Nuclear Phenomenology
3C24
Nuclear and Particle Physics
Tricia Vahle & Simon Dean
(based on Lecture Notes from Ruben Saakyan)
UCL
Nuclear Notation
• Z – atomic number = number of protons
N – neutron number = number of neutrons
A – mass number = number of nucleons
(Z+N)
• Nuclides AX (16O, 40Ca, 55Fe etc…)
– Nuclides with the same A – isobars
– Nuclides with the same Z – isotopes
– Nuclides with the same N – isotones
Masses and binding energies
• Something we know very well:
– Mp = 938.272 MeV/c2, Mn = 939.566 MeV/c2
• One might think that
– M(Z,A) = Z Mp + N Mn - not the case !!!
• In real life
– M(Z,A) < Z Mp + N Mn
• The mass deficit
 DM(Z,A) = M(Z,A) - Z Mp - N Mn
– –DMc2 – the binding energy B.
– B/A – the binding energy per nucleon, the minimum
energy required to remove a nucleon from the
nucleus
Binding energy
Binding energy per nucleon as function of A for stable nuclei
Nuclear Forces
• Existence of stable nuclei suggests
attractive force between nucleons
• But they do not collapse  there must be
a repulsive core at very short ranges
• From pp-scattering, the range of nucleonnucleon force is short which does not
correspond to the exchange of gluons
Nuclear Forces
d
+V0
V(r)
r=R
d<<R
Range~R
B/A ~ V0
0
r
• Charge symmetric pp=nn
• Almost charge –independent
pp=nn=pn
– mirror nuclei, e.g. 11B 11C
• Strongly spin-dependent
– Deutron exists: pn with spin-1
– pn with spin-0 does not
• Nuclear forces saturate
(B/A is not proportional to A)
-V0
Approximate description
of nuclear potential
Nuclei. Shapes and sizes.
• Scattering experiments to find out shapes and
sizes
• Rutherford cross-section:
Z 2 2 ( c) 2
 d 



2
4
d

4
E
sin
( / 2)

 Rutherford
• Taking into account spin: Mott cross-section
 d 
 d 
2
2


1


sin
( / 2)  ,





 d  Mott  d   Rutherford
v

c
Nuclei. Shapes and Sizes.
• Nucleus is not an elementary particle
• Spatial extension must be taken into account
• If f ( x ) – spatial charge distribution, then we define form
factor F (q 2 ) as the Fourier transform of f ( x )
1
iq x /
3
F (q ) 
e
f
(
x
)
d
x,

Ze
2
Ze   f ( x )d 3 x
 d 
 d 
2 2
F (q )

 

 d  exp t  d   Mott
F (q 2 ) can be extracted experimentally, then f ( x ) found from
inverse Fourier
transform
Ze
2
 iq x /
3
f (x) 
F ( q )e
d q
3 
(2 )
In practice d/d falls very rapidly with angle
Shapes and sizes
• Parameterised form is chosen for charge
distribution, form-factor is calculated from
Fourier transform
• A fit made to the data
• Resulting charge distributions can be fitted by
ch (r ) 

1 e
0
ch
( r c ) / a
c = 1.07A1/3 fm
a = 0.54 fm
• Charge density approximately constant in the
nuclear interior and falls rapidly to zero at the
nuclear surface
Radial charge distribution of
various nuclei
Shapes and sizes
• Mean square radius
r
2 1/ 2
~
2
1/ 3
r

(
r
)
dr

0.94
A
fm for medium and heavy nuclei
 ch
• Homogeneous charged sphere is a good
approximation
Rcharge = 1.21 A1/3 fm
• If instead of electrons we will use hadrons to
bombard nuclei, we can probe the nuclear density
of nuclei
nucl ≈ 0.17 nucleons/fm3
Rnuclear ≈ 1.2 A1/3 fm
Liquid drop model: semi-empirical
mass formula
• Semi-empirical formula: theoretical basis
combined with fits to experimental data
• Assumptions
– The interior mass densities are approximately equal
– Total binding energies approximately proportional their
masses
5
M ( Z , A)   f i ( Z , A)
i 0
Semi-empirical mass formula
• “0th“term
f 0 ( Z , A)  ZM p  ( A  Z ) M n
• 1st correction, volume term
f1 ( Z , A)  a1 A
• 2d correction, surface term
f 2 ( Z , A)   a2 A2 / 3
• 3d correction, Coulomb term
Z2
f3 ( Z , A)   a3 1/ 3
A
Semi-empirical mass formula
• 4th correction, asymmetry term
( Z  A / 2)2
f 4 ( Z , A)   a4
A
• Taking into account spins and Pauli principle gives 5th
correction, pairing term
f5 ( Z , A)   f ( A),
if Z even, A - Z  N even
f5 ( Z , A)  0,
if Z even, A - Z  N odd; or Z odd, A - Z  N even
f5 ( Z , A)   f ( A),
if Z odd, A - Z  N odd
f ( A)  a5 A1/ 2  by fitting the data
• Pairing term maximises the binding when both Z and N
are even
Semi-empirical mass formula
Constants
• Commonly used notation
a1 = av, a2 = as, a3 = ac, a4 =aa, a5 = ap
• The constants are obtained by fitting binding energy
data
• Numerical values
av = 15.67, as = 17.23, ac = 0.714, aa = 93.15, ap= 11.2
• All in MeV/c2
Nuclear stability
• n(p) unstable: -(+)
decay
• The maximum binding
energy is around Fe
and Ni
• Fission possible for
heavy nuclei
p-unstable
– One of decay product –
-particle (4He nucleus)
n-unstable
• Spontaneous fission
possible for very heavy
nuclei with Z  110
– Two daughters with
similar masses
-decay. Phenomenology
• Rearranging SEMF
M ( Z , A)   A   Z   Z 
2
d
A1/ 2
as
aa
  M n  av  1/ 3 
A
4
  aa  ( M n  M p  me )
aa
ac
   1/ 3
A A
d  ap
• Odd-mass and even-mass nuclei lie on different
parabolas
Odd-mass nuclei
1)
  : n  p  e  e
Mo  101Tc  e   e
101
M ( Z , A)  M ( Z  1, A)
2)   : p  n  e    e
101
Pd  101Rh  e    e
M ( Z , A)  M ( Z  1, A)  2me
3)
Electron capture
Even-mass nuclei
 emitters lifetimes vary
from ms to 1016 yrs
-decay
 -decay is energetically allowed if
B(2,4) > B(Z,A) – B(Z-2,A-4)
• Using SEMF and assuming that along stability
line Z = N
B(2,4) > B(Z,A) – B(Z-2,A-4) ≈ 4 dB/dA
28.3 ≈ 4(B/A – 7.7×10-3 A)
• Above A=151 -decay becomes energetically
possible
-decay
TUNELLING:
T = exp(-2G) G – Gamow factor
G≈2(Z-2)/ ~ Z/E
Small differences in E, strong effect
on lifetime
Lifetimes vary from 10ns to
1017 yrs (tunneling effect)
Spontaneous fission
• Two daughter nuclei are approximately
equal mass (A > 100)
• Example: 238U  145La + 90Br + 3n (156
MeV energy release)
• Spontaneous fission becomes dominant
only for very heavy elements A  270
• SEMF: if shape is not spherical it will
increase surface term and decrease
Coulomb term
Deformed nuclei
If nucleus deformed we can parameterise deformation by
R
b
a  R(1   )
1 
which preserves the volume
4
4
3
V   R   ab 2
3
3
To find new surface and Coulomb terms we have to
find expression for the surface of ellipsoid in in terms of
a and b and expand it in a power series in  . The result:
2 2
Es  as A (1    ...) and
5
2/3
Ec  ac Z A
2
1/ 3
1 2
(1    ...)
5
Spontaneous fission
• The change in total energy due to
deformation:
DE = (1/5)  2 (2as A2/3 – ac Z2 A-1/3)
• If DE < 0, the deformation is energetically
favourable and fission can occur
• This happens if Z2/A  2as/ac ≈ 48 which
happens for nuclei with Z > 114 and
A  270