Basic Concepts in
Nuclear Physics
Paolo Finelli
Literature/Bibliography
Some useful texts are available at the Library:
Wong,
Nuclear Physics
Krane,
Introductory Nuclear Physics
Basdevant, Rich and Spiro,
Fundamentals in Nuclear Physics
Bertulani,
Nuclear Physics in a Nutshell
Introduction
Purpose of these introductory notes is recollecting few basic notions of
Nuclear Physics. For more details, the reader is referred to the literature.
Binding energy and Liquid Drop Model
Nuclear dimensions
Saturation of nuclear forces
Fermi gas
Shell model
Isospin
Several arguments will not be covered but, of course, are extremely
important: pairing, deformations, single and collective excitations,
α decay, β decay, γ decay, fusion process, fission process,...
The Nuclear Landscape
The scope of nuclear physics is
Improve the knowledge of all nuclei
Understand the stellar nucleosynthesis
© Basdevant, Rich and Spiro
Stellar Nucleosynthesis
The evolution of the nuclear abundances. Each square is a nucleus. The colors indicate the
abundance of the nucleus:
≥ e−4
e−5
e−6
e−7
Dynamical r-process calculation assuming an
expansion with an initial density of 0.029e4 g/cm3, an
initial temperature of 1.5 GK and an expansion
timescale of 0.83 s.
The r-process is responsible for the
origin of about half of the elements
heavier than iron that are found in
nature, including elements such as
gold or uranium. Shown is the result of
a model calculation for this process
that might occur in a supernova
explosion. Iron is bombarded with a
huge flux of neutrons and a sequence
of neutron captures and beta decays is
then creating heavy elements.
©JINA
Binding energy
Atomic Mass
mN c2 = mA c2 − Zme c2 +
Electrons Mass (~Z)
Z
�
i=1
Electrons Binding Energies
(negligible)
Bi � mA c2 − Zme c2
© Basdevant, Rich and Spiro
B = (Zmp + N mn ) c2 − mN c2 � [Zmp + N mn − (mA − Zme )] c2
�
� 2
1
A
B = Zm( H) + N mn − m( X) c
E/A (Binding Energy per nucleon)
Binding energy
Fe
Nuclear Fission Energy
The most bound
isotopes
Nuclear
Fusion
Energy
Average mass of fission
fragments is 118
235U
© Gianluca Usai
A (Mass Number)
Binding energy and Liquid Drop Model
Volume term, proportional to R3 (or A): saturation
Surface term, proportional to R2 (or A2/3)
Coulomb term, proportional to Z2/A1/3
Asymmetry term, neutron-rich nuclei are favored
Pairing term, nucleon pairs
coupled to JΠ=0+ are favored
© Basdevant, Rich and Spiro
Binding energy and Liquid Drop Model
Comparison with empirical data
Contributions to B/A as function of A
© Gianluca Usai
Nuclear Dimensions
Excited States (~eV)
Ground state
Excited States (~ MeV)
Ground state
Excited States (~ GeV)
Ground state
© Gianluca Usai
Nuclear Dimensions: energy scales
Nuclear Dimensions
Fermi distribution
ρ(0)
ρ(r) =
1 + e(r−R)/s
© Basdevant, Rich and Spiro
R : 1/2 density radius
s : skin thickness
Nuclear forces saturation
An old (but still good) definition:
© E. Fermi, Nuclear Physics
Mean potential method: Fermi gas model
In this model, nuclei are considered to be composed of two fermion gases,
a neutron gas and a proton gas. The particles do not interact, but they are
confined in a sphere which has the dimension of the nucleus. The
interaction appear implicitly through the assumption that the nucleons are
confined in the sphere. If the liquid drop model is based on the saturation
of nuclear forces, on the other hand the Fermi model is based on the
quantum statistics effects.
The Fermi model could provide a way to calculate
the basic constants in the Bethe-Weizsäcker formula
Fermi gas model (I)
Hamiltonian
H=
A
�
i=1
Ti ≡
A �
�
i=1
�
∇2i
2M
2
−
�
Hψ(�r1 , �r2 , . . .) = Eψ(�r1 , �r2 , . . .)
Wavefunction factorization
ψ(�r1 , �r2 , . . .) = φ1 (�r1 )φ2 (�r2 ) . . .
�2 2
−
∇i φ(�ri ) = Eφ(�ri )
2M
E = E 1 + E2 + E3 + . . . =
A
�
Ei
i=1
2M Ei
2
2
2
2
Boundary conditions �
ki ≡ (kix + kiy + kiz ) =
>0
2
�
Gasiorowicz, p.58
φi (�r) ≡ φi (x, y, z) = N sin(kix x) sin(kiy y) sin(kiz z)
Separable equations
d2 φi (x)
2
=
−k
ix φi (x)
2
dx
Fermi gas model (II)
φi (x) = B sin(kix x)
Solution
Normalization
1=
�
L
dx|φi (x)|2 = B 2
0
�
L
dx sin2 (kix x) = B 2
0
L
2
B=
� �3/2
2
φi (�r) =
sin(kix x) sin(kiy y) sin(kiz z)
L
kix
π
π
π
= n1i , kiy = n2i , kiz = n3i
L
L
L
(n1i , n2i , n3i = positive integers)
�2�ki2
�2 2
2
2
Ei =
=
(kix + kiy
+ kiz
)
2M
2M
�2 π 2 2
2
2
Ei (n1i , n2i , n3i ) =
(n
+
n
+
n
1i
2i
3i )
2
2M L
�
2
L
Fermi gas model (III)
∆kx,y,z
Density of states
π
π
= (n1,2,3 + 1 − n1,2,3 ) =
L
L
1
1
Ω �
2
dn(k) = 4πk dk
≡
dk
3
3
8
(π/L)
(2π)
n(k̄) =
�
k̄
0
Ω ≡ L3
Ω 4π 3
dn(k) =
k̄
3
(2π) 3
spin-isospin
Number
of particles
Density
of particles
A=4
�
kF
0
Ω
4π 3
2kF3
dn(k) =
4 k =Ω 2
3
(2π) 3
3π
Fermi momentum
2kF3
ρ0 =
3π 2
ρ0 = A/Ω
Fermi gas model (IV)
Fermi gas distribution:
N(k)
Step function
θ(kF − k)
1
filled
0
empty
kF
k
Fermi gas model (V)
Ω
�k
4dn(k)N (k) = 4
θ(k
−
k)d
F
(2π)3
2
T =Ω 2
π
�
ρ0 = 0.17 fm−3
�2 k 2 2
2kF3 3 �2 kF2
3
k dkθ(kF − k) = Ω 2
= A �F
2M
3π 5 2M
5
kF = 1.36 fm−1
The fermi level is
the last level occupied
�T � = 23 MeV
�2 kF2
�F =
= 38.35 MeV
2M
(BE)vol = −bvol A (bvol = 15.56 MeV)
< U >= −15.56− < T >� −39 MeV
Evidences of Shell Structure in Nuclei
© Basdevant, Rich and Spiro
Mean potential method: Shell model
The shell model, in its most simple
version, is composed of a mean
field potential (maybe a harmonic
oscillator) plus a spin-orbit
potential in order to reproduce the
empirical evidences of shell
structure in nuclei
En = (n + 3/2)�ω
H = Vls (r)l · s/�
l·s
�2
© Basdevant, Rich and Spiro
=
=
=
2
j(j+1)−l(l+1)−s(s+1)
2
l/2
j = l + 1/2
−(l + 1)/2 j = l − 1/2
Mean potential method: Shell model
Shell model (I)
A
�
H=
Hi
i=1
�
1 2 1
Hi =
p�i + M ω02 ri2 − V0
2m
2
�
2
p�
1
+ M ω02 r2 ψ(�r) = (E + V0 )ψ(�r)
2M
2
ψ(�r) = Rnl (r)Ylm (θ, φ)
Rnl (r)
=
�
�
2
l + n + 1/2
n
(l + 1/2)!
�
�
2
3
rl e−λr /2 1 F1 −n, l + , λr2
2
(−1)n
�
Shell model (II)
Γ(n + 1)Γ(µ + 1) µ
Ln (z)
1 F1 (−n, µ + 1, z) =
Γ(n + µ + 1)
EN =
d =
2
�
N
�
3
N+
2
Degeneracy
�ω0
N = 2n + l
[N/2]
(2l + 1) = 2
2(2N + 1)
�
n=0
l=0
=
�
�
�
(2(N − 2n) + 1) =
[N/2]
�
N
+1 −8
2
n=0
d = (N + 1)(N + 2)
Shell model (III)
Shell model (IV)
Shell model (V)
Shell model (V)
Isospin
In 1932, Heisenberg suggested that the proton and the neutron
could be seen as two charge states of a single particle.
EM ≠ 0
EM = 0
939.6 MeV
n
938.3 MeV
p
N
Protons and neutrons have almost identical mass
Low energy np scattering and pp scattering below E = 5 MeV, after
correcting for Coulomb effects, is equal within a few percent
Energy spectra of “mirror” nuclei, (N,Z) and (Z,N), are almost identical
Isospin (II)
Isospin is an internal variable that determines the nucleon state
ψN (r, σ, τ ) =
�
ψp (r, σ, 12 )
proton
ψn (r, σ, − 12 ) neutron
One could introduce a (2d) vector space that is mathematical copy of the
usual spin space
�
�
�
�
η 12 , 12 = |π� =
proton state
1
0
η 12 ,− 12 = |ν� =
neutron state
0
1
Isospin
eigenstates of the third component of isospin
In general
τ3 |π� = |π�
τ3 |ν� = −|π�
ψN = a|π� + b|ν� =
The isospin generators
Pauli matrices
�
[ti , tj ] = i�ijk tk
Projectors
Q̂
3
Pp = 1+τ
=
2
e
1−τ3
Pn = 2
a
b
�
τ1 , τ2 , τ3
Fundamental representations
1
ti = τ i
2
Raising and lowering operators
neutron to
proton
t± = t1 ± it2
t+ |ν� = |π�
t− |π� = |ν�
t+ |π� = 0
t− |ν� = 0
proton to
neutron
Isospin for 2 nucleons
T = 0, 1
T� = �t1 + �t2
T =0
T =1


η0,0 =
 η
1,0
√1 (π1 ν2
2
− ν1 π2 )
η1,1 = π1 π2
η1,−1 = ν1 ν2
= √12 (π1 ν2 + π2 ν1 )
Proton-proton state
|T = 1, Tz = 1� = |pp�
Neutron-neutron state
|T = 1, Tz = −1� = |nn�
Proton-neutron state
1
√ [|T = 1, Tz = 0� + |T = 0, Tz = 0�] = |pn�
2
Isospin for 2 nucleons
Wavefunction
a
s
ψ(1, 2) = ψpp (r1 , σ1 , r2 , σ2 )η1,1 + ψnn (r1 , σ1 , r2 , σ2 )η1,−1 + ψnp
(r1 , σ1 , r2 , σ2 )η1,0 + ψnp
(r1 , σ1 , r2 , σ2 )η0,0
antisymmetric
(1)
τ3
(2)
τ3
η1,1
symmetric
η0,0
(1) 2
1
−
�
τ
�τ
1
+
1
+
T =0
T =1
P
=
Pν=1 =
4
2
2
(1)
(2)
1
(1) (2)
1
−
τ
1
−
τ
T =1
(1) (2)
T =1
3
3
P
=
(1
+
�
τ
�
τ
−
2τ
Pν=−1 =
ν=0
3 τ3 )
4
2
2 η1,−1
η1,0
Symmetry for two nucleon states
Ψ(�r, �s1 , �s2 , �t1 , �t2 ) = φ(�r)fσ (�s1 , �s2 )fτ (�t1 , �t2 )
the overall wavefunction must be antisymmetric
(−)
L=0, S=1
L+S+T
T=0
= (−)
3S
1
isospin singlet
ISOSPIN
Sistema di 2
nucleoni identici
(pp,nn)
SPAZIO
L dispari
S=1
ψ(�x) antisimmetrica
ψ(�σ )
(no onda S)
Tz = ±1
Funzione simmetrica
(tripletto T=1)
Tz = 0
Funzione simmetrica
(tripletto T=1)
Sistema di 2
nucleoni distinti
(pn)
L pari
ψ(�x)
simmetrica
L dispari
ψ(�x) antisimmetrica
(no onda S)
L pari
ψ(�x) simmetrica
S=0
ψ(�σ )
simmetrica
1
S0
antisimmetrica
S=1
ψ(�σ )
S=0
ψ(�σ )
L dispari
S=0
ψ(�x) antisimmetrica
ψ(�σ )
simmetrica
1
S0
antisimmetrica
antisimmetrica
(no onda S)
Tz = 0
Funzione antisimmetrica
(singoletto T=0)
SPIN
L pari
ψ(�x)
simmetrica
S=1
ψ(�σ )
3
S1
simmetrica
1S
0
60 eV
(T=1)
1S
0
(T=1)
3S
1
(T=0)
1S
0
(T=1)
Coulomb
0.0
-2.23 MeV
pp
np
nn
Additional slides
...many open questions
Mean potential method
The concept of mean potential (or mean field) strongly relies on the basic assumption
of independent particle motion, i.e. even if we know that the “real” nuclear potential
is complicated and nucleons are strongly correlated, some basic properties can be
adequately described assuming individual nucleons moving in an average potential (it
means that all the nucleons experience the same field).
V (r) =
a rough approximation could be
�
dr � v(r − r � )ρ(r � )
v(r − r � ) = −v0 δ(r − r � )
where v0 can be phenomenologically
estimated to be
�
dr v(r) ∼ 200 MeV fm3
Then one can use a simple guess for V: harmonic oscillator, square well,
Woods-Saxon shape...
V (r) =
V0
1 + e(r−R)/R