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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 ¡
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 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 ▪  V0~40MeV §  In practice potential smooth at boundaries + must add Coulomb for pp 18/05/14 F. Ould-­‐Saada 3 ¡
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 ¡
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 ¡
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 ¡
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For a given ground state, energy levels fill up from bottom of well Fermi level (EF, pF) = energy of highest level that is completely filled pF = 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 ¡
Density of states factor n(p)dp (! App. A) §
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¡
Number of states with momentum between p and p+dp Every state contains up to 2 fermions of same species Number of neutrons and protons within nuclear volume V V=
¡
¡
pF~250MeV n€
ucleons move freely within nucleus with large momenta n = 2 ∫ n( p)dp
0
N=
§
V0 and EF ~independent of A Heavy nuclei generally have surplus of neutrons 18/05/14 F. Ould-­‐Saada V ( pFn )
3
; Z=
3π 2! 3
V ( pFp )
3
3π 2! 3
1/ 3
! $ 9π '
⇒ pF = p = p = & )
R0 % 8 (
Difference between top of well V0 and Fermi level EF : constant €
for most heavy nuclei (~B/A, figure previous slide) §
4 πV 2
3 p dp
(2π!)
pF
4 3 4 3
πR = πR0 A ; R0 = 1.21 fm
3
3
Assume depths of n and p wells same (Z=N=A/2) €
§
§
n( p)dp =
n
F
p
F
≈ 250MeV /c
pF2
EF =
≈ 33MeV
2M
B˜ ≡ B / A = 7 − 8MeV
€
V0 = E F + B˜ ≈ 40MeV
8 ¡
¡
¡
Theoretical expression for some of the dependence of Binding Energy on surplus of neutrons Δ=N-­‐Z Average kinetic energy per nucleon Total kinetic energy of nucleus €
Ekin(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 ≡
∫
pF
E kin p 2 dp
0
∫
pF
0
p 2 dp
3 pF2
=
≈ 20MeV
5 2M
E kin (N,Z) = N E n + Z E p
3
n 2
p 2
=
N ( pF ) + Z ( pF )
10M
2/3
3 ! 2 # 9π & ) N 5 / 3 + Z 5 / 3 ,
E kin (N,Z) =
( +
.
2%
10M R0 $ 4 ' *
A2 / 3
-
[
]
18/05/14 F. Ould-­‐Saada €
9 N=
Dependence on neutron excess ¡
A+Δ
2
Z=
A−Δ
2
Δ ≡N −Z
2
2/3
0
3 ! 2 ' 9π * 5 (N − Z )
⇒ E kin (N,Z) =
+ ...2
) , /A +
10M R02 ( 8 + .
9
A
1
Various terms € to volume term of SEMF §  1st term contributes §  2nd term describes correction resulting from N≠Z (Δ>0) ▪  Contribution to asymmetry term ▪  Contribution to asymmetry coefficient (2.54) is ~44MeV/c2 compared to empirical value (2.57): 93MeV/c2 §  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 ¡
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,ml,ms) Principal quantum number (QN)
n = 1,2,3,4,...
Orbital angular momentum QN
l = 0,1,2,3,...,(n -1)
Magnetic QN
ml = -l,-l +1,...,0,1,...,l -1,l
Spin projection QN
1
ms = ± 2
n -1
§  nd degenerate energy states 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 ml and ms ▪  Spin-­‐orbit coupling à correction to €
energy levels – fine structure (constant α) 18/05/14 F. Ould-­‐Saada 11 §  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 ¡
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 ¡
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) Vcentral (r) =
§  Corresponding potential: Woods-­‐Saxon form ▪  Only works for low magic numbers §  Introduce spin-­‐orbit term (analogy to atomic physics) ! !
Vtotal = Vcentral (r) + Vls (r) L⋅ S
(
)
Splitting between the 2 levels §  Experimentally Vls(r)<0 §  State j=l+1/2 has lower energy than j=l-­‐1/2 §  €Opposite to situation in atoms 18/05/14 )
! ! !
!
! !
! !
! ! 1 !
! !
€
J = L + S ⇒ J 2 = L2 + S 2 + 2 L⋅ S ⇒ L⋅ S = J 2 − L2 − S 2
2
€
% l /2
j = l +1/2
"2
2
ls = [ j( j +1) − l(l +1) − s(s +1)] = " &
j = l −1/2
2
' −(l +1) /2
(
¡
−V0
1+ e(r −R )/ a
F. Ould-­‐Saada ΔE ls =
2l +1 2
! Vls
2
14 ¡
Woods-­‐Saxon potential plus spin-­‐orbit term §
§
§
Low-­‐lying energy levels à (nlj)k k: occupancy of given sub-­‐shell Max occupancy per sub-­‐shell: 2j+1 17
8
O⇒
()
5
2
+
p : (1s1/ 2 ) 2 (1p3 / 2 ) 4 (1p1/ 2 ) 2
n : (1s1/ 2 ) 2 (1p3 / 2 ) 4 (1p1/ 2 ) 2 (1d5 / 2 )1
€
Most ground state properties of 178O Just by stating (1d5/2)1 18/05/14 F. Ould-­‐Saada 15 ¡
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 ¡
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 mj there is another with -­‐mj §  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: |jp-­‐jn| to |jp+jn| 18/05/14 F. Ould-­‐Saada 17 ¡
!
!
!
Pˆ Ψlmn ( r ) = PΨlmn (− r ) = P(−1) l Ψlmn ( r )
!
⇒ Ψlmn ( r )is eigenstate of parity with eigenvalue P(−1) l
PN = +1
Pˆ Y m (θ , φ) = (−1) l Y m (θ , φ) ⇒ P = (−1) l
Nuclear parities l
13
6
O & 137 N : j =€1 / 2,l = 1 ⇒
(1s1/ 2 ) 2 (1p3 / 2 ) 4 (1p1/ 2 )1
33
16
€
l
1 −
2
()
S⇒
3 +
2
()
: (1s1/ 2 ) 2 (1p3 / 2 ) 4 (1p1/ 2 ) 2 (1d5 / 2 ) 6 (2s1/ 2 ) 2 (1d3 / 2 )1
47
22
Ti ⇒
18/05/14 F. Ould-­‐Saada l
7 −
2
()
: ...(1 f 7 / 2 ) 5 BUT exp :
5 −
2
()
18 ¡
Magnetic moment for even-­‐odd nuclei à expect paired nucleons contribute zero §  Consider single nucleon §  Combine gl l and gs s !
e !
e!
µ=g
s ; µB =
= 5.79 ×10 −11 MeV / T
2mc
2mc
e!
µN =
≈ µ B / 2000; µ p ≈ 2.79µ N ; µ n ≈ −1.91µ N
2m p c
µ = gjµ N
µ N : nuclear magneton; g: g-factor
j( j +1) + l(l +1) − s(s +1)
j( j +1) − l(l +1) + s(s +1)
g=
gl +
gs
2 j( j +1)
2 j( j +1)
18/05/14 F. Ould-­‐Saada See reference in book for derivation 19 j( j +1) + l(l +1) − s(s +1)
j( j +1) − l(l +1) + s(s +1)
gl +
gs
2 j( j +1)
2 j( j +1)
1
* jg = g l + g /2
j
=
l
+
l
s
2
,
1
j = l± 2 ⇒+
$
$ 1 '
1 '
1
j =l−2
) − gs j&
)
, jg = gl j&1+
% 2l +1(
% 2l +1(
-
g=
gl ( p) = 1; gl (n) = 0; gs ( p) ≈ +5.6; gs (n) ≈ −3.8
1
* l + 2.8 = j + 2.3
j = l+ 2
,
jg p = +
2.3 j
1
j
−
j =l−2
,
j +1
* −1.9
,
jgn = +1.9 j
,
- j +1
18/05/14 F. Ould-­‐Saada 1
j = l+ 2
1
j =l−2
20 ¡
¡
¡
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 ¡
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
8
¡
€
O⇒
()
5
2
+
:
# p : (1s1/ 2 ) 2 (1p3 / 2 ) 4 (1p1/ 2 ) 2
$
2
4
2
1
% n : (1s1/ 2 ) (1p3 / 2 ) (1p1/ 2 ) (1d5 / 2 )
To excite 178O: 1.
2.
3.
Promote one of the 1p1/2 protons to 1d5/2 à Promote one of the 1p1/2 neutrons to 1d5/2 à Promote the 1d5/2 neutron to 2s1/2 à 18/05/14 F. Ould-­‐Saada (1p1/2 )−1 (1d5/2 )1
(1p1/2 )−1 (1d5/2 )2
(2s1/2 )1 or (1d3/2 )1
22 ¡
17 O 8
1.
2.
3.
(1p1/2 )−1 (1d5/2 )1
Promote one of the 1p1/2 protons to 1d5/2 à −1
2
Promote one of the 1p1/2 neutrons to 1d5/2 à (1p1/2 ) (1d5/2 )
Promote the 1d5/2 neutron to 2s1/2 à (2s )1 or (1d )1
1/2
3/2
§  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 ¡
§
¡
Non-­‐sphericity can mean new phenomena §
§
18/05/14 F. Ould-­‐Saada à quadruple moments large §
additional modes of excitation non-­‐zero EQM More in book 23 ¡
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: Qintrinsic= value of EQM for an ellipsoid at rest (charge Ze) )
+Qintrinsic = 25 Z ( a 2 − b 2 )
+
!
6
2
2
3!
eQ ≡ ∫ ρ ( r )( 3z − r ) d r ⇒ *
≈ 5 ZR 2ε (small deformations)
+
+ a = R(1+ ε ); b = R
,
1+ ε
18/05/14 F. Ould-­‐Saada prolate oblate 24 ¡
Quantum mechanically (no proofs) !
EQM : eQ = ∑ ∫ ψ *qi ( 3zi2 − r 2 )ψd 3 r
§  EQM depends on j and m of nucleus §  EQM is value of Q for which M has maximum value along z-­‐axis i
!
EDM : dz = ∑ ∫ ψ *qi ziψd 3 r ⇒ 0
i
¡
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) §  Odd-­‐A, Odd-­‐Z with single proton with j outside closed sub-­‐shells 18/05/14 F. Ould-­‐Saada €
Q ≈ −R 2
2 j −1
; Q = 0 for j = 1/2
2( j +1)
25 ¡
¡
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~-­‐R2 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 Rainwater ¡
§  EQM in terms of non-­‐spherical nuclei §  Expression (in fission model) assumed to hold only for closed-­‐shell nuclei §  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 18/05/14 ΔE B = −αε 2
1
α = (2as A 2 / 3 − ac Z 2 A −1/ 3 )
5
ΔE B = −αε 2 − βε ⇒ min : β 2 /4 α for ε = −β /2α
j(2 j −1)
Q=
Q
( j +1)(2 j + 3) int rinsic
€
F. Ould-­‐Saada 27 ¡
¡
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 ¡
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 EJ =
2I
§  rotational (I: moment of inertia) §  vibrational energy levels due €to shape oscillations between prolate and oblate ellipsoids ▪  Energy spacing: (simple harmonic oscillator) €ΔE
¡
18/05/14 = !ω
Many prediction of CM confirmed experimentally F. Ould-­‐Saada 29 ¡
¡
¡
¡
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 §
§
Nucleons move independently in a net nucleon potential Quantum statistics of Fermi gas à depth of potential and asymmetry term of SEMF §
§
§
§
§
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 §
§
§
§
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) Fermi gas model Shell model Collective model 18/05/14 F. Ould-­‐Saada 30 ¡
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
4
dB
dA
$ d(B / A) B '
dB
= 4&A
+ )
%
dA
dA
A(
€
19/05/14 F. Ould-­‐Saada 31 B / A plot
d(B / A)
⇒
≈ −7.7 × 10 −3 MeV for A ≥120
dA
B(2,4) = 28.3MeV
'B
*
⇒ 28.3MeV ≈ 4 ) − 7.7 × 10 −3 A ,
(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 ¡
Lifetime of α emitters §  span 10ns – 1017 years -­‐ due to tunnelling effect ▪  p&n (B~7-­‐8MeV in nucleus) cannot escape ▪  Whereas bound states α can escape Potential energy of an α particle ▪  r<R: strong nuclear potential ▪  r>R: Coulomb potential of daughter nucleus, Vc(r) ▪  α can escape à Eα>0 – energy released in the decay 18/05/14 F. Ould-­‐Saada 33 ¡
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
T =e
−G
;
€
m=
€
€
Δr
!κ = 2m VC − Eα
;
2
G=
!
rC
∫
2m VC (r) − Eα dr
R
mα mD
≈ mα
mα + mD
Eα = VC (rC ) ⇒ rC =
2Zα!c
Eα
rC Eα
r
2 2mEα rC % rC (
G=
∫ '& r −1*) dr
!
R
⇒ VC (r) =
18/05/14 F. Ould-­‐Saada 34 ¡
Evaluate integral *
2mc 2 ,$ −1 R '
R $ R '/
G = 4 αZ
&cos
)−
&1 − )
Eα ,+%
rC (
rC % rC ( /.
Eα ≈ 5MeV,V ≈ 40MeV,rC >> R ⇒ G ≈
¡
4 παZ
v
; β= α
β
c
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 vα/2R §  Transition probability vα −G
e
2R
Z
Z
G∝
∝
β
Eα
λ = w(α )
¡
'
))
(
)
)*
log10 t1/ 2 = a + b
Z
Eα
Small differences in Eα have strong effects on lifetime 18/05/14 €
F. Ould-­‐Saada 35 log10 t1/ 2 = a + b
Z
Eα
Geiger-­‐Nuttall empirical relation € (1911) before theoretical derivation (1928) ¡
§  For fixed Z log of ½ lifetime of α
emitters varies linearly with Eα-­‐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 Eαby factor 2.5 leads to lifetime change of 20 orders of magnitude! 18/05/14 F. Ould-­‐Saada 36 ¡
Nuclear β decay: 1st successful theory by Fermi, 1934 §  Analogy with QED §  Transition amplitude Μ fi = ∫ ψ *f (gOˆ )ψ i dV
▪  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 18/05/14 F. Ould-­‐Saada 37 ¡
Transition rate ω=1/τ
−
n → p + e + νe
§  Second Golden Rule 2
2π
ω=
Μ fi n(E)
!
GF
M fi ≡
Μ fi
V
GF 4 π (!c) 3 α w
=
2 2
2
( MW c )
▪  density of states n(E): number of quantum states available to the final system per unit interval of total energy (see Appendix A.2) ▪  Mfi: transition amplitude = matrix element ▪  GF: universal in nuclear theory – related to the weak coupling constant αW €
(
¡
Lifetime of muon – τµ
§  à GF 18/05/14 F. Ould-­‐Saada 2
)
5
mµ c
1
2
=
3
6 GF
τ µ 192€π !(!c)
GF ≈ 90 eV . fm 3
GF
−5
−2
=
1.166
×
10
GeV
(!c) 3
38 ¡
Consider p and n to be heavy (Tp, Tn ~0)
§  Total energy released carried by electron and neutrino: E 0 = E e + Eν = Δm.c 2
§  Transition rate for decays where electron has energy in range E and E+dE 2
2π
dω =
Μ fi nν (E 0 − E e )n e (E e )dE e
!
€
€
§  Density of state factors for e (and nu) €
§  Setting in transition probability a€
nd density of state factors à 18/05/14 n → p + e− + νe
F. Ould-­‐Saada 4 πV 2
n( pe )dpe =
3 pe dpe
(2π!)
dp
E
4 πV
= 2 ⇒ n(E e )dE e =
3 2 pe E e dE e
dE pc
(2π!) c
2
F
2
dω G M fi
=
3 7 4 pe E e pν Eν
dE e 2π ! c
39 §  Change of variable pν c = Eυ2 − mυ2c 4 = (E 0 − E e ) 2 − mυ2c 4
GF2 M fi
2
dω dE e dω
2
=
=
p
pE
dpe dpe dE e 2π 3! 7c 2 e ν ν
§  Neutrino exactly massless à €
2
F
2
2
2
2 p (E − E )
dω G M fi 2 2
2
e
0
e
=
3 7 3 pe Eν = GF M fi
dpe 2π ! c
2π 3! 7c 3
€
18/05/14 F. Ould-­‐Saada 40 ¡
¡
Matrix element ψi,,ψf: nuclear wave functions Μ fi = ∫ ψ *f (gOˆ )ψ i dV
€
¡
Μ fi = ∫ ψ *f ψ *eψυ* Hψ i dV
If lepton WF as free particles §  Mfi depends on kinematical e
variables via density of state €
factor, not on pe §  Approximation good for allowed transitions E e ≈ 1MeV ; p = 1.4 MeV / c ;
€
p / ! ≈ 0.007 / fm → pr / ! << 1
⇒
18/05/14 F. Ould-­‐Saada !
%
! ! r(
' −i( p e + pυ )⋅ *
")
&
e[ ...] ≈ 1+ ε
41 ¡
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=|Ji-­‐Jf|=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 €
18/05/14 F. Ould-­‐Saada Fermi
14
O (0 + )→14 N * (0 + ) + e + + ν e
Gamow − Teller
6
He (0 + )→6 Li (1+ ) + e − + ν e
Mixed
1+
2
1+
2
n ( ) → p ( ) + e− + νe
42 ¡
2
2
2 p (E − E )
dω
2
e
0
e
= F(Z, E e )GF M fi
dpe
2π 3! 7c 3
Kurie Plots §  Test of pt distribution of electrons §  Allowed transitions €
▪  Mfi independent of electron kinematic variables ▪  Z=0, without Fermi screening factors F(Z,Ee), effects of EM force neglected 18/05/14 F. Ould-­‐Saada 43 Allowed transitions
$ dω '
1
H(E e ) ≡ &
∝ E0 − Ee
) 2
dp
p
F(Z,
p
)
% e( e
e
18/05/14 F. Ould-­‐Saada §  For forbidden transitions Kurie plot not a straight line Te = E e − me c 2
44 ¡
Study of precise shape of momentum distribution near upper end-­‐point §  Way of finding mνe §  If mνe≠0 à $ dω '
1
H(E e ) ≡ &
∝ (E 0 − E e ) (E 0 − E e ) 2 − mυ2c 4
) 2
% dpe ( pe F(Z, pe )
{
{
2
2 4
υ
∝ (T0 − Te ) (T0 − Te ) − m c
€
18/05/14 3
F. Ould-­‐Saada 1/ 2
}
1/ 2
}
H →3 He + e − + υ e
45 3
18/05/14 2
H →3 He + eF. −O+
υ
;
T
=
18.6keV
⇒
m
<
2
−
3eV
/c
0
υe
uld-­‐Seaada 46 ¡
Total decay rate from integration of dω/dpe 2
0
2 4
e
cpmax = E − m c
¡
€
For allowed transitions §  Total width §  Comparative half-­‐life f €
t1/2 is a direct measurement of the matrix element ω=
2
F
G M fi
2
2π ! c
3 7 3
pc≈E
F. O
e uld-­‐Saada e
∫ F(Z, p )p (E
e
2
e
0
− E e ) 2 dpe
0
1
f (Z, E) ≡
(me c) 3 (me c 2 ) 2
p max
∫ F(Z, p )p (E
e
2
e
2
−
E
)
dpe
0
e
0
2
GF2 me5c 4
ω=
M fi f (Z, E 0 )
2π 3! 7
ln2
(2ln2)π 3! 7 1
ω=
⇒ ft1/ 2 =
t1/ 2
GF2 me5c 4 M 2
fi
log10 ( ft1/ 2 ) = (5.5 ± 1.5)s
= (7.5 ± 1.5)s
€
= 3−4
18/05/14 p max
allowed
first forbidden
super allowed
⇒ ω ∝ E 5 (Sargent&s rule)
47 ¡
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), … 18/05/14 F. Ould-­‐Saada Lγ = 1 →Lz = ±1
! ! !
Ji = J f + L ⇒ Ji − J f ≤ L ≤ Ji + J f
! ! ! !
J i = 0 → J f = 0 forbidden
48 ¡
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 +1
€
1+ → 0 + : L = 1: ΔP = + ≡"no"= (−1) L +1 ⇒ M1
1− → 0 + : L = 1: ΔP = − ≡" yes"= (−1) L ⇒ E1
18/05/14 F. Ould-­‐Saada 49 0 →0 : Lγz = ±1 ⇒ L ≥1 ⇒ not allowed
El : ΔP = (−1) L
Ml : ΔP = (−1) L +1
1+ → 0 + : L = 1: ΔP = + ≡"no"= (−1) L +1 ⇒ M1
1− → 0 + : L = 1: ΔP = − ≡" yes"= (−1) L ⇒ E1
3+
2
+
→ 12 :L=1 or 2 :ΔP=+≡"no"⇒M1 or E2
€
¡
Although transitions Ji=0àJf=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 ¡
Transition probability per unit time – emission rate TfiE,M(L) §
§
Semi-­‐classical radiation theory Reduced transition probability BE,M(L) §  Contains all nuclear information 1
8π (L +1)
TfiE,M (L) =
4 πε 0 L[(2L +1)!!] 2
2 #
L &2
e
3R
B E (L) =
% €(
4π $ L + 3'
[e
# !c & 2
( E
B M (L) = 10%%
2 ( B (L)
$ m pc R '
[
2
(µN / c )
2
fm 2L −2 ]
2
e2 # 3 &
B (L) =
%
( (R0 ) 2L A 2L / 3
4π $ L + 3'
# e!c & 2 # 3 & 2
10
(
B M (L) = %%
( (R0 ) 2L −2 A(2L −2)/ 3
2( %
π $ 2m p c ' $ L + 3 '
€
F. Ould-­‐Saada reduced transition
probability
fm 2L ]
E
18/05/14 2L +1
1 % Eγ (
B Efi ,M (L)
' *
"
$#$
%
! & !c )
R = R0 A1/ 3 ; R0 = 1.21 fm
51 ¡
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 Gg Γγ (E1) = 0.068Eγ3 A 2 / 3
Γγ (M1) = 0.021Eγ3
Γγ (E2) = (4.9 × 10 −8 )Eγ5 A 4 / 3
€
18/05/14 F. Ould-­‐Saada 52 ¡  Pages 250-­‐251 §  7.1 – 7.4 §  6.6, 6.2 (chapter 6) 18/05/14 F. Ould-­‐Saada 53