From atomic nuclei to From atomic nuclei to neutron starsneutron stars

Piotr Magierski

(Warsaw University of Technology)

Atomic nucleus

5th International Student Conference of Balkan Physical Union

pro

ton

s

neutrons

Nuclear LandscapeNuclear Landscape

82

50

28

28

50

82

2082

28

20

Stable nucleiStable nuclei

known nucleiknown nuclei

neutron starsneutron stars

126

r-process

rp-process

superheavynuclei

neutron drip

line

TerraincognitaTerraincognitapro

ton d

rip lin

e

What are the basic degrees of freedom of a nuclear system?

It depends on the energy scale we are interested?

g

g

g

u d_

g

g

g

gg

g

gg Quarks and gluons

QCD energy scale: 1000MeV

p n

Baryons and mesonsEnergy scale: 100MeV

NucleonsEnergy scale: 10MeV

Collective degrees offreedom: 1MeV

LOW ENERGY NUCLEAR PHYSICS-LOW ENERGY NUCLEAR PHYSICS-- PHYSICS OF ATOMIC NUCLEI- PHYSICS OF ATOMIC NUCLEI

Nucleon-nucleon (N-N) interaction is an effective interaction

3central spin tensor spin orbit bodyV V V V V V 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-bodyterms included.Red – two-bodyand 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: 1/ 30 0, 1.2R r A r fm

In order to make a reliable calculation of the wave function wehave to consider a volume of the order of 3 3

0(2 ) 8V R r 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 momentumof the nucleon at the Fermi level:

1/ 32 33

/ , 0.16 -nuclear saturation density2F Fp k fm

2 FF

Fermi wavelengthk

Fmax 2

Fx

Maximumdistance between points

maxx

Therefore the values of the wave function has to be known at least in

30

3

8( )3

max

Fk rVA A

x

points

But the wave function depends on A variables (disregarding spin):

1 2( , ,..., )Ar r r

Hence to store the wave function we need to store complex numbers. AA

For it means 100A 20010 complex numbers!!!

It can be shown that instead of wave function one may use a density distribution:3 3 2

2 2( , ) ( ) ... | ( , ,..., ) |A Ar r r d r d r r r r

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 requirementthat the functional reaches the minimum value for the ground state density distribution.

Nuclear wave function contains too much information

Not possible now and never will be!!!

Towards the Universal Nuclear Energy Density Functional

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

s 1r

r ;

r '

' ' isovector spin density

s 0r

r ;

r '

' ' isoscalar spin density

j Tr i

2

'

T

r ,r '

r 'r

�J Tr i

2

'

s T

r ,r '

r 'r

T

r

'T

r ,r '

r 'r

T Tr

's Tr ,r '

r 'r

current density

spin-current tensor density

kinetic density

kinetic spin density

Local densities

and currents+ pairing…

In nuclear systems we have to generalize the density functional taking into account also spin and isospin.

HT

r CT

T2 CT

s sT2 CT

TT CTss T

s T

+CT TT jT

2 CTT s T

T T

�J T

2 CTJ T

J T

s T

j T

E tot 2

2m0 + H0

r H1

r

d3r Total ground-

state energy

Example: Skyrme

Functional

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

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

, , ,...

2 22 / 3

1/ 3

Liquid drop formula: Bethe, Weizsacker (40's)

( )( , ) ( , )

where ( , ) is the binding energy of the nucleus with neutrons

and protons ( )

V S C Sym pairZ N Z

B N Z a A a A a a a N ZAA

B N Z N

Z A N Z

Accurate upto 1-2%

Shell model is born (40’s):Inside atomic nuclei nucleons movelike independent particles in some average potential.It explains enhanced stability of ‘magic’nuclei: Together with liquid drop formula shellmodel was able to predict binding energies up to 0.1% accuracy!Problem: Liquid drop formula and shellmodel are incompatible.

40 132 208, , ,...Ca Sn Pb60’s-70’s:- More accurate average potentials have been introduced: Nilsson potential, Woods-Saxon potential.- Liquid drop 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 methodMany successes in interpreting experimentalspectroscopic data in terms of single-nucleonexcitations, rotations of the whole nucleus,vibrations and mutual coupling between thesemodes.

Further work on the theoryof 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

Effective field theory (EFT) is developed (80’s-90’s): Allows to consistently formulatethe effective quantum theory at low energies using the experimentalinformation as well as information from more fundamental theory (symmetries).

Progress in computationalabilities:Properties of heavier nuclei (A<10) were calculated using EFT input.

80’s-90’sThe shell model and liquid dropformula reached the limit of theirusefulness: too many parameters, too much phenomenology, too little physical insight. - More sophisticated phenomenologicalinteractions were used not only to generatean average potential, but also to calculateproperties 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?

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 discoveryNeutron 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 pulsarsNature of the pulsars Pulsar in the Crab Nebula

Conclusion: the pulses are produced by rotation!

Calculated energy lossdue to rotation of a possible

neutron starEnergy radiated

pulse rate = 30/secondslowing down rate = 38 nanoseconds/day

Basic facts about neutron stars:Basic facts about neutron stars:

Radius: 10 kmMass: 1-2 solar massesAverage density:Magnetic field: GMagnetars: GRotation period: 1.5 msec. – 5 sec.

8 1210 10

14 310 /g cm

1510

Gravitational energyof a nucleon at the surface of neutron star

100 MeV

Binding energy per nucleon in an atomic nucleus: 8 MeVNeutron star is bound by gravitational force

Number of known pulsars: > 1000Number of pulsars in our Galaxy:

810

Crust

CoreCore e

e

e

e

TT

TT

corecore

surfsurf

Tcore < Tsurf

For < 100 years:

Cooling waveCooling wave

Thermal evolution of a neutron star:Thermal evolution of a neutron star:

Temperature: 50 MeV 0.1 MeV URCA process:

e

e

p e n

n p e

Temperature: 0.1 MeV 100eV MURCA process:

e

e

e

e

p p e p n

n p e n n

p n p p e

n n n p e

Energy transfer between coreand surface:

2 ;v

TD T D

t C

URCA & MURCA

1km

( min .)t

5( 10 .)t yr

What are the basic degrees of freedom of nuclear matter at various densities?

Why the neutron star is made of neutrons?

Neutron gas Proton gas

Electron gas

Let’s assume that the star consistsof 3 types of noninteracting Fermi gases:

Since electron are about 2000 lighter than nucleons the density of states of electron gas is much smaller.

EnergyNeutron gas Proton gas

Electron gas

Converting protons and electrons to neutrons we minimize the total energy. Equilibrium condition: n p e

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:The total energy of the star:

int

3int

3

( ) ( , ) internal energy of the matter

( )gravitational energy (newtonian)

grav

grav

E E E

E d r r T

M rE d rG r

r

Hydrostatic stability condition:Hydrostatic stability condition:

Consider the uniform contraction or expansion of the spherical star (1 )r r

3 2 3

2

0

2

2

Then ( ) (0) is equal to:

( ) 9 ( )( )= 3 3 ( )

2

where is the energy per particle and

1 - pressure, ; ( ) 4 ' ' ( ')

-

r

E E E

M r M rE d r p G p p G O

r t

p v M r r dr rv

v

pv

adiabatic index, p

v

C

C

3 3

3

( )Stability requires: 3 - Virial theorem

4 4 0

3 3

GM rd r pd r

r

pd r

Ideal Fermi gas nonrelativistic (T=0K):

Ideal ultrarelativistic Fermi gas (T=0K):

2 / 3 5 / 3

1/ 3 4 / 3

5( )

34

( )3

p

p

The equation of state: determines the size and the mass ofthe star through the requirement:

( , )p p T

22

( )( ), 4 ( ) - for ideal spherically symmetric stars

p M r dMG r r r

r drr

The equation of state of nuclear matter for thedensity range up to 10 nuclear densities is needed!

Let us consider the simplest version of the liquid drop formula2 2

2 / 31/ 3

( )( , ) ( , )

where ( , ) is the binding energy of the nucleus with neutrons

and protons ( )

V S C S pairZ N Z

B N Z a A a A a a a N ZAA

B N Z N

Z 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).

What information do we need from physics of atomic nuclei?

Outercrust Inner

crust

Nuclei. .. .. ..

..

. .. .

...

Electrons

Neutrons

Exotic nuclear shapes„pasta” phase

..

..

... .

.

.

.

..

.

CoreUniform

nuclear matter

Quark-Quark-gluongluon plasmaplasma??

14 3ρ≈10 g/cm

56∼ Fe

6 3ρ 10 g/cm

Nuclei

Crystalline solid

11 3ρ≈4×10 g/cm