Transcript Nuclear structure and dynamics in the inner crust of

From atomic nuclei to neutron
stars
Atomic nucleus
Piotr Magierski
(Warsaw University of Technology)
5th International Student Conference of Balkan Physical Union
Nuclear Landscape
126
Stable
nuclei
superheavy
nuclei
82
protons
known
nuclei
Terra
incognita
50
82
28
20
50
8
28
2
20
2 8
neutrons
neutron stars
What are the basic degrees of freedom of a nuclear system?
It depends on the energy scale we are interested?
u
g
g
p
g
g
g

g
g
g
g
g
g

n
_
d
Quarks and gluons
QCD energy scale: 1000MeV
Baryons and mesons
Energy scale: 100MeV
Nucleons
Energy scale: 10MeV
LOW ENERGY NUCLEAR PHYSICS- PHYSICS OF ATOMIC NUCLEI
Collective degrees of
freedom: 1MeV
Nucleon-nucleon (N-N) interaction is an effective interaction
V  Vcentral  Vspin  Vtensor  Vspinorbit  V3body
N-N force can be determined (except for the three-body term)
from the proton-proton and proton-neutron scattering experiments.
Results of solving
Schroedinger eq.
with N-N potential.
Blue – only two-body
terms included.
Red – two-body
and three-body terms.
Green – experiment.
3-body interaction is important!
Can we solve Schroedinger eq. for medium or heavy nuclei?
Can we calculate the wave function for medium and heavy nuclei?
Consider a nucleus of mass number A (number of nucleons).
Its radius is of the order of:
R  r0 A1/ 3 , r0  1.2 fm
In order to make a reliable calculation of the wave function we
have to consider a volume of the order of V  (2 R)3  8r03 A
(In practice it has to be much larger as the wave function has a tale.)
How many points inside the volume V do we need?
From the Fermi gas model we may estimate the momentum
of the nucleon at the Fermi level:
1/ 3
pF /
3

 kF    2  
2

,   0.16 fm 3 -nuclear saturation density
xmax
F  2 k  Fermi wavelength
F

xmax  F Maximum
2 distance between points
Therefore the values of
the wave function has to be known at least in
F
V
 xmax 
3

8( k F r0 )3
3
A  A points
But the wave function depends on A variables (disregarding spin):
(r1 , r2 ,..., rA )
Hence to store the wave function we need to store
200
complex numbers!!!
For A  100 it means 10
A
A
complex numbers.
Not possible now and
never will be!!!
Nuclear wave function contains too much information
It can be shown that instead of wave function one may use a density distribution:
 (r , r )   (r )  d 3r2 ...d 3rA | (r , r2 ,..., rA ) |2

Theorem (Hohenberg & Kohn):
The energy of the nondegenerate ground state of the Fermi system
is uniquely determined by its density distribution.
It is suffcient to search for the density functional: E[  ( r )]
The ground state energy is obtained through the requirement
that the functional reaches the minimum value for the ground
state density distribution.
Towards the Universal Nuclear Energy Density Functional
In nuclear systems we have to generalize the density functional taking into account also spin and isospin.
0 r   0 r ,r    r  ;r  
isoscalar (T=0) density 0  n   p 
1r   1 r ,r    r  ;r  
isovector (T=1) density 1  n   p 





s0 r    r  ;r  '   '
s1r  
isoscalar spin density
 ' 
 r ;r  '    

isovector spin density
'
 '





i
'  T r ,r ' r ' r
2
i
JT r   '   sT r ,r ' r ' r
2

 T r    ' T r,r ' r ' r
kinetic density

TT r    ' sT r,r ' r ' r
kinetic spin density



jT r  
Local densities
and currents
+ pairing…
current density
spin-current tensor density
H T r   CT T2  CTs sT2  CT T T  CTssT sT




+CT T  T  jT2  CTT sT  TT  JT2  CTJ T  JT  sT   jT
 2
 3
E tot     0 + H0 r   H1r d r
2m


Example: Skyrme
Functional
Total groundstate energy
We would like to have the Nuclear Energy Density Functional which is able to give right nuclear
binding energies and equation of state up to about twice the nuclear saturation density.
Why now?
What nuclear theorists have been doing for more than half century?
Short history of nuclear theory
Liquid drop formula: Bethe, Weizsacker (40's)
Z2
( N  Z )2
B( N , Z )  aV A  aS A  aC 1/ 3  aSym
 a pair ( N , Z )
A
A
where B( N , Z ) is the binding energy of the nucleus with N neutrons
2/3
Accurate up
to 1-2%
and Z protons (A  N  Z )
Meson theory of strong interaction: Yukawa (50’s)
Pions are responsible for long range part of nuclear interaction.
Problem I: short range part of N-N interaction requires theory
with many mesons (many coupling constants needed):
Problem II: coupling constants were not small, so perturbation theory failed
 ,  ,  ,...
Shell model is born (40’s):
Inside atomic nuclei nucleons move
like independent particles in some
average potential.
It explains enhanced stability of ‘magic’
nuclei: 40Ca, 132 Sn, 208 Pb,...
60’s-70’s:
Together with liquid drop formula shell
- Morewas
accurate
average
potentials
model
able to predict
binding
have been
Nilsson potential,
energies
up tointroduced:
0.1% accuracy!
Woods-Saxon
potential.
Problem:
Liquid drop
formula and shell
model
are drop
incompatible.
- Liquid
formula has been improved
- more terms added (more parameters).
-First attempts to derive the average potential
from some phenomenological N-N interaction
(density dependent, no hard core)
– Hartree-Fock method
Many successes in interpreting experimental
spectroscopic data in terms of single-nucleon
excitations, rotations of the whole nucleus,
vibrations and mutual coupling between these
modes.
Further work on the theory
of N-N interaction (60’s-70’s)
-Semiphenomenological potentials:
Bonn potential, Paris potential.
-Calculations for deuteron, triton, helium.
-Problems with short range.
70’s-80’s:
-Quantum Chromodynamics (QCD)
is born: strong interaction is mediated
by gluons (8) between quarks.
Meson theory is an effective low
energy theory.
Problem: QCD is nonperturbative at low
energies
80’s-90’s
The shell model and liquid drop
formula reached the limit of their
usefulness:
too many parameters,
too much phenomenology,
too little physical insight.
- More sophisticated phenomenological
interactions were used not only to generate
an average potential, but also to calculate
properties of excited states of heavy nuclei
(effective many-body methods:
RPA,GCM,TDHF).
Problem: how to link this phenomenological
N-N interaction with real N-N interaction?
Effective field theory (EFT) is
developed (80’s-90’s):
Allows to consistently formulate
the effective quantum theory at low
energies using the experimental
information as well as
information from more
fundamental theory (symmetries).
Progress in computational
abilities:
Properties of heavier nuclei (A<10) were
calculated using EFT input.
EFT provides a missing link between real N-N interaction and
phenomenological N-N interaction! Eventually it will help to
construct the Universal Energy Density Functional for nuclear systems
Neutron star discovery
-The existence of neutron stars was predicted by Landau (1932), Baade & Zwicky (1934) and
Oppenheimer& Volkoff (1939).
- On November 28, 1967, Cambridge graduate student Jocelyn Bell (now Burnell) and her advisor,
Anthony Hewish discovered a source with an exceptionally regular pattern of radio flashes. These
radio flashes occurred every 1 1/3 seconds like clockwork. After a few weeks, however, three more
rapidly pulsating sources were detected, all with different periods. They were dubbed "pulsars."
Nature of the pulsars
Pulsar in the Crab Nebula
pulse rate = 30/second
slowing down rate = 38 nanoseconds/day
Calculated energy loss
due to rotation of a possible
neutron star

Energy radiated
Conclusion: the pulses are produced by
Basic facts about neutron stars:
Radius:
10 km
Mass:
1-2 solar masses
14
3
10
g
/
cm
Average density:
8
12
Magnetic field: 10 10 G
1015 G
Magnetars:
Rotation period: 1.5 msec. – 5 sec.
Number of known pulsars: > 1000
Number of pulsars in our Galaxy:
Gravitational energy
of a nucleon at the surface
of neutron star
8
10
100 MeV
Binding energy per nucleon in an atomic nucleus:
Neutron star is bound by gravitational force
8 MeV
Thermal evolution of a neutron star:
Temperature: 50 MeV 0.1 MeV (t
URCA process: p  e  n  n
min .)
g
e
n  p  e n e
Temperature: 0.1 MeV
MURCA process:
g
Crust
5
100eV (t 10 yr.)
p  p  e  p  n n e
ne
n pe
n  nare
 n ethe
What
pn
basic degrees of freedom of
p  nuclear
p  e  n e matter at various densities?
n  n  n  p  e n e
ne
Energy transfer between core
and surface:
T

 D 2T ; D 
t
Cv
For  < 100 years:
Tcore < Tsurf
URCA &
MURCA
Tcore
Core
1km
ne
ne
g
Tsurf
Cooling wave
g
Why the neutron star is made of neutrons?
Electron gas
Let’s assume that the star consists
of 3 types of noninteracting Fermi gases:
Neutron gas
Proton gas
gas
Proton
Energy
Converting
protons
and
electrons
to neutrons
Since electron
are
about
2000 lighter
than we
nucleons
 n   p  e
minimize
the total
energy.
the density
of states
of Equilibrium
electron gas iscondition:
much smaller.
Structure of the neutron star
The stability of the neutron star is a result of the balance between
the gravitational attraction and the pressure of the matter forming the star.
The total energy of the star: E  Eint  Egrav
Eint   d 3r  (r ) (  , T )
Egrav    d 3rG
 internal energy of the matter
M (r )
  r   gravitational energy (newtonian)
r
Hydrostatic stability condition:
Consider the uniform contraction or expansion of the spherical star r  (1   )r
Then  E ( )  E    E (0) is equal to:
 
 M (r ) 
9
 M (r )  
2
3



3
p

g
p

G


   O ( )
r 
2
t


 
where  is the energy per particle and
 E ( )=  d 3r   3 p  G

p
v
- pressure, v 
1

r
; M ( r )  4  r '2 dr '  ( r ')
Cp
 2 v
g 2
- adiabatic index, g 
p
Cv
v
0
GM (r )  3
Stability requires: 
d r  3 pd 3r - Virial theorem
r
4 3
4

  g  3  pd r  0  g  3
Ideal Fermi gas nonrelativistic (T=0K): 
2/3( p
Ideal ultrarelativistic Fermi gas (T=0K): 
 1/ 3 ( p
5
3
4
4/3
 )g 
3
5/3)  g 
The equation of state: p  p(  , T ) determines the size and the mass of
the star through the requirement:
p
M (r )
dM
 G 2  (r ),
 4 r 2  (r ) - for ideal spherically symmetric stars
r
dr
r
The equation of state of nuclear matter for the
density range up to 10 nuclear densities is needed!
What information do we need from physics of atomic nuclei?
Let us consider the simplest version of the liquid drop formula
Z2
( N  Z )2
B( N , Z )  aV A  aS A  aC 1/ 3  aS
 a pair ( N , Z )
A
A
where B( N , Z ) is the binding energy of the nucleus with N neutrons
2/3
and Z protons (A  N  Z )
Which terms are important in the context of neutron stars?
volume energy symmetry energy
pairing energy
Volume energy determines the energy of saturated nuclear matter.
Symmetry energy determines the proton fraction.
Pairing influences the specific heat and mechanical properties
(moment of inertia).
Crystalline solid
.
.. . .. .
.. .
. . . . ..
.
Electrons
. . Nuclei
.
.
.
..
.
.
.
.
.
.
Neutrons
Nuclei
∼
56
ρ 10 6 g/cm 3
ρ ≈ 4 ×1011 g/cm 3
Fe
ρ ≈ 1014 g/cm 3
Outer
crust Inner
crust
Core
Uniform
nuclear matter
Exotic nuclear shapes
„pasta” phase
Quarkgluon
plasma?