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Basic Concepts inNuclear Physics
Corso diTeoria 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 extremelyimportant: pairing, deformations, single and collective excitations, decay, decay, decay, fusion process, fission process,...
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The Nuclear Landscape
The scope of nuclear physics is
Improve the knowledge ofall nucleiUnderstand the stellar nucleosynthesis
Basdevant, Rich and Spiro
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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
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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)
Basdevant, Rich and Spiro
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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
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Binding energy and Liquid Drop Model
Basdevant, Rich and Spiro
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
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Binding energy and Liquid Drop Model
Gianluca Usai
Contributions to B/A as function of A
Comparison with empirical data
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Nuclear Dimensions
Ground state
Excited States (~eV)
Gianluca Usai
Ground state
Ground state
Excited States (~ MeV)
Excited States (~ GeV)
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Nuclear Dimensions: energy scales
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(r) =(0)
1 + e(rR)/s
R : 1/2 density radiuss : skin thickness
Nuclear Dimensions
Basdevant, Rich and Spiro
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. 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
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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
Numberof particles
Densityof particles
spin-isospin
Fermi momentum
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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
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Evidences of Shell Structure in Nuclei
Basdevant, Rich and Spiro
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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
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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 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
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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)
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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
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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
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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
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Additional slides
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...many open questions
M i l h d
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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...