Lecture 5
Nuclear
Cross Sections
In astrophysics we are typically interested in a
reaction rate for two reactants, I and j as in e.g., I (j,k) L :
nI n j  I j v
which has units “reactions cm-3 s-1”
It is generally more convenient to write things
in terms of the mole fractions,
YI 
XI
AI
nI   N A YI
so that the rate becomes
(  N A )2 YI Y j  I j v
and a term in a rate equation might be
dYI
  YI Y j N A  I j v +....
dt
Here
denotes a suitable average over energies and angles
and the reactants are usually assumed to be in thermal equilibrium
We define a cross section for reaction in the usual way:

number reactions/nucleus /second
number incident particles/cm 2 / second
 clearly has units of area (cm 2 )
For a Maxwell-Boltzmann distribution of reactant energies the
average of the cross section times velocity is
 8 
 I jv   
  
1/ 2
 1 
 kT 
where  is the "reduced mass"
Mm
= I j
M I m j
for the reaction I (j, k) L
3/ 2 

0
I j
(E) E e E / kT dE
Center of mass system – that coordinate system in which the
total initial momenta of the reactants is zero.
The energy implied by the motion of the center of mass
is not available to cause reactions.
Replace mass by the "reduced mass"
M1M 2
=
M1  M 2
For T in 109 K,  in barns (1 barn = 10-24 cm2), E6 in MeV, and
k = 1/11.6045 MeV/GK

6.197 x 10-14
11.6045 E6 / T9
 jk v =

(
E
)
E
e
dE6
jk
6
6

1/2 3/ 2
ˆ
A T9
0
AIA j
ˆ
A=
AI  A j
for reaction I(j,k)L
If you know jk from the lab, or a calculation, in the center of mass
frame, just put it in and integrate.
The actual form of  may be very complicated and depends upon the
presence or absence of resonances and the reaction mechanism.
In general, however, it is of the form …
The Cross Section
h
1
D =
p
k
How much I+j looks like
the target nucleus with j
sitting at its surface. Liklihood
of staying inside R once you get
there.
Area subtended by a
Compton wavelength
in the c/m system
 (E) = 
2
geometry
term
 Pl (E)
penetration
factor
(E, A)
nuclear
structure
probability a flux of
particles with energy E
at infinity will reach the
nuclear surface. Must account
for charges and QM reflection.
see Clayton Chapter 4
where D is the Compton wavelenth in the c/m system
 h2  h2 0.656 barns
D  2 2 

 v 2 E Aö E(MeV)
2
where 1 barn = 10-24 cm 2 is large for a nuclear cross section.

M1M 2
M1  M 2
AA
Aˆ  1 2 ~ 1 for neutrons and protons
A1  A2
~ 4 for  -particles
Clayton p. 319ff shows that the solution to Schroedinger's
equation for two interacting particles in a radial potential is given by
 (r) m
(r,  , ) = l
Yl ( , )
r
where (r) satisfies
 -h2 d 2

l(l  1)h2

V (r)  E   l (r) 0

2
2
2

dr
2

r


potential
V (r)
Z I Z j e2
r
V (r) Vnuc
The barrier penetration function Pl is then given by
(for interacting particles with both charge and angular momentum)
2

()
Fl 2 (   )  Gl2 (   )
1
l
This is the solution for
Pl 


2
Fl 2 (,  )  Gl2 (,  )
Fl 2 (,  )  Gl2 (,  )
R<r <
 l ( R)
where F and G, the regular and irregular Coulomb functions
are the solutions of the differential equation
d 2w
2 L( L  1)

(1


)w  0
2
2
d


w  C1 Fl ( ,  )  C2 Gl ( ,  )
rR
rR
Angular momentum term
Classically, centrifugal force goes like
mv2 m 2 v2 r 2
L2
Fc =


3
R
mR
mR 3
One can associate a centrifugal potential with this,
L2
Fc R =
mR 2
Expressing things in the center of mass system and
taking the usual QM eigenvaluens for the operator L2
one has
l(l  1)
R 2
2
 Pl is the barrier penetration function to the nuclear
radius R with angular momentum l. In general,
 Pl 

Fl (,  )  G (,  )
2
2
l
where Fl is the regular Coulomb function
and Gl is the irregular Coulomb function
See Abramowitcz and Stegun, Handbook of Mathematical Functions, p. 537
These are functions of the dimensionless variables
=

Z I Z j e2
hv
 0.1575Z I Z j Aö / E
2 E
ö R
R  0.2187 AE
fm
2
h
Physical meaning of  
Z I Z j e2
v
The classical turning radius, r0 , is given by
2
1 2 ZI Z je
v =
2
r0
The Compton wavelength on the other hand is
=
Hence
p
=

r0
2
v
ro 
2Z I Z j e2
v 2

2
v
 2
nb., both  and  are dimensionless.
The probability of finding the particle inside of its classical turning
radius decreases exponentially with this ratio.
On the other hand,
2 E
R
=
R
2
D
h

h
h


p v
h
1

2   v2 
2

is just the size of the nucleus measured in Compton
wavelengths.
This enters in, even when the angular momentum and
charges are zero, because an abrupt change in potential
at the nuclear surface leads to reflection of the wave
function.
For low interaction energy, (2 >> ) and Z j  0,  Pl
has the interesting limit

2l(l  1) 
 Pl  2 exp  2  4 2 


2 
where

ö
2  0.2625 Z I Z j AR
fm
Abramowitz and Stegun,
14.6.7

1/ 2
which is independent of energy but depends on nuclear size.
Note:
rapid decrease with energy and charge
rapid decrease with increasing angular momentum
The leading order term for l = 0 is
 Pl  exp 2 
=

Z I Z j e2
hv
 0.1575Z I Z j Aö / E
2 E
ö R
R  0.2187 AE
fm
2
h
There exist other interesting limits for  Pl ,
for example when  is small - as for neutrons where it is 0
  E1/2
 P0  
3
 P1
1  2
  1 for cases of interest
fo neutron capture
5
 P2 
9  3 2   4
This implies that for l = 0 neutrons
the cross section will go as 1/v.
i.e., 
1/ 2
2
E
 P0 
 E 1/ 2
E
For low energy neutron induced reactions, the
cross section times velocity, i.e., the reaction rate
term, is approximately a constant
=

Z I Z j e2
hv
 0
2 E
ö R
R  0.2187 AE
fm
2
h
So providing X(A,E) does not vary rapidly with
energy (exception to come), i.e., the nucleus is "structureless"
e2
 (E) =  D  Pl X ( A, E) 
E
2
This motivates the definition of an "S-factor"
S(E)   (E) E exp(2 )
  0.1575Z I Z j Aö / E
Aö 
AI Aj
AI  Aj
This S-factor should vary slowly with energy. The first order
effects of the Coulomb barrier and Compton wavelength have been
factored out.
ZIZj
Again, for accurate calculations we would just enter the
energy variation of S(E) and do the integral numerically.
However, Clayton shows (p. 301 - 306) that
 E

exp 
 2  can be replaced to good accuracy by
 kT

  E  E 2 
0
C exp 
 , i.e. a Gaussian
  / 2 2 
where E o is the Gamow Energy
E o = 0.122


2 2 ˆ 2 1/3
Z I Z j AT9
MeV
and  is its full width at half maximum

4
1/2
ˆ 5
=
Eo kT  = 0.237 Z I2 Z 2j AT
9
3

1/6
MeV
f   2 e 

d
A

  4/3  
dT
3T
3T
A
T 1/ 3
df
d
d
 2 e 
  2 e 
dT
dT
dT
T
f
T

T

 df 

2 

(2

e
)(

)

(

e
)(

)
 
2 
2 
3T
 e
3T
 d   e
 d ln f    2


3
 d ln T 
 f  Tn
n
 2
3
For example, 12C + 12C at 8 x 108 K
1/ 3
 2 2 12  12 
 6 6 12  12 
  4.248 

0.8




 90.66
n
90.66  2
 20.5
3
p + p at 1.5 x 107 K
1 1 

1

1


11 
  4.248 

0.015




 13.67
1/ 3
n=
13.67 - 2
 3.89
3
Thus “non-resonant” reaction rates will have a
temperature dependence of the form

Constant
constant
exp(

)
2/3
1/ 3
T
T
 2 e 
This is all predicated upon S(Eo ) being constant, or at
least slowly varying. This will be the case provided:
i) E << ECoul
ii) All narrow resonances, if any, lie well outside the
Gamow “window”
Eo   / 2
That is there are no resonances or there are
very many overlapping resonances
iii) No competing reactions (e.g., (p,n), (p,a) vs (p,g))
open up in the Gamow window
In general, there are four categories of strong and electromagnetic
reactions determined by the properties of resonances through which
each proceeds
S(E) ~ const
•
S(E) ~ const
• Reactions that proceed through the tails of broad
S(E) highly
variable
S(E)~ const
Truly non-resonant reactions (direct capture and
the like)
distant resonances
• Reactions that proceed through one or a few
“narrow” resonances
• Reactions that have a very large number of
resonances in the “Gamow window”
Reaction Mechanisms
1) Direct Capture - an analogue of atomic radiative capture
The target nucleus and incident nucleon (or nucleus) react
without a sharing of energy among all the nucleons. An example
be the direct radiative capture of a neutron or proton and
the immediate ejection of one or more photons. The ejected photons
are strongly peaked along the trajectory of the incident projectile.
The reaction time is very short, ~ R/c ~10-21 s.
This sort of mechanism dominates at high energy (greater than
about 20 MeV, or when there are no strong resonances in or near the
Gamow window. It is especially important at low energies in light
nuclei where the density of resonances is very low.
The S-factor for direct capture is smooth and featureless.
Examples:
3
He( , )7 Be, 2 H(p, )3He, 3He(3He, 2p)4 He
12
C(n, )13C,
48
Ca(n, )49Ca
Treating the incoming
particle as a plane wave
distorted by the nuclear
potential results in the
“Distorted Wave Born
Approximation” often used
to calculate direct reactions.
Here the incoming particle
is represented as a plane wave
which goes directly to a standing
wave with orbital angular
momentum l in the final nucleus.
The process involves a a single matrix element and is thus
a single step process. Direct capture is analogous to
bremsstrahlung in atoms.
Direct capture provides a mechanism for reaction in
the absence of resonances. Usually DC cross sections are
much smaller than resonant cross sections on similar
nuclei - if a resonance is present.
2) Resonant Reaction:
A two step reaction in which a relatively long-lived excited
state of the “compound nucleus” is formed – the “resonance”.
This state decays statistically without any memory (other than energy
and quantum numbers) of how it was produced. The outgoing
particles are not peaked along the trajectory of the incident particle.
(This is called the “Bohr hypothesis” or the “hypothesis of
nuclear amnesia”). The presence of a resonance says that the
internal structure of the nucleus is important and that a “long-lived”
state is being formed.
Resonances may be broad or narrow. The width is given by the
(inverse of the ) lifetime of the state and the uncertainty principle.
E t
Generally states that can decay by emitting a neutron or proton will
be broad (if the proton has energy greater than the Coulomb barrier.
Resonances will be narrow if they can only decay by emitting a
photon or if the charged particle has energy << the Coulomb barrier..
I. Direct reactions (for example, direct radiative capture)
En
Sn
direct transition into bound states

A+n
B
II. Resonant reactions (for example, resonant capture)
Step 1: Coumpound nucleus formation
(in an unbound state)
G
En
Sn
A+n
B
Step 2: Compound nucleus decay
G

B
Step 1: Compound nucleus formation
(in an unbound state)
Sn
Step 2: Compound nucleus decay

G
En
I+n
K
Nucleus I + n
S (K)
Nucleus I + n
Not all reactions emit radiation and stay within the original compound nucleus.
One may temporarily form a highly excited state that decays by ejecting
e.g., a n, p, or alpha-particle. E.g., I(n,)K:
One or more resonances may be present in the Gamow energy
window, in which case their contributions are added, or there
may be a broad resonance just outside the Gamow energy
window, either above or below.
The S-factor will be smooth in this latter case. In the case
of one or a few narrow resonances it will definitely not be
smooth. In the case of many broad overlapping resonances,
it will be smooth again.
2.366
- 1.944
0.422 MeV
13
(422)  457
12
What is the energy range of astrophysical interest?
Say hydrogen burning at 2 x 107 K, or T9 = 0.020
12
C(p, )13N
1/ 3
12  1


EGamow  0.122  62 12
0.022 
12  1



 = 0.237  62 12
12  1

12  1
 0.0289 MeV = 28.9 keV
1/ 6
0.02
5



 0.0163 MeV = 16.3 keV
Note on the previous page, there is no data at energies this low.
As is generally the case, one must extrapolate the experimental
date to lower energies than are experimentally accessible. The
S-factor is useful for this.
Another Example:
RESONANT PLUS
Resonance contributions are on top of direct capture cross sections
This reaction might be of interest either in hot hydrogen burning
at 30 million K or in carbon burning at 800 million K. Consider the
latter.
24
Mg(p, )25 Al
 2 2 24  1
2
EGamow  0.122  12 1
0.8 
24  1


 2 2 24  1
 = 0.237  12 1
25  1

5
0.8


1/ 3
 0.543 MeV
1/ 6
 0.447 MeV
Now two resonances and direct capture contribute.
… and the corresponding S-factor
Not constant S-factor
for resonances
(log scale !!!!)
~ constant S-factor
for direct capture
Note varying widths !
If a reaction is dominated by narrow resonances, its
cross section will be given by the Breit-Wigner equation
(see page 347 Clayton, also probs. 3-7 and 3-103).
 jk ( E )   
2
G j Gk
 E  r 
2
 Gtot 2 / 4
2 J r 1

(2 J I  1)(2 J j  1)
The G’s are the partial widths (like a probability but with
dimensions of energy) for the resonance to break up into
various channels. These now contain the penetration factors.
The lifetime of a resonance is
r 
Gtot
Gtot   Gk
 6.582 1022 MeVsec
This cross section will be sharply peaked around r, with a width Gtot
The cross section contribution due to a single resonance is given by the
Breit-Wigner formula:
 (E)   D
2
Usual geometric factor

656 .6 1
barn
A E
Spin factor:

2J r 1
(2 J1  1)(2 J 2  1)
G1G 2
  
(E  Er )2  (G / 2)2
 G1
 G2
Partial width for decay of resonance
by emission of particle 1
= Rate for formation of Compund
nucleus state
Partial width for decay of resonance
by emission of particle 2
= Rate for decay of Compund nucleus
into the right exit channel
G Total width is in the denominator as
a large total width reduces the relative
probabilities for formation and decay into
specific channels.
G tot   G i
Rate of reaction through a narrow resonance
Narrow means:
G  E
In this case, the resonance energy must be “near” the relevant energy range
E to contribute to the stellar reaction rate.
Recall:

E

1
 v 
 ( E ) E e kT dE
3/ 2 
 (kT ) 0
8
G1 (E)G 2 (E)
 (E)   D 
(E  Er )2  (G(E) / 2)2
2
and
For a narrow resonance assume:
M.B. distribution ( E )  E e

E
kT
This term contains
most of the energy
dependence
constant over resonance
All widths GE
constant over resonance
2
constant over resonance
( E)  ( Er )
Gi ( E)  Gi ( Er )
Then one can carry out the integration analytically (Clayton 4-193) and finds:
For the contribution of a single narrow resonance to the stellar reaction rate:
11.605 E r [ MeV]
N A  v  1.5410 ( AT9 )
11
3 / 2
 [MeV]e
The rate is entirely determined by the “resonance strength”
T9
cm3
s mole

2J r 1
G1G2
 
(2 J1  1)(2 J T  1) G
Which in turn depends mainly on the total and partial widths of the resonance at
resonance energies.
Often
G  G1  G2
Then for
G1G2
 G1
G
GG
G2  G1 
 G  G1 
 1 2  G2
G
G1  G2 
 G  G2 

And reaction rate is determined by the smaller one of the widths !
Here
 =
G G
G  G
at low energy
 G
Again as example: (Herndl et al. PRC52(95)1078)
Direct
Sp=3.34 MeV
Res.
Resonance
strengths
As one goes up in
excitation energy many
more states and many
more reactions become
accessible.
As one goes to heavier nuclei and/or to higher excitation
energy in the nucleus, the number of excited states, and hence
the number of potential resonances increases exponentially.
The thermal energy of a non-relativistic, nearly degenerate
gas (i.e., the nucleus) has a leading term that goes as T2 where
T is the “nuclear temperature. The energy of a degenerate gas
from an expansion of Fermi integrals is:
E = f( ) + a(kT)2  b (kT)4 + ....
here  is the
density
One definition of temperature is
1
 ln
=
kT
E
where  is the number of states (i.e., the partition function)
 ln   ln  E

T
E T
d ln  ~
1  E 
1
2
dT
~
2ak
T


kT T
kT

 dT
ln  ~ 2ak  dT = 2akT + const
 ~ C exp 2akT
and if we identify the excitation energy, E x  a(kT)2 ,
i.e., the first order thermal correction to the internal energy, then
E
The number of excited states
kT2 ~ x
(resonances) per unit excitation
a
energy increases exponentialy

 = C exp 2 aE x

with excitation energy.
Empirically a  A/9. There are corrections to a for shell
and pairing effects. In one model (back-shifted Fermi gas)
0.482
C = 5/6 3/2
A Ex
Take N (>>1) equally spaced identical resonances in an energy interval E
Generate an energy averaged cross section
D
E E
E
 

 ( E ) dE
E
E
D << E

G j G k dE
E E
1

E

E
G j Gk
E
( E   r )2  G r 2 / 4

N
dE
 ( E   r )2  G r 2 / 4


dE
2

 ( E   r )2  Gr 2 / 4 Gr
  2 2 2
G j Gk
Gr D
where T j  2

Gj
D

2
N
1

E D
T jTk
Ttot
Transmission Functions:
Tj Is a dimensionless number between 0 and 1 that gives the
probability that a particle of type j will cross the nuclear surface
(with angular momentum l)
Tj is related to the “strength function”, < G/D>, which
describes how width is distributed smoothly in the nucleus.
These functions still contain the barrier penetration functions.
This gives the Hauser-Feshbach formula for estimating
cross sections where the density of resonances is high.
 jk (E) =
 D2
2J
2J I  12J j  1 all
Jr
r
 1
T jl (J  , E)Tkl (J  , E)
Ttot (J  , E)
Expressions for the transmission functions for n, p,  and 
are given in Woosley et al, ADNDT, 22, 378, (1978). See also
the appendix here.
This formula has been used to generate thousands of cross sections
for nuclei with A greater than about 24. The general requirement
is many ( > 10) resonances in the Gamow window.
This equation assumes that, for the reaction I(j,k)L, and
for I having a given spin and parity, resonances of all spin
and parity exist within the compound nucleus for all
possible l of interaction.
Target
½+
l
0
1
2
3
Possible states
1/2+
1/2-,3/23/2+, 5/2+
5/2-, 7/2-
For j-projectile = 0+.
For other incident particles,
their spin must be included
in the balance.
etc.
The sum is over J in the compound nucleus – the right hand column.
If the state can be made by more than one value of l, then the T’s
are summed over l for each J.
More levels to make
transitions to at higher
Q and also, more
phase space for the
outgoing photon.
Q2
Q1
E3 for electric dipole
T (Q 2 )  T (Q1 )
and as a result
 n 
Tn T
Tn + T
is larger if Q is larger
 T
The Q-value for capture on nuclei that are tightly
bound (e.g., even-even nuclei, closed shell nuclei)
than for nuclei that are less tightly bound (e.g., odd A
nuclei, odd-odd nuclei).
As a result, nuclear stability translates into smaller
cross sections for destruction - most obviously for
nuclei made by neutron capture, but also to some
extent for charged particle capture as well.
This is perhaps the chief reason that tightly bound
nuclei are more abundant in nature than their less
abundant neighbors (not true for the iron group
or in general for why abundances above iron are rare).
Summary of reaction mechanisms
I(j,k)L
Summary of reaction mechanisms
I(j,k)L
• Add to Q-value and look inside nucleus I+j
• Any resonances nearby or in window
No
Yes
Right spin and parity?
No
Tail of
Broad
Direct
Capture
Extrapolate
S-factor
Yes
A few
Narrow
BreitWigner
Many
Overlapping
HauserFeshbach
Special Complications in Astrophysics
• Low energy = small cross section – experiments are hard.
• Very many nuclei to deal with (our networks often include
1600 nuclei; more if one includes the r-process)
•
The targets are often radioactive and short lived so that
the cross sections cannot be measured in the laboratory
(56Ni, 44Ti, 26Al, etc)
• Sometimes even the basic nuclear properties are not know
- binding energy, lifetime. E.g., the r-process and the rpprocess which transpire near the neutron and protondrip lines respectively.
• Unknown resonances in many situations
• Target in excited state effects – in the laboratory the
target is always in its ground state. In a star, it may not be
• Electron screening
Nuclei are always completely ionized – or almost
completely ionized at temperature in stars where
nuclear fusion occurs. But the density may be
sufficiently high that two fusing nuclei do not
experience each others full Coulomb repulsion.
This is particularly significant in Type Ia supernova
ignition.
Appendix:
Barrier Penetration
and Transmission Functions
Reflection at a Potential Change
For simplicity consider the case where the incident particle has no
charge, i.e., a neutron
In QM there exists reflection
whether V increases or
decreases
Energy
incident
E
reflected
0
x<0
E+Vo
Perfectly
absorbing –
what gets in
stays in
-Vo
p2
E=
2
2 E
V(x)
x>0
Wave number for incident particles
inside well
k
K
2 E
x0
2 ( E  Vo )

2Vo

p

2

1
 k
 ( x)  Aeikx  Be ikx
 CeiKx
x0
Incident wave plus reflected wave
x0
Wave traveling to the right
 ( x),  ( x) continuous implies at x=0, A+B=C
ikA  ikB  iKC
K
B
k

A 1 K
k
1

2
T 1 
B

A
(1 
K 2
K
)  (1  ) 2
k
k  4 K / k  4 Kk
K 2
K 2 (k  K ) 2
(1  )
(1  )
k
k
The fraction that “penetrates”
to the region with the new
potential.
and if E  Vo
T
4k 4 kR 4


 4 S  P0
K  KR  KR
recall  P0    kR
1
where S 
is the "black nucleus strength function"
 KR
f corrects empiricaly for the fact that the nucleus is
not purely absorptive at radius R
Though for simplicity we took the case
l = 0 and Z = 0 here, the result can be generalized
to reactants with charge and angular momentum
For Z= 0
 P0  
l 0
3
 P1
1  2
l 1
5
 P2 
9  3 2   4
l2
For Z > 0


ZI Z j e
2
v
2 E
2
 0.1575 Z I Z j
Aˆ
E ( MeV )
ˆ R ( fm)
R0  0.2187 AE
0
Pl 

Fl ( ,  )  Gl 2 ( ,  )
2
It is customary to define the transmission function for particles
(not photons) as
T  4 S f (  Pl )
where S is the strength function defined as
S
3
2
 j2
 R2 D
which is a constant provided that  j2  D, the level spacing.
This is consistent with the definition
T  2
G
D
Here “f” is the “reflection factor”, empirically 2.7 for n and p
and 4.8 for alpha-particles, which accounts for the fact that the
reflection is less when the potential does not have infinitely
sharp edges at R. Hence the transmission is increased.
The transmission functions are parameterized in terms of
the radius of an “equivalent square well” and its potential.
For nuclei A < 65
R = 1.25 A1/3 + 0.1 fm
1.09 A1/3 + 2.3 fm
for n,p
for alpha particles
1
S
 KR
2 Vo
K
2
Vo  60 MeV
This is expounded more in the next two pages.
Analogously the photon transmission function is defined as:
T  2
G
D
Phase space
 Strength function * phase space factor
E 3 for dipole radiation
E 5 for quadrupole radiation
The strength function is usually taken to be a constant
or else given a ``Giant Dipole” (Lorentzian) form.
The transmission functions to the ground state and each excited
state are calculated separately and added together to get a total photon
transmission function.
Typically G ~ eV – larger for large E in the transition; smaller if
a large J is required or E is small.
For nucleons and alpha particles it can be shown (Clayton 330 – 333)
that
3 2  2
125.41 MeV 2
Gj  
 j  Pl 
 j  Pl
2 
2
ˆ
AR ( fm)
 R 
l
where j2 is the “dimensionless reduced width” which must be
evaluated experimentally, but is between 0 and 1 (typically 0.1).
The resulting widths are obviously very energy sensitive (via Pl)
but for neutrons and protons not too much less than the Coulomb
energy, they are typically keV to MeV.
The decay rate of the state is qualitatively given by (Clayton p 331)
aside:
  probability/sec for particle from decaying system to cross large
spherical shell
1
 =  velocity at infinity * penetration factor * probability per unit dr

that the particle is at the nuclear
radius  dr
=
G
= v  Pl
3 2
3 v
3
2
2
 
R
P



P

l
R
 R2
 R2 l
3 4 R 2 dr
where

is the probability per unit radius
3
R 4 / 3 R
for finding the nucleon if the density is constant
 2  dimensionless constant < 1
 = kR 
v
R
2 E
2
R
d (volume)
volume