Basic Concepts in Nuclear Physics

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Basic Concepts in Nuclear Physics

Corso di

Teoria delle Forze Nucleari 2011

Paolo Finelli

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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

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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,...

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The Nuclear Landscape

The scope of nuclear physics is

Improve the knowledge of all nuclei

Understand the stellar nucleosynthesis

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e

⁻⁵

e

⁻⁶

e

⁻⁷

≥ e

⁻⁴

Stellar Nucleosynthesis

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.

The evolution of the nuclear abundances. Each square is a nucleus. The colors indicate the abundance of the nucleus:

©JINA

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m

_N

c

²

= m

_A

c

²

− Zm

^e

c

²

+

�

Z i=1

B

_i

� m

^A

c

²

− Zm

^e

c

²

B = (Zm

_p

+ Nm

_n

) c

²

− m

^N

c

²

� [Zm

^p

+ Nm

_n

− (m

^A

− Zm

^e

)] c

²

B = �

Zm(

¹

H) + N m

_n

− m(

^A

X) � c

²

Binding energy

Electrons Mass (~Z)

Atomic Mass Electrons Binding Energies

(negligible)

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E/A (Binding Energy per nucleon)

A (Mass Number)

Average mass of fission fragments is 118

Fe Nuclear Fission Energy

Nuclear Fusion Energy

235U

isotopes

Binding energy

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Binding energy and Liquid Drop Model

Volume term, proportional to R³ (or A): saturation Surface term, proportional to R² (or A^2/3)

Coulomb term, proportional to Z²/A^1/3

Pairing term, nucleon pairs coupled to J^Π=0⁺ are favored

Asymmetry term, neutron-rich nuclei are favored

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Binding energy and Liquid Drop Model

Contributions to B/A as function of A

Comparison with empirical data

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Nuclear Dimensions

Ground state

Excited States (~eV)

Ground state

Ground state Excited States (~ MeV)

Excited States (~ GeV)

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Nuclear Dimensions: energy scales

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ρ(r) = ρ(0) 1 + e^(r^−R)/s

R : 1/2 density radius s : skin thickness

Nuclear Dimensions

Fermi distribution

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Nuclear forces saturation

An old (but still good) definition:

© E. Fermi, Nuclear Physics

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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 provides a way to calculate the basic constants in the Bethe-Weizsäcker formula

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Fermi gas model (I)

Hamiltonian

Wavefunction factorization

Boundary conditions

Separable equations

Gasiorowicz, p.58

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Fermi gas model (II)

Solution

Normalization

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Fermi gas model (III)

Density of states

Number of particles

Density

of particles

spin-isospin

Fermi momentum

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ρ

₀

= 0.17 fm

⁻³

k

_F

= 1.36 fm

⁻¹

�

_F

= �

²

k

_F²

2M = 38.35 MeV

�T � = 23 MeV

Fermi gas model (IV)

The fermi level is

the last level occupied

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Evidences of Shell Structure in Nuclei

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E

_n

= (n + 3/2)�ω

H = V _ls (r)l · s/� ²

l·s

�²

=

^j(j+1)−l(l+1)−s(s+1)

= l/2

2

j = l + 1/2

= −(l + 1)/2 j = l − 1/2

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

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Shell model (I)

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Shell model (II)

Degeneracy

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Shell model (III)

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Shell model (IV)

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Shell model (V)

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Shell model (V)

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Isospin

In 1932, Heisenberg suggested that the proton and the neutron could be seen as two charge states of a single particle.

939.6 MeV 938.3 MeV

EM ≠ 0 EM = 0

n

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

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ψ_N(r, σ, τ) =

� ψ_p(r, σ, ¹₂) proton ψ_n(r, σ, −¹₂) neutron

η

¹

2,¹₂

= |π� =

� 1 0

�

η

¹

2,−¹₂

= |ν� =

� 0 1

�

Isospin is an internal variable that determines the nucleon state

One could introduce a (2d) vector space that is mathematical copy of the usual spin space

proton state neutron state

Isospin (II)

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τ

₃

|π� = |π�

τ

₃

|ν� = −|π�

ψ

_N

= a|π� + b|ν� =

� a b

�

[t

_i

, t

_j

] = i�

_ijk

t

_k

P

_p

=

^1+τ₂ ³

=

^Q_e^ˆ

P

_n

=

¹^−τ₂ ³

τ

₁

, τ

₂

, τ

₃

t

_i

= 1 2 τ

_i

t

₊

|ν� = |π�

t

₋

|π� = |ν�

t

₊

|π� = 0 t

₋

|ν� = 0 t

_±

= t

₁

± it

²

Isospin

eigenstates of the third component of isospin

In general

The isospin generators

Projectors Raising and lowering operators

Pauli matrices

neutron to

proton proton to

neutron Fundamental representations

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T = �t �

₁

+ �t

₂

T = 0, 1

T = 0 η

_0,0

=

^√¹

2

(π

₁

ν

₂

− ν

¹

π

₂

) T = 1

 



η

_1,1

= π

₁

π

₂

η

_1,₋₁

= ν

₁

ν

₂

η

_1,0

=

^√¹₂

(π

₁

ν

₂

+ π

₂

ν

₁

)

Isospin for 2 nucleons

|T = 1, T

^z

= 1� = |pp�

|T = 1, T

^z

= −1� = |nn�

√ 1

2 [|T = 1, T

^z

= 0� + |T = 0, T

^z

= 0�] = |pn�

Proton-proton state Neutron-neutron state

Proton-neutron state

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Isospin for 2 nucleons

ψ(1, 2) = ψ_pp(r1, σ₁, r₂, σ₂)η1,1 + ψnn(r1, σ₁, r₂, σ₂)η1,−1 + ψ_np^a (r1, σ₁, r₂, σ₂)η1,0 + ψ_np^s (r1, σ₁, r₂, σ₂)η0,0

P

^{T =0}

= 1 − �τ

⁽¹⁾

�τ

²

P

_ν=1^{T =1}

= 1 + τ

₃⁽¹⁾

4

2 1 + τ

₃⁽²⁾

2 P

_ν=0^{T =1}

= 1

4 (1 + �τ

⁽¹⁾

�τ

⁽²⁾

− 2τ

₃⁽¹⁾

τ

₃⁽²⁾

)

η

_0,0

η

_1,1

P

_ν=^{T =1}₋₁

= 1 − τ

3⁽¹⁾

2 1 − τ

3⁽²⁾

2 η

_1,₋₁

η

_1,0

antisymmetric symmetric

Wavefunction

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Additional slides

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...many open questions

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v(r − r

^�

) = −v

⁰

δ(r − r

^�

) V (r) =

�

dr

^�

v(r − r

^�

)ρ(r

^�

)

� dr v(r) ∼ 200 MeV fm

³

V (r) = V

₀

1 + e

^(r^−R)/R

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).

a rough approximation could be

where v0 can be phenomenologically estimated to be

Then one can use a simple guess for V: harmonic oscillator, square well, Woods-Saxon shape...

Basic Concepts in Nuclear Physics