semi-empirical mass formula applications – ii nucleon separation energies and fission [sec. 4.4 +...
TRANSCRIPT
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
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
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.
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.
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