Presentación de proyectos de investigación en física atómica y

Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Presentación de proyectos de investigación en
física atómica y molecular
Jorge Mahecha
Grupo de Física Atómica y Molecular
Instituto de Física, Universidad de Antioquia
Medellín, Colombia
Medellín, 25 de marzo, 2009
Publications
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
Resumen
Se describen los proyectos en que participa el profesor Jorge
Mahecha G. del Grupo de Física Atómica y Molecular.
• La compuerta de fase controlada con RMN (Camilo Estrada).
• Dinámica molecular con transiciones cuánticas (Beatriz
Londoño).
• Dinámica clásica de la predisociación del complejo NeBr2
(Ricardo Smith, Jesús Rubayo, Maykel González).
• Classical dynamics of the NeBr2 complex (Fernando Blesa,
Manuel Iñarrea, J. Pablo Salas).
• Resonant Coupling Effects on the Photoassociation of Ultracold
Rb and Cs Atoms (Beatriz Londoño, Eliane Luc-Koenig, Anne
Crubellier, Françoise Masnou-Seeuws).
• Shape resonances in ground state diatomic molecules: general
trends and RbCs example (Beatriz Londoño, Eliane Luc-Koenig,
Anne Crubellier).
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Contents
Resumen
Magnetoassociation
Predissociation of NeBr2
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
Photoassociation
Shape resonances
Publications
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
Magnetoassociation of Two H Atoms: Classical
Dynamics with Quantum Transitions
Idea of the MD with electronic transitions
Nuclear dynamics is described by means of the Newton equations of
motion. At every step of the numerical integration one must to choose
the PES to be used. In the present work, this method is applied to the
study of the magnetoassociation of two H atoms with a model of two
PES. From this, one can, in principle, to deduce parameters of the
Feshbach resonance in the colision of two hidrogen atoms. Then, one
can to determine the time of life of the molecular state as a function
magnetic field. This study can be of utility in the analysis of the
adiabatic magnetoassociation of ultracold atoms, and also of
photoassociation processes.
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
✭
✭
✏❞e2
✭
✏✏
r2A ✭✭✭✭✭
✏
✍✂
✭
✏
r2B
✭✭✭✭ ✏✏✏ r12
✭
✭
✂
✏
A ✭✭✭✭✭
R
✏
t✭
✭✭✭ t✂
✏✏
❙
✏✏ ✭✭✭✭✭✭
B
✏
r1A ❙
✏✏✭✭✭✭
✭✭✭
✇ ❞✏
❙
r1B
✭✏
e1
Publications
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
s2 ❞✁✁✕
✁✁
iA
t✁✁✕ iB
✁
✁
t✒
s1
✠
Atom A
❞
Atom B
Publications
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
The Hamiltonian of collision H-H is,
H=−
~2 2
∇ + HA + HB + V
2µ R
where HA corresponds to the Hamiltonian of the atom formed by the
nucleus A and an electron and similarly HB . The states and the
corresponding energies are |αi and Eα , where α = A, B.
F =1
1s1/2
n=1
Bohr
❅
❅
1s1/2
Dirac
❅
Lamb ❅
F =0
Hyperfine
Structure
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
At big distances the most important terms in the hyperfine interaction
come from magnetic dipole and electrical quadrupole moments. The
hyperfine interaction together with the Zeeman interaction determine
the internal Hamiltonian of each atom,
a
Hiint = µe B · σie − µp B · σip + σie · σip
4
where i=A,B.
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
The potentials at B = 0.1 T
Potentials of open and closed channels and coupling in the collision
of two hydrogen atoms.
V(r) (u.a)2
1.5
1
0.5
acople
cerrado
2
-0.5
-1
4
abierto
6
8
10
r (u.a)
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
Molecular dynamics
Molecules are formed with electrons and nuclei that interact through
Coulomb forces. The big mobility of the electrons gives place to the
appearance of effective non-Coulombic forces between the nuclei.
The potential interaction energies depend on the quantum state of the
electrons. There are named Potential Energy Surfaces (PES).
The Molecular Dynamics (MD) is used to describe the motions of the
nuclei of a molecule by means of the equations of the Newton
classical mechanics. The interactions are described by means of
PES.
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
MD with quantum transitions
It can exist strong couplings that give place to changes of the
electronic state.
In J. C. Tully, JCP 93 (1990) 1062 the MD is used when there can
occur a change of the electronic state.
The classic equations of the motion for nucleus are solved together
with the time dependentl Schrödinger equation that describes the
evolution of the electrons.
At the begining of each step of the procedure of numerical integration
it must to be chosen the PES that will be used in the integration
during that step.
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
Coordinates and Hamiltonian
ǫ = {r1 , ...rN }
ν = {R1 , ...RM }
are the coordinates of the N electrons and the M nuclei. The
trajectory of the nuclei is ν(t).
H = Tν + H0 (ǫ, ν)
H0 (ǫ, ν) is the Hamiltonian of the electrons when the nuclei are fixed.
Tν it is the kinetic energy of the nuclei.
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Electronic states
Basis of electronic states: {φj (ǫ, ν)}.
Matrix elements of the electronic Hamiltonian in this basis,
Hij (ν) = hφi (ǫ, ν)|H0 (ǫ, ν)|φj (ǫ, ν)i
“Non adiabatic coupling vector”,
dij (ν) = hφi (ǫ, ν)|∇ν |φj (ǫ, ν)i
Model with two PES,
ψ(ǫ, ν, t) =
2
X
j=1
cj (t)φj (ǫ, ν)
Publications
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Evolution of the ci (t)
Schrödinger equation,
i~ċk =
2
X
j=1
cj (Hjk − i~ν̇ · dkj )
Density matrix,
ρkj = ck c∗j
Populations and coherence of the two electronic states,
ρ11 , ρ22 , ρ12 = ρ∗21
Publications
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Evolution of the density matrix
Time dependent Schrödinger equation for the density matrix,
i~ρ̇kj
=
2
P
l=1
[ρlj (Hkl − i~ν̇ · dkl )
− ρkl (Hlj + i~ν̇ · dlj )]
X
ρ̇ii =
bij
j6=i
bij =
2
Im(ρ∗ij Hij ) + 2Re(ρ∗ij ν̇ · dij )
~
Publications
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
Changing the PES
N initial conditions are considered. At time t there will be
N1 (t) = N ρ11 (t) trajectories of the nuclei in the PES 1 and
N2 (t) = N ρ22 (t) in the PES 2.
If N1 (t + ∆t)N2 (t), necessarily N2 (t + ∆t) > N2 (t).
Probability that a trajectory that at time t is in the PES 1 pass at time
t + ∆t to the PES 2,
ρ̇22 (t)
b21 (t)
N1 (t) − N1 (t + ∆t)
≈
∆t =
∆t
N1 (t)
ρ11 (t)
ρ11 (t)
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
Algorithm for the changes of PES
A number ζ is generated at random, with 0 < ζ < 1.
If the trajectory at t in the PES 1, a change is realized from the PES 1
to the PES 2 if
b21 (t)
∆t > ζ
ρ11 (t)
If trajectory is at time t in PES 2, it is realizad a change from the PES
2 to the PES 1 if
b12 (t)
∆t > ζ
ρ22 (t)
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Trajectories
0.02
0.015
0.01
0.005
0
-0.005
-0.01
-0.015
-0.02
1
1.5
2
2.5
3
3.5
4
4.5
Publications
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Times of life of the trajectories
1000
’Tiempos-B8-rndmR-rndmCan.dat’
900
800
700
600
500
400
300
200
100
0
0
50
100
150
200
250
300
350
400
450
500
Publications
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Histogram with the times of life
0.09
’Histograma.dat’
f(x,200.)
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0
200
400
600
800
1000
Publications
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Histogram with the times of life
80
60
Out[147]=
40
20
200
400
600
800
Publications
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Fitting to a Poisson distribution
Show@p2, poissonD
0.0035
0.0030
0.0025
0.0020
Out[148]=
0.0015
0.0010
0.0005
200
In[129]:=
Out[129]=
300 * 2.4 * 10 ^ -17
7.2 ´ 10-15
400
600
800
Publications
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
Conclusion
• The method of the molecular dynamics with quantum transitions
applied to the two-channel model allows to understand the
formation of a molecular state by means of Feshbach resonance.
• The model predicts that for a value of B ≈ 0,1T a resonance
occurs.
• The model predicts a life time of only τ ≈ 10−14 s, corresponding
to a width of Γ ≈ 2 · 10−4 au.
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
Perspectives
To realize an exhaustive study with a fine change of the magnetic
field.
To implement it for realistic systems, such alkaline dimers.
To apply the method of the MD with quantum transitions to the
multichannel model of the photoassociation of cold alkaline atoms.
The method of the MD with quantum transitions can be useful in the
preparation of photoassociation experiments with chirp pulses. At
first, it allows the optimization of the parameters of the pulses using
control algorithms.
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
Vibrational predissociation of NeBr2
Experiments with pumping and probe laser pulses, accompanied by
resolved in time and frequency measurements, are used to have
access to the vibrational predissociation mechanisms and the
intramolecular relaxation vibrational dynamics of systems such as the
NeBr2 . It was found that the lifetimes are of some few tens ps.
[Cabrera, J. Chem. Phys., 123, 054311, 2005]. In the present
investigation theoretical ab initio semiclassical calculations about the
dynamics of the vibrational predissociation of the NeBr2 are realized.
The used potential energy surfaces (PES) were obtained from
parameters successfully used in the literature [Garcia-Vela, J. Chem.
Phys., 124, 034905, 2006; Gonzalez, Phys. Chem. Chem. Phys., 8,
4550, 2006; Roncero, J. Chem. Phys., 115, 2566, 2001]. Starting
from a semiclassical approximation to the autocorrelation function,
followed by an analysis of the complex frequency contents of the time
signal, the predissociation times can be calculated, for different
vibrational levels.
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
NeBr2 . Excited state potential energy.
VHR,Θ,r0=2.667L
6
4
2
Out[119]=
0
-2
-4
-6
-5
0
5
Publications
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
NeBr2 . Trajectory of the Ne for v = 27. r(t).
3.4
3.2
3.0
Out[192]=
2.8
2.6
2.´10-13
4.´10-13
6.´10-13
8.´10-13
Publications
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
NeBr2 . Trajectory of the Ne for v = 27. R(t).
30
25
20
Out[194]=
15
10
2.´10-13
4.´10-13
6.´10-13
8.´10-13
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
NeBr2 . Trajectory of the Ne for v = 27, dissociation
channel ∆v = 1.
6.0
5.5
5.0
4.5
Out[196]=
4.0
3.5
3.0
2.5
2.0
2.0
2.5
3.0
3.5
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
Classical model of the NeBr2 predissociation.
Out[1829]=
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
Classical Dynamics of the NeBr2 complex
In the above problem, the motion of the Ne in the excited potential
surface was considered. That surface has no a minimum. This
causes vibrational predissociation of the Ne atom. The potential
surface of the ground state, allows two stable configurations, the
linear and the T-shape. These configurations are separated by saddle
points. The movements of the Ne can have regular and chaotic
dynamics, and isomerization of the complex is possible. In this work a
complete study of the phase space and the classical dynamics of this
system is realized.
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
The Hamiltonian
We consider the motion of a neon atom around a Br2 molecule which
bond coordinate r is frozen at its equilibrium distance re ∼ 2,281. By
considering the total angular momentum of the molecule is zero, the
dynamics of the Ne atom is described by the two-dimensional
Hamiltonian
1
1
1
p2
p2θ + V (R; θ, re )
+
H= R +
2µ2
2 2µ2 R2
µ1 re2
V (R; θ, re ) is the potential energy surface describing the interaction of
the neon atom with the Br2 molecule.
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
The potential energy surface
The potential energy is given by
V (R; θ, re ) =
" 5
X X
λ
i=1
αiλ e−2βi (R−Ri ) − 2e−βi (R−Ri )
#
δλ
ηλ
− 6 − 8 Pλ (cos θ
R
R
with λ = 0, 2, 4, 6, 8 and the parameters are taken from a paper of
Prosmiti et al.
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
NeBr2 . Ground state potential energy.
VHR,Θ,r0 =2.281L
6
4
2
Out[105]=
0
-2
-4
-6
-5
0
5
Publications
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
Curvas equipotenciales de la energía potencial
V (R, θ, re)
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
Superficies de Poincaré calculadas con la energía
E = −0,00039 u.a. (a) PR = 0. (b) θ = π.
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
Órbitas cuasiperiódicas alrededor de las órbitas
periódicas rectilíneas L1 (azul) y L2 (rojo).
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
(a) y (b) son respectivamente las superficies de
sección de Poincaré para PR = 0 del isómero en
forma de T con θ = π/2 y el isómero lineal con
θ = π. (c) muestra la superficie de sección de
Poincaré para θ = π del isómero lineal. Todas se
calcularon para E = −0,00037 u.a.
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
PHOTOASSOCIATION
The photoassociation (PA) with continuous laser has demonstrated to
be an efficient way to form short lived diatomic molecules in an
excited electronic state, starting from a set of free ultracold atoms,
with temperature near 50 µK. In a second step, the stabilization, by
using the spontaneous emission, gives place to the formation of a set
of molecules distributed in many vibrational states of the lowest
electronic curve [Masnou, Adv. At. Mol. Opt. Phys. 47, 53, 2001].
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
PHOTOASSOCIATION
The Cold Atoms Group at LAC is analyzing theoretically schemes of
two color photoassociation, with pumping and probe laser pulses.
Chirped laser pulses, both in the photoassociation and stabilization
steps are used [PRA 70, 033414, 2004; Koch, PRA 73, 033408,
2006)].
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
Photoassociation
The achievement of samples formed with stable cold polar molecules
in the ground rovibrational energy level is needed by applications in
quantum computing, in the study of many-body highly correlated
systems, in ultracold chemistry and other applications. Recently
molecules in the X1 Σ+ (v=0) state by photoassociation of Rb and Cs
with continuous laser, followed by a transference stimulated with
laser, were obtained.
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
Photoassociation
To investigate the possibility of doing such cold stable molecules with
pumping and probe lasers we have chosen the photoassociation by
means of resonant coupling of Rb and Cs molecules in the [Rb(5s) +
Cs(6p1/2 )]0+ state. A previous condition for this project is the analysis
of the spectroscopic properties of the system, which is being done in
this collaboration.
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Resonant Coupling Photoassociation
Publications
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
Shape resonances in ground state diatomic
molecules: general trends and RbCs example
We present a systematic study of the universal trends of Shape
resonances in ground state of diatomic molecules, using reduced
units and R−6 model. Our interest are the resonances by tunneling
through the centrifugal barrier, and we show results for l=1 to 6. We
also calculated resonances for 85 RbCs using numerical potential
X 1 Σ+ .
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
Reduced units and R−6 model
The ground state potential of the alkali diatomic molecules to large
distance is leading for −C6 /R6 term of the multipolar expansion van
der Waals, which is followed by −C8 /R8 , −C10 /R10 , etc. Restricting
the asymptotic potential to the −Cn /Rn term is generally valid. Then
one can introduce scaling factors depends only on the rank n.
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
Radial motion
The Schrödinger radial equation for a particle of mass µ, and angular
moment l moving in an asymptotic potential −Cn /Rn can be reduced
to a universal equation, where the length and energy scaling factors
are σ and ǫ,
−
~2 d2
Cn
~2 l(l + 1)
−
+
−
E
F (R) = 0
2µ dR2
Rn 2µ R2
l(l + 1)
1
2
y = 0
y ′′ + n −
+
k
x
x2
1/(n−2)
2µCn
σ =
~2
2
~
ǫ =
2µσ 2
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Scattering length
As it is well known, the scattering length L is a crucial parameter in
the elastic collision at ultracold temperatures. The scattering length
es a property of the whole potential and for real atoms is very
sensitive to small uncertainties in the short range, chemical
boundaring region, which is for instance given by the zero energy
semiclassical phase of the threshold wavefunction, calculated from
inner turning point Rt , where V (Rt ) = 0 to infinity.
Publications
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
General trends
In the study of the general trends, we use the simplest asymptotic
model, which consists in a −1/xn potential limited by a hard-core
potential wall at a distance x0 . The relationship between the
parameter x0 and the reduced scattering length a = L/σ is obtained
from an analytic formula.
The choise of a scattering length a interval fixed the x0 values, we
choise a interval that let us consider the region near the threshold,
and below the centrifugal barrier.
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
The rotational constant
Each vibrational state of the diatomic molecule has a value of
rotational constant Bv ,
1 e
~2
e
Bv =
Ψv (R) 2 Ψv (R)
2µ
R
where e denotes the electronic state.
The rotational constant has a large value for low vibrational states,
and as one approaches the threshold the value Bv is becoming
smaller. If one considers l 6= 0 in equation, the presence of the
centrifugal barrier creates quasi-bound states, and we could see
increases in the value of Bv for states above the threshold. In a cold
collision of two atoms one can tune a resonant state by tunneling
through the centrifugal barrier.
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
Trends of asymptotic model
In this paper we present systematic results for l = 1 to 6. Considering
only the R−6 term in the expansion of van der Waals and centrifugal
term, one can find analytically expresion for the position and
maximum of the barrier.
1/4
6µC6
Rmax =
l(l + 1)~2
3/2
2~3 l(l + 1)
Vmax = √
6µ
C6
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
The barrier
l
1
2
3
4
5
6
Rmax
1.106680
0.840897
0.707107
0.622333
0.562342
0.516973
Vmax
1.08866
5.65685
16.000
34.4265
63.2456
104.766
W
0,983812
0,747535
0,6286
0,553238
0,499907
0,459576
Cuadro: Position, maximum and width of the centrifugal barrier in reduced
units. The width W is calculated by finding the positions to the left R1 , and to
the right R2 of Rmax for which the potential is Vmax /2, then the width is
estimated to be R2 − R1
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
Effective potentials
In the figure are the potentials, and the table shows the values in
reduced units for the position, maximum and width of centrifugal
barrier.
VHxL
100
50
x
1
2
3
4
-50
Figura: Asymtotic potential in reduced units V (x) = −1/x6 + l(l + 1)/x2 for
l = 1 to l = 6.
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Parameters
The calculations are performed with Mathematica, we explore
intervals to allow us to study the region close to the threshold. The
intervals used are reported in table.
l
1
2
3
4
5
6
(amin , amax )
(0,7, 2,0)
(x0min , x0max )
(0,1681838, 0,1788823)
(−1, 3)
(−5, 5)
(0,1659801, 0,1795555)
(0,1649578, 0,1800319)
(0,434, 3,434)
(0,1715827, 0,1799449)
Cuadro: The a and respective x0 intervals in reduces units.
Publications
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
Energy
For each value of the scattering length a, we calculate the phase shift
as a function of the energy δ(E). A resonance appears as a jump in
the phase shift, then one stimated dδ/dE close to the jump to find the
position Er and width Γ of the resonance. The figure shows the
energy as function of scattering length. The region with E < 0
correspond to bound levels, which are calculated by imposing the
conditions on wavefunction: y(x0 ) = 0 and decay exponentially, y → 0
for x → ∞.
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
Resonances
l=1
l=3
l=6
l=4
2.0
50
120
40
100
30
1.5
0.5
10
0
-10
0.0
1.0
1.2
1.4
a Hreduced unitsL
1.6
1.8
2.0
30
20
10
0
-20
0.8
E Hreduced unitsL
1.0
E Hreduced unitsL
E Hreduced unitsL
E Hreduced unitsL
20
0
1
a Hreduced unitsL
2
3
60
40
20
0
-20
-10
-1
80
-5
-4
-3
-2
a Hreduced unitsL
-1
0
1
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
a Hreduced unitsL
Figura: Energy (E) vs. Scattering length (a) in reduced units for l = 1 to l = 6.
For all cases the horizontal line is Vmax . Shape resonances are characterized
by a position Er and width Γ, the central curve corresponds to Er , the bottom
to Er − Γ and up to Er + Γ. The region with E < 0 correspond to bound
levels.
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
Threshold
In the region with E > 0 and very close to the threshold there are
Sharp resonances, the barrier creates states with larger lifetime, then
to calculate resonance with very small widths we used Siegert states,
these states have complex energy E = Er + iΓ, where the real part is
the position and complex part is the width of the resonance. In the
figure for E > 0 in the region where we obseved a single curve the
calculations were made with Siegert conditions, this region joins
smoothly with the region of Shape resonances, as the width of the
resonance increases.
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
Threshold
l
1
2
3
4
5
6
Ebl
-0.0525427
-0.838138
-0.133446
-0.271423
abl
0.9665
xbl
0
0.176251
xbl
out
0.837071
0.06
-3.05
0.168503
0.165172
0.534636
0.472695
0.634
0.173677
0.392717
Cuadro: Last bound level for the interval x0 in the table, using the conditions
of bound level (bl), y(x0 ) = 0, y → 0 for x → ∞. The table shows the value of
the energy Ebl , the scattering lenght abl , the correspond value xbl
0 , and finally
we calculate the outer turning point at Ebl , xbl
.
out
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
Shape resonances in 85RbCs
The values of reduced mass for 85 RbCs and 87 RbCs in atomic units
are µ85 = 94444au and µ87 = 95788au, and using the value
C6 = 5284au reported in the literature, one can calculate with the
radial equation the scaling factors for length σ and energy ǫ of the
molecule RbCs.
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
Shape resonances in 85RbCs
σ85
=
177, 743au
ǫ85
σ87
=
=
1, 67576(10−10)au = 3, 67786(10−5)cm−1
178, 372au
ǫ87
=
1, 64062(10−10)au = 3,60073(10−5)cm−1
(1)
Resumen
Magnetoassociation
Predisociación
Isomerization of NeBr2
Photoassociation
Shape resonances
Publications
• S. Bhatnagar, J. Mahecha. Phys. Rev. D. 2009 (submitted). ICTP
Preprint Serial Number IC/2008/081.
• B. E. Londoño, J. Mahecha, E. Luc-Koenig, A. Crubellier, F.
Masnou-Seews. Phys. Rev. A. 2009 (in submission).
• Ricardo Smith, Jorge Mahecha. Rev. Col. Quím., Mayo 2008 (in
submission).
• J. Macana, J. Mahecha. Ingenium (2007). (submitted)
• C. Castro, J. Mahecha. Chapter in Quantization in Astrophysics,
Brownian Motion, and Supersymmetry. F. Smarandache, V.
Christianto (editors). MathTiger, Chennai, Tamil Nadu, India, 2007.
ISBN: 978-81-902190-9-9
Publications