Transcript Semi-Empirical Mass Formula Applications
Semi-Empirical Mass Formula Applications – II Nucleon Separation Energies and Fission [Sec. 4.4 + 12.1 Dunlap]
The Semi-Empirical Mass Formula M
Z
M (
A
,
Z
) (
A
Z
)
m n
Z
(
m p
m e
) 1
c
2
B
A
,
Z
B
(
A
,
Z
)
a V A
a S A
2 / 3
a C Z
2
A
1 / 3
a A
(
A
2
Z
) 2
A
a P
1
A
3 / 4 Let us see how this equation can be applied to (i) Neutron Separation Energies (ii) Alpha Particle Decay Energies (iii) Fission
Single neutron separation energies Fig 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 energies In an earlier lecture we found that the separation energy for a neutron was:
S n
M
(
A
1 ,
Z
)
m n
M
(
A
,
Z
)
c
2 This can be written in terms of mass of constituents and binding energies
S n
(
B
(
A A
1 1 ,
Z Z
) )
m n
B
(
Zm H A
,
Z
) 1
c
2
B
(
A
1 ,
Z
)
m n
(
A
Z
)
m n
B
Zm H
1
c
2
B
(
A
,
Z
)
c
2 to o-o
S n
Even
B
A
Odd
a P A
3 / 4 to e-e Apply the SEMF assuming B(A,Z) is continuous in A.
A 1
B
(
A
)
B
(
A
1 )
B
A
.
A
Single neutron separation energies
S n
B
A
a P A
3 / 4 Now apply the SEMF:
S n
A
a V A
a S A
2 / 3
a C
A
a V A
a S A
2 / 3
a C Z A
1 / 2 3
a A
(
A
2
Z
) 2
A
a P A
3 / 4
Z
2
A
1 / 3
a A A
4
a A Z
4
Z
2
A
a P A
3 / 4
a V
a V
2 3
a S A
1 / 3
a A
2 3
a S
1 3
a C A
1 / 3
Z
2
A
4 / 3
a A
4
a A Z
2
A
2
a P A
3 / 4
a C
3
A
4 / 3 4
a A A
2
Z
2
a P A
3 / 4 This is an interesting result because it can give us an equation for the “neutron drip” line by putting S n =0
Z
drip 1 3
a A a C
2
A
3 4
a S
/ 3
A
1 / 3 4
a A
A a
V
2
Mass Parabolas Proton number Z=N Neutron drip line Neutron number
Alpha Particle Decay Q We saw in a previous lecture that the Q-value (energy released) in -decay is:
Q
M
B
(
A
,
B
(
A
,
B Z
(
Z
) )
A
,
Z M
)
B
( (
A A B
( 4 ,
A
4 ,
Z Z
4 ,
Z
2 ) 2 )
m
2 )
B
c
2
B
28 .
B
A
.
4 3
MeV
4
B
Z
B A
.
2 2
B
Z
where
B
(
A
,
Z
)
a V A
a S A
2 / 3
a C Z
2
A
1 / 3
a A
(
A
2
Z
) 2
A
From which:
B
A
B
Z
a V
2
a C
2 3
a S A
2 / 3 1 3
a C Z A
1 / 3 4
a A
8
a A A Z
4 2
Z
/ 3
A
a A
4
a A Z
2
A
2
Q
28 .
3
MeV
4
a V
8 3
a S
1
A
1 / 3 4
a C Z A
1 / 3
Z
3
A
4
a A
2
Z A
2
Alpha Particle Decay Q
Energy released in Fission The diagram shows the Q (energy released) from the fission of 236 U 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 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.
Understanding the Fission Barrier Consider the stability of an Ellipsoidal Deformation, =eccentricity of ellipse How do B S and B C vary on deformation?
Understanding the Fission Barrier SURFACE ENERGY The surface area increases on deformation and so does B S. The nucleus becomes LESS bound
Surface tenstion
a S
The mass energy increases with deformation – This produces a potential that seeks to keep =0, I.E. the nucleus in SPHERICAL condition
Understanding the Fission Barrier COULOMB 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
a C Z
2
A
1 / 3 1 5 2
a S A
2 / 3 2 5 2
The fission barrier on the SEMF To 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 Z 2 /A as a function of A. Note that the fastest decaying man-made transuranics still have F<45
The rate of spontaneous fission NOTE log of the decay rate (period) is approximately proportional to the fissionability Z 2 /A