Hyperfine structure of mesomolecules pdμ, ptμ, dtμ in variational method The XXIV International Workshop High Energy Physics and Quantum Field Theory September 23rd, Sochi Sorokin V.V. In collaboration with: A.V. Eskin, A.P. Martynenko, F.A. Martynenko and O.S. Sukhorukova
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Hyperfine structure of mesomolecules pdμ,
ptμ, dtμ in variational method
The XXIV International Workshop
High Energy Physics and Quantum Field Theory
September 23rd, Sochi
Sorokin V.V.
In collaboration with: A.V. Eskin, A.P.
Martynenko, F.A. Martynenko and O.S.
Sukhorukova
INTRODUCTION AND OVERVIEW
2
The investigation of energy spectra of hydrogen muonic molecules is
important for muonic catalysis of nuclear fusion reactions. A calculation
of fine and hyperfine structure of muonic molecular ions as well as
higher order QED corrections allows us to predict the rates of reactions
of their formation and other parameters of the μCF cycle.
Korobov V.I., Puzynin I.V. and Vinitsky S.I. Physics Letters B 196 (1987)
272-276.
Frolov A.M. and Wardlaw D.M. Eur. Phys. J. D 63 (2011) 339–350.
Aznabayev D.T., Bekbaev A. K., Ishmukhamedov I. S., and Korobov V. I.
Physics of Particles and Nuclei Letters 12 (2015) 689–694.
3
PURPOSE
The aim of this work is to study hyperfine splitting of three-particle tdμ,
dpμ and tpμ muonic molecular ions on the basis of variational approach.
Tasks:
1. Analytical calculation of diagonal and off-diagonal matrix elements of
kinetic energy, potential energy and overlap for basis functions;
2. Writing computer code to solve bound state problem for three particles using
stochastic variational method with correlated Gaussian basis;
3. Calculation of the energy of the ground state of tdμ, dpμ and tpμ muonic
molecular ions and their hyperfine structure.
4
GENERAL FORMALISM
N
ji
ijij
N
i
cm
i
i VVTm
H1
2
2
ptwo-body interaction potentials
EH
Let us consider a system of 3 particles with masses m1, m2 and m3 and charges z1, z2
and z3 respectively. The Schrodinger equation in the Jacobi coordinates has the form:
In variational method the wave function of the system is presented as follows
1
( , )K
i i
i
c A
x
An upper bound for the energy of the ground state is given by the lowest eigenvalue
of the generalized eigenvalue problem:
( , ) | | ( , )
( , ) | ( , )
K
ij i j
ij i j
HC E BC
H A H A
B A A
x x
x x
5
GAUSSIAN BASIS FUNCTIONS
In variational approach with correlated Gaussian basis wave functions have the form:
1 2 12 3 123
/2
1 1
1 1
1 2 3
( , ) ( )
( ) ,
.
.
( ),
( ) [[[ ( ) ( )] ( )] ...]
A
A
A
N N
ij i j
i
L
L l l L LM
j
l L
A G
G e
A A x x
~
x x
~
x x
x
x
x
x x
x
x x
N
N
x
xx ),...,(11
x Jacobi coordinates
center of mass coordinate
The diagonal elements of the (N-1)×(N-1) dimensional symmetric, positive definite
matrix A correspond to the nonlinear parameters of Gaussian expansion, and the off-
diagonal elements connect different relative coordinates thus representing the
correlations between particles.
The angular part of the basis wave function has the following form:
6
We use the following order of particles:
JACOBI COORDINATES
0
3
21
2211
21
R
rrr
λ
rrρ
mm
mm
ρλrrr
ρλrrr
ρrrr
11
2112
21
13223
21
233
mm
m
mm
m
ρ
λ
The Jacobi coordinates are related to the relative
particle coordinates as follows:
For the interparticle coordinates:
. 321321321
μtp and μpd, μdt
GROUND STATE L=0
7
}2
ћ
2
ћ{
det
24|| 00
2
200
1
2
5.2
300
^'
II
BT
'
112222
2
122212122222
2
12
00
221111
2
121112121111
2
12
00
))((2
))((2
ijijij AAB
BBABABAABAI
BBABABAABAI
2 211 22 12
1[ 2 ( )]
200 ( , , )
A A A
A e
ρ λ
ρ λ
Kinetic energy operator:
where:
2
2
22
1
2^
2
ћ
2
ћλρ
T
321
3212
21
211
)(,
mmm
mmm
mm
mm
Matrix elements of kinetic energy:
))((
28
det
28
||
223,13
2
23,13
122
23,13
1
5.200
23,13
22
5.200
12
00
2332
00
1331
00
1221
^'
FFBFI
BBI
IeeIeeIeeV
21
23,13
1,2
2212
23,13
2
21
23,13
1,2
12
2
21
23,13
1,2
2211
23,13
1
2
2)(
mm
mBBF
mm
mB
mm
mBBF
5.1
300'
det
8|
B
GROUND STATE L=0
Potential energy operator:
Matrix elements of potential energy:
Overlap matrix elements:
||||||
21
2
32
21
2
3121
ρλρλρ
mm
m
ee
mm
m
eeeeV
8
HYPERFINE STRUCTURE
9
Hyperfine structure can be described with the following interaction potential:
Where indices 1, 2 denote nuclei of hydrogen isotopes, index 3 denotes muon.
10
MATRIX ELEMENTS FOR HFS
K
ji
ji
K
ji
ji
K
ji
ji
m
mB
m
mBB
cc
m
mB
m
mBB
cc
Bcc
1,2/3
2
12
122
12
11211
2/3
1,2/3
2
12
222
12
21211
2/3
1,2/3
22
2/3
)2(
)2()(
)2(
)2()(
)(
)2()(
23
13
12
r
r
r
ji
ji
ji
AAB
AAB
AAB
mmm
222222
121212
111111
2112
11
AVERAGING OVER SPIN FUNCTIONS
Averaging over spin functions can be performed using the following relations:
1212 12
12 12 1 2 3
12 1 212
12 12 1 1 1 3 3 312 12
12 3 1 21 12
3 12 112
1212 1
, | ( ) | , ( ) ,
, | ( ) | , (2 1)(2 1)(2 1)( 1) (2 1)( 1)
( 1) ,1 ; 1
, | ( ) | , (2
{ }{ }max min
S SS
S S S S S S
S S S S S S
S S S S S S S S S S S S
S S S S S S
S S S S
S S S S S
1 2
1 3
2 3
S S
S S
S S
12 1 2 3
12 2 2 2 3 3 32
12 3 2 12 1 12
3 12 212
1)(2 1)(2 1)( 1) (2 1)( 1)
( 1) .1 1
{ }{ }maxS S S S S
S S S S S S S
S S S S S S
S S S S
12
ENERGY MATRIX
After averaging over spin functions energy matrix takes form:
c
acb
bcba
cba
4
3
4
3
4
3
4
30
4
3
2
1
2
1
4
10
00)(4
1
All particles have spins 1/2
After diagonalization eigenvalues can be obtained:
)(4
1
)(4
1
3
222
2,1
cba
acbcabcbacba
13
cba
cbacb
cbcba
cba
4
1000
012
1
3
1
3
2
3
20
03
2
3
2
12
5
6
5
2
10
0004
1
2
1
2
1
Particles have spins 1, 1/2, 1/2
ENERGY MATRIX
After averaging over spin functions energy matrix takes form:
After diagonalization eigenvalues can be obtained:
cbaacbcabcba
cba
cba
44144994
1
)22(4
1
4
1
222
4,3
2
1
14
ADDITIONAL CORRECTIONS, VACUUM POLARIZATION
To improve the accuracy of our calculations we take into account vacuum polarization
corrections:
Where indices a, b correspond to nuclei of hydrogen isotopes, µ corresponds to muon.
15
ADDITIONAL CORRECTIONS, NUCLEAR STRUCTURE
Nuclear structure corrections of the leading order take the form:
16
THE PROGRAM
For the numerical calculation of energy levels of three-particle Coulomb bound
states the code in MATLAB is written. The program uses stochastic variational
approach with random optimization procedure for nonlinear variational parameters;
The program is based on the Fortran program by K.Varga and Y.Suzuki;
A number of changes is made compared to the Fortran program, including the
ability to calculate states with nonzero L, more convenient generation of variational
parameters and various optimization changes;
The main results of the calculation include energies of ground and excited states
along with variational wave functions for each state. The program is capable of
calculating L=0 and L=1;
We are now working on calculation of L=1 hyperfine structure.
K. Varga, Y. Suzuki// Computer Physics Communications 106 (1997) 157-168
NUMERICAL RESULTS
17
For ground state and hyperfine structure of tdμ, dpμ and tpμ the following numerical results
are obtained:
A. M. Frolov // Eur. Phys. J. D. –– 2012. –– Vol. 66. –– P. 212––223.