Lesson 12

Fission

Importance of Fission

• Technological importance (reactors, bombs)

• Socio-political importance• Role of chemists• Very difficult problem

Overview of fission

Probability of Fission

• Divide study of fission into two parts, the gssaddle point (probability of fission) and the saddlescission point (distribution of fission products)

• Use liquid drop model to study gssaddle point

Liquid drop model

€

R(ϑ ) = R0(1+α2P2(cosϑ ))

€

Es = Es0(1+

2

5α2

2 )

Ec = Ec0(1−

1

5α2

2 )

€

Ec0

2Es0 = 1

fissionability parameter.

€

x =Ec

0

2Es0 =

1

2

Coulomb energy of a charged sphere

Surface energy of the sphere

⎛

⎝ ⎜

⎞

⎠ ⎟

€

Ec0 =

3

5

Z 2e2

R0A1/ 3

= ac

Z 2

A1/ 3

⎛

⎝ ⎜

⎞

⎠ ⎟

€

Es0 = 4πR0

2SA2 / 3 = asA2 / 3

€

x =ac

2as

⎛

⎝ ⎜

⎞

⎠ ⎟Z 2

A

⎛

⎝ ⎜

⎞

⎠ ⎟=

Z 2

A

⎛

⎝ ⎜

⎞

⎠ ⎟/

Z 2

A

⎛

⎝ ⎜

⎞

⎠ ⎟critical

€

Z 2 / A( )critical

= 50.883 1−1.7826( N − Z )

A

⎛

⎝ ⎜

⎞

⎠ ⎟

2 ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

Limits on the Periodic Table

• Notice that x (ac/2as)

• Zlimit=2(as/ac)Alimit

• Zlimit ~ 125

• For all stable nuclei, x < 1• As nucleus deforms, pot. energy increases by

€

1

5α 2

2(2E so − Ec

0)

•Eventually Coulomb energy will cause deformation energyto decrease, ie, get fission barrier.

Fission Barriers

Shell Effects

€

E = ELDM + (δS +δP)p, n

∑

Consequences of Double Humped Fission Barriers

• Spontaneously fissioning isomers• Superdeformed nuclei• Subthreshold resonances

Spontaneous fission

€

t1/ 2SF =

ln(2)

fP

€

P = (1+ exp[2π (Bf )/ hω ])−1

€

t1/ 2SF = 2.77x10−21 exp[2πBf / hω ]

Understanding spontaneous fission

lifetimes

Spontaneously fissioning isomers

Spontaneously fissioning isomers

• Spntaneously fissioning isomers are nuclei caught in states in the second minimum of the fission potential energy surface. Their sf decay is enhanced relative to gs sf.

• Lifetimes are 10-9 - 10-3s• Typically c/a =2:1

Sub-threshold fission resonances

“Normal fission”--the fission transition state nucleus

• Has the same role as the transition state in chemical reactions.

• prob. of fission =Aexp(-Bf/T)

• Bn > Bf (235U); Bn < Bf (238U)

• Big Three (233U, 235U, 239Pu)

Fission probability

• Fission probablity (Nf/(Nf+Nn+Ngamma+Nch.p.))

Multiple chance fission

n/f

n/f

€

n

Γf

=gμr0

2

h2

4A2 / 3af (E * −Bn )

an{2af1/ 2(E * −Bf )1/ 2 −1}

exp[2an1/ 2(E * −Bf )1/ 2 − 2af

1/ 2(E * −Bf )1/ 2 ]

€

n

Γf

=2TA 2 / 3

10exp[(Bf − Bn )/T ]

Fission Product Distributions

TKE Distribution

€

TKE =Z1Z2e

2

1.8(A11/ 3 + A2

1/ 3 )MeV

Fission Mass Distributions

Fission Mass Distributions

Fission Mass Distributions

Fission Product Charge Distributions

€

P(Z ) =1

cπexp[−(Z − Zp )2 / c]

Energetics of Fission

• Q value ~ 200 MeV• TKE ~172 MeV• Neutrons ~18 MeV• Gammas ~ 7.5 MeV, etc ~2.5 MeV

Prompt Neutrons

Prompt Neutrons

Prompt Neutron Spectra

• Average neutron energy ~ 2 MeV• Spectrum:frame of moving fragment; Maxwellian

P (E)=Enexp (-En/T)

lab frame; Watt spectrum

€

P(En ) = e−En / T sinh(4EnEf /T 2 )1/ 2

Fission Fragment Angular Distributions

Fission Fragment Angular Distributions

Fission Fragment Angular Distributions

€

PM, KJ (θ ) = [(2J +1)][

2πR2 sinθdθ

4πR2] dM, k

J (θ )2

(K) ∝ exp (-K2/K20) K<J

=0 K > J

€

K02 =

ℑ effT

h2

€

W (θ )∝ (2J +1)TJ

(2J +1) dM =0, KJ (θ )

2exp(−K / 2K0

2 )

exp(−K 2 / 2K02 )

K =−J

J

∑K =−J

J

∑J =0

∞

∑

€

W (θ )∝(2J +1)2TJ exp[−(J + 0.5)2 sin2θ / 4K0

2 ]J0[ i(J + 0.5)2 sin2θ / 4K02 ]

erf [(J + 0.5)/(2K02 )1/ 2 ]J =0

∞

∑

€

erf (x) = (2/ π 1/ 2 ) exp(− t2 )dt0

x

∫