Probing nuclear structure by cold emission processes

1
Probing nuclear structure by cold emission
processes

• D.S. Delion (Bucharest)
• J. Suhonen S.Peltonen (Jyvaskyla)
• R.J. Liotta R. Wyss (Stockholm)
• A. Sandulescu (Bucharest)

2
Contents

• Classification of cold emission processes
• Emission theory from deformed nuclei
• Decay rules in alpha decay
• Fine structure in alpha decay
• Double fine structure in cold fission
• Decay rules in proton emission
• Conclusions

3
The family of cold decay processes

• Cold emission
• Splitting of a parent nucleus (P) into two or
  more
• fragments (12) close to their ground states,
• which energetically are more bound
• Q MPc2 ( M1c2 M2c2 ) gt 0
• predicted /
  measured
• Proton emission 1960 / 1970
• Two-proton emission 1960 / 1995
• Alpha decay
  1896
• Heavy-cluster emission 1980 / 1984
• Cold (neutronless) fission 1962

4
Cold emission processesare magic decays

• Cold emission of heavy fragments
• is connected with magic nuclei
• Alpha decay He
  emission
• Heavy-cluster emission
  (C,O,Ne,Mg,Si)
• 
  Pb emission
• Cold fission Sn
  emission

5
Why is important the investigationof cold
emission processes ?

• The nuclei close to proton/neutron drip lines are
  very unstable and decay through emission channels
• Neutron rich nuclei are produced by fission
• Superheavy nuclei are investigated by using alpha
  decay chains and fission channels

6
Which are the measured quantities?

• 1. Emission energy Q - Ec
• where EcE1E2 is the sum of
• excitation fragment energies
• in some channel (partition) c
• 2. Partial decay width (intensity) in the channel
  c
• or partial half-life
• 3. Angular distribution of emitted fragments

7
Emission theory from deformed nuclei

• A decaying nucleus is an open system
• Half life (gt10-12s) gtgt nuclear time (10-22s), or
• decay width (lt10-10 MeV) ltlt Q-value (1 MeV)
• The system is described by outgoing solutions
• (Gamow states) of the stationary Scrodinger
  equation
• with some potential
• between emitted fragments

8
Geometry of the binaryemission process

• 
  

z
Z2
Z1
r1
r2
R
9
Stationary Schrodinger equationfor binary
emission

• distance between
  fragments
• internal coordinates of
  fragments
• reduced mass

10
Double folding potentialbetween emitted fragments

• One supposes that the two fragments have
• the probability 1 to exist at any distance R
• nuclear densities
• nucleon-nucleon interaction

11
Wave function is a superpositionof different
emission channels

• where we introduced channel index
• fragments variables (Euler angles for rotations)
• fragment angular harmonics

12
Fragment angular harmonics

• for a given angular momentum J and projection M
• are orthonormal

13
Fragment angular harmonicsfor various emission
processes

• Cold fission
• ground state ? rotational states
• Alpha (cluster) decay
• ground state ? rotational state
• Proton emission
• odd-proton nucleus with spin I ? rotational state

14
Coupled channels procedurefor radial components,
channel c(lJ1J2)
15
The system of radial Schrodinger equationsby
denoting the channel c? l

• at small distances (lt0.1 fm) becomes Laplacian
• at large distances (gt15 fm) becomes Coulombian
• where we defined Coulomb
  parameter

16
How to solve the system ?

• 1) One uses asymptotic conditions
• in order to find
• one matrix of fundamental solutions
• for the internal region
• and one for the external region
• 2) The two matrices and their derivatives are
  matched
• in order to obtain
• the eigenvalues and eigenstates

17
Fundamental matrix of solutionsat small distances

• has the following asymptotic form
• l labels the component
• k labels the solution number

18
Fundamental matrix of solutionsat large distances

• has the following outgoing asymptotic form

19
The general solution is a superpositionof the
fundamental matrix

• The matching condition at some radius inside
  barrier
• gives the secular equation

20
Resonant outgoing states orGamow states

• are solutions of the secular equation
• with complex energies
• and correspond to the poles of the S-matrix
• because the incoming flux vanishes

21
Normalisation condition

• over the internal region 0,Rext,
• where Rext is the external turning point (EV)
• fully determines the unknown coefficients Mk and
  Nk

22
Outgoing resonant Coulomb wave

• with complex energy (Gamow state)
• has the following asymptotic form
• The scattering amplitude Nl is determined
• by internal radial components fk

23
Decay width

• By using the continuity equation
• for the wave function
• one obtains the total decay width

24
Partial decay width

• One obtains the factorisation between
• the penetrability and deformed spectroscopic
  probability

25
Spectroscopic function is a superposition of
radial wave function components

• where we introduced the propagator matrix
• which becomes unity matrix for spherical emitters

26
Outgoing Coulomb waveWKB approximation inside
the barrier

• where

27
Gamow decay rule for the half life
28
Geiger-Nuttal decay law for even-even alpha
emitters(Viola-Seaborg decay rule)
29
Alpha decay spectroscopic probability versus N
for even-even emitters
30
Geiger-Nuttall decay lawfor heavy cluster
emission
31
Alpha decay from ground stateto an alpha
particle rotating daughter

• Alpha-nucleus potential
• Wave function with the channel index c(l,l,0)
• and total angular momentum J0

32
Alpha nucleus potential NuclearCoulomb
folding potential Pauli repulsion
33
Spectroscopic factor

• is given by the ratio
• because alpha-particle exists
• with the probability lt 1 inside the potential
• Solution is to correct nucleon-nucleon
  interaction
• by a screening factor valt1

34
Fine structure in alpha decay

• One defines the following quantities
• Intensity
• Hindrance factor

35
Energy systematics forZlt82 (a) and Zgt82 (b)
36
Experimental intensities versusexcitation energy
(a) and neutron number (b)
37
Experimental hindrance factors versusexcitation
energy (a) and neutron numbers (b)
38
The influence of the screening factor vaon
Q-value, half life and intensities
39
Alpha decay datafor rotational nuclei
40
Q-value, repulsive depth (a) and half-life (b)
versus the decay number
41
Screening factor

• explains the Viola-Seaborg decay rule
• which generalizes the Gamow decay rule

42
Radial wave function components (a) anddiagonal
components of the potential (b)
43
Deformation parameters (a) and intensities
(b)versus the decay number
44
Double fine structurein cold fission

• Nuclear density in the intrinsic system of the
  fragment
• where we introduced the difusivity a(k)
• and skin parameter w(k)

45
Total potential (dot-dashed)Pole-pole potential
(solid)Coulomb potential (dashed)
46
Structure of the wave function

47
Dependence of radial componentson neutron
density parameters
48
Dependence of fragments yieldson neutron density
parametersfor 252Cf?108Mo146Ba

49
Proton emission

• Wave function of the proton core system
• where the channel index is c(ljJ)
• For transitions between ground states
• one obtains Nilsson wave function
• with the channel index c(lj)

50
Proton - nucleus potential Nuclear mean field
Coulomb interaction
51
Experimental proton half-lives versus Coulomb
parameter
52
Half-life for proton emitters

• The reduced half-life depends on the angular
  momentum
• only via the spectroscopic part

53
Proton reduced half-livesversus Coulomb parameter
54
Deformation (a) and proton reduced width (b)
versus charge number
55
Proton reduced width versus deformation
56
Geiger-Nuttal decay law for proton emitters

• is given by two lines
• for the reduced half-life
• log10 Tred ak?bk
• where
• k Z ak bk sk
• 1 lt68 1.31 -2.44 0.26
• 2 gt68 1.25 -4.71 0.23
• This pattern is given by a discontinuity
• in deformation at Z68

57
Conclusions

• Reduced half life (reduced width) and
• and fine structure in emission processes
• are sensitive tools
• to probe nuclear structure details