensional problem now reduces to a one dimensional hyperradius problem, with a set of effective potential obtained by solving a two dimensional equation
[27, 28]. Pekeris coordinates are especially convenient when describing the wave function and
the relative importance of different configurations of the system. The most suitable choice of the
coordinate system also depends on which physical system is studied, and may be important for
the efficiency of the numerical treatment.
Jacobi coordinates are one of the suitable choices for describing scattering processes and
work equally well in bound state calculations for three-body systems. To define a Jacobi coordinate, R = (x, y) has been used to describe the internal motion of the system shown in figure
10
Figure 3.4: The Jacobi coordinates for a three body system.
(3.4).
Here, x is the vector between particle 2 and 3, y is the vector between particle 1 and center
of mass of the pair (2,3), θ is the angle between the vectors x, y. The inter-particle distances, ri j
are associated to the Jacobi coordinates (in one possible configuration).
r23 = x,
r12 =
y2 −
r13 =
y2 +
m3
2m3
xy cos θ + (
x)2
m2 + m3
m2 + m3
1/2
2m2
m2
xy cos θ + (
x)2
m2 + m3
m2 + m3
1/2
,
,
Here m2 and m3 are the masses of particles 2 and 3 respectively. An advantage of using Jacobi
coordinates is their orthogonality, i.e. the kinetic energy operator is diagonal, meaning that there
is no mixing of the different derivatives.
3.3.2
Mass Polarization
To understand the phenomena of mass polarization [47] consider an atom or ion containing a
nucleus of mass M and charge Ze and N electrons of mass m and charge −e. The coordinate of
the nucleus with respect to the fixed origin O is denoted by R0 , and R1 , R2 .....RN those of the
electrons. In the absence of external fields, and neglecting all but the coulomb interactions, the
non-relativistic Hamiltonian operator of this system is given by
H = T +V
(3.19)
11
Figure 3.5: Coordinate System for two-electron atoms.
Where, the K.E operator, T is
T =−
h̄2 2
h̄2 N
∇R0 −
∇2Ri
∑
2M
2m i=1
(3.20)
and the Coulomb energy V is the sum of all the (N + 1) particles of the system.
In order to separate the motion of the center of mass, we change our coordinates from (R0 , R1 , R2 ....RN )
to (R, r1 ....rN ) where
R=
1
(MR0 + mR1 + ...... + mRN )
M + Nm
(3.21)
is the coordinate of the center of mass and
ri = Ri − R0 ,
i = 1, 2, ...N
(3.22)
are the relative coordinate of the electron w.r.t, the nucleus. It can be shown from eq.(3.21) and
eq.(3.22) that using the chain rule
∇R0 =
12
N
M
∇R − ∑ ∇ri
M + Nm
i=1
(3.23)
m
∇R + ∇ri
M + Nm
∇Ri =
(3.24)
Hence,
∇2R0 =
M
M + Nm
2
N
2M
∇2R −
∑ ∇R · ∇ri +
M + Nm i=1
!2
N
∑ ∇r
i
(3.25)
i=1
and
∇2Ri
=
m
M + Nm
2
∇2R +
2m
∇R · ∇ri + ∇2ri
M + Nm
(3.26)
Inserting the expression eq.(3.25) and (3.26) and eq.(3.20), we find the kinetic energy operator
in the new coordinates becomes
T =−
h̄2
h̄2
∇2R −
2(M + Nm)
2µ
h̄2
N
∑ ∇2r − M ∑ ∇r · ∇r
i
i
i=1
j
(3.27)
i> j
where
µ=
mM
m+M
(3.28)
is the reduced mass of the electron with respect to the nucleus. The Hamiltonian eq.(3.19) may
therefore be written as
H =−
h̄2
h̄2
∇2R −
2(M + Nm)
2µ
N
∑ ∇2ri −
i=1
h̄2
∑ ∇ri · ∇r j +V (r1 , r2 ...rN )(3.29)
M i>
j
The only term containing the coordinate R in eq.(3.29) is the first one, which shows the
kinetic energy operator of the center of mass. The next represents the sum of the kinetic energy
operator of the N electrons. The third term is due to nuclear motion, which is often called the
"mass polarization" term. Note that this term is smaller than the electronic kinetic energy by a
factor µ/M. Thus for normal atoms this is a small (though sometimes important) correction.
For exotic systems involving heavier particles, such as muons, this term is larger and cannot be
neglected.
Now, we solve this three-particle problem using Jacobi coordinates defined by
1
(MR0 + mR1 + mR2 ),
M + 2m
x = R1 − R0 ,
1
y = R2 −
(MR0 + mR1 ),
M+m
R=
(3.30)
(3.31)
(3.32)
Here R is the position of the center of mass of the atom, x is the position of electron relative
to the nucleus, and y is the position of the second electron relative to the center of mass of the
13
other two particles.
The kinetic energy operator becomes
T =−
h̄2 2
h̄2 2
h̄2 2
∇R0 −
∇R1 −
∇
2M
2m
2m R2
(3.33)
The derivatives in eq.(3.33) transform according to
M
M
∇y +
∇R ),
M+m
M + 2m
m
m
= (∇x −
∇y +
∇R ),
M+m
M + 2m
m
= (∇y +
∇R ),
M + 2m
∇R0 = (−∇x −
∇R1
∇R2
Now, the kinetic energy operator gives
2
h̄ M + m
M + 2m
h̄2
h̄2
T= −
∇2x −
∇2y −
∇2R
2
Mm
2 m(M + m)
2(M + 2m)
And, the total Hamiltonian becomes
2
h̄2
M + 2m
h̄2
h̄ M + m
2
2
2
∇x −
∇y −
∇ +V (x, y)
H= −
2
Mm
2 m(M + m)
2(M + 2m) R
(3.34)
(3.35)
Equation (3.35) has an important advantage that there is no mass-polarization term in the kinetic
energy part, but at the expense of making the expression for the potential more complicated
as we do not use inter-particle coordinates. We note though that for pairwise interactions the
potential can be written as
V (xi , yi ) = V (x1 ) +V (x2 ) +V (x3 )
(3.36)
where i in xi denotes the three different ways (or rearrangement channels) in which the Jacobi
coordinates can be defined. To evaluate the potential matrix elements, we thus need to be able
to transform between different rearrangement channels.
3.3.3
The Couple Rearrangement Channel method
In order to solve the many-body bound state problem accurately different methods are used. The
Couple Rearrangement Channel (CRC) Method is one of them, and was first introduced by M.
Kamimura [41] in 1988. This method has been applied to few-body problems of different types
in atomic and nuclear physics. The main advantage of this method is that it includes all three
possible sets of Jacobi coordinates and can thus describe different rearrangement channels. The
detailed computational explanation of the CRC method is discussed below.
The total three body wave function is expanded in terms of basis functions spanning the three
rearrangement channels in the Jacobian coordinate system α = a, b, c shown in figure (3.6).
ΨJM = ∑ cµ φµ
µ
14
(3.37)
r ,l
a a
R
R ,L
b b
R ,L
a a
r
c
,
l
b b
,
L
c
r
,
l
c c
Figure 3.6: The three arrangement channels of three body system and their Jacobi coordinates.
2
2
φµ = Nα rαlα RLαα e(−rα /rαi ) e(−Rα /RαI ) Ylα (r̂α ) ⊗YLα (R̂α ) JM
(3.38)
where
Ylα (r̂α ) ⊗YLα (R̂α ) JM ≡
∑ hlα m Lα M|J MJ iYl
αm
(r̂α )YLα M (R̂α )
(3.39)
m,M
In eq.(3.38) lα (Lα ) stands for the angular momentum of the relative motion associated with
the coordinate rα (Rα ) and the [ ]JMJ represent the vector coupling of the spherical harmonics.
Also lα (Lα ) are limited as 0 ≤ lα ≤ lαmax , |J − lα | ≤ Lα ≤ J − lα . In the term on the right hand
side of eq.(3.39) h i is the Clebsch-Gordan coefficient.
The non-linear variational parameters rαi and RαI are chosen as a geometric progression
rαi = rα1
rαn
rα1
i−1
n−1
,
i = 1, · · · , n
(3.40)
RαI = Rα1
Rαn
Rα1
I−1
N−1
,
I = 1, · · · , N
This choice gives a good balance between many basis functions in the inner short-range region,
and a few diffuse Gaussian which capture the long-range behaviour. Also, an advantage of
Gaussian basis functions is that they allow analytical calculation of the kinetic, potential energy
and overlap matrix elements in the Jacobian coordinate system. The requirement < φµ |φµ > =1
gives the normalization constant Nα
"
2lα +2
Nα = √
π(2lα + 1)!!
2
rαi
lα +3/2
2Lα +2
√
π(2Lα + 1)!!
2
RαI
Lα +3/2 #1/2
(3.41)
Note however that the basis functions are not orthogonal < φµ |φν >6= 0 , which makes it necessary to, in addition to H, also calculate the overlap matrix S, and solve a generalized eigenvalue
problem.
15
The main difficulty in constructing the Hamiltonian and overlap matrix lies in the transformation between different sets of Jacobi coordinates, as both functions in a matrix element must
be expressed in the same coordinate system. For the Coulomb potential (and more generally
potential on the form rN ) and Gaussian potentials, all integrals can be evaluated analytically.
In order to calculate matrix elements between basis functions of different channels α, one
of the functions is projected onto the channel coordinates of the other [42, 43].
We use a coordinate transformation (rβ , Rβ ) →(rα , Rα ) in the form
rβ = γβ α rα + δβ α Rα ,
0
(3.42)
0
Rβ = γβ α rα + δβ α Rα
0
0
Here γ, δ , γ , δ are kinetic mass factors. The formula below transforms the factor rl RL [Yl (r̂) ⊗
YL (R̂)]JM appearing in the basis function (3.38) from the channel β to channel α [44]:
p
4π(2L + 1)!
rLYLM (θ , φ ) =
rL−λ rλ
L
2
1
∑ (−1)λ p(2λ + 1)!(2(L
− λ + 1)!)
λ =0
× [YL−λ (Ω1 ) ⊗Yλ (Ω2 )]LM
(3.43)
where
r = r1 − r2 ·
Thus using eq.(3.42), we have
q
4π(2lβ + 1)!
=
l
rββ Ylβ mβ (r̂β )
l −λ
lβ
l −λ
γββα δβλα rαβ
∑q
Rλα
(2λ + 1)!(2(lβ − λ ) + 1)!
h
i
× Ylβ −λ (r̂α ) ⊗Yλ (R̂α )
λ =0
lβ mβ
q
4π(2Lβ + 1)!
=
L
Rββ YLβ Mβ (R̂β )
0
Lβ
∑q
(3.44)
L −Λ Λ
Rα
0
γβ α Lβ −Λ δβ α Λ rαβ
(2Λ + 1)!(2(lβ − Λ) + 1)!
h
i
× YLβ −Λ (r̂α ) ⊗YΛ (R̂α )
Λ=0
Lβ Mβ
(3.45)
The product of these two expressions forms the angular part of the basis-function. We can
combine them and using from section (5.16) in [44]:
h
i
0
0
Yl (Ω1 ) ⊗Yl (Ω2 )
1
0
00
0
2
00
∑ hL L M M
LM
16
0
LM
h
i
00
00
Yl (Ω1 ) ⊗Yl (Ω2 )
0
1
|LMi ∑ Bl10 l20 L0
l1 l2
2
00 00 00
l1 l2 L l1 l2 L
00
L M
00
=
[Yl1 (Ω1 ) ⊗Yl2 (Ω2 )]LM
(3.46)
Bl10 l20 L0
1
4π
q
0
00
0
00
0
00
(2l1 + 1)(2l1 + 1)(2l2 + 1)(2l2 + 1)(2L1 + 1)(2L1 + 1)
 0

00
l1 l1 l1 

0 00
0 00
0
00
×hl1 l1 0 0|l1 0ihl2 l2 0 0|l2 0i
·
l l l
 20 200 2 
L L L
=
00 00 00
l1 l2 L l1 l2 L
(3.47)
Thus:
i
h
l L
rββ Rββ Ylβ (r̂β ) ⊗YLβ (R̂β )
=
JM
=
∑
l
mβ Mβ
L
hlβ Lβ mβ Mβ |JMirββ Ylβ (r̂β )Rββ YLβ (R̂β )
∑
q
hlβ Lβ mβ Mβ |JMi4π (2lβ + 1)!(2Lβ + 1)!
lβ
Lβ
mβ Mβ
×
1
∑∑q
λ =0 Λ=0
l −λ
(2λ + 1)!(2Λ + 1)!(2(lβ − λ ) + 1)!(2(Lβ − Λ) + 1)!
0
l +Lβ −λ −Λ λ +Λ
Rα
0
×γββα γβ α Lβ −Λ δβλα δβ α Λ rαβ
×
L L
∑ hlβ Lβ mβ Mβ |L ML i ∑ Bll −λ
λl
α
L ML
lα Lα
β
α
β
Lβ −Λ Λ Lβ
Ylα (r̂α ) ⊗YLα (R̂α ) LM
(3.48)
We now use the orthogonality of the Clebsch-Gordan coefficients
∑
hlβ Lβ mβ Mβ |J Mihlβ Lβ mβ Mβ |L ML i = δJL δMML
mβ Mβ
which gives
h
i
l L
rββ Rββ Ylβ (r̂β ) ⊗YLβ (R̂β )
=
JM
lb
(2lβ + 1)(2Lβ + 1)
Lb
∑∑
λ =0 Λ=0
lβ −λ
0
Lβ −Λ
s
(2lβ )!
(2λ )!(2lβ − 2λ )!
0
s
(2Lβ )!
(2Λ)!(2Lβ − 2Λ)!
l +L −λ −Λ
×(γβ α )
(γβ α )
(δβ α )λ (δβ α )Λ rαβ β
Rλα+Λ


 lβ − λ Lβ − Λ lα 
λ
Λ
Lα
×∑
hl − λ Lβ − Λ 0 0|lα 0ihλ Λ 0 0|Lα 0i

 β
lα Lα
lβ
Lβ
J
× Ylα (r̂α ) ⊗YLα (R̂α ) JM
(3.49)
where the nine − j symbol {} gives the coupling between the four angular momenta lα Lα , lβ Lβ .
This expression can be summarized in the form defining T ≡ λ + Λ
17
i
h
ˆ
Ylβ (rˆβ ) ⊗YLβ (Rβ )
l L
rββ Rββ
∑
≡
JM
l +Lβ −T
< lβ Lβ J |lα Lα T J >β α rαβ
RTα Ylα (r̂α ) ⊗YLα (R̂α ) JM
(3.50)
lα ,Lα ,T
and the transformation coefficients < lβ Lβ J |lα Lα T J >β α can calculated and stored prior to
the computation. This equation is key to the transformation between channels.
Now we can write the product of two different channel’s trial wave-functions in the coordinates of a single channel (channel α) by applying the transformation above to one of the basis
function.
φµ∗ (rα , Rα )φν (rβ , Rβ ) =
†
Nα rαlα RLαα exp[−(rα /rαi )2 − (Rα /RαI )2 ] Ylα (r̂α ) ⊗YLα (R̂α ) JM
0
1 0
1
2
2
×Nβ exp − 2 (γβ α rα + δβ α Rα ) − 2 (γβ α rα + δβ α Rα )
rαi
RαI
h
i
0
0
l +L +T
× ∑ hlβ Lβ J |lα Lα T Jiβ α rαβ β RTα Yl 0 (r̂α ) ⊗YL0 (R̂α )
0
α
0
α
lα ,Lα ,T
0
JM
(3.51)
0
where lα and Lα denote the auxiliary angular momenta in channel α that are used in the transformation (3.50). If we separate the radial and angular components of the above expression, the
argument of the transformation Gaussian is rewritten as
0
R2α
1
rα2
1 0
−
− (γ rα + δβ α Rα )2 − 2 (γβ α rα + δβ α Rα )2 ≡ −ηrα2 − 2ξ rα · Rα − ζ R2α
rα2 i R2αI rα2 i β α
RαI
(3.52)
and using the expansion
−
∞
exp(i r1 · r2 ) = 4π
∑ iλ jλ (r1 r2 ) [Yλ (r̂1 ) ⊗Yλ (r̂2 )]0 0
λ =0
∞
exp[−2ξ rα · Rα ] =
∑
4πiλ
p
2λ + 1 jλ (2iξ rR) Yλ (r̂α ) ⊗Yλ (R̂α ) 0
(3.53)
λ =0
where jλ (z) are the spherical Bessel functions. Inserting into eq.(3.51), the angular dependence
is given by
∑
0
0
lα ,Lα
18
i
† h
Ylα (r̂α ) ⊗YLα (R̂α ) JM Yλ (r̂α ) ⊗Yλ (R̂α ) 0 Yl 0 (r̂α ) ⊗YL0 (R̂α )
α
α
JM
(3.54)
which can be simplified, again using from section (5.16.2) in [44]
i
h
Yλ (r̂α ) ⊗Yλ (R̂α ) 0 Yl 0 (r̂α ) ⊗YL0 (R̂α )
α

 λ
0
× lα

Σ
JM
α
λ
0
Lα
Λ
=
2λ + 1
∑
4π Σ,Λ
q
0
0
(2J + 1)(2lα + 1)(2Lα + 1)

(3.55)
0 
0
0
hλ lα 0 0|Σ 0ihλ Lα 0 0|Λ 0i YΣ (r̂α ) ⊗YΛ (R̂α ) JM
J

J
Relation (3.51) then becomes
φµ∗ (rα , Rα )φν (rβ , Rβ ) =
Nα Nβ rαlα RLαα exp[−ηrα2 − ζ R2α ]
0
×
∑
0
l +Lβ +T
hlβ Lβ J |lα Lα T Jiβ α rαβ
RTα
0
0
lα ,Lα ,T
q
0
0
× ∑ (2J + 1)(2lα + 1)(2Lα + 1)(2λ + 1)3/2 jλ (2iξ rR) iλ
Σ


 λ λ 0 
0
0
0
0
×∑
hλ lα 0 0|Σ 0ihλ Lα 0 0|Λ 0i
lα Lα J


Σ,Λ
Σ Λ J
†
× YΣ (r̂α ) ⊗YΛ (R̂α ) JM Ylα (r̂α ) ⊗YLα (R̂α ) JM
(3.56)
Now, this can be directly used for analytical calculation of the matrix elements of the kinetic
energy operator and the interaction potential.
As an example, we derive the overlap integral: we start with the angular integration which gives,
Z
Z
† dΩr dΩR Ylα (rˆα ) ⊗YLα (Rˆα ) JM Yλ (rˆα ) ⊗YΛ (Rˆα ) 00
q
h
i†
2λ + 1
0
0
ˆ
× Yl 0 (rˆα ) ⊗YL0 (Rα )
(2J + 1)(2lα + 1)(2Lα + 1)
=
∑
α
α
4π Σ,Λ
JM


 λ λ 0 
0
0
0
0
× lα Lα J
hλ lα 0 0|Σ 0ihλ Lα 0 0|Λ 0i


Σ Λ J
Z
Z
† × dΩrα dΩRα Ylα (rˆα ) ⊗YLα (Rˆα ) JM Yλ (rˆα ) ⊗YΣ (RˆΛ ) JM
q
2λ + 1
0
0
=
(2J + 1)(2lα + 1)(2Lα + 1)
4π


 λ λ 0 
0
0
0
0
× lα Lα J
hλ lα 0 0|Σ 0ihλ Lα 0 0|Λ 0i


lα Lα J
(3.57)
The remaining radial integrations have the form:
19
Z
I=
Z
dr
2
2
dR rlβ +Lβ +lα −T RLα +T +2 e−ηr e−ζ R iλ jλ (2iξ rR)
(3.58)
We start with the integration over R and use,
r
jλ (z) =
π 1
√ J 1 (z)
2 z λ+2
(3.59)
where Jλ (z) is the usual Bessel function.
1
Further, let x = ζ 2 R and M = (Lα + T − λ )/2, which gives
λ + 12
I=i
r Z
Z
λ
5
3
1
2
2
2
π
dr rlβ +Lβ +lα −T + 3 e−ηr ζ −(M+ 2 + 4 ) dx x2M+λ + 2 e−x Jλ + 1 (2iξ ζ − 2 rx) (3.60)
2
4
The integral over x can be found in [45]. The result is
λ − 21
r
I=i
π −ηr2 −(M+ λ + 5 )
2
4
e
ζ
4
Z
dr r
lβ +Lβ +lα −T + 23
M! ξ 2ζr2
e
2
iξ r
p
ζ
!λ + 12
(λ + 1 )
LM 2
ξ 2 r2
ζ
(3.61)
(λ + 21 )
where LM
is the Laguerre polynomial.
Introducing N =
mial [46]
lβ +Lβ +lβ −T −λ
,
2
Θ = ηζ − ξ 2 and using the expansion for the Laguerre polyno-
M
α
LM
(x) =
∑ (−1)k
k=0
1 M+α k
x
k! M − k
(3.62)
gives
√
Z
M
π
M! M + λ + 12
− Θ r2
λ λ −(M+λ + 23 )
2k −k
I=
(−1) ξ ζ
ξ ζ
dr r2(N+λ +k+1) e ζ
∑
M−k
4
k=0 k!
(3.63)
The remaining r-integral is standard and gives finally
√
π
I = (−1)
8
λ
M
3
M! M + λ + 21
3
∑ k! M − k Γ(N + k + λ + 2 )ξ λ +2k ζ N−M Θ−N−λ −k− 2
k=0
(3.64)
The product of the angular integrals, the radial integral and the pre-factors in the big bracket
( ) gives the overlap matrix. The potential and kinetic energies can be derived in similar, but
slightly more complicated ways.
20
e-
+
e+
+ ...
Figure 3.7: Vacuum polarization insertion in the photon propagator.
3.4
Vacuum Polarization
In the mid 1930’s the quantum electro-dynamical notion of vacuum polarization emerged from
the work of Dirac, Furry and Oppenheimer, Serber and Uehling [48–51]. Vacuum polarization is
a distortion of the Coulombic interaction due to the production of virtual electron-positron pairs
by a strong electromagnetic field. It can be expressed as a correction to the photon propagator
as in figure (3.7).
Muonic hydrogen differs from ordinary hydrogen atoms in that the effect of vacuum polarization is much larger and is the prime QED contribution. The QED shift for an ordinary
hydrogen atom is called Lamb shift. The total Lamb shift of the 2S level of ordinary H is 1058
MHz, while the vacuum polarization is only −27 MHz. Thus for ordinary H, the vacuum polarization is a small part of the total QED effect. For muonic hydrogen the vacuum polarization is
the dominating QED correction for the 2S-2P splitting of µ p by −0.2 eV. This is precisely the
splitting which has been measured to obtain the proton radius.
The importance of vacuum polarization for the energy spectrum of µ p atoms has increased
with the reduction of atomic radius. The Bohr radius is 200 times smaller, hence, S-state muonic
wave-functions overlap strongly with the charge distribution of the virtual e+ e− pairs. The first
order Uehling potential for µ p gives a value of 205.001 meV [15] for the 2P-2S Lamb shift. The
question arises: why are the p, d and f states less shifted? The reason is that only in S-states is the
electron wave-function different from 0 at r = 0 (i.e. close to nucleus). The shift of the P-state
is given in table (3.1). If we consider a charge distribution of the nucleus which is symmetric,
the effective Uehling potential [51] giving the first order correction to the Coulomb potential
V (r) = Z1 Z2 /r reads as [52]
2α
Vpol (r) =
3
Z ∞
1
×
p
1
2
2
dt t − 1 2 + 4
t
t
"Z
0
r
0
sinh(2tcr ) e−2ctr
dr r ρ(r )
+
0
r
ctr
0 02
0
Z ∞
r
sinh(2tcr) e−2ctr
dr r ρ(r)
0
0
ctr
r
0 02
0
#
(3.65)
0
where ρ(r ) is the charge distribution of the nucleus, α the fine-structure constant and c the
0
0
speed of light in vacuum. For hydrogen nuclei we can with decent accuracy write ρ(r ) = δ 3 (r ),
yielding
21
Table 3.1: Corrections due to vacuum polarization of the 2s and 2p states in muonic hydrogen,
calculated in first-order perturbation theory. Energies in meV. The experimental 2p-2s splitting is
dominated by vacuum polarization, but also includes various other small terms, including terms
dependent on the proton radius.
experimental
Uehling potential
Fitted Uehling potential
E(2s) meV
E(2p) meV
−219.58
−219.57
−14.58
−14.57
Z1 Z2 α
Vpol (r) =
r 3π
Z 1
0
E(2p) − E(2s) meV
206.295
205.01
205.00
s
1
−2cr/x
2
e
(2 + x )
− 1 dx.
x2
(3.66)
This form of the potential is very difficult to treat numerically. Instead we use the approach from
[43] and fit it to a sum of 20 Gaussians. We find that our fitted potential gives results agreeing
very well with the results using eq.(3.66). The Vpol (r) causes a splitting ∆E pol ≡ E2p − E2s =
205.00 meV of n = 2 levels of µ p as shown in table (3.1).
22
4. Negative Hydrogen Ion
To explore the pµe hypothesis, we start with the negative hydrogen ion pe− e− and change the
mass of one e− to make an exotic particle. While gradually scaling the mass of the exotic particle
towards the muon mass, we follow the binding energy and life times of resonances in the threebody system. Exotic particles are usually unstable and thus have short lifetimes. Here, though,
we treat all particles as having infinite lifetimes, which is motivated since e.g. the muon lifetime
is much longer than typical atomic time scales. In addition, exotic systems serve to examine
the general theory of three body systems and analyze their inter-particle correlation. Because
all these exotic particles (except the positron), are heavier than the electron and therefore more
strongly bound to the nucleus than electrons, their transitions during the de-excitation are considerably more energetic than those of electrons which we will discuss later in the chapter.
Negative hydrogen ions are essential in astrophysics especially for the description of the
opacity of the sun’s atmosphere [66]. These resonances in H− studied both experimentally and
theoretically from the first identification of the presence of the resonances in electron-hydrogen
scattering studies by Burke and Schey [54], Smith and others [55–59].
Like most negative atomic ions, H− has only one stable bound state. In this ground state of
−
H the correlation between the two electrons is already strong. Negative hydrogen ions have no
singly excited states, but there exist doubly excited states, which are embedded in the continuous
part of hydrogen spectrum and can be observed as resonances.
The solution to the non-relativistic Schrödinger equation for the three body problem was
found using the Coupled Rearrangement Channel method (discussed in section (3.3.3)) [41],
including the Complex Scaling method which was disscused in section (3.2.2). In the Complex
Scaling method, the separation of bound state, resonant states and the continuum states can be
performed without any ambiguity by the scaling angle θ as shown in fig. (4.1). The results for
the ground states of H− are shown in Table (4.1), which shows the convergence and accuracy
of the code. At low values of the real scaling parameter α (discussed in section 3.2.1) the
deviation of our results from literature value is 10−9 a.u. [60]. For the resonances which have
more extended wave functions, the best accuracy is achieved for α ∼ 1. We are interested in
the resonance below the n = 2 threshold. Figure (4.2) shows the convergence using different
basis functions for the n = 2 state. It shows that within the stabilized plateau the rotational paths
(changes E as a function of θ ) meet each other at the position of a pole. Hence, at the position
of a pole, the change in energy with respect to θ is minimized. In this case, their position and
widths are obtained by the condition ∂ E/∂ θ ≈ 0.
In Table (4.2), we compare our present results with the experimental results of Warner et al
[63], Williams [64], Sanchez and Burrow [65] and theoretical results of Bürgers and Lindroth
[60], Ho and Bhatia [61] and Chen [62]. Our results for the resonance energy (ER ) agree quite
well with those of [60]. The deviation in the energies is about 10−7 a.u. and about 10−6 a.u. for
23
Figure 4.1: The spectrum of H θ for H− .The only ground state is marked with a circle and filled
colour circles show a pseudo-continuum energies, rotated downwards by 2θ in the complex energy
plane. The resonance state, however has a θ -independent complex energy shown by the square.
24
−3
0.5
x 10
2826
2985
3024
3798
5105
5319
0
Im(E)(a.u)
−0.5
−1
−1.5
−2
−2.5
−3
−0.15
−0.1495
−0.149
−0.1485
Re(E)(a.u)
−0.148
−0.1475
−0.147
Figure 4.2: Using different basis sets for the state n=2, J=0, showing the convergence. The value
of complex scaling parameter θ changes with steps of 0.01 rad. The resonance is positioned E =
0.014877625373 − i0.00086617931.
25
Table 4.1: Study of the rate of convergence for the different basis functions for the system p+ e− e− .
N is the number of configurations, α is the scaling length and the maximum value used for angular
momenta L and l are 4. In the last row we show the best ground state eigenvalue of the system from
the reference [60] and the present work.
N
α
15
1
0.9
0.3
No. of Basis functions
Infinite Proton mass
2985
3798
5101
5700
5101
5101
Exact
15
Exact
1
Real Proton mass
5300
−ER
-0.52775067690
-0.52775074550
-0.52775094624
-0.52775101621
-0.52775101472
-0.52775101560
-0.52775101654
-0.52744588108
-0.52744584392
the width.
Table 4.2: Comparison of non-relativistic resonance widths of H− below n = 2 hydrogen threshold
in (a.u).
States
1s2 1 Se
2s2 1 Se
2s3s 1 Se
2s3s 3 Se
2s4s 1 Se
2s4s 3 Se
2s3p 1 Pe
2s4p 1 Pe
Present results
−ER
0.52775101401
0.14877625373
0.12602006146
0.127104276
0.125059737
0.125118188
0.1260498047
0.1250351840
Bürgers et al [60]
−ER
0.52775101654
0.14877625394
0.12602006374
0.12505785
0.12604985948
0.125035052
Ho et al [61]
−ER
0.148775
0.126021
0.127104
0.1250580
0.12511818
0.126049
-
Chen [62]
−ER
0.148782
0.126021
0.1271042
0.1250579
0.1260499
0.1250349
Warner et al [63]
−ER
0.14908(47)
Williams [64]
−ER
0.14879(37)
Sanchez et al [65]
Burrow, −ER
0.14875(37)
We thus conclude that our numerical method can represent the n = 2 resonances in H− with
very good accuracy.
Our next step is to start scaling the mass of one of the electrons to higher values. In this
way we want to find out whether the pµe system may support any resonances under the µ p(2S)
threshold. We replace the mass of one of the orbital electrons in negative hydrogen system and
increase it steadily and call it the pex system, where x is a negatively charged particle with mass
mx = xme . The atomic orbit of the x-particle is therefore closer to the proton than the electron’s
orbit in an ordinary hydrogen atom. Since we are interested in the resonance which lie under
the pµ(n = 2) threshold, we focus on resonances under the px(n = 2) threshold. We find that
the resonance disappears at mx = 3.8me . We also see some extra resonances which appear under
the H(1s) threshold around mx = 3me shown, in paper I. The energy and width of the resonances
increase when we increase the mass of the x particle, when the and px(n = 2) thresholds cross
at mx = 4me , the resonance follows to the H(1s) and px(n = 2) threshold instead. We detect a
resonance at binding energy 0.14 meV at mass mx = 6.7me but the width of the resonances above
mx = 6me is too small to be determined accurately. The figure and table are shown in Paper I.
26
Table 4.3: Comparison of non-relativistic resonance widths of H− below n = 2 hydrogen threshold
in (a.u).
States
1s2 1 Se
2s2 1 Se
2s3s 1 Se
2s3s 3 Se
2s4s 1 Se
2s4s 3 Se
2s3p 1 Pe
2s4p 1 Pe
4.1
Present results
Γ/2
0.000866617931
0.000045254
0.000000332
0.00000260
0.00000003
0.0000005029
0.000000058
Bürgers et al[60]
Γ/2
0.00086661817
0.00004526486
0.00000261
0.0000006841
0.000000039
Ho et al[61]
Γ/2
0.00086
0.000044
0.000000335
0.00000254
0.00000066
-
Chen [62]
Γ/2
0.00086
0.0000447
0.000000342
0.00000258
0.00000002
0.00000061
0.0.000000035
Warner et al[63]
Γ/2
0.00116
Williams [64]
Γ/2
0.000825
Sanchez et al [65]
Burrow, Γ/2
-
The pµe Hypothesis
We also employed another way to investigate this pµe hypothesis, starting with pµ µ system
and decreasing the mass of one of the muon and observing where the resonance state at n = 2
disappears as we did in the pex system. To do this, we first calculated the ground state energy
for pµ µ giving a good agreement with the literature values [66, 67] shown in Table (4.4). For
the resonance energy at n = 2 threshold the binding energy Eb = 120.4716543 eV and width
Γ = 7.146072583 eV which is slightly larger binding energy than [21, 68]. We compare our
results to [21] in paper I, they used a different numerical technique. However, the results agree
very well with and without including the correction due to vacuum polarization given in table
(3.1).
Table 4.4: The ground state energy of pµ µ (a.u).
Present result
-97.5669903006
Frolov [66]
-97.5669834
Ancarani [67]
-97.3747607
Because the muon’s orbit is close to the proton, the proton charge pµe is shielded from the
outer electron. However, the muon and proton in the 2S-2P state form a neutral core which
interacts with the electron via dipole interactions. As expected, when we decrease the mass of
the third particle, its orbital around the pµ(n = 2) core becomes larger and its binding energy
decreases. The resonances then follow the pµ(n = 2) threshold, until it disappears at mx ' 30me .
On the other hand, we also find that some new resonances appear under the px(n = 1) threshold
at mx ∼ 65me . For the pµx system the mechanism leading to an additional resonance below the
px(n = 1) threshold is similar to pex. The figure and tables are discussed in Paper I.
27
28
5. Positron-Hydrogen Scattering (e+H)
We also searched for quasi-bound states in (e− , e+ , p), a three body system including a positron,
an electron, and a proton. This problem is of great interest [9–13]. The positron is the antiparticle of the electron with the same mass however having opposite charge. The electron and
positron can form a bound state like an electron and a proton. The bound states of e− and e+
are called positronium states. The electromagnetic interaction is accountable for binding; thus,
positronium shares many features with the hydrogen atom. An important difference, however,
is the possibility of annihilation of the electron when it meets the positron, giving gamma rays.
Positronium is a short-lived system.
Resonance phenomena have been studied extensively in positron-hydrogen scattering. According to Mittlemen [69], an infinite sequence of resonances should exist below the n = 2 excitation threshold in positron-hydrogen scattering. Investigation of the phenomena of resonance
in positron-hydrogen scattering remained an important concern for research over the last few
decades. Prior research argued the persistence of resonances below the excitation of the n = 2
threshold and a variety of theoretical approximations have been used to predict the positions and
widths of resonances [9–13], even though there are not yet any experimental observations [5–7].
In calculation of resonances, we use the same complex scaling method described in section
(3.2.2). We find resonances for natural and unnatural parity states. We all know about natural
parity, π = (−1)L where L is the total angular momentum, but in case of unnatural parity states
we have, π = (−1)L+1 . When a positron collides with an excited hydrogen in the vicinity of the
Ps (n = 2) threshold, the positron may pick up the electron from the H and form real or virtual
positronium in its n = 2 states. Here we should mention that the S-state component does not
contribute to the states with unnatural parities of π = (−1)L+1 . The states interacting with the
scattering continua will manifest themselves as resonances in positron scattering with excited
hydrogen.
Resonance states for the e+ H system with natural parity have been calculated for S-, Pand D-waves both for physical and infinite proton mass. Most of the resonances are similar to
previous literature values, a few of them have better values and some resonances not previously
reported are presented in Paper II. Also resonances for unnatural parity states for P- and D-waves
are calculated and discussed shown in Paper II.
29
30
Summary and Outlook
In this work, resonance positions and widths have been studied for three-body systems. We have
shown that for arbitrary mass values accurate results can be obtained using the CRC method
with complex scaling. In addition we have been able to incorporate corrections to the Coulomb
potential by expressing them as a sum of Gaussians, and applying complex scaling also to these
corrections.
For outlook, we are working on resonances in positron-alkali atom systems. Alkali atoms
can be expressed as quasi-one-electron targets, the positron-alkali-atom scattering is very different from the positron-hydrogen scattering in different ways. The alkali atoms have large dipole
polarizabilities that is strong enough to produce bound states [70]. In addition, the hydrogen
atom has degeneracy in its spectrum while the alkali atoms do not have such characteristics.
Another important feature of a positron-alkali system is rearrangement process, i.e. positronium(Ps) formation, which is energetically possible even at zero impact energy and is expected to
have a significant effect on the scattering process. The positron-electron interaction is of course
only the usual Coulomb interaction. For the positron-core interaction we can use the HartreeFock approximation for the core (considered inert), and calculate an effective shielding due to
the core electrons. For the electron-core interaction we do not use the shielding given by the
Hartree-Fock approximation because (i) it does not incorporate electron exchange (ii) it is very
important to get the atomic threshold energies to very high accuracy. For this purpose there are
instead model potentials which have been used in literature [71].
Another project is to develop our codes so that we can calculate scattering cross sections.
We have shown that positron-hydrogen resonances can be calculated to very high accuracy. It
would therefore be natural to look at positron scattering on hydrogen in the first instance. Our
three-body code, including all sets of Jacobi coordinates, is especially well suited to calculate
rearrangement processes such as positronium formation. Of particular interest is the reverse
process, hydrogen formation in positronium scattering on a proton. This process (or rather its
equivalent charge conjugate process) could be a way of making anti-hydrogen.
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