ABSTRACT
Basic units Energies are expressed in MeV, distances in fm and times in fm/c. This unit system is well suited to the physical situations considered here. – 1 MeV = 106 eV ' 1:6 1013 J – 1 fm = 1015 m – 1 fm=c ' 1015 m=(3 108 m s1) ' 3:3 1024 s
Fundamental constants – ~c ' 197 MeV fm – ~
=2m ' 20:73 MeV fm2 (for a nucleon of mass mc2 940 MeV) – e
) = 1:44 MeV fm Auxiliary units
– nucleon density []! fm3 – temperature [T ]! kT in MeV (with Boltzmann constant k = 1) – pressure [P ]! MeV fm3 – cross-section []! 1 barn = 100 fm2
Throughout this book we use the following notations:
vectors and operators are denoted with bold characters (for example the Hamiltonian operator is denotedH , and position r)
in formal developments we usually take ~ = 1 Boltzmann constant k = 1 we use standard Dirac notation for kets, j i; the corresponding wavefunction
10.3.1 Properties of nuclei
10.3.1.1 The nucleon-nucleon interaction
average nucleon-nucleon cross-section 40 mb ' 4 fm2 typical range of the nucleon-nucleon interaction d 1 fm hard core radius r
0:6 fm typical mean free path in ground-state nuclei ' 5-10 fm
10.3.1.2 Ground-state nuclei
Gross properties – nuclear radius R ' r
' 1:12 fm so that R 2-8 fm – surface width 1-2 fm – typical nuclear binding energyE=A ' 8 MeV – typical de Broglie wavelength in ground-state nuclei
' 5 fm Nuclear potential well
– typical nuclear potential depth V ' 50 MeV – typical nuclear chemical potential ' 8 MeV – typical Fermi energy
' 40 MeV Fermi gas properties (zero temperature, spin-isospin degenerate)
– Density =
– Fermi energy
– values for saturation density = 0:17 fm3
10.3.1.3 Thermodynamics of nuclei relation between thermal excitation energy and temperature (low-
temperature limit) E
typical value of level density parameter (from experiments, overlooking shell effects) a ' A=8
relation between entropy and temperature (low temperature limit)
10.3.1.4 Low-energy dynamics
Giant resonances – parametrization of frequencies
– typical giant resonance period GR
50-150 fm/c Fission
– fissility parameter
(sphere) 2E
(sphere) =
– typical fission barrier height: heavy nuclei B
' 40 MeV – typical fission time
–10
20 s
10.3.2 The nucleonic equation of state
Saturation point (symmetric nuclear matter) – saturation density
' 0:17 fm3 – saturation energyE=A
' 16 MeV – incompressibility modulus K
' 220 MeV – parabolic form of the nuclear matter equation of state in the vicinity of
saturation point
Thermodynamical properties (symmetric nuclear matter) – typical range of nucleonic physics:
– spinodale density (zero temperature) s
' 0:10 fm3 – critical density (liquid-gas transition)
' 0:06-0.07 fm3 – critical temperature (liquid-gas transition) T
' 16-18 MeV
10.3.3 Kinematics and cross-sections
Available centre-of-mass energy
Nuclear collisions in the Fermi energy range consist of bombarding a projectile on a fixed target nucleus. Hence part of the incident energy is spent as recoil energy of the centre-of-mass. Thus, the available centre-ofmass (CM) energy, i.e. the energy available for the collision E
is only a fraction of the laboratory energy of the projectile:
: (10.1)
This expression is a non-relativistic approximation;E lab
is the beam energy, A
) the projectile (target) mass number. The centre-of-mass energy per nucleon writes as:
: (10.2)
The corresponding centre-of-mass velocity v CM
can be written as
(10.3)
is the beam velocity in the laboratory frame. In the nonrelativistic approximatiom, the conversion from the centre-of-mass frame to the laboratory frame is realized using the law of vectorial addition of the velocities: v
where v is the velocity of a given particle in the centre-of-mass frame, v
the corresponding velocity in the laboratory frame.
