Download Probing nuclear correlations by two proton emission

Probing nuclear correlations
by two proton emission
D.S. Delion, L. Ixaru, V.V. Baran
Horia Hulubei National Institute of Physics
and Nuclear Engineering (IFIN-HH)
Bucharest, Romania
Outline
1. Coupled channel description of two proton emission
2. Two body interaction outside and inside nucleus
3. Half life versus two body interaction parameters
4. Conclusions
Nuclear chart of decay modes
1. Coupled channel description
of two proton emission
We describe emission of two particles by solving
the Schrödinger equation with
1) pairing two-proton wave function on the nuclear surface
as an initial condition and
2) outgoing boundary conditions at large distance.
The main purpose is:
to investigate the dependence of the
half-life versus the parameters of the inter-proton interaction.
Geometry of the two proton emission
is described by proton radii r1, r2 and
the angle between them θ
Relative inter-proton radius is:
p
We neglect the recoil effects
of the daughter nucleus (D)
r1
r12
ϑ
p
D
Spherical BCS parent wave function
can be expanded in terms of proton monopole pairs
acting on the daughter BCS wave function
where the proton creation operator corresponds
to a single particle orbital with quantum numbers εljm
in a Woods-Saxon mean field reproducing
the proton energy ε=E/2,
where E is the two proton Q-value
Two proton amplitude
is proportional to the pairing density
in terms of BCS amplitudes
BCS equations
are solved for bound states
and narrow resonances in continuum
Number of particle condition
where
Level density in continuum
where δ(e) is the phase shift induced by potential and
Rb=100 fm is the dimension of the box
Initial condition is given by
two proton wave function on the nuclear surface
Singlet component is defined by:
in terms of:
- single particle radial wave function φεlj ,
- two-proton amplitude Xεlj ,
- two-proton azimuthal harmonics Y.
Two proton azimuthal harmonics
are proportional to Legendre polinomial
depending on the angle θ between emitted protons
Two proton Schrődinger equation
beyond the nuclear surface
can be written in terms of r1, r2 and θ as follows
in terms of the angular momentum operator squared
proton-daughter potentials V(rk) and inter-proton potential v(r12).
We neglect the motion of the daughter nucleus.
The wave function beyond the nuclear surface
has a similar ansatz
By integrating over the angle θ one obtains
a system of coupled radial equations
where the matrix elements of the inter-proton interaction v(r12) are given by:
Scattering conditions a better described
by using the system of polar coordinates
r
r2=r sinφ
φ
r1=r cosφ
If φ=π/4 then
r1=r2=r/√2
for any angle ϑ
between radii
By introducing a new wave function
one obtains the system of equations
in terms of the following l-dependent potential
describing interaction between protons and daughter nucleus
Matrix elements of the inter proton interaction
v0 = -35 MeV
r0 = 2 fm
Solution can be expanded
in terms of two proton sectorial harmonics
Two proton sectorial harmonics
satisfy the following eigenvalue problem
Sectorial harmonics at r=8 fm
In order to solve the propagation problem
we introduce the matrix of fundamental solutions
which is defined by the following boundary condition
where Hl(+) is the outgoing Coulomb-Hankel wave function
describing one-proton motion
General solution
is a superposition of fundamental solutions
Matching condition
on the nuclear surface determines
scattering amplitudes Nl
where ε,l,j are quantum numbers of the initial state
Ratio between computed and exact decoupled solution
(without inter-proton interaction) at r=50 fm
for k=15 (dots) , 20 (dashes) and 25 (solid line) sectorial harmonics
Asymptotics of the two proton wave function
at large distances
leads to the following expression
of the current
Total decay width
is given by integrating the current
over angle θ and φ
where
2. Two body interaction outside and inside nucleus
Inter proton potential beyond nuclear surface is different
from the effective interaction inside nucleus
V(r12)
Two-proton emission
45Fe
->
43Cr
+ 2p
Ṽ(r12)
Coherence length
is defined as the mean value of the relative radius squared
(
≡ ∫ ξ ( R, r )r dr
with respect to the paring density κ=X
2
)
1/ 2
Coherence length versus cm radius in 45Fe
has a strong dependence with respect to cm radius
The integrand of the coherence length
beyond the nuclear surface versus
the relative radius has a peak
around 2 fm, i.e. close to the radial
width of the inter proton interaction r0
7 while inside the nucleus has a
larger value, around 7 fm,
thus leading to a larger radial
width of the effective interaction.
The result is close to
the HFB calculation in
N. Pillet, N. Sandulescu, P. Schuck,
Phys. Rev. C76, 024310 (2007).
The integrand of the coherence length
versus relative and cm radius
Effective radial width of the interaction
is determined by a selfconsistent condition
< ξ >= r0( eff )
where inter proton effective radius defines
the potential inside the nucleus
Experimental gaps are reproduced
for the following effective radii…
b=r0eff ≈ 3 r0 for 45Fe
…and strengths
3. Half life versus two body interaction parameters
Pairing density versus single particle levels in 45Fe
Two proton wave function at the nuclear surface
Initial solution at r=6 fm versus φ
for v0=35 MeV
Solution at r=10 rm for versus φ
l=0: long dashes,
l=1: short dashes,
l=2: dots,
l=3: solid line,
l=4: dot-dashes
Solution at r=20 rm versus φ
for l=0,1,2,3,4
The width of the distribution is unchanged:
thus the ratio between proton radii is
weakly changes during propagation
and the motion along angle φ
can be neglected
Solution at r=50 rm versus φ
l=0: long dashes,
l=1: short dashes,
l=2: dots,
l=3: solid line,
l=4: dot-dashes
Pairing gap versus “bare” inter proton strength
Half life weakly depends on the matching radius,
but strongly depends on the inter proton strength
Half life versus “bare” inter proton strength
4. Conclusions
1. The ratio between proton radii weakly
changes during propagation: thus, the motion
along angle φ can be neglected.
2. Half life weakly depends upon the matching
radius, but strongly depends upon the
inter proton interaction strength.
3. Thus, two proton emission is an important tool
to probe the two-body interaction
THANK YOU!
Extreme Light Infrastructure (ELI):
the future laser facility @ Bucharest-Măgurele
ELI will afford new investigations in particle physics, nuclear physics,
gravitational physics, nonlinear field theory, ultrahigh-pressure physics,
astrophysics and cosmology (generating intensities exceeding 10²³ W/cm²).