Fundamentals in Nuclear Physics

018/ Fundamentals in Nuclear Physics Kenichi Ishikawa ( ) http://ishiken.free.fr/english/lecture.html ishiken@n.t.u-tokyo.ac.jp 1

Schedule Nuclear reactions 5/1 Nuclear decays and fundamental interactions Small test during each session References Basdevant, Rich, and Spiro, Fundamentals in Nuclear Physics (Springer, 005) Krane, Introductory Nuclear Physics (Wiley, 1987) (, 1971) II Material downloadable from: http://ishiken.free.fr/english/lecture.html

Nuclear reactions free particle (photon, electron, positron, neutron, proton, ) Examples projectile scattering target nuclear reactions α + 14 N 17 O + p (Rutherford, 1919) p + 7 Li 4 He + α (Cockcroft and Walton, 1930) 3

Typical nuclear reactions projectile a + X Y + b target X(a,b)Y Important nuclear reactions for thermal energy generation Fission Fusion 4

Energetics a + X Y + b m X c + T X + m a c + T a = m Y c + T Y + m b c + T b rest mass kinetic energy reaction Q value Q =(m initial m final )c =(m X + m a m Y m b )c = T Y + T b T X T a excess kinetic energy Q > 0 : exothermic Q < 0 : endothermic 5

Cross section. L dz number density radius reaction probability n r dp = ( r )n(l dz) L = r ndz ndz r cross section 6

Cross section can be used to define a probability for any type of reaction Probability P proportional to. number density of target particles n target thickness dz dz L dp = ndz Unit of cross section dimension of area m, cm size of nucleus ~ a few fm 1 barn (b) = 10-8 m = 10-4 cm 7

Different types of target objects number density cross section ni i dp = dz i in i 8

Example Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo) Thomson scattering Elastic scattering of electromagnetic radiation (photon) by a free charged particle (often electron) photon e - E(t) =E 0 cos(kz t) incident energy flux 0cE 0 equation of motion mẍ = ee 0 cos t radiated power from an oscillating charge e 6 0c 3 ẍ = 4 3 0 ce 0 e 4 0mc = radiated power incident energy flux = 8 3 e 4 0mc =0.665 b 9

Differential cross section angular dependence ( ) target detector d Probability that the incident particle is scattered to a solid angle d dp, = d d ndzd for isotropic scattering ( ) d d = 4 differential cross section ( ) total cross section = d d d = 0 d 0 d d (, ) sin d 10

Differential cross section reaction creating N particles a b x1 x x3 xn probability to create the particles xi in the momentum ranges d 3 p i around p i dp = d d 3 p 1 d 3 p N n b dz d 3 p 1 d 3 p N differential cross section ( ) total probability for the reaction dp ab x1 x N = ab x1 x N n b dz reaction cross section ab x 1 x N = d 3 p 1 d 3 p N d d 3 p 1 d 3 p N d 3 p 1 d 3 p N if there are more than one reactions dp = tot n b dz tot = i i 11

Mean free path and reaction rate. flux F df = F ndz L df dz = F n dz F (z) =F (0)e nz = F (0)e z macroscopic cross section ( ) = n [1/length] mean free path if there are different types of target objects (nuclei) reaction rate l =1/ l =1/ n i v l = n in i v F(z)/F(0) 1.0 0.8 0.6 0.4 1/e = 0.368 0. 0.0 1/ n also distribution of free path z 1

Rutherford scattering scattering of a charged particles by a Coulomb potential impact parameter classical scattering in general b (b) d b+db = bdb (b + db) = + d = (b)+ d db db d = sin d d d = b( ) sin db d 13

Example: hard sphere with a radius R b = R cos d d = R 4 = R geometrical cross section 14

Rutherford scattering scattering of a charged particles by a Coulomb potential impact parameter V (r) = Z 1Z e 4 0r b = Z 1Z e d cot 8 0E k d = Coulomb force is long-range = Z 1Z e 16 0E k 1 sin 4 The same result is obtained by the quantum theory. Incident particle is scattered no matter how large the impact parameter may be. Practically, the Coulomb potential is screened at large distances by oppositely charged particles 15

Rutherford scattering Geiger-Marsden experiment (1909) Ernest Rutherford 1871 1937 electron 1911 http://mishiewishieblogg.blogspot.jp/ nucleus Rutherford (or planetary) model 16

General characteristics of cross-sections Elastic scattering The internal states of the projectile and target (scatterer) do not change before and after the scattering. Rutherford scattering, (n,n), (p,p), etc. Inelastic scattering (n,γ), (p,γ), (n,α), (n,p), (n,d), (n,t), etc. fission, fusion 17

10 4 10 3 Elastic neutron scattering relevant to (neutron) moderator in nuclear reactors due to the short-range strong interaction JENDL flat region range of the strong interaction el 0 b (fm) 0.1b Cross section (barn) 10 10 1 10 0 10-1 1H(n,n) H(n,n) 6Li(n,n) 10-5 10-3 10-1 10 1 10 3 10 5 10 7 Energy (ev) resonance p > h/r (p /m n > 00 MeV) 18

Elastic neutron scattering 10 9.6 Cross section (barn) 10 4 10 3 10 10 1 10 0 10-1 1H(n,n) H(n,n) 6Li(n,n) JENDL resonance 10-5 10-3 10-1 10 1 10 3 10 5 10 7 Energy (ev) Energy (MeV) 9 8 7 6 5 4 3 1 0 0 7.53 n + 6 Li 4 7.47 6.54 4.63 0.478 0 7 Li The energy levels of 7 Li and two dissociated states n- 6 Li and 3 H- 4 He (t-α) 6.466 3 H + 4 He 8 10 n + 6 Li 7 Li * n + 6 Li 19

Nuclear data libraries ENDF (Evaluated Nuclear Data File, USA) JENDL (Japanese Evaluated Nuclear Data Library, Japan) JEFF (Joint Evaluated Fission and Fusion file, Europe) CENDL (Chinese Evaluated Nuclear Data Library, China) ROSFOND (Russia) BROND (Russia) http://www-nds.iaea.org/exfor/endf.htm 0

Differential cross section for elastic neutron scattering 5 d d cos = d d 1MeV 800 MeV (x50) 1 H θ (barn) 0 5 100MeV (x0) isotropic for dσ /dcos 0.1 MeV 1 MeV 9 Be p< Rnucleus 0 5 10 MeV quantization of angular momentum L = pr nucleus < 0.1 MeV 1 MeV (x0.5) 10 MeV (x0.) 0 1 1 cosθ 08 Pb The differential cross-section, d d cos = d d, for elastic scatter- s-wave scattering 1

Inelastic scattering Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo) Neutron capture neutron binding energy = ca. 8 MeV activation exothermic reaction in most cases Highly excited states formed, which subsequently decay. Radiative capture AX(n,γ) A+1 X emits a gamma ray 113 Cd(n,γ) 114 Cd neutron shield Other neutron capture reactions 10 B(n,α) 7 Li, 3 He(n,p) 3 H, 6 Li(n,t) 4 He Applications: neutron detector, shield, neutron capture therapy for cancer

Inelastic scattering Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo) neutron radiative capture 10 1 No threshold exothermic, no Coulomb barrier JENDL 10 0 Cross section (barn) 10-1 10-10 -3 10-4 10-5 1H(n,gamma) 6Li(n,gamma) H(n,gamma) JENDL ENDF 10-6 10-5 10-3 10-1 10 1 10 3 10 5 10 7 Energy (ev) E 1/ 1/v discrepancy between JENDL and ENDF 1/v law Energy-independent reaction rate v 3

Neutron capture reactions with large cross section 113 Cd(n,γ) 114 Cd : shield 157 Gd(n,γ) 158 Gd : neutron absorber in nuclear fuel, cancer therapy 10 B(n,α) 7 Li : detector, cancer therapy 3 He(n,p) 3 H : detector 6 Li(n,t) 4 He : shield, filter, detector 4

Inelastic scattering 157Gd(n,γ) 158 Gd Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo) Heavy nuclei have many excited states. Complicated resonance structure Cross section (barn) 10 7 10 6 10 5 10 4 10 3 10 10 1 10 0 10-1 10-10 -3 1/v law JENDL 10-5 10-3 10-1 10 1 10 3 10 5 10 7 Energy (ev) thermal neutron = 0.05 ev (4) Auger effect () internal conversion http://www.nuclear.kth.se/courses/lab/latex/internal/internal.html Applications Burnable poison GdO3 (neutron absorber in nuclear fuel) Gadolinium neutron capture therapy (GdNCT) for cancer Possible channels (1) ()+(3) ()+(4) 5

Inelastic scattering Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo) 10B(n,α) 7 Li Cross section (barn) 10 5 10 4 10 3 10 10 1 10 0 10-1 1/v law 10B(n,alpha)7Li JENDL 10-5 10-3 10-1 10 1 10 3 10 5 10 7 Energy (ev) Applications BF3 proportional counter Boron neutron capture therapy (BNCT) for cancer 3He(n,p) 3 H Helium-3 proportional counter 6

Inelastic scattering Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo) Neutron capture by 6 Li 10 9.6 Cross section (barn) 10 4 10 3 10 10 1 10 0 10-1 10-10 -3 10-4 1/v law (n,n) (n,t) (n,gamma) 6Li JENDL resonance (n,p) 10-5 10-3 10-1 10 1 10 3 10 5 10 7 Energy (ev) Energy (MeV) 9 8 7 6 5 4 3 1 0 0 7.53 n + 6 Li 4 7.47 6.54 4.63 0.478 0 7 Li 6.466 3 H + 4 He 8 10 6Li(n,t) 4 He LiF neutron shield and filter Neutron detector The energy levels of 7 Li and two dissociated states n- 6 Li and 3 H- 4 He (t-α) threshold endothermic reaction 7

Inelastic proton scattering Coulomb barrier The low-energy cross-section for inelastic reactions are strongly affected (suppressed) by Coulomb barriers through which a particle must tunnel. 10 - Cross section (barn) 10-3 10-4 10-5 H(n,gamma)3H strong only 8 MeV 10-6 strong + Coulomb H(p,gamma)3He 10-5 10-3 10-1 10 1 10 3 10 5 10 7 Energy (ev) proton energy > the Coulomb potential energy at the 6 Li surface 3e 4 0 1.4fm 1.8MeV 8

High-energy inelastic nucleus-nucleus collision Coulomb barrier ineffective for E cm > Z 1Z e sum of the radii of the two nuclei 4 0R Energy < 1 GeV/u (GeV/nucleon) Total inelastic cross section ~ order of R Break up of one or both of the nuclei Fragmentation reaction - for medium-a nuclei Collision-induced fission - for heavy nuclei Spallation - fragmentation by protons or neutrons application: production of unstable (radioactive) nuclides issue in carbon-ion cancer therapy 9

Fusion evaporation reaction Occasionally, the target and projectile may fuse to form a much heavier nucleus. The produced excited nucleus emits neutrons until a bound nucleus is produced. used to produce trans-uranium elements Energy > 1 GeV/u http://www.phy.ornl.gov/hribf/science/abc/fusion-evap.shtml Production of pions and other hadrons cosmic-ray protons upper-atmosphere nuclei pions + µ + + µ µ + µ muons - primary component of cosmic rays on the ground 30

Photo-nuclear reaction Excitation and break-up (dissociation) through photo-absorption Analog of the photoelectric effect.5x10-3.0 H 0.7 0.6 08Pb 08Pb(gamma,n)07Pb Cross section (barn) 1.5 1.0 H(gamma,n)1H Cross section (barn) 0.5 0.4 0.3 0. 0.5 0.1 0.0 0 5 10 15 0 5 30 Energy (MeV) threshold (. MeV) = binding energy of H 0.0 0 5 10 15 Energy (MeV) giant resonance collective oscillation of protons in the nucleus 0 5 30 31

Neutrino reaction ν e e ν e e ν e e ν e e Only weak interactions Cross section ~ 10-48 m ν µ e ν µ e ν µ e ν µ e ν e n e p ν e p e + n 3

Energy (MeV) 10 9 8 7 6 5 4 3 1 0 0 Cross section (barn) 10 4 10 3 10 10 1 10 0 10-1 10-10 -3 10-4 1/v law (n,n) (n,t) Resonance (n,gamma) 6Li JENDL resonance (n,p) 10-5 10-3 10-1 10 1 10 3 10 5 10 7 Energy (ev) 7.53 n + 6 Li 4 9.6 7.47 6.54 4.63 0.478 0 7 Li 6.466 3 H + 4 He 8 10 cross section (barn) 10 10 10 elastic (x10) (n,γ) (/100) 10 elastic 10 10 1 4 1 4 (n,fission) (/105) (n,γ) (/10 4 ) Many excited states for heavy nuclei complicated resonance structure 35 U 38 U 3 1 10 10 10 Excited states of 39 U E (ev) 33

Resonance line shape (E) Resonance A (E E 0 ) + ( /).0 1.5 1.0 0.5 full width at half maximum (FWHM) long tail Doppler effect 1.0 0.8 0.6 0.4-3 - -1 1 3 Lorentzian Life time = / Decay rate 1 = / = uncertainty principle E : natural width homogeneous width 0. -3 - -1 1 3 inhomogeneous width (E E 0 ) exp E 34

Time-dependent wave function of an excited state (r,t)= (r)e ie 0t/ To be consistent with the exponential decay law (r,t) = (r) does not decay (r,t) = (r) e t/ (r,t)= (r)e ie 0t/ e t/ Energy spectrum (by Fourier transform) P (E) 0 e iet/ e ie 0t/ e t/ dt 1 (E E 0 ) + ( /) 35

- Quantum treatment of nucleon-nucleus scattering 10 4 JENDL Cross section (barn) 10 3 10 10 1 10 0 1H(n,n) H(n,n) 6Li(n,n) 10-1 10-5 10-3 10-1 10 1 10 3 10 5 10 7 Energy (ev) 36

V(r) isotropic scattering R R 1.A 1/3 fm r angular momentum L = kr kr 1 V 0 for neutron scattering E = p ( c) m n m n c R 13 MeV A /3 Schrödinger equation z e ikr r m + V (r) k(r) = k m k (r) e ikz k(r) =e ikz + feikr r (r >R) 37

e ikz = l=0 (l + 1)i l j l (kr)p l (cos ) spherical Bessel function Legendre polynomial P 0 (cos )=1 P 1 (cos ) = cos P (cos )= 3 cos 1 : ( ) = 0(1)4 0 apple apple 1. j l (kr) (kr) l /(l + 1)!! 5 j 0 (z) = sin z z j 1 (z) = sin z z cos z z θ (barn) dσ /dcos 0 5 0 5 800 MeV (x50) 1MeV 100MeV (x0) 0.1 MeV 1 MeV 10 MeV 1 H 9 Be 0.1 MeV 1 MeV (x0.5) 10 MeV (x0.) 0 1 1 cosθ 08 Pb 38

k(r) = e ikz sin kr kr anisotropic sin kr kr r isotropic + feikr + sin kr + feikr kr r isotropic near the boundary (r >R) kr 1 k(r) isotropic also at r < R u k (r) =r k (r) m d u k dr + V (r) u k(r) = k m u k(r) u k (r) = sin kr k + fe ikr (r>r) 39

Solution at r < R u k (r) =A sin Kr (r <R) Boundary condition kr 1 f = R Cross section Scattering length u k (r) and u k (r) continuous at r = R tan KR KR =4 f =4 R tan KR KR a = f(k = 0) (k 0) = 4 a mv 1 0 K low-energy scattering 1 40

=4 f =4 R tan KR KR 1 R 30 5 0 15 10 5 1 3 4 5 6 KR R mv 0 ~ depth of attractive potential kr = / infinite scattering length R no scattering Ramsauer-Townsend effect 41

Cross section (barn) 10 4 10 3 10 10 1 10 0 10-1 Nucleon-nucleon effect 1H(n,n) H(n,n) 6Li(n,n) 10-5 10-3 10-1 10 1 10 3 10 5 10 7 Energy (ev) - JENDL flat region range of the strong interaction el 0 b (fm) 0.1b u kr / V 0 R ~109 MeV fm u(r) = sin kr r=r u(r) = exp( κ r) deutron V(r) V(r)= V0 V(r)=0 Fig. 1.10. A square-well potential and the square of the wavefunction ( ) = r 4

Nucleon-nucleon effect - f R V 0 V 0 R (fm) (fm) (MeV) (MeV fm ) n p (s=1, T=0) +5.43 ± 0.005 1.73 ± 0.0 46.7 139.6 n p (s=0, T=1) 3.715 ± 0.015.73 ± 0.03 1.55 93.5 p p (s=0, T=1) 17.1 ± 0..794 ± 0.015 11.6 90.5 n n (s=0, T=1) 16.6 ± 0.6.84 ± 0.03 11.1 89.5 n p = 3 4 4 f s=1 + 1 4 4 f s=0 0 b 43