intro nuclear e 2011

7/29/2019 Intro Nuclear e 2011

1/34

Basic Concepts inNuclear Physics

Corso diTeoria delle Forze Nucleari

2011

Paolo Finelli


2/34

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


3/34

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 extremelyimportant: pairing, deformations, single and collective excitations, decay, decay, decay, fusion process, fission process,...


4/34

The Nuclear Landscape

The scope of nuclear physics is

Improve the knowledge ofall nucleiUnderstand the stellar nucleosynthesis

Basdevant, Rich and Spiro


5/34

e5

e6

e7

e4

Stellar Nucleosynthesis

Dynamical r-process calculation assuming anexpansion with an initial density of 0.029e4 g/cm3, aninitial temperature of 1.5 GK and an expansiontimescale of 0.83 s.

The r-process is responsible for theorigin of about half of the elementsheavier than iron that are found innature, including elements such asgold or uranium. Shown is the result of

a model calculation for this processthat might occur in a supernovaexplosion. Iron is bombarded with ahuge flux of neutrons and a sequenceof neutron captures and beta decays isthen creating heavy elements.

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

JINA


6/34

mNc2

= mAc2 Zmec

2

+i=1

Bi mAc2 Zmec

2

B = (Zmp +Nmn) c2mNc

2 [Zmp +Nmn (mA Zme)] c

2

B =Zm( 1H) + Nmn m(

AX)

c2

Binding energy

Electrons Mass (~Z)

Atomic Mass Electrons Binding Energies(negligible)



7/34

E/A

(BindingEnergy

pernucleon)

A (Mass Number)

Average mass of fissionfragments is 118

Fe Nuclear Fission Energy

NuclearFusionEnergy

235

U

Gianluca Usai

The most boundisotopes

Binding energy


8/34



Volume term, proportional to R3 (or A): saturation

Surface term, proportional to R2 (or A2/3)

Coulomb term, proportional to Z2/A1/3

Pairing term, nucleon pairscoupled to J=0+ are favored

Asymmetry term, neutron-rich nuclei are favored


9/34


Gianluca Usai

Contributions to B/A as function of A

Comparison with empirical data


10/34

Nuclear Dimensions

Ground state

Excited States (~eV)

Gianluca Usai

Ground state

Ground state

Excited States (~ MeV)

Excited States (~ GeV)


11/34

Nuclear Dimensions: energy scales


12/34

(r) =(0)

1 + e(rR)/s

R : 1/2 density radiuss : skin thickness

Nuclear Dimensions


Fermi distribution


13/34

Nuclear forces saturation

An old (but still good) definition:

E. Fermi, Nuclear Physics


14/34

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. Theinteraction appear implicitly through the assumption that the nucleons areconfined in the sphere. If the liquid drop model is based on the saturationof nuclear forces, on the other hand the Fermi model is based on thequantum statistics effects.

The Fermi model provides a way to calculate thebasic constants in the Bethe-Weizscker formula


15/34

Fermi gas model (I)

Hamiltonian

Wavefunction factorization

Boundary conditions

Separable equations

Gasiorowicz, p.58


16/34

Fermi gas model (II)

Solution

Normalization


17/34

Fermi gas model (III)

Density of states

Numberof particles

Densityof particles

spin-isospin

Fermi momentum


18/34

0 = 0.17 fm3

kF = 1.36 fm1

F = kF

2M= 38.35 MeV

T = 23 MeV

Fermi gas model (IV)

The fermi level isthe last level occupied


19/34

Evidences of Shell Structure in Nuclei



20/34

En = (n + 3/2)

H= Vls(r)l s/2

ls

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

The shell model, in its most simpleversion, is composed of a mean

field potential (maybe a harmonicoscillator) plus a spin-orbitpotential in order to reproduce theempirical evidences of shellstructure in nuclei



21/34

Shell model (I)


22/34

Shell model (II)

Degeneracy


23/34

Shell model (III)


24/34

Shell model (IV)


25/34

Shell model (V)


26/34

Shell model (V)


27/34

Isospin

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

939.6 MeV

938.3 MeV

EM 0 EM = 0

n

pN

Protons and neutrons have almost identical mass

Low energy np scattering and pp scattering below E = 5 MeV, aftercorrecting for Coulomb effects, is equal within a few percent

Energy spectra of mirror nuclei, (N,Z) and (Z,N), are almost identical


28/34

N(r,, ) =

p(r,,

1

2) proton

n(r,,1

2) neutron

12,1

2

= | =

1

0

12,

1

2

= | =

0

1

Isospin is an internal variable that determines the nucleon state

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

proton state neutron state

Isospin (II)


29/34

3| = |3| = |

N = a|+ b| =

a

b

[ti, tj ] = iijktk

Pp = 1+3

2= Q

e

Pn =132

1, 2, 3

ti=1

2i

t+| = |t

| = |t+| = 0t

| = 0

t = t1 it2

Isospin

eigenstates ofthe third component of isospin

In general

The isospin generators

Projectors Raising and lowering operators

Pauli matrices

neutron toproton proton to

neutron

Fundamental representations


30/34

T = t1 + t2 T = 0, 1

T = 0 0,0 =12

(12 12)

T = 1

1,1 = 121,1 = 12

1,0 =12 (12 + 21)

Isospin for 2 nucleons

|T = 1, Tz = 1 = |pp

|T = 1, Tz =

1

=

|nn

12

[|T = 1, Tz = 0 + |T = 0, Tz = 0] = |pn

Proton-proton state

Neutron-neutron state

Proton-neutron state


31/34

Isospin for 2 nucleons

(1, 2) = pp(r1,1,r2,2)1,1 + nn(r1,1,r2,2)1,1 + a

np(r1,1,r2,2)1,0 +

s

np(r1,1,r2,2)0,0

PT=0

=1

(1)2

4P

T=1=1 =

1 + (1)3

2

1 + (2)3

2

PT=1=0 =

1

4 (1 +(1)

(2)

2(1)

3(2)

3 )

0,01,1

P

T=1=1 =

1 (1)3

2

1 (2)3

2 1,1 1,0

antisymmetric symmetric

Wavefunction


32/34

Additional slides


33/34

...many open questions

M i l h d


34/34

v(r r) = v0(r r)

V(r) =

dr v(r r)(r)

dr v

(r

)

200 MeV fm3

V(r) =0

1 + e(rR)/R

Mean potential method

The concept ofmean potential(ormean field) strongly relies on the basic assumptionof 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 beadequately described assuming individual nucleons moving in an average potential (itmeans that all the nucleons experience the same field).

a rough approximation could be

where v0can be phenomenologically estimated to be

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

intro nuclear e 2011

Documents