Basic Concepts in Nuclear Physics. Paolo Finelli

Transcription

1 Basic Concepts in Nuclear Physics Paolo Finelli

2 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

3 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 extremely important: pairing, deformations, single and collective excitations, α decay, β decay, γ decay, fusion process, fission process,...

4 The Nuclear Landscape The scope of nuclear physics is Improve the knowledge of all nuclei Understand the stellar nucleosynthesis Basdevant, Rich and Spiro

5 Stellar Nucleosynthesis The evolution of the nuclear abundances. Each square is a nucleus. The colors indicate the abundance of the nucleus: e 4 e 5 e 6 e 7 Dynamical r-process calculation assuming an expansion with an initial density of 0.029e4 g/cm3, an initial temperature of 1.5 GK and an expansion timescale of 0.83 s. The r-process is responsible for the origin of about half of the elements heavier than iron that are found in nature, including elements such as gold or uranium. Shown is the result of a model calculation for this process that might occur in a supernova explosion. Iron is bombarded with a huge flux of neutrons and a sequence of neutron captures and beta decays is then creating heavy elements. JINA

6 Binding energy Atomic Mass m N c 2 = m A c 2 Zm e c 2 + Electrons Mass (~Z) Z i=1 Electrons Binding Energies (negligible) B i m A c 2 Zm e c 2 Basdevant, Rich and Spiro B =(Zm p + Nm n ) c 2 m N c 2 [Zm p + Nm n (m A Zm e )] c 2 B = Zm( 1 H)+Nm n m( A X) c 2

7 Binding energy E/A (Binding Energy per nucleon) Nuclear Fusion Energy Fe The most bound isotopes Average mass of fission fragments is 118 A (Mass Number) Nuclear Fission Energy 235 U Gianluca Usai

8 Binding energy and Liquid Drop Model Volume term, proportional to R 3 (or A): saturation Surface term, proportional to R 2 (or A 2/3 ) Coulomb term, proportional to Z 2 /A 1/3 Asymmetry term, neutron-rich nuclei are favored Pairing term, nucleon pairs coupled to J Π =0 + are favored Basdevant, Rich and Spiro

9 Binding energy and Liquid Drop Model Comparison with empirical data Contributions to B/A as function of A Gianluca Usai

10 Nuclear Dimensions Excited States (~ev) Ground state Excited States (~ MeV) Ground state Excited States (~ GeV) Ground state Gianluca Usai

11 Nuclear Dimensions: energy scales

12 Nuclear Dimensions Fermi distribution ρ(0) ρ(r) = 1+e (r R)/s Basdevant, Rich and Spiro R : 1/2 density radius s : skin thickness

13 Nuclear forces saturation An old (but still good) definition: E. Fermi, Nuclear Physics

14 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. The interaction appear implicitly through the assumption that the nucleons are confined in the sphere. If the liquid drop model is based on the saturation of nuclear forces, on the other hand the Fermi model is based on the quantum statistics effects. The Fermi model could provide a way to calculate the basic constants in the Bethe-Weizsäcker formula

15 Fermi gas model (I) Hamiltonian H = A T i i=1 A i=1 2 2M 2 i Hψ(r 1,r 2,...)=Eψ(r 1,r 2,...) Wavefunction factorization ψ(r 1,r 2,...)=φ 1 (r 1 )φ 2 (r 2 ) M 2 i φ(r i )=Eφ(r i ) E = E 1 + E 2 + E = A i=1 E i Boundary conditions Separable equations k 2 i (k 2 ix + k 2 iy + k 2 iz) = 2ME i 2 > 0 φ i (r ) φ i (x, y, z) =N sin(k ix x)sin(k iy y)sin(k iz z) d 2 φ i (x) dx 2 = k 2 ixφ i (x) Gasiorowicz, p.58

16 Fermi gas model (II) Solution φ i (x) =B sin(k ix x) Normalization 1= L 0 dx φ i (x) 2 = B 2 L 0 dx sin 2 (k ix x)=b 2 L 2 B = 2 L φ i (r )= 3/2 2 sin(k ixx)sin(k iyy)sin(k izz) L k ix = π L n 1i,k iy = π L n 2i,k iz = π L n 3i (n 1i,n 2i,n 3i = positive integers) E i = 2 k 2 i 2M = 2 2M (k2 ix + k 2 iy + k 2 iz) E i (n 1i,n 2i,n 3i )= 2 π 2 2ML 2 (n2 1i + n 2 2i + n 2 3i)

17 Fermi gas model (III) k x,y,z = π L (n 1,2,3 +1 n 1,2,3 )= π L Density of states dn(k) = 1 8 4πk2 dk 1 (π/l) 3 Ω (2π) 3 d k Ω L 3 n( k) = k 0 dn(k) = Ω 4π (2π) 3 3 k 3 Number of particles spin-isospin kf A =4 0 dn(k) = Ω (2π) 3 44π 3 k3 =Ω 2k3 F 3π 2 Fermi momentum Density of particles ρ 0 = 2k3 F 3π 2 ρ 0 = A/Ω

18 Fermi gas model (IV) Fermi gas distribution: N(k) 1 Step function θ(k F k) filled empty 0 kf k

19 Fermi gas model (V) 4dn(k)N(k) =4 Ω (2π) 3 θ(k F k)d k T =Ω 2 π 2 2 k 2 2M k2 dkθ(k F k) =Ω 2k3 F 3 3π k 2 F 2M = A3 5 F ρ 0 =0.17 fm 3 k F =1.36 fm 1 T = 23 MeV The fermi level is the last level occupied F = 2 k 2 F 2M (BE) vol = b vol A = MeV (b vol = MeV) <U>= <T > 39 MeV

20 Basdevant, Rich and Spiro Evidences of Shell Structure in Nuclei

21 Mean potential method: Shell model The shell model, in its most simple version, is composed of a mean field potential (maybe a harmonic oscillator) plus a spin-orbit potential in order to reproduce the empirical evidences of shell structure in nuclei E n =(n +3/2)ω H = V ls (r)l s/ 2 l s = 2 j(j+1) l(l+1) s(s+1) 2 = l/2 j = l +1/2 = (l + 1)/2 j = l 1/2 Basdevant, Rich and Spiro

22 Mean potential method: Shell model

23 Shell model (I) H = A i=1 H i H i = 1 2m p 2 i Mω2 0ri 2 V 0 p 2 2M Mω2 0r 2 ψ(r )=(E + V 0 )ψ(r ) ψ(r )=R nl (r)y lm (θ, φ) R nl (r) = ( 1) n 2 l + n +1/2 (l +1/2)! n r l e λr2 /2 1 F 1 n, l + 3 2,λr2

24 1F 1 ( n, µ +1,z)= E N = Shell model (II) N Γ(n + 1)Γ(µ + 1) Γ(n + µ + 1) ω 0 N =2n + l L µ n(z) d = 2 N (2l + 1) = 2 l=0 = 2(2N + 1) [N/2] n=0 N 2 +1 (2(N 2n) + 1) = 8 [N/2] n=0 Degeneracy d =(N + 1)(N + 2)

25 Shell model (III)

26 Shell model (IV)

27 Shell model (V)

28 Shell model (V)

29 Isospin In 1932, Heisenberg suggested that the proton and the neutron could be seen as two charge states of a single particle MeV MeV EM 0 EM = 0 n p N Protons and neutrons have almost identical mass Low energy np scattering and pp scattering below E = 5 MeV, after correcting for Coulomb effects, is equal within a few percent Energy spectra of mirror nuclei, (N,Z) and (Z,N), are almost identical

30 Isospin (II) Isospin is an internal variable that determines the nucleon state ψp (r, σ, 1 ψ N (r, σ, τ) = 2 ) proton ψ n (r, σ, 1 2 ) neutron One could introduce a (2d) vector space that is mathematical copy of the usual spin space η 1 2, 1 2 = π = 1 0 proton state η 1 2, 1 2 = ν = 0 1 neutron state

31 Isospin eigenstates of the third component of isospin In general The isospin generators τ 3 π = π τ 3 ν = π a ψ N = a π + b ν = b [t i,t j ]=i ijk t k Pauli matrices τ 1, τ 2, τ 3 Fundamental representations t i = 1 2 τ i Projectors Raising and lowering operators P p = 1+τ 3 P n = 1 τ = ˆQ e neutron to proton t ± = t 1 ± it 2 t + ν = π t π = ν t + π =0 t ν =0 proton to neutron

32 Isospin for 2 nucleons T = t 1 + t 2 T =0, 1 T =0 η 0,0 = 1 2 (π 1 ν 2 ν 1 π 2 ) η 1,1 = π 1 π 2 T =1 η 1, 1 = ν 1 ν 2 η 1,0 = 1 2 (π 1 ν 2 + π 2 ν 1 ) Proton-proton state Neutron-neutron state Proton-neutron state T =1,T z =1 = pp T =1,T z = 1 = nn 1 [ T =1,T z =0 + T =0,T z =0] = pn 2

33 Isospin for 2 nucleons ψ(1, 2) = ψ pp (r 1, σ 1, r 2, σ 2 )η 1,1 + ψ nn (r 1, σ 1, r 2, σ 2 )η 1, 1 + ψ a np(r 1, σ 1, r 2, σ 2 )η 1,0 + ψ s np(r 1, σ 1, r 2, σ 2 )η 0,0 (1) Pν=1 T =1 1+τ 3 = 2 Pν= 1 T =1 = 1 τ (1) τ (2) τ (2) 3 Wavefunction η 1,1 P T =1 antisymmetric P T =0 = 1 τ (1) τ 2 ν=0 = 1 4 (1 + τ (1) τ (2) 2τ (1) 4 symmetric η 0,0 2 η 1, 1 η 1,0 3 τ (2) 3 )

34 Symmetry for two nucleon states Ψ(r,s 1,s 2,t 1,t 2 )=φ(r )f σ (s 1,s 2 )f τ (t 1,t 2 ) the overall wavefunction must be antisymmetric ( ) L+S+T =( ) L=0, S=1 T=0 3 S1 isospin singlet

35 Sistema di 2 nucleoni identici (pp,nn) ISOSPIN SPAZIO SPIN T z = ±1 Funzione simmetrica (tripletto T=1) ψ(x) ψ(x) L dispari antisimmetrica (no onda S) L pari simmetrica S=1 ψ(σ ) S=0 simmetrica 1 S 0 ψ(σ ) antisimmetrica T z =0 Funzione simmetrica (tripletto T=1) ψ(x) ψ(x) L dispari antisimmetrica (no onda S) L pari simmetrica S=1 ψ(σ ) S=0 simmetrica 1 S 0 ψ(σ ) antisimmetrica Sistema di 2 nucleoni distinti (pn) T z =0 Funzione antisimmetrica (singoletto T=0) L dispari ψ(x) antisimmetrica (no onda S) L pari ψ(x) simmetrica S=0 ψ(σ ) antisimmetrica S=1 3 S 1 ψ(σ ) simmetrica

36 60 ev S0 (T=1) Coulomb 1 S0 (T=1) 1 S0 (T=1) MeV 3 S1 (T=0) pp np nn

37 Additional slides

38 ...many open questions

39 Mean potential method The concept of mean potential (or mean field) strongly relies on the basic assumption of 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 be adequately described assuming individual nucleons moving in an average potential (it means that all the nucleons experience the same field). V (r) = a rough approximation could be dr v(r r )ρ(r ) v(r r )= v 0 δ(r r ) where v0 can be phenomenologically estimated to be V (r) = dr v(r) 200 MeV fm 3 Then one can use a simple guess for V: harmonic oscillator, square well, Woods-Saxon shape... V 0 1+e (r R)/R