alphafis

8/6/2019 alphafis

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Alpha DecayAlpha DecayEnergy relationsEnergy relations

experimental bindingenergy of 4He

Q a ( A, Z ) = B( A - 4, Z ) + 28.3MeV - B( A, Z )

Q a Q a = T a + T d =

T a

M D + M a

M D

ÊËÁÁ

ˆ¯˜ ª T a

A

A - 4

ÊËÁ ˆ

¯˜

recoil term effect

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Alpha DecayAlpha DecayTheory of Alpha decayTheory of Alpha decay

Coulomb potential

Attractive nuclearpotential

1928 Gamow

Time-dependent approach-

h 2

—2 +V (rr )

È

ÎÍÍ

˘y

rr , t ( )= ih

∂ y rr , t ( )

U(r)

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∂

∂ t r

r

r , t ( )+r

—r

j r

r , t ( )= 0 Continuity equation

Since a current of alpha-particles is leaving the nucleus, the densityr has to be a decreasing function of time. If we want to approximatethe initial wave function through a stationary state, we have toassume complex E:

E E -i

2

G G = D E ( )

r rr , t ( )= y

rr , t ( )2

= y rr ( )2

e- l t (l ≡ G /h )

Line width

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Realisticpotential

Square well

ApproximationsApproximations

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• square-well potential• spherical symmetry• l=0

c ' ' +

2 M

h 2 ( E - V ) c = 0 c = j r( ) Radial Schroedinger equation

c I = A sin pr , p 2 = 2 ME

h 2Region I:

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…In almost all cases | c III| is much larger than | c I|. We are nowinterested in those situations where | c III| is as small as possible.

c + = 0 tan( pa ) = - p

q

defines “virtual” levels in region I:alpha particle is well localized; verysmall penetrability through the barrier

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c III 2

c I

2 =1

41 +

q 2

p2

Ê

ËÁÁ

ˆ

¯˜ c +

2eG + c-

2e- G( )+

1

21 -

q 2

p2

Ê

ËÁÁ

ˆ

¯˜c +c -

When c+=0, the penetrability becomes proportional to exp(-G), i.e.,

c III 2

c I

2 µ exp -2h

2 M (V b - E )(b - a )

ÈÎÍ

˘˚˙

This is the semi-classical WKB result: P =c III

2

c I

2 µ exp - 2 k ( r )drr1

r 2

Ú È

ÎÍÍ

˘

˚˙˙

In the case of the Coulomb

barrier, the above integral can beevaluated exactly. The decayconstant can be obtained by

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DataData ……

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Fine structure in alpha decayFine structure in alpha decay

238Pu

centrifugal barrier effectcentrifugal barrier effect

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Alpha DecayAlpha DecayParity violation in strong (alpha) decayParity violation in strong (alpha) decay

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elongationnecking

split

FissionFission

N,ZN,Z

NN22 ,Z,Z 22NN11 ,Z,Z 11

N=NN=N 11+N+N 22

Z=ZZ=Z 11+Z+Z 22

240 Pu

1938 - Hahn & Strassmann1939 - Meitner & Frisch1939 - Bohr & Wheeler1940 - Petrzhak & Flerov

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Global Properties of Atomic NucleiGlobal Properties of Atomic Nucleifission and fusionfission and fusion

¥ All elements heavier than A=110-120 are fission

unstable!¥

But É the fission process is unimportant fornuclei with A<230. Why?

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Neutron multiplicitiesNeutron multiplicities

9 2

2 3 5U + n "

3 6

9 2Kr +

5 6

1 4 2 Ba + 2 n

+180 MeV

235 U

238 U

Cross sectionsCross sections

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The classical droplet stays

stable and spherical for x<1.For x>1, it fissions immediately.For 238 U, x=0.8.

The classical droplet stays

stable and spherical for x<1.For x>1, it fissions immediately.For 238 U, x=0.8.

¢ ¡¤ £ ( ) =¥ ( ) ¦ ( ) - +

§ ( ) -( )[ ]¨

( )=

© ( )( ) ( )

=

( )( )

=

( )( )

=

( ) " !$ #

»

%

fissibilityparameterfissibility

parameter

Global Properties of Atomic NucleiGlobal Properties of Atomic Nucleispontaneous fissionspontaneous fission

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Adiabatic Approaches to Fission

q

E

multidimensional spaceof collective parameters

collective inertia(mass parameter)

V(q)

WKB:

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Static description (micro-macro)Möller et al., Nature 409, 785 (2001)

228

Ra

• 5D landscapes• Bimodal fission• “Flooding” algorithm• Saddle points• Outer barrier and

scission shapes

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A. Staszczak, J. DobaczewskiW. Nazarewicz, in preparation

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Kinetic Energy and Mass Distributionsin HFB+TDGCM(GOA)

one-dimensionaldynamicalWahl (experiment)

HFB + Gogny D1S + Time-Dependent GOAH. Goutte, P. Casoli, J.-F. Berger, D. Gogny, Phys. Rev. C71, 024316 (2005)

• Time-dependent microscopiccollective Schroedinger equation

• Two collective degrees of freedom• TKE and mass distributions

reproduced• Dynamical effects are responsible

for the large widths of the massdistributions

• No free parameters

238

U

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