Semi-Empirical Mass Formula

Applications – II

Nucleon Separation Energies and

Fission

[Sec. 4.4 + 12.1 Dunlap]

The Semi-Empirical Mass Formula

4/3

2

3/1

23/2 1)2(

),(A

aA

ZAa

A

ZaAaAaZAB PACSV

ZABc

mmZmZAZAX epnAZ ,

1)()(),(MM

2

Let us see how this equation can be applied to

(i) Neutron Separation Energies

(ii) Alpha Particle Decay Energies

(iii) Fission

Single neutron separation energiesFig 4.8 PULLING NEUTRONS OUT OF ODD-A NUCLIDES

The arrows show the transitions from the odd A parabola to the even (A-1) parabolas for the two cases of

(e,o)(o,o) breaking pairing on neutron side

(o,e)(e,e) breaking no pairing bond

Single neutron separation energiesIn an earlier lecture we found that the separation energy for a neutron was:

2),(),1( cZAMmZAMS nn

This can be written in terms of mass of constituents and binding energies

),(),1(

),(1

)(),1(1

)1( 222

ZABZAB

cZABc

ZmmZAmZABc

ZmmZAS HnnHnn

OddEven

Apply the SEMF assuming B(A,Z) is continuous in A.

A

B

1

4/3A

a

A

BS Pn

AA

BABAB

.)1()(

to o-o

to e-e

Single neutron separation energies

4/3A

a

A

BS Pn

Now apply the SEMF:

4/32

23/43/1

4/32

2

3/4

23/1

4/3

2

3/1

23/2

4/3

2

3/1

23/2

4

33

2

43

1

3

2

44

)2(

A

aZ

A

a

A

aAaaa

A

a

A

Zaa

A

ZaAaa

A

a

A

ZZaAa

A

ZaAaAa

A

A

a

A

ZAa

A

ZaAaAa

AS

PACSAV

PAACSV

PAACSV

PACSVn

This is an interesting result because it can give us an equation for the “neutron drip” line

23/431

3/132

drip 4

AaAa

aAaaZ

AC

VSA

by putting Sn=0

Mass Parabolas

Neutron number

Proton number Z=N

Neutron drip line

Alpha Particle Decay QWe saw in a previous lecture that the Q-value (energy released) in -decay is:

Z

B

A

BMeV

Z

B

A

BB

ZABZABB

BZABZAB

cmZAMZAMQ

243.28

2.4.

)2,4(),(

)2,4(),(-

)2,4(),( 2

A

ZAa

A

ZaAaAaZAB ACSV

2

3/1

23/2 )2(

),(

where

From which:

A

Zaa

A

Za

Z

BA

Zaa

A

ZaAaa

A

B

AAC

AACSV

842

43

1

3

2

3/1

2

2

3/4

23/2

2

3/13/1214

314

1

3

843.28

A

Za

A

Z

A

Za

AaaMeVQ ACSV

Alpha Particle Decay Q

Energy released in Fission

The diagram shows the Q (energy released) from the fission of 236U as a function of the A of one of the fragments (as obtained from the SEMF). Note that maximum energy release is 210MeV/Fission for the nucleus splitting into equal fragments.

Energy released in Fission

This figure shows the prediction of the SEMF for the energy released in FISSION when two equal fragments are formed.

Energy released in Fission

The Fission Barrier

The origin of the fission barrier can be seen by reversing the fission process. Two fission fragments approach with (1/r) potential – consider the fragments equal. When r decreases until the two fragments are nearly touching the nuclear attractive strong force takes over – the potential energy is less than that calculated by Coulomb law.

Fission barrier

Understanding the Fission BarrierConsider the stability of an Ellipsoidal Deformation, =eccentricity of ellipse

How do BS and BC vary on deformation?

Understanding the Fission BarrierSURFACE ENERGY The surface area increases on deformation and

so does BS. The nucleus becomes LESS bound

The mass energy increases with deformation – This produces a potential that seeks to keep =0, I.E. the nucleus in SPHERICAL condition

Surface tenstion

Sa

Understanding the Fission BarrierCOULOMB ENERGY The Coulomb energy has the opposite tendency. On

deformation the charge in the nucleus is less condensed – the electrostatic “blow apart” energy is less

Nuclear deformation makes the nucleus MORE BOUND.

Understanding the Fission Barrier

23/22

3/1

2

5

2

5

1 AaA

Za SC

The fission barrier on the SEMFTo calculate the height of the fission barrier using the SEMF is fairly complex, but can be done as seen in this study – Fig12.3 Dunlap.

The dotted lines show variations that are understood on the shell model.

Note that the barrier is only small ~3MeV for A>250.

The Fissionability

The Fissionability parameter Z2/A as a function of A. Note that the fastest decaying man-made transuranics still have F<45

The rate of spontaneous fissionNOTE log of the decay rate (period) is approximately proportional to the fissionability Z2/A