From atomic nuclei to neutron stars

1. From atomic nuclei to neutron stars Atomic nucleus Piotr Magierski (Warsaw University of Technology) 5th International Student Conference of Balkan Physical Union

2. superheavy nuclei proton drip line neutron drip line Nuclear Landscape 126 Stablenuclei 82 r-process known nuclei Terra incognita 50 protons 82 rp-process neutron stars 28 20 50 8 28 neutrons 2 20 8 2

3. Quarks and gluons QCD energy scale: 1000MeV Baryons and mesons Energy scale: 100MeV g g g g g g g g g g g  Nucleons Energy scale: 10MeV _ p n  d u What are the basic degrees of freedom of a nuclear system? It depends on the energy scale we are interested? LOW ENERGY NUCLEAR PHYSICS- - PHYSICS OF ATOMIC NUCLEI Collective degrees of freedom: 1MeV

4. Nucleon-nucleon (N-N) interaction is an effective interaction 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!

5. Consider a nucleus of mass number A (number of nucleons). Its radius is of the order of: In order to make a reliable calculation of the wave function we have to consider a volume of the order of (In practice it has to be much larger as the wave function has a tale.) From the Fermi gas model we may estimate the momentum of the nucleon at the Fermi level: Maximum distance between points Therefore the values of the wave function has to be known at least in points Can we solve Schroedinger eq. for medium or heavy nuclei? Can we calculate the wave function for medium and heavy nuclei? How many points inside the volume V do we need?

6. But the wave function depends on A variables (disregarding spin): Hence to store the wave function we need to store complex numbers. complex numbers!!! For it means It can be shown that instead of wave function one may use a density distribution: 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: The ground state energy is obtained through the requirement that the functional reaches the minimum value for the ground state density distribution. Not possible now and never will be!!! Nuclear wave function contains too much information

7. Towards the Universal Nuclear Energy Density Functional In nuclear systems we have to generalize the density functional taking into account also spin and isospin. isoscalar (T=0) density isovector (T=1) density isoscalar spin density Local densities and currents + pairing… isovector spin density current density spin-current tensor density kinetic density kinetic spin density Example: Skyrme Functional Total ground-state 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.

8. Accurate up to 1-2% 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 Why now? What nuclear theorists have been doing for more than half century? Short history of nuclear theory

9. 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: Together with liquid drop formula shell model was able to predict binding energies up to 0.1% accuracy! Problem: Liquid drop formula and shell model are incompatible. • 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. • 60’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 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. • 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

10. 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

11. 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 rotation!

12. Basic facts about neutron stars: Radius: 10 km Mass: 1-2 solar masses Average density: Magnetic field: G Magnetars: G 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 100 MeV Binding energy per nucleon in an atomic nucleus: 8 MeV Neutron star is bound by gravitational force

13. Thermal evolution of a neutron star: Temperature: 50 MeV 0.1 MeV URCA process: g g Crust Temperature: 0.1 MeV 100eV MURCA process: ne What are the basic degrees of freedom of nuclear matter at various densities? URCA & MURCA ne Energy transfer between core and surface: ne Core T core ne g For t < 100 years: T g surf Tcore < Tsurf Cooling wave

14. Neutron gas Proton gas Neutron gas Electron gas Electron gas Proton gas Converting protons and electrons to neutrons we minimize the total energy. Equilibrium condition: Why the neutron star is made of neutrons? Let’s assume that the star consists of 3 types of noninteracting Fermi gases: Energy Since electron are about 2000 lighter than nucleons the density of states of electron gas is much smaller.

15. The total energy of the star: Consider the uniform contraction or expansion of the spherical star 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. Hydrostatic stability condition:

16. Ideal Fermi gas nonrelativistic (T=0K): Ideal ultrarelativistic Fermi gas (T=0K): The equation of state: determines the size and the mass of the star through the requirement: The equation of state of nuclear matter for the density range up to 10 nuclear densities is needed!

17. 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? Let us consider the simplest version of the liquid drop formula Which terms are important in the context of neutron stars?

18. Crystalline solid . . . . . Nuclei . . . . . . . . . . . . Electrons . . . . . Nuclei . . Outer crust . . . Inner crust . . . Core Uniform nuclear matter Neutrons Exotic nuclear shapes „pasta” phase Quark-gluon plasma?