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Nomenclature
• Consider the reaction4He + 14N 17O + 1H
• Can write this asProjectile P + Target T Residual Nucleus
R and Emitted Particle x• Or
T(P,x)R 14N(4He,p)17O
Conservation Laws
• Conservation of neutrons, protons and nucleons
59Co(p,n)?Protons 27 + 1 =0 + xNeutrons 32 +0=1 + y
Product is 59Ni
Conservation Laws (cont.)
• Conservation of energyConsider 12C(4He,2H)14N
Q=(masses of reactants)-(masses of products)
Q=M(12C)+M(4He)-M(2H)-M14N)Q=+ exothermicQ=- endothermic
Conservation Laws(cont.)
• Conservation of momentummv=(m+M)V
TR=Ti*m/(m+M)
• Suppose we want to observe a reaction where Q=-.
-Q=T-TR=T*M/(M+m)
T=-Q*(M+m)/M
Center of Mass System
pi=pT
ptot=0
velocity of cm =Vcm
velocity of incident particle = v-Vcm
velocity of target nucleus =Vcm
m(v-Vcm)=MV
m(v-Vcm)-MV=0
Vcm=mv/(m+M)
T’=(1/2)m(v-Vcm)2+(1/2)MV2
T’=Ti*m/(m+M)
Kinematics
€
Q = Tx 1+mx
mR
⎛
⎝ ⎜
⎞
⎠ ⎟− TP 1−
mp
mR
⎛
⎝ ⎜
⎞
⎠ ⎟−
2
mR
mPTPmxTx( )1/ 2
cosθ
€
Tx1/ 2 =
mPmxTP( )1/ 2
cosθ ±{mPmxTP cos2θ + mR +mx( )[mRQ +(mR − mP )TP ]}1/ 2
mR +mx
Nuclear Reaction Cross Sections
fraction of beam particles that react=fraction of A covered by nuclei
a(area covered by nuclei)=n(atoms/cm3)*x(cm)*((effective area of one nucleus, cm2))
fraction=a/A=nx-d=nx
trans=initiale-nx
initial- trans= initial(1-e-nx)
Factoids about
• Units of are area (cm2).• Unit of area=10-24 cm2= 1 barn• Many reactions may occur, thus we
divide into partial cross sections for a given process, with no implication with respect to area.
• Total cross section is sum of partial cross sections.
Charged Particles vs Neutrons
In reactors, particles traveling in all directionsNumber of reactions/s=Number of target atomsx x (particles/cm2/s)
I=particles/s
What if the product is radioactive?
€
dN
dt= (rate of production) − (rate of decay)
dN
dt= nσΔxφ − λN
dN
nσΔxφ − λN= dt
d(λN)
λN − nσΔxφ= −λdt
ln(λN − nσΔxφ) |0N = −λ t |0
t
λN − nσΔxφ
−nσΔxφ= e−λ t
A = λN = nσΔxφ(1− e−λ t )
Neutron Cross Sections-General Considerations
≈ ( + R r')2 = r02(AP + AT)2
€
l = r x p = pb
€
p =h
D
€
lh=hbD
€
b = lD
€
l =π l +1( )2D2 − πl 2D2
€
l =πD2(l 2 + 2l +1− l 2 )
€
l =πD2(2l +1)
€
total = σ l = πD2(2l +1) = πD2 (2l +1) = πD2(l max +1)2
l =0
l max
∑l =0
l max
∑l
∑
€
l max
€
lmax =R
D
l max +1=R + D
D
σ total = π (R + D)2
Semi-classicalQM
€
total = πD2 (2l +1)Tl
l = 0
∞
∑
The transmission coefficient
€
Tl
• Sharp cutoff model (higher energy neutrons)
€
Tl =1 for l ≤ l max
Tl = 0 for l > l max
• Low energy neutrons
€
total ∝ πD2 ε ∝ πh2
2mεε ∝
1
ε
Charged Particle Cross Sections-General Considerations
B = Z1Z2e2/R
p=(2mT)1/2 = (2)1/2(-B)1/2 = (2)1/2(1-B/)1/2
reduced mass =A1A2/(A1+A2)
€
l = rxp
l max = R 2με( )1/ 2
1−B
ε
⎛
⎝ ⎜
⎞
⎠ ⎟
1/ 2
Semi-classicalQM
€
l → lh
€
total = πD2 l max +1( )2
≈ πD2l max2 = πD2R2 2με
h21−
B
ε
⎛
⎝ ⎜
⎞
⎠ ⎟= πD2R2 1
D21−
B
ε
⎛
⎝ ⎜
⎞
⎠ ⎟
€
total = πR2 1−B
ε
⎛
⎝ ⎜
⎞
⎠ ⎟
Barriers for charged particle induced reactions
Vtot(R)=VC(R)+Vnucl(R)+Vcent(R)
VC(r) = Z1Z2/r for r < RC
VC(r) = (Z1Z2/RC) (3/2 – (r 2/RC
2)) for r > RC
Vnucl(r) = V0/(1 + exp((r-R/a)))
€
Vcent(r ) =h2
2μ
l (l +1)
r 2
€
1
2mv2 =
1
2mv0
2 +Z1Z2e
2
d
€
v0
v
⎛
⎝ ⎜
⎞
⎠ ⎟2
= 1−d0
d
€
d0 =2Z1Z2e
2
mv2=Z1Z2e
2
Tp
mvb = mv0d )(02202 ddddvvb −=⎟⎠⎞⎜⎝⎛=
d = b cot( /2)
tan = 2b/d0
022cotdb=⎟⎠⎞⎜⎝⎛θ
Consider I0 particles/unit area incident on a plane normal to the beam.Flux of particles passing through a ring of width dbbetween b and b+db is
dI = (Flux/unit area)(area of ring)dI = I0 (2πb db)
€
dI =1
4πI0d0
2
cosθ
2
⎛
⎝ ⎜
⎞
⎠ ⎟
sin3 θ
2
⎛
⎝ ⎜
⎞
⎠ ⎟dθ
€
dσ
dΩ=dI
I0
1
dΩ=
d0
4
⎛
⎝ ⎜
⎞
⎠ ⎟2
1
sin4 θ
2
⎛
⎝ ⎜
⎞
⎠ ⎟=
Z1Z2e2
4Tpcm
⎛
⎝ ⎜
⎞
⎠ ⎟
2
1
sin4 θ
2
⎛
⎝ ⎜
⎞
⎠ ⎟
Elastic (Q=0) and inelastic(Q<0) scattering
• To represent elastic and inelastic scattering, need to represent the nuclear potential as having a real part and an imaginary part.
This is called the optical model
Direct Reactions
• Direct reactions are reactions in which one of the participants in the initial two body interaction leaves the nucleus without interacting with another particle.
• Classes of direct reactions include stripping and pickup reactions.
• Examples include (d,p), (p,d), etc.