18/05/14 F. Ould- Saada 1

Transcription

1 1. Basic Concepts 2. Nuclear Phenomenology 3. Particle Phenomenology 4. Experimental Methods 5. Quark Dynamics: The Strong Interaction 6. Weak Interactions And Electroweak Unification 7. Models And Theories Of Nuclear Physics 8. Applications Of Nuclear Physics 9. Outstanding Questions and Future Prospects 18/05/14 F. Ould- Saada 1

2 Nuclei Held together by strong nuclear force between nucleons Form of force: more complicated than simple 1- particle exchange à Phenomenological evidence from low energy NN scattering experiments Interpretation of results in terms of fundamental strong interaction between quarks Various models and theories constructed to explain nuclear data in particular domains 18/05/14 F. Ould- Saada 2

3 Stable nuclei Overall net NN force attractive + much stronger than Coulomb force Repulsive core at very short distances (negligible at low energies) Resulting potential Idealised square well R: range of force δ<<r : short range repulsion important V 0 ~40MeV In practice potential smooth at boundaries + must add Coulomb for pp 18/05/14 F. Ould- Saada 3

4 Comparison nn & pp scattering data Nuclear Force charge- symmetric: pp=nn NF almost charge- independent: pp=nn=pn Nucleon- nucleon forces are spin- dependent pn in an overall spin- 1 state à deuteron spin- 0 pn (antiparralel spins) leads to no bound state Nuclear Forces saturate Nucleon in nucleus experiences attractive interaction only with a limited # nucleons (short- range nature of NF) (2.3.1) Difficult to interpret N- N potential in terms of fundamental q- q interactions 18/05/14 F. Ould- Saada 4

5 Difficult to interpret N- N potential in terms of fundamental q- q interactions Nucleons are colourless objects Pion exchange plays a role, heavier mesons (di- quarks) also possible Each exchange would give a contribution to the overall N- N- potential (in analogy with the Yukawa potential resulting from exchange of spin- 0 meson) à various couplings Boson exchange models cannot give a fundamental explanation of the repulsion Instead, specific models are used to describe phenomena in different areas of nuclear physics 18/05/14 F. Ould- Saada 5

6 n and p making up nucleus 2 independent systems of nucleons (spin1/2 fermions) freely moving within nuclear volume subject to constraints of Pauli principle Potential felt by every nucleon superposition of potential due to all the other nucleons Assumed to be a finite- depth square well, modified by Coulomb potential in case of protons 18/05/14 F. Ould- Saada 6

7 For a given ground state, energy levels fill up from bottom of well Fermi level (E F, p F ) = energy of highest level that is completely filled p F = 2ME F Fermi levels of protons and neutrons in a stable nucleus have to be equal (otherwise nucleus can become more stable by β- decay) à Depth of well for neutron- gas deeper than that for proton- gas 18/05/14 F. Ould- Saada 7

8 Density of states factor n(p)dp (! App. A) Number of states with momentum between p and p+dp n( p)dp = 4πV p 2 dp ( 2π! ) 3 Every state contains up to 2 fermions of same species Number of neutrons and protons within nuclear volume V V = 4 3 πr3 = 4 3 πr 3 0 A ; R 0 =1.21fm n = 2 p F 0 n(p)dp ( ) 3 ( ) 3 N = V p n F 3π 2! ; Z = V p p F 3 3π 2! 3 Assume depths of n and p wells same (Z=N=A/2) p F ~250MeV nucleons move freely within nucleus with large momenta Difference between top of well V 0 and Fermi level E F : constant for most heavy nuclei (~B/A, figure previous slide) V 0 and E F ~independent of A Heavy nuclei generally have surplus of neutrons p F = p n F = p p F =! $ 9π ' & ) R 0 % 8 ( 1/ 3 B B / A = 7 8MeV V 0 = E F + B 40MeV 250MeV /c 2 E F = p F 2M 33MeV 18/05/14 F. Ould- Saada 8

9 Theoretical expression for some of the dependence of Binding Energy on surplus of neutrons Δ=N- Z Average kinetic energy per nucleon E kin 0 p F 0 E kin p F p 2 dp p 2 dp = 3 5 p F 2 2M 20MeV Total kinetic energy of nucleus E kin (N,Z) Expression for fixed A but varying N Has minimum at N=Z Power series in Δ/A à dependence on neutron excess E kin (N,Z) = N E n + Z E p = 3 10M N p n F E kin (N,Z) = 3 10M [ ( ) 2 p + Z( p F ) 2 ]! 2 R 0 2 # 9π & % ( $ 4 ' 2 / 3 ) N 5 / 3 + Z 5 / 3, + * A 2 / /05/14 F. Ould- Saada 9

10 Dependence on neutron excess N = A + Δ 2 Z = A Δ 2 E kin (N,Z) = 3 10M! 2 R 0 2 ' 9π * ), ( 8 + Δ N Z 2 / 3 - / A ( N Z) 2 A Various terms 1st term contributes to volume term of SEMF 2nd term describes correction resulting from N Z (Δ>0) Contribution to asymmetry term Contribution to asymmetry coefficient (2.54) is ~44MeV/c 2 compared to empirical value (2.57): 93MeV/c 2 In practice to obtain actual term more accurately, need to take into account change in potential energy for N Z 18/05/14 F. Ould- Saada 10

11 Binding Energy of electrons in atoms: Primarily due to central Coulomb potential of nucleus Complications due to Coulomb field of other electrons Atomic energy levels characterised by quantum numbers (QNs) Any energy eigenstate in hydrogen atom is labelled by QNs (n,l,m l,m s ) Principal quantum number (QN) n =1,2,3,4,... Orbital angular momentum QN l = 0,1,2,3,...,(n -1) Magnetic QN m l = -l,-l +1,...,0,1,...,l -1,l Spin projection QN m s = ± 1 2 n d degenerate energy states n -1 n d = 2 (2l +1) = 2n 2 l =0 High degree of degeneracy can be broken in case of preferred direction in space Magnetic field à dependence on m l and m s Spin- orbit coupling à correction to energy levels fine structure (constant α) 18/05/14 F. Ould- Saada 11

12 Beyond hydrogen Electron- electron Coulomb interaction à splitting in any energy level n according to l If shell or sub- shell filled then m s = 0 ; m l = 0 Strong pairing effect for closed shells Pauli principle è! L = S! = 0! ;! J = L! + S! = 0! No valence electrons available à inert atoms Z= 2, 10, 18, 36, 54, à atomic magic numbers He, Ne, Ar, Kr, Xe, Ar: closed shells for n=1,2; closed sub- shells n=3, l=0,1 18/05/14 F. Ould- Saada 12

13 Evidence for magic numbers in NP Values of Z and N at which nuclear binding particularly strong B/A curve (Fig. 2.8) + à Where data lie above SEMF predictions Nuclear magic numbers N=2,8,20,28,50,82,126 Z=2,8,20,28,50,82 Doubly magic if both Z and N (ex: α) Magic nuclei have more stable isotopes than other nuclei very small electric dipole moments (almost spherical) Neutron capture cross sections show strong drops Sharp changes in nucleon separation energies What about effective potential? 18/05/14 F. Ould- Saada 13

14 Simple Coulomb potential not appropriate Need some form to describe effective potential of all other nucleons Strong Nuclear force short- range Expect potential to follow form of nucleon density function in nucleus Fermi distribution fits data (Chap.2) Corresponding potential: Woods- Saxon form Only works for low magic numbers Introduce spin- orbit term (analogy to atomic physics)! J = L! + S! J! 2 = L! 2 + S! L! S! ls = "2 2 [ j( j +1) l(l +1) s(s +1) ] = "! L S! = 1 2 V V central (r) = 0 (r R )/ a 1+ e V total = V central (r) +V ls (r) L! S!! J 2 L! 2! ( S 2 ) % l /2 j = l +1/2 2 & ' (l +1) /2 j = l 1/2 ( ) Splitting between the 2 levels Experimentally V ls (r)<0 State j=l+1/2 has lower energy than j=l- 1/2 Opposite to situation in atoms ΔE ls = 2l +1 2!2 V ls 18/05/14 F. Ould- Saada 14

15 Woods- Saxon potential plus spin- orbit term Low- lying energy levels à (nl j ) k k: occupancy of given sub- shell Max occupancy per sub- shell: 2j O ( 5 2) + p : (1s 1/ 2 ) 2 (1p 3 / 2 ) 4 (1p 1/ 2 ) 2 n : (1s 1/ 2 ) 2 (1p 3 / 2 ) 4 (1p 1/ 2 ) 2 (1d 5 / 2 ) 1 Most ground state properties of 17 8 O Just by stating (1d 5/2 ) 1 18/05/14 F. Ould- Saada 15

16 Energy levels and magic number sequences obtained by solving using a harmonic oscillator potential, the Woods Saxon potential and the potential with a spin orbit coupling term The last one reproduces the observed sequence of magic numbers 16

17 Nuclear shell model predictions about spins of ground states A filled sub- shell must have J=0 2j+1 always even for filled sub- shells, for eqch nucleon with m j there is another with - m j Magic Z/N predicted to have zero nuclear spin, as observed experimentally Same holds for all even- Z/even- N nuclei Pairing hypothesis Last proton and/or neutron determines net nuclear spin For even- A odd- Z/odd- N nuclides, both unpaired p and unpaired n net spin: j p - j n to j p +j n 18/05/14 F. Ould- Saada 17

18 Nuclear parities P ˆ Ψ lmn ( r! ) = PΨ lmn ( r! ) = P( 1) l Ψ lmn ( r! ) Ψ lmn ( r! )is eigenstate of parity with eigenvalue P( 1) l P N = +1 ˆ P Y l m (θ,φ) = ( 1) l Y l m (θ,φ) P l = ( 1) l 13 6O & 13 7 N : j = 1 / 2,l =1 ( 1 2) (1s 1/ 2 ) 2 (1p 3 / 2 ) 4 (1p 1/ 2 ) S 3 2 ( ) + : (1s 1/ 2 ) 2 (1p 3 / 2 ) 4 (1p 1/ 2 ) 2 (1d 5 / 2 ) 6 (2s 1/ 2 ) 2 (1d 3 / 2 ) Ti 7 2 ( ) :...(1f 7 / 2 ) 5 BUT exp : 5 ( 2) 18/05/14 F. Ould- Saada 18

19 Magnetic moment for even- odd nuclei à expect paired nucleons contribute zero Consider single nucleon Combine g l l and g s s! µ = g e 2mc! s ; µ B = e! 2mc = MeV / T µ N = e! 2m p c µ B / 2000; µ p 2.79µ N ; µ n 1.91µ N µ = gjµ N µ N : nuclear magneton; g: g-factor g = j( j +1)+ l(l +1) s(s +1) 2 j( j +1) g l + j( j +1) l(l +1)+ s(s +1) 2 j( j +1) g s See reference in book for derivation 18/05/14 F. Ould- Saada 19

20 g = j( j +1) + l(l +1) s(s +1) 2 j( j +1) g l + j( j +1) l(l +1) + s(s +1) 2 j( j +1) * jg = g l l + g s /2 j = l j = l ± 1, + 2 $ jg = g l j& 1+ 1 ' $ 1 ', ) g % 2l +1( s j& ) j = l % 2l +1( g l ( p) =1; g l (n) = 0; g s ( p) +5.6; g s (n) 3.8 * l = j j = l + 1, 2 jg p = +, j 2.3j j = l j +1 * 1.9 j = l + 1, 2 jg n = j, j = l j +1 18/05/14 F. Ould- Saada 20 g s

21 Measured Magnetic moments for Odd- N, even- Z (top) and Odd- Z even- N (bottom) as function of nuclear spin Compared to prediction of single- particle shell model For a given j, measured moments lie between j=l- 1/2 and j=l+1/2 (Schmidt) lines No accurate predictions of moments Exception: a few low- A nuclei where number of nucleons close to magic values Reason? Nucleons inside nuclei may have effective intrinsic moments different from the free- particle values 18/05/14 F. Ould- Saada 21

22 In principle, ShM s energy level structure can be used to predict nuclear excited states Works ok for first one or 2 excited states when only one possible configuration of nucleus For higher states situation more complicated Several nucleons can get excited simultaneously into a superposition of different configs to produce a nuclear spin & parity Excited energy levels involve either Moving an unpaired nucleon to the next high level Or moving a nucleon from sub- shell below unpaired nucleon up one level to pair with it 17 8O ( 5 2) + : # p : (1s 1/ 2 ) 2 (1p 3 / 2 ) 4 (1p 1/ 2 ) 2 $ % n : (1s 1/ 2 ) 2 (1p 3 / 2 ) 4 (1p 1/ 2 ) 2 (1d 5 / 2 ) 1 To excite 17 8 O: 1. Promote one of the 1p 1/2 protons to 1d 5/2 à 2. Promote one of the 1p 1/2 neutrons to 1d 5/2 à 3. Promote the 1d 5/2 neutron to 2s 1/2 à (1p 1/2 ) 1 (1d 5/2 ) 1 (1p 1/2 ) 1 (1d 5/2 ) 2 (2s 1/2 ) 1 or (1d 3/2 ) 1 18/05/14 F. Ould- Saada 22

23 17 8 O 1. Promote one of the 1p 1/2 protons to 1d 5/2 à 2. Promote one of the 1p 1/2 neutrons to 1d 5/2 à 3. Promote the 1d 5/2 neutron to 2s 1/2 à (1p 1/2 ) 1 (1d 5/2 ) 1 (1p 1/2 ) 1 (1d 5/2 ) 2 (2s 1/2 ) 1 or (1d 3/2 ) 1 Fig à possibility 3 above corresponds to smallest energy shift and should be favored Then 2 (keeps excited neutron paired with another), then 1 (creates 2 unpaired protons) Expected excited states exist but not necessarily in predicted order ShM has limitations because Nucleons assumed to move independently in a spherical symmetric potential (only valid for nuclei close to having doubly filled magic shells à quadruple moment is zero) In practice many nuclei deformed à quadruple moments large additional modes of excitation non- zero EQM 18/05/14 F. Ould- Saada More in book 23 Non- sphericity can mean new phenomena

24 Non- sphericity can mean new phenomena additional modes of excitation non- zero EQM Charge distribution in nucleus in terms of electric multi- pole moments (EDM vanishes if WI neglected) Axis of symmetry is z- axis Classically: Q intrinsic = value of EQM for an ellipsoid at rest (charge Ze) eq ρ( r!! )( 3z 2 r 2 )d 3 r ) + Q intrinsic = 2 Z 5 ( a2 b 2 ) + * 6 5 ZR2 ε (small deformations) + + R a = R(1+ε); b =, 1+ε prolate oblate 18/05/14 F. Ould- Saada 24

25 Quantum mechanically (no proofs) EQM depends on j and m of nucleus EQM is value of Q for which M has maximum value along z- axis Predictions 1 particle ShM Odd- A, Odd- N: single neutron outside closed sub- shells à Q=0 (no e- charge) Idem for all Even- Z, odd- N à Q=0 (pairing effect)! EQM : eq = ψ * q i ( 3z 2 i r 2 )ψd 3 r i! EDM : d z = ψ * q i z i ψd 3 r 0 i Odd- A, Odd- Z with single proton with j outside closed sub- shells Q R 2 2 j 1 2( j +1) ; Q = 0 for j =1/2 18/05/14 F. Ould- Saada 25

26 Odd- N: grey, odd- Z: black Arrows: position of major closed shells Change of sign of Q One proton less than a closed shell behaves like a hole with negative charge Features from diagram Q~- R 2 for odd- Z with only few nucleons outside closed shell In general EQM larger by 2-3 (up to 10) Odd- N nuclei also have non- zero EQM Spherical symmetry not good approximation 18/05/14 F. Ould- Saada 26

27 Rainwater EQM in terms of non- spherical nuclei Expression (in fission model) assumed to hold only for closed- shell nuclei ΔE B = αε 2 α = 1 ( 5 2a sa 2 / 3 a c Z 2 A 1/ 3 ) Additional term linear in ε Values of EQM of correct sign but overestimation by typically factor of 2 Better estimates of β allow better agreement Collective model ΔE B = αε 2 βε min : β 2 /4α for ε = β/2α Q = j(2 j 1) ( j +1)(2 j + 3) Q int rinsic 18/05/14 F. Ould- Saada 27

28 Rainwater model equivalent to assuming a spherical liquid drop Å. Bohr, Mottelsen Many properties of heavy nuclei could be ascribed to the surface motion of a drop However single particle Shell Model explains general features of nuclear structure Collective Model (CM): Reconcile shell and liquid drop models 18/05/14 F. Ould- Saada 28

29 Collective Model (CM) Reconciles shell and liquid drop models CM views nucleus as having Hard core of nucleons in filled shells With outer valence nucleons behaving like surface molecules of a drop model Motions of valence nucleons introduce non- sphericity in the core à perturbation of quantum states J(J +1)!2 Nucleus can rotate and vibrate E J = 2I rotational (I: moment of inertia) vibrational energy levels due to shape oscillations between prolate and oblate ellipsoids Energy spacing: (simple harmonic oscillator) ΔE =!ω Many prediction of CM confirmed experimentally 18/05/14 F. Ould- Saada 29

30 Liquid drop model All nuclei have similar mass densities B/A proportional to M SEMF good description of <M> and B/A Largely classical + some QM (asymmetry, pairing terms in ad- hoc way) Input from experiment needed to extract SEMF coefficients Fermi gas model Nucleons move independently in a net nucleon potential Quantum statistics of Fermi gas à depth of potential and asymmetry term of SEMF Shell model Quantum mechanics model Shrödinger eq. with specific spherical nuclear potential Strong spin- orbit term Predicts nuclear magic numbers, spins, parities of ground state nuclei; pairing term of SEMF Less successful in predicting magnetic moments Collective model QM model Potential allowed to undergo deformation from strictly spherical form used in shell model Can predict MDM and EQM with some success Additional modes of excitation (vibrational, rotational) 18/05/14 F. Ould- Saada 30

31 α decay (assume line of stability is N=Z=A/2) Z=N à one independent variable, A energetically possible if B(2,4) > B(Z,A) B(Z 2, A 4) 4 db da 4 db $ d(b / A) = 4 A da da + B ' % & A ( ) 19/05/14 F. Ould- Saada 31

32 B / A plot d(b / A) da MeV for A 120 B(2,4) = 28.3MeV ' 28.3MeV 4 B A 7.7 * 10 3 A ( ) +, Straight line on B/A vs A plot cutting plot at A~151, above which α decay energetically possible (inequality satisfied) 18/05/14 F. Ould- Saada 32

33 Lifetime of α emitters span 10ns years - due to tunnelling effect p&n (B~7-8MeV in nucleus) cannot escape Whereas bound states α can escape r<r: strong nuclear potential r>r: Coulomb potential of daughter nucleus, V c (r) α can escape à E α >0 energy released in the decay Potential energy of an α particle 18/05/14 F. Ould- Saada 33

34 QM tunneling through barrier (App. A) T: Transmission probability through (V,Δr) Coulomb barrier as successive thin barriers à Gamow factor G (see App. A) m: reduced mass (α, daughter) T e 2κ Δr ;!κ = 2mV C E α T = e G ; G = 2! m = r C R m α m D m α + m D m α 2mV C (r) E α dr E α = V C (r C ) r C = 2Zα!c E α V C (r) = r C E α r G = 2 2mE α! % r ' C & r 1 ( * ) 18/05/14 F. Ould- Saada 34 r C R dr

35 Evaluate integral G = 4αZ 2mc 2 * $ R ' & cos 1 ) R $ 1 R '-, & )/ E α +,% r C ( r C % r C (./ E α 5MeV,V 40MeV,r C >> R G 4παZ β Probability per unit time λ of α escaping from nucleus proportional to Probability w(α) of finding α in nucleus Frequency of collision of α with barrier v α /2R Transition probability λ = w(α) v α 2R e G G Z β Z E α ' ) ( ) *) log 10 t 1/ 2 = a + b Small differences in E α have strong effects on lifetime Z E α ; β = v α c 18/05/14 F. Ould- Saada 35

36 log 10 t 1/ 2 = a + b Z E α Geiger- Nuttall empirical relation (1911) before theoretical derivation (1928) For fixed Z log of ½ lifetime of α emitters varies linearly with E α - 1/2 a: depends on w(α); is function of nucleus b: constant ~1.7 è Simple barrier penetration model capable of explaining wide range of lifetimes Changing E α by factor 2.5 leads to lifetime change of 20 orders of magnitude! 18/05/14 F. Ould- Saada 36

37 Nuclear β decay: 1 st successful theory by Fermi, 1934 Analogy with QED Transition amplitude Μ fi = Ô: combination of Lorentz invariant forms: S, PS, V, A, T Correct helicity properties with V- A combination (for purely leptonic decays) Relative strength from experiment in case of nuclei (extended objects) V.V combinations: Fermi transitions A.A combinations: Gamow- Teller transitions ψ f * (g ˆ O )ψ i dv 18/05/14 F. Ould- Saada 37

38 Transition rate ω=1/τ Second Golden Rule density of states n(e): number of quantum states available to the final system per unit interval of total energy (see Appendix A.2) M fi : transition amplitude = matrix element G F : universal in nuclear theory related to the weak coupling constant α W n p + e +ν e ω = 2π! Μ fi 2 n(e) M fi G F V Μ fi G F 2 = 4π(!c)3 α w ( M W c 2 ) 2 Lifetime of muon τ µ à G F 1 τ µ = ( m µ c 2 ) 5 192π 3!(!c) G 2 6 F G F 90 ev.fm 3 18/05/14 F. Ould- Saada 38 G F (!c) 3 = GeV 2

39 Consider p and n to be heavy (T p, T n ~0) Total energy released carried by electron and neutrino: E 0 = E e + E ν = Δm.c 2 Transition rate for decays where electron has energy in range E and E+dE dω = 2π! Μ fi n p + e +ν e 2 nν (E 0 E e )n e (E e )de e Density of state factors for e (and nu) n( p e )dp e = 4πV ( 2π! ) p 2 3 edp e dp de = E pc n(e )de = 2 e e 4πV ( 2π! ) 3 c p E de 2 e e e Setting in transition probability and density of state factors à dω de e = G 2 2 F M fi 2π 3! 7 c 4 p e E e p ν E ν 18/05/14 F. Ould- Saada 39

40 Change of variable p ν c = E υ 2 m υ 2 c 4 = (E 0 E e ) 2 m υ 2 c 4 dω dp e = de e dp e dω de e = G 2 2 F M fi 2π 3! 7 c p 2 2 e p ν E ν Neutrino exactly massless à dω dp e = G 2 2 F M fi 2π 3! 7 c p 2 3 e E 2 ν = G 2 2 p 2 F M e (E 0 E e ) 2 fi 2π 3! 7 c 3 18/05/14 F. Ould- Saada 40

41 Matrix element ψ i,,ψ f : nuclear wave functions Μ fi = ψ f * (g ˆ O )ψ i dv Μ fi = ψ f * ψ e * ψ υ * Hψ i dv If lepton WF as free particles M fi depends on kinematical variables via density of state factor, not on p e Approximation good for allowed transitions e i! p e + p!! % r ( ' ( υ ) * & " ) E e 1MeV ; p = 1.4MeV / c ; p /! / fm pr /! << 1 e [...] 1+ε 18/05/14 F. Ould- Saada 41

42 Need to consider spin effects Not present in simple Fermi theory Fermi transitions L=0 S(e) and s(ν) antiparallel à no change in nuclear spin ΔJ= J i - J f =0 and no change in parity ΔP=0; and isospin ΔI=0 Gamow- Teller transitions S(e) and s(ν) parallel ΔJ=0,1 and ΔP=0; and ΔI=0,1 (nuclear WF can change) 0+à 0+ forbidden since lepton- pair carries 1 unit angular momentum Mixed transitions Weighted average of Fermi and Gamow- Teller ME Relative weight through experiment Fermi 14 O (0 + ) 14 N * (0 + ) + e + +ν e Gamow Teller 6 He (0 + ) 6 Li (1 + ) + e +ν e Mixed n ( ) p ( ) + e +ν e 18/05/14 F. Ould- Saada 42

43 Kurie Plots Test of p t distribution of electrons Allowed transitions M fi independent of electron kinematic variables Z=0, without Fermi screening factors F(Z,E e ), effects of EM force neglected dω dp e = F(Z,E e )G F 2 M fi 2 p e 2 (E 0 E e ) 2 2π 3! 7 c 3 18/05/14 F. Ould- Saada 43

44 Allowed transitions H(E e ) $ dω ' & ) % dp e ( 1 p e 2 F(Z, p e ) E 0 E e For forbidden transitions Kurie plot not a straight line 18/05/14 F. Ould- Saada T e = E e m e c 2 44

45 Study of precise shape of momentum distribution near upper end- point Way of finding m νe If m νe 0 à H(E e ) $ dω ' & ) % dp e ( 1 p 2 e F(Z, p e ) { (E E ) (E E 0 e 0 e )2 m 2 υ c 4 } 1/ 2 {(T 0 T e ) (T 0 T e ) 2 m 2 υ c 4 } 1/ 2 3 H 3 He + e +υ e 18/05/14 F. Ould- Saada 45

46 3 H 3 He + e +υ e ; T 0 =18.6keV m υ e < 2 3eV /c 2 18/05/14 F. Ould- Saada 46

47 Total decay rate from integration of dω/dp e cp max = E 0 2 m e 2 c 4 For allowed transitions Total width Comparative half- life f t 1/2 is a direct measurement of the matrix element ω = G 2 2 p F M max fi F(Z, p 2π 3! 7 c 3 e ) p 2 e (E 0 E e ) 2 dp e f (Z,E) ω = G 2 Fm 5 e c 4 2π 3! 7 log 10 ( ft 1/ 2 ) = (5.5 ±1.5)s 0 M fi 2 f (Z,E 0 ) ω = ln2 t 1/ 2 ft 1/ 2 = (2ln2)π 3! 7 G F 2 m e 5 c 4 1 allowed p max 1 F(Z, p (m e c) 3 (m e c 2 ) 2 e ) p 2 e (E 0 E e ) 2 dp e = (7.5 ± 1.5)s first forbidden = 3 4 super allowed 0 M fi 2 p e c E e ω E 5 (Sargent&s rule) 18/05/14 F. Ould- Saada 47

48 Excited states of nuclei decay to lower states down to ground state by photon emission de- excite by ejecting electron from a low- lying atomic orbit Gamma emission in form of EM radiation Caused by changing E- field inducing M- field 2 possibilities Electric (E) radiation and magnetic (M) radiation Total angular momentum and parity conserved in EM γ (spin-1) carries away angular momentum (multipolarity) L L=1: dipole, L=2 (quadrupole), L=3 (octupole), L γ =1 L z = ±1! J i = J! f + L! J i J f L J i + J f! J i = 0! J! f = 0! forbidden 18/05/14 F. Ould- Saada 48

49 General result: E- multipole: P=(- 1) L ; M- multipole: P=(- 1) L+1 (E- dipole: qr à P=- 1; M- dipole:qrxvà P=+1) El : ΔP = ( 1) L Ml : ΔP = ( 1) L : L =1: ΔP = + "no"= ( 1) L +1 M : L =1: ΔP = " yes"= ( 1) L E1 18/05/14 F. Ould- Saada 49

50 El : ΔP = ( 1) L Ml : ΔP = ( 1) L +1 γ 0 0 : L z = ±1 L 1 not allowed : L =1: ΔP = + "no"= ( 1) L +1 M : L =1: ΔP = " yes"= ( 1) L E :L=1 or 2 :ΔP=+ "no" M1 or E2 2 2 Although transitions J i =0à J f =0 forbidden if photon is real Can occur for virtual photons (transverse polarisation), provided parity does not change Internal conversion or internal pair production 18/05/14 F. Ould- Saada 50

51 Transition probability per unit time emission rate T fi E,M (L) Semi- classical radiation theory Reduced transition probability B E,M (L) Contains all nuclear information T E,M fi (L) = 1 8π(L +1) 4πε 0 L (2L +1)!! 1 [ ] 2! % E γ ( ' * &!c ) 2L +1 B E,M fi (L) " $ # $ % reduced transition probability # B E (L) = e2 3R L & % ( 4π $ L + 3' #!c & B M (L) =10% $ m p c 2 R ( ' 2 2 B E (L) B E (L) = e2 2 # 3 & % ( (R 4π $ L + 3' 0 ) 2L A 2L / 3 B M (L) = 10 # e!c & π % $ 2m p c 2 ( ' 2 # 3 & % ( $ L + 3' [ e 2 fm 2L ] (µ N / c) 2 2L 2 [ fm ] 2 (R 0 ) 2L 2 (2L 2)/ 3 A R = R 0 A 1/ 3 ; R 0 =1.21fm 18/05/14 F. Ould- Saada 51

52 Transition rates using single- particle shell model formulas of Weisskopf: A=60 assuming radiation results from transition from a single proton from initial state of shell model to final state of J=0 Decrease in decay rate with increasing L E- transitions ~100 times larger than M- transitions à Radiation widths G g Γ γ (E1) = 0.068E 3 γ A 2 / 3 3 Γ γ (M1) = 0.021E γ Γ γ (E2) = ( )E 5 γ A 4 / 3 18/05/14 F. Ould- Saada 52

53 Pages , 6.2 (chapter 6) 18/05/14 F. Ould- Saada 53