Adiabatic expansion approximation solutions
for the three-body problem
M.R. Pahlavani and S.M. Motevalli
Abstract. The motion of a muon in two centers coulomb field is one of
the interesting problems of quantum mechanics. The adiabatic expansion
method is powerful approach to study the muonic three-body system.
In this investigation the three-body problem is studied for short-range
interactions. Bound states and energy levels of this system were calculated
and compared with their Born-Oppenheimer method counterparts. The
obtained results are in good agreement with the previous calculations.
M.S.C. 2000: 81Q05, 70F07.
Key words: three-body system, adiabatic expansion method, bound states, energy
levels, Born-Oppenheimer method.
1
Introduction
A negative muon is a lepton of the second generation with mass number about times
heavier than that of electrons, and has a finite lifetime of τµ = 2.197µsec. Nuclear
fusion reactions can be catalyzed in a suitable fusion fuel by muons [1, 2, 3]. Energetic
negative Muons, after stopping in a hydrogen isotope H2 /D2 /T2 mixture, fuse with
different hydrogen isotopes to form the muonic atoms pµ, dµ and tµ in excited states.
The size of a muonic atom is about 209 times smaller than a normal atom. After
a sequence of cascade transitions lasting about 10−11 sec at liquid hydrogen density
(L.H.D ≈ 4.25 × 1022 atoms/cm3 ), muonic atoms are formed in the ground 1S-state
with kinetic energy in the range of 10−3 −102 eV , depending on the temperature of the
target and the prehistory of cascade transitions [4, 5]. Muonic molecular formation
may take place in collisions of the muonic atom in its ground state with ordinary
hydrogen molecules.
The crucial process is the resonant formation of the muonic molecule dtµ into the
loosely bound rotational-vibrational state J = 1, ν = 1 (10−8 − 10−10 sec). This
state quickly de-excites to J = 0 levels, from which the two nuclei fuse (≈ 10−12 ).
Usually the muon is free to start the next cycle, but occasionally (with probability
ωs ' 0.555% in a D-T target at density φ = 1.2 L.H.D [6]) it is removed from
the active cycle by ”sticking” to the helium nucleus. Resonant formation of the
ddµ molecule at very low temperatures was observed in solid and liquid D2 targets
Applied Sciences, Vol.10, 2008, pp.
199-206.
c Balkan Society of Geometers, Geometry Balkan Press 2008.
°
200
M.R. Pahlavani and S.M. Motevalli
[7, 8]. Energy levels of muonic molecules in muon catalyzed fusion (µCF ) systems
have been extensively studied with various calculational methods [9, 10, 11, 12]. The
solution of the non-relativistic Schrödinger equation for the three-body system was
found using two different methods: by: the adiabatic expansion method and BornOppenheimer method. The Born-Oppenheimer approach assumes the nuclei to be
infinitely heavy with respect to the negatively charged particle. It should be kept in
mind that the following Born-Oppenheimer approach is the simplest solution to the
three-body coulomb system. This approach is expected to be a poor approximation
for calculations of muonic molecule eigenvalues. In the present paper, we calculate
binding energies of the bound states of the ddµ muonic molecule using the adiabatic
expansion method.
2
Theoretical calculations
The exact Hamiltonian that describes muonic three-body system can be shown by
following relation:
1
1
1
∇2R1 −
∇2R2 −
∇2
2m1
2m2
2mµ Rµ
z1 zµ
z2 zµ
z1 z2
−
−
+
,
|rµ − R1 | |rµ − R2 | |R1 − R2 |
H=−
(2.1)
where 1 and 2 denote the two nuclei, their position is given by R1 and R2 , and the
muon coordinate is rµ . The center of mass coordinate RCM is given by
(2.2)
RCM =
m1 R1 + m2 R2 + mµ rµ
.
m1 + m2 + mµ
It is convenient to define Jacobi coordinate r and R as follow:
R1 + R2
2
R = R2 − R1 ,
(2.3)
r = rµ −
where R is the internuclear coordinate and r is the muon coordinate to midpoint
between the two nuclei. In these coordinates (R, r), the Hamiltonian denoted by
equation (1) is change to the following operator:
H=−
(2.4)
µ
¶2
1
λ
1
1 2
∇2RCM −
∇R + ∇r −
∇
2MT
2M
2
2m r
z1 zµ
z2 zµ
z1 z2
−
−
+
,
|rµ − R1 | |rµ − R2 | |R1 − R2 |
Adiabatic expansion approximation solutions
201
where
(2.5)
MT = m1 + m2 + mµ
1
1
1
=
+
M
m1
m2
1
1
1
=
+
m
m1 + m2
mµ
m1 − m2
.
λ=
m1 + m2
After separation of variables, the non-relativistic Hamiltonian in units of e = ~ =
mµ = 1, can be given by
µ
¶2
1
λ
z1 z2
(2.6)
H(ô, R) = −
∇ R + ∇ r − H1 +
,
2M
2
R
where
(2.7)
H1 = −
1 2 z1
z2
∇r −
− ,
2m
r1
r2
where ô represent the five dimensional variable. We use the set ô = (Θ, Φ, ξ, η, ϕ)
where (Θ, Φ, ϕ) define the Euler rotation specifying the body-fixed frame with its
unit vectors to coincide with the principal axes of the inertia tensor of a three-body
system. The hyper-spheroidal coordinates ξ and η are easily expressed by the muonnucleus distances r1 , r2 and the internuclear distance R,
r1 + r2
, (1 ≤ ξ < ∞)
R
r1 − r2
η=
, (−1 ≤ η ≤ 1).
R
ξ=
(2.8)
The three-body Hamiltonian (6) commutes with the total angular momentum operator for the three particle system, J, its projection on z-axis, Jz , and the total parity
operator, P (R −→ − R, r −→ − r). Eigenfunctions of the Hamiltonian in the total
angular momentum representation reads:
(2.9)
J
X
ΨpJ,M (R, r) =
Jp
Jp
Fm
(R, ξ, η)DM
m (Φ, Θ, ϕ).
m=−J
Jp
Adiabatic expansion of radial function, Fm
(R, ξ, η) is usually written in the form:
Jp
Fm
(R, ξ, η) =
∞ N
−1
X
X
−1
ψN lm (R; ξ, η)χJp
N lm (R)R
N =1 l=0
∞
(2.10)
+
∞ Z
X
l=0
0
−1
dkψklm (R; ξ, η)χJp
,
klm (R)R
where χJp
i (R) describe relative motion of the nuclei. Let us consider the Wigner funcJp
2
tion, DM
m (Φ, Θ, ϕ) which is the eigenstates of J , Jz and R.J/R with the eigenvlues
202
M.R. Pahlavani and S.M. Motevalli
J(J + 1), M and m [13]. It can be transformed under the inversion as follow:
J
P DM
m (Φ, Θ, ϕ)
(2.11)
J
= DM
m (Φ + π, π − Θ, π − ϕ)
J
= (−1)J−m DM,−m
(Φ, Θ, ϕ).
If m 6= 0, the resultant Wigner functions would be different, and the angular functions
consist both even and odd combinations. It is convenient to specify these combinations
as follows:
√
2J + 1
Jp
J
[(−1)m DM
DM m (Φ, Θ, ϕ) =
m (Φ, Θ, ϕ)
4π
J
(2.12)
+p(−1)J DM,−m
(Φ, Θ, ϕ)],
where P = ±(−1)J is the eigenvalue of the parity operator:
Jp
Jp
P DM
m = pDM m .
(2.13)
The functions presented in equation (12)(in bracket) are satisfy the following orthonormality condition:
Z π
Z 2π
Z 2π
h
i∗
0 0
Jp
J p
sinΘdΘ
dΦ
dϕ DM
DM
(Φ, Θ, ϕ)
0
m (Φ, Θ, ϕ)
m0
0
0
0
= δJJ 0 δpp0 δM M 0 δmm0 .
(2.14)
If m = 0, both the Wigner functions in (11) are reduced to the ordinary spherical
function YJM (Θ, Φ) so that the dependence of ϕ disappears and the angular functions
satisfying the conditions (13) and (14) are:
(2.15)
Jp
DM,m=0
(Φ, Θ, ϕ) =
YJM (Θ, Φ)
√
.
2π
In this case the parity is unambiguously specified by the quantum number J: p =
+(−1)J . So, our basis functions have the following structure:
(2.16)
Jp
ΨJp
M jm (R, Θ, Φ, ξ, η, ϕ) = DM,m (Φ, Θ, ϕ)ψjm (ξ, η; R)
χJp
jm (R)
.
R
The wave functions ΨJp
M jm (R, Θ, Φ, ξ, η, ϕ) describing reactions hµ+h, h = (p, d, t) can
be decomposed over the solutions ψjm (ξ, η; R) of the Coulomb two-center problem.
ψjm (ξ, η; R) is the complete set of solutions of the Coulomb two-center problem,
therefore
(2.17)
H1 ψi (ξ, η; R)F (ϕ) = Ei (R)ψi (ξ, η; R)F (ϕ),
describing the muon motion around fixed nuclei separated by a distance R. Ei (R) is
the energy of a muon in the state i as a function of R. Here we show how to separate
the variables through the use of the ellipsoidal (or, prolate spheroidal) coordinates
Rp 2
(2.18)
(ξ − 1)(1 − η 2 )cosϕ
x=
2
Rp 2
(ξ − 1)(1 − η 2 )sinϕ
y=
2
R
z = ξη.
2
Adiabatic expansion approximation solutions
203
Note that the coordinates ξ, η and ϕ are orthogonal, and we have the first fundamental
form
ds2 = dx2 + dy 2 + dz 2 = h2ξ dξ 2 + h2η dη 2 + h2ϕ dϕ2 ,
(2.19)
where
µ
(2.20)
h2ξ
=
µ
∂x
∂ξ
¶2
µ
+
Thus
(2.21)
∂x
∂η
µ
¶2
µ
+
∂z
∂ξ
¶2
R2
=
4
µ
1 − η2
ξ2 − 1
¶
¶2 µ ¶2
µ
¶
R2 ξ 2 − 1
∂y
∂z
+
=
∂η
∂η
4
1 − η2
µ ¶2 µ ¶2 µ ¶2
¢¡
¢
∂y
∂z
R2 ¡ 2
∂x
+
+
=
ξ − 1 1 − η2 .
h2ϕ =
∂ϕ
∂ϕ
∂ϕ
4
h2η =
¶2
∂y
∂ξ
+
· µ
¶
µ
¶
µ
¶¸
1
∂ hη hϕ ∂
∂ hξ hϕ ∂
∂
hξ hη ∂
∇2 =
+
+
hξ hη hϕ ∂ξ
hξ ∂ξ
∂η
hη ∂η
∂ϕ
hϕ ∂ϕ
½ ·
¸
·
¸
¾
2
2
∂
∂
∂
ξ −η
∂2
2
2 ∂
=
(ξ − 1)
+
(1 − η )
+ 2
∂ξ
∂ξ
∂η
∂η
(ξ − 1)(1 − η 2 ) ∂ϕ2
4
× 2 2
.
R (ξ − η 2 )
Note that through the coordinate transformation (18), we have
(2.22)
R
(ξ + η)
2
R
r2 = (ξ − η).
2
r1 =
Writing the wave function as ψi (ξ, η; R)F (ϕ) = G(ξ)H(η)F (ϕ) and changing the
variable to spheroidal coordinates, equation (17) can be separated into following three
one-dimensional equations:
(2.23)
d2 F (ϕ)
+ m2 F (ϕ) = 0
dϕ2
(2.24)
·
¸ µ
¶
d
dG(ξ)
m2
(ξ 2 − 1)
+ −A + αqξ − q 2 ξ 2 − 2
G(ξ) = 0
dξ
dξ
ξ −1
(2.25)
·
¸ µ
¶
m2
d
2 dH(η)
2 2
(1 − η )
+ +A − βqη + q η −
H(η) = 0,
dη
dη
1 − η2
where
(2.26)
R
(z1 + z2 )
q
R
β = (z1 − z2 ).
q
α=
204
M.R. Pahlavani and S.M. Motevalli
Note that A and q are unknown parameters and should be obtained from (24) and
(25) as eigenvalues of the coupled system. Once A and q are obtained, then E can be
obtained from q 2 = −R2 E/2. By substitution of expression (16) into the Schrödinger
equation with Hamiltonian (6) and after averaging over spherical angles (Θ, Φ) and
the muon state, one obtains the radial equation
·
¸
1 d2 J
J(J + 1) J
χ
(R)
+
ε
−
V
(R)
−
U
(R)
−
χi (R) = 0,
B−O
i
2M dR2 i
2M R2
(2.27)
where ε = E − Ei (∞) is the collision energy and E is the total energy of the system
and Ei (∞) is the ground state energy of muonic atom. VB−O (R) = E − E(∞) + z1Rz2
is the potential
¡ corresponding
¢ to the Born-Oppenheimer (B-O) approximation and
Ui (R) = hi| H − H1 − z1Rz2 |ii is the adiabatic correction. The adiabatic potential
VAd (R) is:
(2.28)
VAd (R) = VB−O (R) + Ui (R).
The adiabatic potential VAd (R) for the (ddµ) muonic molecule is calculated in the
adiabatic expansion method. The adiabatic potential curves and qualitatively similar
for each of muonic molecules and are displayed for the (ddµ) muonic molecule in
Figure 1. Results of the calculations of binding energies of the bound states (J, ν) of
the (ddµ) muonic molecule are compared with the results of the other methods used
in Refs. [14, 15] and are given in Table 1.
0.75
J=0
J=1
J=2
0.25
Adiabatic potential, V
Ad
(m a.u.)
0.50
0.00
-0.25
-0.50
-0.75
0
2
4
6
8
10
12
14
16
Internuclear separation, R(a.u.)
18
20
Adiabatic expansion approximation solutions
205
Table 1: Binding energies (eV ) of the states (J, ν) for the (ddµ).
(J, ν)
(0, 0)
(0, 1)
(1, 0)
(1, 1)
(2, 0)
Ad (This work)
325.06
35.79
226.62
1.73
86.20
Ref. [14]
325.0735
35.8436
226.6815
1.97475
86.4936
Ref. [15]
325.070 540
35.844 227
226.679 792
1.974 985
—
Figure 1: Adiabatic potential curves, VAd (J = 0, 1 and 2), corresponding
to d − d − µ system.
3
Conclusions
In this paper, we have presented the calculation of the binding energies of the
symmetric muonic molecular ion. Early calculations were unable to demonstrate the
existence of the crucial weakly bound states. For example, in the Born-Oppenheimer
(fixed nuclei) approximation the state is much too bound, but if adiabatic corrections
are included it is not bound at all. The results are obtained by the three-body
Hamiltonian in the adiabatic expansion method. The calculated binding energies are
in good agreement with the previous calculations by other authors using different
methods.
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Authors’ addresses:
M. R. Pahlavani
Department of Physics, Faculty of Science,
Mazandaran University, P.O.Box 47415-416, Babolsar, Iran.
E-mail: m.pahlavani@umz.ac.ir
S. M. Motevalli
Department of Physics, Faculty of Science,
Mazandaran University, P.O.Box 47415-416, Babolsar, Iran.
Department of Physics, Islamic Azad University,
Ayatollah Amoli Branch, P.O.Box 678, Amol, Mazandaran, Iran.
E-mail: motavali@umz.ac.ir