The Henryk Niewodniczański
INSTITUTE OF NUCLEAR PHYSICS
Polish Academy of Sciences
Department of Strong Interactions
and Mechanisms of Nuclear Reactions
MUONIC ATOM SCATTERING FROM ATOMS
AND MOLECULES
Jakub Gronowski
A thesis submitted for the Doctor of Philosophy in Physics degree
prepared under the supervision of dr hab. Andrzej Adamczak
Acknowledgments
First of all, I want to express my gratitude to my adviser Andrzej Adamczak for his in-valuable patience, help and continuous support. I would like to thank Wilhelm Czapliński from the AGH University of Science and Technology for inspiring and encouraging me to pursue a scientiﬁc career and interesting me in the ﬁeld of muonic physics.
I am grateful to Prof. Antoni Szczurek, Head of Department of Strong Interactions and Mechanisms of Nuclear Reactions, for a very friendly and creative atmosphere during my PhD studies.
I also wish to thank my colleagues from the PhD studies at the Institute of Nuclear Physics Polish Academy of Sciences and other people I had opportunity to meet during my work on the Thesis.
Abstract
Theoretical study of the three-body systems that consist of the hydrogen and helium isotopes and the muon is a subject of this thesis. Also, this work deals with the muonic hydrogen atom scattering from hydrogenic molecules, especially at collision energies com-parable and greater than the dissociation thresholds of these molecules.
The adiabatic representation of the three-body problem with the Coulomb interaction and the phase-function method are applied to the hydrogen-helium-muon systems. In particular, the elastic cross sections for muonic hydrogen scattering from helium nuclei, the energy levels of the hydrogen-helium muonic molecules, and the cross sections and rates for creation and deexcitation of such molecules are calculated and discussed.
The diﬀerential cross sections for muonic hydrogen atom scattering from hydrogenic molecules are calculated using the Morse potential for internuclear interaction in these molecules. Numerical estimations are performed for various rotational, vibrational, and spin states of the considered systems. Dissociation of the hydrogenic molecules in collision with the muonic atoms and corresponding contributions of the discrete and continuous energy spectra of these molecules to the scattering cross sections are also studied.
A role of the considered muonic atomic and muonic molecular processes in various experiments in low-energy muon physics is discussed. In particular, a study of diﬀusion of the muonic atoms pµ and dµ in H2 and D2 gases is performed by means of the Monte
Contents
1 Introduction 1
2 Hydrogen-helium muonic molecules 9
2.1 Adiabatic expansion of a three-body system . . . 9
2.2 Matrix elements for hydrogen-helium muonic system . . . 14
2.3 Molecular terms and two-center wave functions . . . 16
2.4 One-level adiabatic approximation for 2pσ state . . . 20
2.5 Phase-function method for the one-level problem . . . 21
2.6 Elastic scattering of muonic hydrogen on helium . . . 24
2.7 Inﬂuence of the adiabatic corrections on rovibrational levels . . . 28
2.8 Resonant enhancement of the hydrogen-helium muonic molecule formation 32 2.9 Rotational deexcitation of the hydrogen-helium muonic molecules . . . 37
3 Muonic hydrogen atom scattering from hydrogenic molecules 45 3.1 Scattering problem in the crossover region . . . 45
3.2 The Morse potential . . . 46
3.3 Nuclear amplitudes and cross sections . . . 50
3.4 Molecular scattering . . . 56
3.5 Electron screening corrections . . . 62
3.6 Numerical implementation . . . 64
3.7 Examples of the calculated cross sections . . . 66
3.8 Monte Carlo simulations . . . 77
4 Summary and conclusions 81
Chapter 1
Introduction
A detailed knowledge of various muonic-atomic and muonic-molecular processes is im-portant in low-energy muon science. In particular, this knowledge is indispensable in spectroscopy of muonic atoms, in studies of the weak interaction via µ− _{nuclear capture}
in muonic atoms and molecules, and in muon-catalyzed fusion of light isotopes. The negative muon µ−_{, which is a product the negative pion decay}
π− _{→ µ}−+ νµ, (1.1)
is an unstable particle with the lifetime τµ= 2.2 × 10−6s. Its mass mµ equals about 207
electron masses meand, therefore, the muonic atoms are about 207 times smaller than the
corresponding electronic atoms. As a result, the spectrometry of muonic atoms enables an accurate determination of the electromagnetic structure of nuclei. A common method of observing the excited states of muonic atoms and subsequent radiative transitions is the muon exchange between muonic hydrogen atoms and atoms of heavier elements. For example, this mechanism is employed in a novel method for studying properties of short-living radioisotopes, which is now being developed [1]. A great progress in laser technology gave rise to the laser spectroscopy of muonic atoms. A spectacular example is the recent high-precision determination of the root-mean-square charge radius of the proton [2], which is a test of the bound-state quantum electrodynamics. This has been achieved via the laser resonance measurement of the Lamb shift (diﬀerence between the binding energies of the 23_{S}
1/2 and 25P3/2 states of hydrogen-like atoms) in the muonic
hydrogen atom, many relativistic corrections to the bound states are enhanced, which enables high-precision experiments.
Observations of muonic atoms are also important in studies of the weak interaction in low-energy limit. A high-precision measurement of the nuclear muon capture rate in the (pµ)1s atom
µ−+ p → n + νµ, (1.2)
has lead to the accurate determination of the induced pseudoscalar coupling constant gP
in the weak form factor of the nucleon [3] (MuCap experiment at PSI). A new MuSun experiment — a high-precision measurement of the nuclear muon capture rate in the muonic deuterium atom (dµ)1s is underway [4]. This is the ﬁrst precise measurement of
the basic electroweak reaction in the two-nucleon system
µ−_{+ d → n + n + ν}µ, (1.3)
which will have an impact on our understanding of the solar pp fusion and the ν + d scattering of solar neutrinos.
The muonic hydrogen atoms in collisions with hydrogenic molecules or helium atoms can form muonic molecular ions, usually simply called muonic molecules, e.g., (pµp)+_{,}
(pµd)+_{, (dµd)}+_{, or (Heµp)}++_{. Such molecules are about 207 times smaller than the }
ordi-nary molecular ion H+
2. A small internuclear distance and a lowered Coulomb barrier lead
to a high probability of nuclear fusion in these molecules, even at the room temperature. This phenomenon, called muon-catalyzed fusion (µCF), was ﬁrst considered by Frank [5]. Muon-catalyzed fusion was ﬁrst observed by Alvarez in a natural-hydrogen bubble cham-ber [6]. In this case, muons from cosmic rays catalyzed the following fusion reactions in the pµd molecule: pµd −→ 3_{He + µ + 5.5 MeV} µ3_{He + γ + 5.5 MeV .} (1.4) In the ﬁrst reaction, the muon is released and can catalyze a next fusion event. The muonic helium atom is created in the second reaction. As a result, the muon cannot catalyze next hydrogen-isotope fusions, unless it is released in further collisions with surrounding molecules. The Alvarez discovery was followed by the pioneering theoretical
paper by Jackson [7], who also considered µCF in deuterium-tritium targets dµt −→ 4_{He + n + µ + 17.6 MeV} µ4_{He + n + 17.6 MeV .} (1.5)
These investigations launched extensive experimental and theoretical studies on possible application of µCF to energy production. An eﬀective energy production via µCF requires catalysis of many hundreds of fusion events during the muon lifetime and very eﬃcient and dense muon beams. The muonic-molecule formation rate and the probability of muonic helium creation after the fusion are the crucial parameters that limit the eﬀectiveness of energy production. A muonic molecular ion is formed as a heavy center of a muonic-molecular complex, as a product of the reaction
dµ+ H2 −→ [(pµd)pe]++ e , (1.6)
in which the energy of muonic-molecular bond (∼ 30–300 eV) is taken by the conversion electron. It was proved that this nonresonant mechanism of formation is not suﬃciently fast for an eﬀective µCF. The experimental and theoretical research at JINR in Dubna resulted in discovery of a faster resonance mechanism of dµd and dµt formation, due to the existence of weakly-bound (≈ 1 eV) states in these muonic molecules [8–10]. In this mechanism, a relatively small sum of the binding energy and collision energy is transferred to rovibrational excitations of the muonic-molecular complex
dµ+ D2 −→
h
(dµd) deei∗. (1.7)
Then it was proved that µCF is the most optimal in D/T targets, where one muon can catalyze more than 100 fusions [11]. In experiments, up to 130 fusions per muon have already been observed. Since this factor is too small for an eﬀective energy produc-tion from pure µCF, a hybrid (fusion-ﬁssion) muon-catalytic reactor has been proposed by Petrov [12]. The optimal conditions for energy production are predicted for high-pressure (up to 1500 bar) D/T at temperatures 1000–2000 K. Experimental studies of µCF in such conditions are planned in future muon facilities. Another possible practi-cal application of µCF in D/T, which does not require a positive energy output, is an
intense source (1014 _{n /}_{cm}2_{· s) of 14-MeV neutrons for research in materials science and}
nuclear-waste incineration [13]. Details of the µCF basics are given in reviews [14–16] and references therein.
Independently of possible practical applications, µCF is the unique phenomenon that enables studies of nuclear fusion in a very low-energy limit (. 1 keV). Here, the fusion takes places in bound states of muonic-molecular ions with deﬁnite angular momenta and spins, e.g., in (dµd)+ _{or (Heµp)}++_{. This is of particular interest for astrophysics (nuclear}
reactions in stars) and for studies of lightest nuclei [17]. The recent experiment concerning the dd radiative fusion in dµd in Dubna is an example of such studies. In the accelerator-driven fusion, such very low energies are hardly reachable, so that the fusion cross sections are often extrapolated down from the MeV energy region. An accurate description of the µCF processes is also important in the precision measurements of muon nuclear capture, which are mentioned above. In such experiments, the muon capture takes place not only in the pµ and dµ atoms, but also in the muonic molecules pµp and dµd. This eﬀect is especially important in the MuSun experiment since the process (1.7) is resonant.
Intense muon beams are produced in the so-called meson factories. They usually employ proton beams, which produce pions in collisions with nuclei of a suitable target. The pions are then collected and formed into a pion beam. Finally, the muons are created after the in-ﬂight decay of pions (1.1). The most intense muon beams are now available at PSI (Switzerland), at RIKEN-RAL Muon facility (UK), and at TRIUMF (Canada). A new muon facility MUSE has been recently built at J-PARC (Japan).
Hydrogenic molecules are the most common targets in various experiments in low-energy muon science. Such experiments have been performed in gaseous, liquid, and solid targets at temperatures ranging from 3 K to 800 K. Sometimes heavier elements are added in order to study the muon transfer between muonic hydrogen atoms and heavier atoms. Certain elements, such as oxygen, carbon or nitrogen, are always present in experimental targets as a small contamination. The presence of helium in such targets is of particular importance. Helium nuclei are always produced in muon-catalyzed fusion and the muon sticking to helium is a main process that limits the number fusions per one muon. Moreover, helium is permanently created in targets containing tritium, as a result of the decay process
Free muons injected into a gaseous H/D/T mixture are decelerated. When kinetic energy of the muon approaches the electron binding energy of about 10 eV, the electron in a hydrogen-isotope molecule is replaced by the muon. As a result, the excited complexes, such as H+
2µ and D+2µ, are formed. Then the complex decays and the muonic hydrogen
atom is created in an exited state with the principal quantum number n ≈qmµ/me≈ 14
and with initial kinetic energy of about 1 eV. Formation of the muonic atoms is followed by atomic cascade, i.e., a large set of various processes such as the external Auger eﬀect, Stark transitions, Coulomb transitions, radiative deexcitation, and elastic scattering [15, 18–20]. During the cascade, a fraction of the initially formed muonic atoms reach the ground 1S state. Under special conditions, in low-pressure H2, about 1% of all stopped muons
form the metastable (pµ)2S atoms [21]. A suﬃciently long existence of these metastable
states has been crucial for the above-mentioned determination of the proton radius using a tunable laser. The excited muonic atoms can also form the muonic molecular complexes or loose their muons via the transfer to heavier elements, in collisions with surrounding matter. Owing to various cascade processes, the initial energy of the ground-state muonic atoms can extend up to 100–1000 eV [21–23]. In particular, the presence of a very energetic and long-lived tail in the energy spectrum of (pµ)1s atoms established an indirect proof
of an appreciable population of the metastable 2S state [21].
Theoretical studies of the three-body systems, which consist of the hydrogen and helium isotopes and the muon, and the muonic hydrogen scattering from hydrogenic molecules are the subjects of this thesis. In contrary to the systems that consist of two hydrogen-isotope nuclei and the muon, the hydrogen-helium-muon systems have not been extensively studied yet, although they are very important in µCF and experimen-tal investigation of the weak and strong interactions involving helium isotopes [24–28]. In Chapter 2, the adiabatic representation of the three-body problem with the Coulomb interaction [29] and the phase-function method [30] are applied to the hydrogen-helium-muon systems. In particular, the elastic cross sections for hydrogen-helium-muonic hydrogen scattering from helium nuclei, the energy levels of the hydrogen-helium muonic molecules (Heµh)++
(h denotes here a hydrogen isotope), and the cross sections for (Heµh)++ _{creation and}
deexcitation are calculated and discussed. Chapter 3 is devoted to calculation of the dif-ferential cross sections for muonic hydrogen atom scattering from hydrogenic molecules, with the use of the realistic Morse potential for internuclear interaction in these molecules.
Such calculations were previously performed using only the harmonic potential [31], which is a good approximation at collision energies below a few eV. Since in many experiments, especially in the above-mentioned laser spectroscopy in low-pressure hydrogen targets, a fraction of muonic atoms has much larger energies, it is necessary to use a better ap-proximation. In particular, ﬁnite numbers of the rotational and vibrational energy levels of hydrogenic molecules, the rotational-vibrational coupling, and molecular dissociation at higher collision energies should be taken into account. The Morse potential enables a satisfactory description of these molecular features. The numerical calculations are performed for various rotational, vibrational, and spin states of the considered systems. Finally, the obtained diﬀerential cross sections are applied for Monte Carlo simulations of diﬀusion and thermalization of pµ and dµ atoms in speciﬁc gaseous hydrogen and deuterium.
A part of the results presented in this thesis has been published in the following articles:
1. Elastic (hµ)1s+He++scattering and the influence of adiabatic corrections on(Heµh)++
bound states
J. Gronowski, W. Czapliński, Acta Phys. Pol. A 106, 795 (2004), 2. Diffusion radius of muonic hydrogen atoms in H-D gas
A. Adamczak, J. Gronowski, Eur. Phys. J. D 41, 493 (2007).
3. Calculation of cross sections for muonic atom scattering from hydrogenic molecules using the Morse potential
J. Gronowski, A. Adamczak, Phys. Rev. A 78, 054701 (2008), 4. Rotational de-excitation of hydrogen-helium muonic molecules
W. Czapliński, J. Gronowski, N. Popov, J. Phys. B 41, 035101 (2008),
5. Rotational de-excitation of hydrogen-helium muonic molecules in non-ionizing col-lisions with hydrogen atoms
W. Czapliński, J. Gronowski, N. Popov, M. Zegrodnik, J. Phys. B 41, 165202 (2008), 6. Differential cross sections for pµ + H2 and dµ+ D2 scattering using the Morse
potential
7. Resonant enhancement of the formation of hydrogen–helium muonic molecules W. Czapliński, J. Gronowski, W. Kamiński, N. Popov, Phys. Lett. A 375, 155 (2010),
and in the conference proceedings:
8. Muonic atom scattering from hydrogen molecules using the Morse potential
J. Gronowski, A. Adamczak, Proceedings of the International Conference on Muon Catalyzed Fusion and Related Topics (µCF-07) (JINR, Dubna, Russia), p. 276.
Chapter 2
Hydrogen-helium muonic molecules
2.1
Adiabatic expansion of a three-body system
In this section is reviewed the adiabatic-expansion method for a three-body system a-µ-b with Coulombic interaction, which consists of two nuclei a and b and the negative muon (or another particle with the elementary charge −e). The total non-relativistic
(Z
ae, M
a)
(Z
be, M
b)
(−e, m
µ)
R
aR
br
µFigure 2.1: Coordinates of the three-body system consisting of two nuclei a and b and the muon.
Hamiltonian of this system can be written down as follows (in units e = ~ = 1)
H = − 1 2Ma∇ 2 Ra − 1 2Mb∇ 2 Rb − 1 2mµ∇ 2 r µ − Za |rµ − Ra| − Zb |rµ− Rb| + ZaZb |Rb− Ra| . (2.1) The nuclear positions and their charges are denoted by Ra, Rb and Z_{a}, Z_{b},
(see Fig. 2.1). After separation of the center-of-mass motion and splitting of the total Hamiltonian [29], one obtains
H_{= −} 1 2M n [∇R+ (κ/2) ∇r]2− [(1 + κ) /2]2∇2r o + ZaZb R + Hµ, (2.2) Hµ = − 1 2µaµ∇ 2 r− Za ra − Zb rb , (2.3)
where M is the reduced mass of the nuclei a and b, and µaµ is the reduced mass of the
aµ atom
M−1 = M_{a}−1+ M_{b}−1, µ−1_{aµ} = M_{a}−1+ m−1_{µ} , κ= Ma− Mb Ma+ Mb
. (2.4)
All distances in the above equations are expressed in units of the muonic Bohr radius (aµ = meae/mµ). In Fig. 2.2 are shown new variables R and r, which are used for a
a
b
µ
X
Y
Z
R
r
r
ar
bΦ
Θ
Figure 2.2: Relative coordinates used for description of three-body system.
description of the system. Vector R = (R, Θ, Φ) connects nucleus a with nucleus b. The muon position r is reckoned from the geometric center of the two-nuclei system. Vectors
ra and rb are the distances between the muon and the nuclei a and b, respectively. Solving
the Schrödinger equation with the muonic Hamiltonian Hµ
2.1 ADIABATIC EXPANSION OF A THREE-BODY SYSTEM
one obtains an adiabatic basis formed by the eigenfunctions Φj(r; R). The index j
de-notes a set of the quantum numbers, which describe the bound-spectrum or continuous-spectrum states of the muon. The considered problem can be simpliﬁed using the prolate spheroidal coordinate system (ξ, η, ϕ), which is deﬁned in terms of the spherical coordinate system by the following relations
ξ = ra+ rb R , η= ra− rb R , ϕ= arctan x y. (2.6)
The variables ξ, η, and ϕ fulﬁll the conditions:
ξ ∈ [1, ∞) , η ∈ [−1, 1] , ϕ ∈ [0, π) . (2.7)
In the new coordinates, the Laplace operator ∇2 _{and the Coulomb attractive potential}
V _{= −Z}a/ra− Zb/rb take the forms
∇2 = −_{R}_{2}_{(ξ}_{2}2 − η2_{)} ( ∂ ∂ξ(ξ 2 − 1)_{∂ξ}∂ + ∂ ∂η(1 − η 2_{)} ∂ ∂η − ξ2_{− η}2 (ξ2_{− 1)(1 − η}2_{)} ∂2 ∂ϕ2 ) , (2.8) V = − 2 R2 sξ+ tη ξ2_{− η}2 . (2.9)
The coeﬃcients s and t depend on the charges of nuclei
s= R(Za+ Zb) , t= R(Zb− Za) . (2.10)
Upon inserting Eqs. (2.8) and (2.9) into Eq. (2.3), one obtains the following Hamiltonian for the motion of muon in the ﬁeld of two ﬁxed nuclei (in muonic atomic units e = ~ = µaµ = 1): hµ= 4 R2_{(ξ}2_{− η}2_{)} ( ∂ ∂ξ(ξ 2 − 1)_{∂ξ}∂ + _{∂η}∂ (1 − η2)_{∂η}∂ − ξ 2_{− η}2 (ξ2_{− 1)(1 − η}2_{)}m 2 ) − _{R}22 sξ+ tη ξ2_{− η}2 . (2.11)
After changing the variables, one can isolate the angular dependence and represent the muon wave function as follows
Φn1qm(r; R) = φn1qm(ξ, η; R) √ 2π (−1)m_{exp(imϕ),} exp(−imϕ), (2.12) and φn1qm(ξ, η; R) = Nn1qm(R)Πmn1(ξ; R)Ξmq(η; R) . (2.13)
The states of the muon are labeled by n1 and q (which denote the zeros of the functions
Π(ξ; R) and Ξ(η; R), respectively) and by the magnetic quantum number m. Factor Nn1qm(R), which is constant for a ﬁxed internuclear distance R, can be calculated using
the normalization constraint
Z d3_{r}_{Φ}∗ j(r; R)Φ′j(r; R) = δjj′, (2.14a) where Z d3 r . . . ≡ _{R} 2 3Z _{∞} 1 dξ Z 1 −1dη Z 2π 0 dϕ (ξ 2 − η2) . . . (2.14b) The Schrödinger equation (2.5) is now separated into the following one-dimensional equa-tions: ( d dξ(ξ2− 1) d dξ + " −p2(ξ2− 1) + sξ + λ − m 2 ξ2_{− 1} #) Πmn1(ξ; R) = 0 (2.15a) and ( d dη(1 − η2) d dη + " −p2_{(1 − η}2_{) + tη − λ −} m2 1 − η2 #) Ξmq(η; R) = 0 , (2.15b)
with the boundary conditions lim ξ→1(ξ 2 − 1)−m/2Πmn1(ξ; R) = 1 , lim ξ→∞Πmn1(ξ; R) = 0 , (2.16a) and lim η→1(1 − η 2_{)}−m/2_{Ξ} mq(η; R) = 1 , |Ξmq(η; R)| < ∞ for η = −1 . (2.16b)
2.1 ADIABATIC EXPANSION OF A THREE-BODY SYSTEM
Parameters p and λ are the eigenvalues of the coupled system p= pn1mq(R) =
R 2
q
−2En1mq(R) , λ= λn1mq(R) . (2.17)
As one can see, the parameter p is directly related to the molecular-state energy En1mq(R)
and λ is a separation constant. In the description of the muonic discrete states in the united-atom classiﬁcation (R → 0), it is convenient to change the set of quantum numbers (n1mq) to the spherical quantum numbers (Nlm). These two sets are connected by the
following relations [29]
N = n1+ l + 1 , l = q + m . (2.18)
Instead of the quantum number N, the continuous part of the spectrum is labeled by the variable k, which is the muon impulse at r → ∞. The three-body Hamiltonian (2.2) commutes with the total-angular-momentum operator _{J}b _{for the three-particle system,}
with the z-axis projection _{J}c_{z} _{of operator} _{J}b_{, and with the total-parity operator} _{P}b_{(R →}
−R, r → −r). b J2DJβ mmJ = J(J + 1)D Jβ mmJ , b JzDmmJβ J = mJD Jβ mmJ , b P D_{mm}Jβ _{J} = βDJβ mmJ , β = ±(−1) J_{.} (2.19) The eigenstates DJβ
mmJ(Φ, Θ, ϕ) of the operators, called the symmetrized Wigner functions,
are expressed by the usual Wigner functions DJβ
mmJ(Φ, Θ, ϕ) in the following way [32]
DJβ mmJ(Φ, Θ, ϕ) = s 2J + 1 16π2_{(1 + δ} m0) h DJ_{mm}_{J}(Φ, Θ, ϕ) exp(imϕ) + (−1)J_{βD}J −mmJ(Φ, Θ, ϕ) exp(−imϕ) i , (2.20)
with m being an eigenvalue of the operator R · J/R. The complete wave function of the a-µ-b system in the total-angular-momentum representation reads as follows
Ψ_{m}Jβ_{J}(r, R) = J X m=0 DJβ mmJ(Φ, Θ, ϕ)F Jβ m (ξ, η, R) , (2.21)
where F_{m}Jβ(ξ, η, R) = ∞ X N =1 N −1_{X} l=0 ΦN lm(ξ, η; R)R−1χJβN lm(R) +X∞ l=0 Z ∞ 0 dk Φlm(ξ, η; R, k)R −1_{χ}Jβ klm(R; k) . (2.22) The functions χJβ N lm and χ Jβ
lm describe the relative motion of the nuclei. Substitution of
the expression (2.21) into the Schrödinger three-body equation with the Hamiltonian (2.2) and further averaging over the spherical angles (Φ, Θ) and over the muon coordinates (r) lead to the system of coupled radial equations
d2 dR2χi(R) + " 2Mε − J(J + 1) R2 # χi(R) = Nj X j Uij(R)χj(R) + Nc X c Z dk Uic(R; k)χc(R; k) , d2 dR2χc(R; k) + " 2Mε − J(J + 1) R2 # χc(R; k) = Nj X j Ucj(R; k)χj(R) +XNc c′ Z dk Ucc′(R; k)χ_{c}′(R; k) , (2.23)
in which ε is the energy of the three-body system. The indices i, j = (Nlm) and c, c′
= (lm) denote the sets of spheroidal quantum numbers of the discrete and continuous spectra of the two-center problem and Uij(R), Uic(R; k), Ucj(R; k), and Ucc′(R; k) are
the eﬀective potentials. These potentials include the molecular terms and adiabatic cor-rections, which describe the coupling between the nuclear motion and diﬀerent muonic states. An inﬂuence of the adiabatic corrections on the bound states and energy levels of the molecular ions (Heµh)++ _{and (Heπh)}++ _{is presented in Sec. 2.7.}
2.2
Matrix elements for hydrogen-helium muonic
sys-tem
The calculations of various two-center matrix elements in the framework of adiabatic approximation have been one of the main tasks of the present work. In this section,
2.2 MATRIX ELEMENTS FOR HYDROGEN-HELIUM MUONIC SYSTEM
a computation scheme for the He++_{-µ-h system is described, where He}++_{denotes a helium}
nuclide (3_{He,} 4_{He) and h stands for a hydrogen nuclide (p, d, t). The molecular terms,}
the corresponding two-center wave functions and, ﬁnally, the matrix elements as functions of the internuclear distance R are presented below. Since the charges Za and Zb are
parameters only, the prepared computer program MatEl2C [33] could be applied for studies of various exotic three-body systems, such as those containing kaons, pions or antiprotons. In the case of considered He++_{-µ-h system, one can choose the nuclei as}
follows a = h and b = He++_{. Consequently, the coeﬃcients (2.10) take the corresponding}
values: s = 3R and t = R.
In order to solve the system of coupled equations (2.15), it is necessary to rewrite the boundary conditions (2.16). This is caused by a singularity of the wave function Ξ at the points with η = ±1 and a singularity of the function Π for ξ = 1. For m = 0, a behavior of the function Ξ(η; R) in the neighborhood of the boundary points is described by the relations [34] Ξ0q(η; R) = exp[−p(1 + η)] ∞ X w=0 cw(1 + η)w, for η → −1 , (2.24a) and Ξ0q(η; R) = exp[−p(1 − η)] ∞ X w=0 c′_{w}(1 − η)w, for η → 1, (2.24b) The coeﬃcients cw+1, cw, cw−1 in the above expansions fulﬁll the following relations [34]
α′_{w}cw+1− α′′wcw+ α′′′wcw−1 = 0 , c−1 = 0, (2.25) where α′_{w} = (w + 1)[R − 2p(w + 1)] 2w + 3 , α′′_{w} _{= w(w + 1) − λ ,} α′′′_{w} = w(R + 2pw) 2w − 1 . (2.26)
The analogous equations can be written down for the coeﬃcients c′
for-mulas (2.25) and (2.26) into Eqs. (2.24a) and (2.24b) leads to Ξ0q(η → −1; R) = c0 " 1 + 2p + R − λ 2 (1 + η) # , Ξ0q(η → 1; R) = c′0 " 1 + 2p − R − λ 2 (1 − η) # , (2.27) where c′
0 is chosen so that Ξ0q(η = 1; R) = 1 and c0 is an arbitrary constant. The
boundary condition for Π0n1(ξ → 1, R) can be obtained similarly, whereas the function
Π0n1 vanishes for ξ → ∞, according to the formula
Π0n1(ξ → ∞) = exp(−pξ) . (2.28)
In order to solve the system of coupled equations (2.15) it is convenient to begin the numerical computation at R = 0. Due to the fact that energy values E(R = 0) are known analytically, obtaining λ(R = 0) for each molecular term is straightforward. Starting at the origin with the unique pairs of E and λ values and continuing the calculations with a very small step size ∆R, it is possible to separate the crossing molecular terms.
2.3
Molecular terms and two-center wave functions
From now on, we will use the spherical quantum numbers (Nlm), deﬁned in Eq. (2.18), to denote the muon states and wave functions. All the results presented in the current section have been computed using the MatEl2C program [33]. Figure 2.3 shows the ﬁrst ﬁfteen σ-type (m = 0) molecular terms. It is easily seen that the values of the plotted functions E(R), in the limit R → 0, correspond to the energy levels of the hydrogen-like atom with Z = Za+ Zb = 3. The most important muon states for further evaluations are
the two lowest states. For R → ∞, these 1sσ and 2pσ states correspond to h + (Heµ)+ 1s
and (hµ)1s+ He++, respectively.
Some examples of the two-center wave functions Π and Ξ of the considered system are shown in Fig. 2.4, for diﬀerent values of the internuclear separation. These functions are calculated with the use of boundary conditions mentioned in the previous section and are not independently normalized.
2.3 MOLECULAR TERMS AND TWO-CENTER WAVE FUNCTIONS
)
µR (a
0 5 10 15 20 25)
_{µ}ε
E (
-1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00 σ 2s σ 2p σ 3s σ 3p σ 3d σ 4s 4pσ σ 4d σ 4f σ 5s 5pσ σ 5d 5fσ 5gσ 0 5 10 15 20 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 σ 1sFigure 2.3: Molecular terms for the Za = 1 and Zb = 2 system. Only the terms with m = 0
ξ
1 10;R)
ξ
(
_{σ} 1sΠ
0.0 0.2 0.4 0.6 0.8 1.0 µ R = 0.1 a µ R = 0.5 a µ R = 3 a (a) Π1sσ(R)ξ
1 10 102;R)
ξ
(
_{σ} 2pΠ
0 1 2 3 4 5 µ R = 0.1 a µ R = 0.5 a µ R = 3 a (b)Π2pσ(R)η
-1.0 -0.6 -0.2 0.2 0.6 1.0;R)
η
(
_{σ} 1sΞ
0.0 0.2 0.4 0.6 0.8 1.0 1.2 µ R = 0.1 a µ R = 0.5 a µ R = 3 a (c) Ξ1sσ(R)η
-1.0 -0.6 -0.2 0.2 0.6 1.0;R)
η
(
_{σ} 2pΞ
-1.0 -0.6 -0.2 0.2 0.6 1.0 µ R = 0.1 a µ R = 3 a µ R = 10 a (d)Ξ2pσ(R)Figure 2.4: Examples of the spheroidal wave functions ΠN lm and ΞN lm versus the variables
ξ and η defined in Eq.(2.6). The functions are plotted for the 1sσ (subfigures (a), (c)) and 2pσ (subfigures (b), (d)) states and for different values of the internuclear distance R.
2.3 MOLECULAR TERMS AND TWO-CENTER WAVE FUNCTIONS
condition (2.14) and calculating the normalization functions N(R) for each value of the internuclear distance R, one obtains the probability density functions. These functions
ξ 1 11 21 31 41 51 η -1.0 -0.5 0.0 0.5 1.0 2 ;R=0.2)| η , ξ ( _{σ} 1s φ | 0 10 20 30 (a)|φ1sσ(ξ, η; R = 0.2)|2 ξ 1 11 21 31 41 51 η -1.0 -0.5 0.0 0.5 1.0 2 ;R=0.2)| η , ξ ( _{σ} 2p φ | 0.0 0.2 0.4 0.6 0.8 1.0 (b)|φ2pσ(ξ, η; R = 0.2)|2 ξ 1.0 1.5 2.0 2.5 3.0 η -1.0 -0.5 0.0 0.5 1.0 2 ;R=2)| η , ξ ( _{σ} 1s φ | 0 5 10 15 (c) _{|φ}1sσ(ξ, η; R = 2)|2 ξ 1.0 1.5 2.0 2.5 3.0 η -1.0 -0.5 0.0 0.5 1.0 2 ;R=5)| η , ξ ( _{σ} 2p φ | 0.0 0.5 1.0 1.5 (d) _{|φ}2pσ(ξ, η; R = 5)|2
Figure 2.5: Probability density of finding muon for the 1sσ and 2pσ states for small (a), (b) and large (c), (d) internuclear distances.
are presented in Fig. 2.5 for the 1sσ and 2pσ muon states. According to the deﬁnition of the prolate spheroidal variables (2.6) and due to the fact that Za = 1 and Zb = 2, the
points with spheroidal coordinates (ξ = 1, η = −1) and (ξ = 1, η = 1) correspond to the positions of the hydrogen and helium nuclei, respectively. Figures 2.5c and 2.5d clearly illustrate diﬀerent asymptotic behavior of the considered states. For the 1sσ molecular term and large values of the internuclear separation, the probability density of ﬁnding
muon grows rapidly for points close to the helium nucleus. For R → ∞, this state corresponds to h + (Heµ)+
1s system. Similarly, the probability density for the 2pσ state,
which describes (hµ)1s+ He++ system, vanishes outside vicinity of the hydrogen nucleus.
The set of computed two-center wave functions has been used to calculate various matrix elements. The numerical code MatEl2C has been validated by comparison our results with available data [34, 35]. The obtained two-center matrix elements are neces-sary for investigation of resonant enhancement of formation the hydrogen-helium muonic molecules and for study of the rotational transitions in the hydrogen-helium molecular ions. The results of these calculations are presented in sections 2.8 and 2.9.
2.4
One-level adiabatic approximation for 2pσ state
In this section, we focus on the inﬂuence of the adiabatic corrections on the rovibrational energy levels εJv of quasistationary states. These states of the three-body systems are
considered in the one-level adiabatic approximation for the 2pσ state. This approxima-tion is quite accurate due to the fact that the coupling terms of the potential matrix, which correspond to the two-level treatment, are much smaller than the diagonal terms. The 2pσ component of the total wave function is localized in region of the internuclear separation R ≥ 4aµ, where overlapping with the wave function of the other states is
rel-atively weak [36, 37]. The corresponding results, which were presented for the ﬁrst time in Ref. [38], diﬀer only by few percent from the later results of the variational method (trial wave functions of many-thousand-terms [39–41]) and of the two-level approxima-tion [36, 37, 42, 43].
After retaining only one term for Nlm = 2pσ in the expansion (2.22), the adiabatic wave function takes the form
ΨJmJ(r, R) = YJmJ(Φ, Θ) Φ2pσ(ξ, η; R) χ
J
2pσ(R)R−1. (2.29)
As a result, the system of coupled equations (2.23) transforms into the one-level radial equations d2 dR2χ J 2pσ(R) + 2M " ε − VA(R) − J(J + 1) 2MR2 # χJ 2pσ(R) = 0 , (2.30)
2.5 PHASE-FUNCTION METHOD FOR THE ONE-LEVEL PROBLEM
of the hµ atom. The adiabatic potential VA(R) reads
VA(R) = VBO(R) + UA(R) 2M , (2.31) where VBO = E2pσ(R) − E2pσ(∞) + 2 R (2.32)
is the potential corresponding to the Born–Oppenheimer approximation and
UA(R) = h2pσ|2MT |2pσi (2.33)
is the adiabatic correction, which is usually written in the following form [29]
UA(R) = H+(R) − H∗(R) + κ[H−(R) − 2H∗(R)]. (2.34)
The functions H±_{(R) and H}∗_{(R) are deﬁned as follows}
H+_{(R) = −} 3 2R2 + 1 R2[4E2pσ(R) + RE ′ 2pσ]hΦ2pσ|r2|Φ2pσi − 3R2hΦ2pσ|r2W |Φ2pσi + hΦ′2pσ|Φ′2pσi, H−_{(R) = −} 1 R[4E2pσ(R) + RE ′ 2pσ]hΦ2pσ|z|Φ2pσi + 3 RhΦ2pσ|zW |Φ2pσi, H∗(R) = − [E2pσ(R) + RE2pσ′ ]/2 , W _{= − 2[(Z}a+ Zb)ξ + (Zb− Za)η]/R(ξ2− η2) , (2.35)
where the “prime” sign denotes the derivative with respect to variable R.
2.5
Phase-function method for the one-level problem
Instead of a direct numerical solution of the Schrödinger equation, the numbers of bound states and the values of energy levels have been calculated in the framework of the phase-function method. Also, the cross sections for elastic scattering of muonic hydrogen on helium nuclei have been obtained in the framework of the same method.
with the following boundary conditions
χJ_{2pσ}(0) = 0 (2.36a)
and
χJ_{2pσ}(R → ∞) = AJ[cos δJjJ(kR) − sin δJnJ(kR)] , (2.36b)
where k = √2Mε, jJ(kR), and nJ(kR) are the Riccati-Bessel functions [44] and δJ
de-notes the phase shift. The phase-function method is based on treating the phase shift as a function of the distance R. As a result, the function δJ(R) is the phase shift obtained by
the cutoﬀ of potential at the distance R. Using the condition (2.36b) for large distances, we make the following assumption for the radial function [30]
χJ2pσ(R) = AJ(R) [cos δJjJ(kR) − sin δJnJ(kR)] . (2.37)
This assumption, along with the following constraint for continuity of the wave-function derivative d dRχJ(R) = AJ(R) " cos δJ d dRjJ(kR) − sin δJ d dRnJ(kR) # , (2.38)
leads to the system of two coupled ﬁrst-order diﬀerential equations for the amplitude AJ(R) and the phase-function δJ(R) [30]
d dRAJ(R) = − 2M k AJ(R)V (R) [cos δJjJ(kR) − sin δJnJ(kR)] × [sin δJjJ(kR) + cos δJnJ(kR)] (2.39) and d dRδJ(R) = − 2M k V(R) [cos δJjJ(kR) − sin δJnJ(kR)] 2 , (2.40)
with the boundary conditions
AJ(R → ∞) = 1 , (2.41)
and
δJ(0) = 0 . (2.42)
The equations (2.39) and (2.40) are called the amplitude and phase-equation, respectively. Together with the conditions (2.41) and (2.42), they are equivalent to the boundary-value
2.5 PHASE-FUNCTION METHOD FOR THE ONE-LEVEL PROBLEM
problems (2.30) and (2.36).
For a given collision energy ε, the total phase shift can be obtained from the large distance condition δJ = δJ(R → ∞). The Levinson theorem
δJ(k = 0) − δJ(k = ∞) = nπ , (2.43)
which relates the zero-energy scattering phase shift to the number of bound states n for angular momentum J, has been used for counting the bound-state of the hydrogen–helium muonic (and pionic) molecules. In this work, we focus only on inﬂuence of the adiabatic corrections on the bound states.
In order to obtain not only the number of bound states but also the values of rovibra-tional energy levels, it is necessary to transform the phase equation (2.40). This transfor-mation is based on the known fact that the bound-state energies correspond to the imagi-nary poles of the scattering S-matrix elements in the complex k-plane, for Im(k) > 0 [45]. After the following substitution
SJ(R) = e2iδJ(R), (2.44)
from Eqs. (2.40) and (2.42) we obtain the corresponding equations for the partial S-matrix function [30] d dRSJ(R, k) = − iM k V(R) h h(−) J (kR) + SJ(R, k)h(+)J (kR) i2 , SJ(0, k) = 1 , (2.45) where h(±)
J (kR) = jJ(kR) ± inJ(kR). The bound-state energies εn can be then obtained
from the requirement SJ(∞; iκn) = ∞, where k = iκn; κn > 0, εn = −~2κ2n/2M, and
n= 0, 1, 2, . . .. For the numerical purposes, however, it is convenient to replace Eqs.(2.45) by the corresponding real equations for the real partial-scattering amplitude function, which is deﬁned as
These equations take the form d dRfJ(R, κ) = − 2M κ V(R) pJ(κR) + 2 πfJ(R, κ)qJ(κR) 2 , fJ(0, κ) = 0 , (2.47)
where pJ(x) = (−i)J+1jJ(ix) and qJ(x) = π iJ+1h(+)J (ix)/2. The bound states are
ob-tained for the condition fJ(∞, κn) = 0. Additionally, a regularization of Eqs.(2.47) by
the substitution fJ(R, κ) = tan[γJ(R, κ)] leads to the following boundary-value problem
d dRγJ(R, κ) = − 2M κ V(R) pJ(κR) cos γJ(R, κ) + 2 πqJ(κR) sin γJ(R, κ) 2 , γJ(0, κ) = 0 (2.48)
with the condition
γJ(∞, κn) = (2n + 1)
π
2 (2.49)
for the bound states.
2.6
Elastic scattering of muonic hydrogen on helium
The phase shifts δJ, calculated in the framework of the method presented in the
previ-ous section, have been used to obtain the cross sections for elastic scattering of muonic hydrogen on helium nuclei (hµ)1s + He++. These cross sections, together with the
cor-responding time delays and scattering lengths, were calculated for a range of collision energies from 0 eV to 50 eV. A part of the results presented below is an extension and development of the author’s master thesis [46]
The total elastic cross section as function of the partial phase shifts δJ is given by the
formula σtot = 4π k2 ∞ X J=0 (2J + 1) sin2_{δ} J. (2.50)
At this point, a comparison between our and previously obtained results is going to be discussed for the process (pµ)1s+3He++. In Fig. 2.6a the low-energy cross sections, which
include the dominating s and p partial waves, are compared with the corresponding two-level results of Ref. [47]. The agreement is quite satisfactory in the whole energy range
2.6 ELASTIC SCATTERING OF MUONIC HYDROGEN ON HELIUM
considered. At the same time, our cross sections signiﬁcantly diﬀer from the correspond-ing results that were obtained in the frame of the same calculation method [48], i.e., in one-level adiabatic approximation. As an illustration of discrepancy, Fig. 2.6b presents a comparison of the dominating s-wave cross sections obtained in the present work with the cross sections calculated in Refs. [47] and [48]. Our results are close to the correspond-ing two-level cross sections of Ref. [47], whereas they signiﬁcantly diﬀer from the results of Ref. [48], especially for the low-energy limit. Also, we have found a signiﬁcant discrep-ancy between our s-wave elastic cross sections and the corresponding ones of Ref. [48], for all the remaining isotope compositions of the (hµ)1s+ He++ system. The total elastic
(a) total cross section (including s and p
par-tial waves
(b)s-wave cross section
Figure 2.6: Comparison of the cross sections for elastic (pµ)1s +3He++ scattering in the
one-level adiabatic approximation calculated in the present paper (solid line) and in Ref. [48] (dashed line). The corresponding cross section calculated in the two-level approximation [47] is represented by the dotted line.
cross sections for (hµ)1s+3He++ and (hµ)1s+4He++(h = p, d, t), calculated by summing
the partial contributions up to J = 10, are presented in Figs. 2.7a and 2.8a, respectively. Very large values of the zero-energy cross sections for the h = d and t cases, which are visible in the insets, are due to the presence of virtual (anti-bound) or weakly-bound states [45]. In order to distinguish between these two possibilities, we have calculated the corresponding scattering lengths α = − limk→∞(tan δ0/k). The results are shown in
Table 2.1. One sees from this table that the low-energy elastic (tµ)1s+4He++ cross
sec-tion is determined by the presence of a loosely bound state (α > 0) of the corresponding muonic molecule, (4_{He}++_{, see Sec. 2.7). At the same time, the low-energy elastic cross}
(a) (b)
Figure 2.7: (a) Energy dependence of the total cross sections for elastic (hµ)1s +3He++
scattering for h = p (solid), h = d (dashed), and h = t (dotted line). (b) Energy dependence of the partial resonance cross sections (σJ), the respective phase
shifts (δJ) and time delays (tJ), for the (hµ)1s+3He++ scattering. The curves
are denoted similarly as in subfigure (a).
sections for the remaining hydrogen and helium isotope compositions of (hµ)1s+ He++
system (excluding h = p) are determined by the presence of virtual states (α < 0). In the case of pµ scattering from helium nuclei, the Ramsauer-Townsend eﬀect is visible. The minima are located at ε ≈ 0.28 eV for (pµ) +3_{He}++ _{and at ε ≈ 0.69 eV for (pµ) +}4_{He}++_{,}
see Figs. 2.6a, 2.7a and 2.8a, respectively.
Similar Ramsauer-Townsend minima are observed in the dµ+H2and tµ+H2scattering,
which is considered in Sec. 3.7. For ε ≫ 0, the total cross sections exhibit distinct maxima, which are due to single partial-wave resonances corresponding to J = 2 for h = p and to J = 3 for h = d, t. The corresponding partial cross sections, phase shifts, and time delays (tJ = dδJ/dE) are presented in Figs. 2.7b and 2.8b. One sees from these ﬁgures that each
2.6 ELASTIC SCATTERING OF MUONIC HYDROGEN ON HELIUM
(a) (b)
Figure 2.8: (a) Energy dependence of the total cross sections for elastic (hµ)1s +4He++
scattering for h = p (solid), h = d (dashed), and h = t (dotted line). (b) Energy dependence of the partial resonance cross sections (σJ), the respective phase
shifts (δJ) and time delays (tJ), for the (hµ)1s+4He++ scattering. The curves
are denoted similarly as in subfigure (a). positions of resonances εr
J correspond to maxima of the time delays tJ. The widths of the
resonances can be calculated using the equation (Ref. [49]) ΓJ = 2
q
−2δ′
J/δJ′′′|ε=εr
J, (2.51)
where the “prime” sign denotes the derivative with respect to ε. Some results are also presented in Table 2.1. As one can see from Figs. 2.7b and 2.8b and Table 2.1, the relation tJ(εrJ) = 4/ΓJ between the time delay tJ and the resonance width ΓJ is almost fulﬁlled for
α[10−10_{cm]} _{ε}r J[eV] ΓJ[eV] 3_{Heµp} _{−0.64} _{12.9 (12.9)} _{6.6 (7.3)} 3_{Heµd} _{−4.9} _{32.7 (32.7)} _{21.7 (26.5)} 3_{Heµt} _{−21.8} _{19.0 (18.9)} _{7.5 (8.1)} 4_{Heµp} _{−1.3} _{7.4 (7.4)} _{2.0 (2.1)} 4_{Heµd} _{−13.0} _{23.1 (22.9)} _{10.1 (11.6)} 4_{Heµt} _{15.7} _{7.9 (7.9)} _{0.8 (0.8)}
Table 2.1: Scattering lengths α, resonance positions εr
J and widths ΓJ, obtained
from Eq. (31) and from fitting (in brackets) for the elastic (hµ)1s+ He++
scat-tering. The resonance partial waves for J = 2 and J = 3 correspond to h = p and h = d, t, respectively.
resonance positions and widths, which are calculated by ﬁtting the formula δJ(ε) = arctan ε − ε r J ΓJ/2 ! + βJ (2.52)
to the corresponding phase shifts, are presented in Table 2.1 in the parentheses. The apparent diﬀerences are due to the energy dependence of βJ, which is neglected in Eqs. 2.51
and 2.52.
The resonances that appear in the elastic scattering (hµ)1s + He++ are the shape
resonances [50]. These resonances are responsible for enhancement of the cross sections in the hydrogen-helium muonic molecules formation. This eﬀect is described in details in Sec. 2.8.
2.7
Influence of the adiabatic corrections on
rovibra-tional levels
The R-dependence of phase shift δJ for J = 0, which corresponds to the elastic scattering
(tµ)1s+4He, is presented in Fig. 2.9 at the three collision energies. The characteristic
jumps of the phase shift δ0(R) by π, which appear at R1 ≈ 3.8 and R2 ≈ 15 when
ε → 0, correspond to the two bound states. An abscissa of a given jump equals to a minimal potential range, at which the respective bound state appears. The bound-state energies have been calculated by a numerical solution of Eqs. (2.48) and (2.49). The function γ0(Rmax, ε), which corresponds to the elastic (tµ)1s +4He scattering, is plotted
2.7 INFLUENCE OF THE ADIABATIC CORRECTIONS ON ROVIBRATIONAL LEVELS 0 5 10 15 20 25 30 0 1 2 3 4 5 6 7 2 R 2 = 15 0 [ r a d ] R [a] R 1 = 3.8
Figure 2.9: R-dependence of the phase shift δ0 for the elastic (tµ)1s +4He scattering at
different collision energies: ε = 10−2 _{eV (dotted), ε = 10}−4 _{eV (dashed), and}
ε= 10−10 _{eV (solid line).}
by π, are equal to the binding energies of the corresponding muonic molecule for J = 0. It is remarkable that the curves from Figs. 2.9 and 2.10 are consistent, i.e., they both indicate the presence of the two bound states. The binding energies of the (Heµh)++ _{molecules for}
all the isotope combinations, with and without the adiabatic corrections, are presented in Table 2.2. The adiabatic results presented in Table 2.2 coincide almost exactly with the results of Ref. [38], which were obtained by a direct numerical solution of the Schrödinger equation (2.30) with the boundary conditions (2.36). Also, the many-channel variational
10 3 10 2 10 1 10 0 10 -1 10 -2 -2 -1 0 1 2 3 4 5 -01 = 0.076 3 /2 0 ( R m a x ) [eV] /2 -00 = 80.5
Figure 2.10: Energy dependence of function γ0(Rmax; ε) for the elastic (tµ)1s+4He
−εJν 3Heµp 4Heµp 3Heµd 4Heµd 3Heµt 4Heµt −ε00 Ad. [46, 51] 67.2 73.85 69.5 77.5 71.6 80.5 B-O [46, 51] 39.7 42.3 59.3 63.5 68.7 73.8 [36, 37] 72.76 80.64 69.37 77.49 [41] 70.976 79.340 72.296 81.335 −ε01 Ad. [46, 51] 0.076 B-O [46, 51] 0.053 [41] 0.134 −ε10 Ad. [46, 51] 34.2 41.6 46.5 55.9 52.4 62.9 B-O [46, 51] 13.8 17.1 37.9 43.6 49.9 56.9 [36, 37] 38.82 46.31 46.41 55.74 [41] 48.419 58.222 53.330 63.958 −ε20 Ad. [46, 51] 7.25 17.7 18.2 30.7 B-O [46, 51] 2.4 9.4 16.8 26.2 [36, 37] 7.11 17.49 [41] 9.434 20.416 19.379 32.063
Table 2.2: The energy levels of (Heµh)++ _{calculated in the one-level adiabatic (Ad.) and in}
the Born-Oppenheimer (B-O) approximation. Also, the results obtained from the two-channel and many-channel calculations are presented for a comparison. All energies are given in eV.
results obtained in Refs. [36, 37, 41] are presented for a comparison. It is remarkable that the numbers of bound states obtained in the frameworks of the adiabatic and the B–O approximations are the same for every J. However, the adiabatic binding energies are shifted below the corresponding B–O energies. An inﬂuence of the adiabatic corrections on the energy value is minimal for the J = 0 ground-state energy ε00 of the (4Heµt)++
molecule, whereas such an inﬂuence is maximal for the J = 2 ground-state energy ε20 of
the (3_{Heµd)}++ _{system.}
In Table 2.3 average sizes of the hydrogen-helium muonic molecules (Heµh)++ _{are}
(J, ν) 3_{Heµp} 4_{Heµp} 3_{Heµd} 4_{Heµd} 3_{Heµt} 4_{Heµt}
(0,0) 5.003 4.888 4.934 4.804 4.886 4.75
(0,1) 32.519
(1,0) 5.488 5.287 5.199 5.015 5.086 4.904
(2,0) 6.2 5.641 5.666 5.296
Table 2.3: Average sizes of the hydrogen-helium muonic molecules (Heµh)++ _{calculated for}
all the existing bound states (J, ν). All sizes are given in aµ.
vi-2.7 INFLUENCE OF THE ADIABATIC CORRECTIONS ON ROVIBRATIONAL LEVELS
brational states ν = 0, all average sizes are similar. Rotational excitation of the molecules do not signiﬁcantly change these sizes. On the other hand, the dimension of the 4_{Heµt}
molecule in the excited vibrational state ν = 1 is much larger. It is clear that the proba-bility of nuclear fusion in this excited state is very small.
It is interesting to examine an inﬂuence of the adiabatic corrections on the bound-state energies of the hydrogen–helium pionic molecules, in which the muon is replaced by the negative pion π−_{. The corresponding binding energies, calculated with and without the}
adiabatic corrections, are presented in Table 2.4. According to our best knowledge, these
−εJν 3Heπp 4Heπp 3Heπd 4Heπd 3Heπt 4Heπt
−ε00 Ad. [51] 79.29 89.32 78.27 90.30 79.61 92.82 B-O [51] 38.22 41.44 62.87 68.53 75.37 82.51 −ε10 Ad. [51] 27.89 38.30 41.86 55.42 48.96 64.21 B-O [51] 2.74 6.19 30.10 37.73 45.74 55.46 −ε20 Ad. [51] 14.59 B-O [51] 9.27
Table 2.4: Energy levels of (Heπh)++ _{calculated in the one-level adiabatic (Ad.) and the}
Born-Oppenheimer (B-O) approximations. All energies are given in eV.
results are presented for the ﬁrst time. Due to the smaller value of the M/ma mass ratio,
inﬂuence of the adiabatic corrections on the energy levels is much more pronounced for the pionic systems than for the muonic ones.
The same conclusion is true for the nuclear wave functions, as one may expect. Fig-ure 2.11 presents the energy levels (top) corresponding to the state J = 0 of a ﬁctitious molecule (3_{Hexp)}++_{, calculated in the adiabatic one-level approximation versus the mass}
mx of a ﬁctitious negative particle x. The mass mx varies from the electron mass to
the antiproton mass. A corresponding relative diﬀerence between the adiabatic and B–O energy levels η = (εA− εBO)/εBO (bottom) increases with increasing mx. For a given
mx, variable η sharply increases with the increasing vibrational quantum number v. The
points of intersections between the vertical lines (corresponding to the muon mass and the pion mass) and the curves in diagram correspond to the binding energies of the respective muonic and pionic molecules with J = 0 and to the relative energy diﬀerences η. These binding energies are presented in Tables 2.2 and 2.4. As one can see in these Tables and in Fig. 2.11, the inﬂuence of adiabatic corrections on the energy levels decreases with
Figure 2.11: Bound state energies (top) corresponding to J = 0 of fictitious (3_{Hexp)}++
molecule calculated in the one-level adiabatic approximation versus the mass of a fictitious negative particle x. Mass dependence of the corresponding relative difference between adiabatic and B-O energy levels η = (εA− εBO)/εBOis also
presented (bottom).
increasing the mass ratio M/ma and with decreasing the coeﬃcient κ. This observation
agrees with the form of potential VA(R) that is expressed by Eqs. (2.31)–(2.34).
2.8
Resonant enhancement of the hydrogen-helium
muonic molecule formation
The muonic molecules (Heµh)++ _{or, strictly speaking, the three-body He}++_{-µ-h }
reso-nances [42, 52, 53] are spontaneously formed in He–H mixtures in collisions of the ground-state muonic atom (hµ)1s with an ordinary helium electronic atom, via the electron
con-2.8 RESONANT ENHANCEMENT OF THE HYDROGEN-HELIUM MUONIC MOLECULE FORMATION
version process [38]
(hµ)1s+ He → [(Heµh)++J , e]++ e, (2.53)
in which J denotes the rotational quantum number of the molecule. The energy levels of these molecules, calculated in the one-level adiabatic approximation for the 2pσ state, have been presented in the previous section (see Table 2.2). At this point, we present the formation cross sections and reaction rates for the muonic molecules (4_{Heµp)}++
J=1 and
(3_{Heµd)}++
J=0,1,2 that are created in the process (2.53), in a broad range of collision energies
between 0.004 eV and 50 eV.
A method of calculation, whose the main steps are present below, is applicable to all the rotational states of the muonic molecules. This is in contrast to the approach considered in Ref. [38], which is valid only for J = 1. The coordinates used in our
He
++h
µ
e
1e
2r
Her
hr
µR
r
r
1r
2Figure 2.12: Coordinates that are used for a description of the formation reaction (hµ)1s+
He → [(Heµh)++1 , e]+e.
calculation are shown in Fig. 2.12. A dominant contribution to the formation process comes from the dipole interaction of the helium electrons with the three-body He++_{-µ-h}
system. The potential of this interaction (which is treated as a perturbation) reads Vdip = −d · r1 r3 1 +r1 r3 1 ! , (2.54)
where d is the dipole moment
d= X c=h,µ,He
For a detailed description of the calculation method for the formation process (2.53) see Ref. [54]. The total cross section for the considered process has the following form
σ_{J}form = 64 3π Z3 i V2 i kik I2(η)(2J + 1)X L (2L + 1) J L 1 0 0 0 2 d2_{JL}, (2.56)
in which Zi = 27/16 denotes the eﬀective charge and function I(η) of the variable η = k/Zi
is deﬁned as
I(η) = 2
s
η
(1 + η2_{)[1 − exp(−2πη)]}exp[−2η tan
−1_{(1/η)].} _{(2.57)}
The momentum of the outgoing electron is denoted by k, while Vi and ki stand for
the collision velocity and the collision momentum, respectively. The expression (. . .)2
represents the squared Wigner 3j symbol, and L indicates the partial waves of the elastic scattering (hµ)1s+He. The coeﬃcient dJLthat appears in Eq. (2.56) denotes the absolute
value of the molecular dipole moment (2.55), averaged over the initial and ﬁnal states of the system. The value of dJL can be obtained using the following relation
dJL= 1 2 − MHe− Mh MHe+ Mh+ mµ ! IJL(1)+ 1 + 2mµ MHe+ Mh+ mµ ! IJL(2). (2.58) The integrals I_{JL}(1) = Z ∞ 0 dR χ J 2pσ(R)χL2pσ(R)R , I_{JL}(2) =Z ∞ 0 dR χ J 2pσ(R)χL2pσ(R)D µ 2pσ(R) (2.59)
include the nuclear wave functions of the initial state χL
2pσ and ﬁnal molecular state χJ2pσ.
The matrix element Dµ2pσ(R) = * φµ2pσ r · R R φ µ 2pσ + = R 2 Z ∞ 1 dξ Z 1 −1dη(ξ 2 −η2)ξη Ξ22pσ(ξ; R)Π22pσ(η; R) . (2.60)
which is shown in Fig. 2.13, has been calculated with the use of the code MatEl2C [33]. The molecule-formation processes (2.53) are induced by the dipole L → J transitions. The radial wave functions χL
2pσ describe with a good accuracy the relative motion of
2.8 RESONANT ENHANCEMENT OF THE HYDROGEN-HELIUM MUONIC MOLECULE FORMATION
) µ R (a 0 2 4 6 8 10 (R) µ σ2p D 0 1 2 3 4 5
Figure 2.13: The matrix element Dµ_{2pσ}(R) as a function of the internuclear distance R.
collision energy (eV)
-2 10 10-1 1 10 ) -1 reaction rate (s 4 10 5 10 6 10 7 10 8 10 9 10 J ++ d) µ He 3 ( 1 ++ p) µ He 4 ( J=0 J=1 J=2
Figure 2.14: Reaction rates for the (3_{Heµd)}++
J=0,1,2 (solid line) and (4Heµp)++J=1 (dashed line)
formation. The dots and triangles represent the corresponding results for J = 1, obtained in Ref. [38].
corresponding elastic scattering (see Sec. 2.6). This is due to the fact that electron screening in the entrance channel (2.53) is negligible at collision energies above 1 eV [48]. In the case of a resonance, the function χL
2pσ has a large amplitude in the region where
the ﬁnal (molecular) state described by χJ
2pσ is localized. The overlap of these functions
results in large values of the integrals (2.59) and thus consequently leads to large values of the formation cross sections (2.56). The resonances that have been found in the elastic (hµ)1s+ He++ scattering for the partial waves L = 2, 3 (see Figs. 2.7 and 2.8) caused the
resonant enhancement of the (Heµh)++
J formation in the rotational states with J = 1, 2.
The results of numerical calculations are shown in Figs. (2.14) and (2.15). The reaction rates for the (3_{Heµd)}++
J=0,1,2 and (4Heµp)++J=1 formation as functions of collision energy
are presented in Fig. (2.14). The formation rate for (4_{Heµp)}++
J=1 reveals a signiﬁcant
collision energy (eV)
0 10 20 30 40 50 ) 2 cm -21 cross section (10 0 1 2 3 4 5 6 J=2 form σ J=1 form σ -2 10 × L=3 elas σ -3 10 × L=2 elas σ
Figure 2.15: Formation cross sections σform
J for the (4Heµp)++J=1(dashed line) and (3Heµd)++J=2
(solid line) molecules compared with the resonant partial cross sections σelas
L for
the corresponding elastic scattering.
enhancement near the collision energy of 7 eV, which is due to the presence of resonance in the elastic (pµ)1s +4He++ scattering for the partial wave L = 2 (see Fig. 2.8). The
analogous enhancement for the molecule (3_{Heµd)}++
J=2 , near the collision energy of 30 eV,
is caused by the resonance for the partial wave L = 3 in the corresponding scattering reaction (see Fig. 2.7). The reaction rate for the (3_{Heµd)}++
J=2 formation has the threshold
at 17.33 eV, which is due to the fact that the binding energy of 7.25 eV for the rotational state J = 2 (see Table 2.2) is smaller than the helium ionization potential of 24.58 eV.
In order to compare our results with the previous calculations, the formation rates for J = 1 that were obtained in Ref. [38] are also shown in Fig. (2.14). The results for (4_{Heµp)}++
J=1 practically coincide beyond the resonant peak, whereas they signiﬁcantly
diﬀer at 10 eV and 20 eV. This is a consequence of the fact that the d-wave contribution (L = 2) to the formation rate was not taken into account in Ref. [38]. This contribution to the sum in Eq. (2.56) dominates the near-resonant energy and is negligible beyond the resonant peak. In the case of (3_{Heµd)}++
2.9 ROTATIONAL DEEXCITATION OF THE HYDROGEN-HELIUM MUONIC MOLECULES
a broad energy range due to the fact there is no resonance in the corresponding elastic scattering for the partial wave L = 2.
The formation cross sections for the molecules (4_{Heµp)}++
J=1 and (3Heµd)++J=2, together
with the resonant partial cross sections for the corresponding elastic scattering, are pre-sented in Fig. (2.15). The nature of resonant enhancement is clearly visible here, espe-cially in the (4_{Heµp)}++
J=1 case. The analogous resonant enhancement of formation of the
molecules (pµd)+
J=1 and (pµt)+J=1 in hydrogen-isotope mixtures was found in Ref.[55].
2.9
Rotational deexcitation of the hydrogen-helium
muonic molecules
In the previous section, formation process of the hydrogen-helium muonic molecules has been considered in a broad range of the collision energies. In room-temperature targets, slow collisions (s-wave) dominate. As a consequence, the rotational state of (Heµh)++
with the total angular momentum J = 1 is mostly populated. Nuclear He-h fusion, which occurs inside the molecule (Heµh)++_{, is favored when J = 0 [38]. For example,}
in the case of (3_{Heµd)}++ _{molecule, the ratio λ}J=0
f /λJ=1f of fusion rates for J = 0 and
J = 1 equals about 300–6000 (see Fig. 2.14). Experimental and theoretical studies of the nuclear fusion from molecular states are motivated mainly by very slow collisions (E ∼ 1 keV) of the nuclei in the (Heµh)++ _{molecule. Such energies are hardly reachable}
in accelerator experiments and the molecules provide a rare opportunity for investigation of strong interactions at low energies, especially studies of their charge symmetry and the PT invariance [56]. Nuclear synthesis of light elements at low energies is of interest for nuclear astrophysics, in particular for understanding the abundances of light elements in the Universe [57].
Investigation of the transitions between the rotational J = 0 and J = 1 states of the (Heµh)++
1 muonic molecule is of crucial importance for studies of the He++-h fusion. In
this work, we consider the 1 → 0 transition in the following processes (3_{Heµh)}++