¡
Nuclei specified by ¡
§  Z – atomic number: number of protons §  N – neutron number: number of neutrons §  A = Z+N – mass/nucleon number: number of nucleons ¡
¡
Nucleus charge: +Ze Nuclides: AZY – Y: chemical symbol for element ¡
¡
§  Same Z à isotopes (12C, 13C, 14C: Z=6) §  Same N à isotones §  Same A à isobars 23/03/14 F. Ould-­‐Saada ¡
Not necessary to consider nuclear physics in terms of quarks and gluons, even if protons and neutrons are made of quarks. In classical nuclear physics, the existence of quarks can be ignored as well as the existence of meson and hadron resonances. A nucleus consists of nucleons that somehow behave as almost free particles, although they are in a high density medium (about 1038 nucleons/cm3). Average kinetic energies of nucleons in the nucleus are of the order of 20MeV << energy scale of elementary particles 1 ¡
Nucleus mass: §  Fundamental measurable quantity uniquely defining nuclide §  As test of nuclear models and models of short-­‐lived exotic nuclei ¡
>1500 unstable nuclei Measure of mass §  Deflection spectrometers §  Kinematic analysis §  Penning Trap measurements 23/03/14 F. Ould-­‐Saada 2 ¡
Mass measurement by passing ion beams through crossed B,E fields §  Isotopes separated and focused onto a detector (photographic plate) !
!
! !
F = qv × B1 + qE
! !
E⊥B1 ⇒ F = qvB1 − qE
E
F =0⇒v =
B1
mv 2
= qvB2
ρ
q
E
E
=
= 2
m B1B2 ρ B ρ
§  In practice, to achieve €
23/03/14 F. Ould-­‐Saada higher accuracy, measure mass differences §  ΔM/M ~ 10-­‐6 3 ¡
!
!
!
!"
2
! p)
a(Ei , pi ) + A(mA c , 0) → a(E f , p f ) + A* ( E,
Masses from kinematics of nuclear reactions Etot (initial) = Ei + ma c 2 + mA c 2
! 2
Etot ( final) = E f + E! + ma c 2 + mc
§  Inelastic reaction A(a,a)A* , short-­‐lived nucleus §  Non-­‐relativistic kinematics à mass difference ΔE 2
2
2
p
!
p
p
f
2
i
ΔE ≡ (m! − mA )c = Ei − E f − E! =
−
−
2ma 2ma 2 m!
p − conservation → pi = p f cosθ + p! x ; 0 = p f sin θ − p! y
% m (
% m ( 2m
ΔE = Ei '1− a * − E f '1+ a * + a Ei E f cosθ
&
&
m! )
m! ) m!
¡
ΔE iteratively from formula from measurements of kinetic energies Ei and Ef è mass of A* 23/03/14 F. Ould-­‐Saada 4 % ma (
% ma ( 2ma
ΔE ≡ ( m˜ − mA )c = E i '1 −
* − E f '1+
*+
&
&
m˜ )
m˜ ) m˜
2
¡
€ ¡
¡
E i E f cos θ
Kinetic energies in formula measured in Laboratory frame In centre-­‐of-­‐mass (See appendix B for CM vs Lab) " ma % −1
E CM = E lab $1+
'
m
#
A&
General reaction A(a,b)B à mass difference with Q kinetic energy released in reaction $ ma '
$ m€
' 2
b
ΔE = E i &1 −
ma mb E i E f cos θ + Q
) − E f &1+
)+
m
m
m
%
%
B(
B(
B
23/03/14 F. Ould-­‐Saada 5 ¡
Shape and size of Nucleus obtained from scattering experiments §  Electrons as projectiles: EM force à Charge distribution §  Hadrons as projectiles: nuclear strong interaction in addition à Matter density ▪  Neutrons à EM effects absent ▪  Let us first derive Rutherford scattering (Appendix C) 23/03/14 F. Ould-­‐Saada 6 Coulomb scattering neglected ¡
Momentum & energy conservation !
!
!
mα vi = mα v f + mt vt !
#
"⇒
mα vi2 = mα v 2f + mt vt2 $
#
! !
mt2 2
m v = mα v +
vt + 2mt ( v f ⋅ vt )
mα
2
α i
2
f
( m +
! !
vt2 *1− t - = 2 v f ⋅ vt
) mα ,
t = e ⇒ mt = me << mα ⇒ no large angle scattering (las)
t = Au ⇒ mt >> mα ⇒ LAS
23/03/14 F. Ould-­‐Saada 7 Coulomb scattering: M>>m Angular momentum conservation : mvb = mr 2
dφ
dt
(5)
Scattering symmetric about y - axis →along y
p i = −mv sin(θ /2) = − p f = p ⇒ Δp = 2mv sin(θ /2)
b: impact parameter €
py due to Coulomb force:[ F=dp/dt ]
zZe 2
⇒ Δp = ∫
cos φ dt
2
4
πε
r
0
−∞
+∞
(5) ⇒
zZe 2 " 1 % +( π −θ )/2
2mvsin(θ / 2) =
$ ' ∫ cos φ dφ
4πε 0 # bv & −( π −θ )/2
23/03/14 F. Ould-­‐Saada "θ %
zZe 2 1
Impact parameter: b =
⋅
cot $ '
#2&
8πε 0 Ekin
8 initial flux of particles : J
intensity between b and b + db : 2πbJ db
equal to rate of scattered particles into dΩ = 2π sin θ dθ
dW = 2πbJ db
dW = 2πbJ db
single target particle : (see 1.60)
dσ
dσ
dW = J
dΩ = 2πJ sin θ dθ
dΩ
dΩ
dσ
b db
=
⋅
dΩ sin θ dθ
&θ )
zZe 2 1
b=
⋅
cot( +
' 2*
8πε 0 E kin
€
d(cot x) = −(sin x)−2 dx
2
2
& zZe 2 ) 2
&
)
& dσ )
&
)
θ
zZe
1
4
⇒ (
=(
+
+ cosec ( + = (
+
&θ )
' dΩ * Rutherford ' 16πε 0 E kin *
' 2 * ' 16πε 0 E kin *
sin 4 ( +
'2*
23/03/14 F. Ould-­‐Saada 9 ¡
¡
Previous (classical) formula adequate for α-­‐scattering For e-­‐ (z=-­‐1) –Nucleus (Z) scattering, quantum mechanics and relativity necessary §  Use eq. 1.69 – neglecting spin ¡
integral diverges à introduce charge screening at large distances through term e-λr §  Integral twice by parts §  and let λ à 0 after integration 23/03/14 F. Ould-­‐Saada dσ
1 p'2
"2 2
=
M(
q
)
dΩ 4 π 2! 4 vv'
" "
q ⋅r
"
" i
"
M(q ) = ∫ V ( r )e ! d 3 r
" " "
q = p − p'
αZ(!c)
"
"
V ( r ) = VC ( r ) = −
r
! !
€
& αZ("c)e − λr ) i q ⋅r 3 !
!
MC (q ) = lim ∫ ( −
e "d r
+
λ→0
r
'
*
!
! !
q along z − axis : q ⋅ r = qr cosθ
∞
4 π ("c)αZ"
qr
!
M C (q ) = −
lim ∫ e − λr sin( )dr
λ→0
q
"
0
4 π ("c)αZ" 2
!
M C (q ) = −
q2
10 ¡
Rutherford formula §  Scattering angle θ small "2 2
dσ
1 p'2
=
M
(
q
)
2 4
dΩ 4π ! vv'
!
4π ("c)α Z" 2
M C (q) = −
q2
2
dσ
2 2
2 p'
⇒
= 4Z α (!c)
dΩ
vv'q 4
p 2 = p'2 = 2mE kin ; v = v' = 2E kin /m ; q = 2 psin(θ /2)
⇒ previous Rutherford formula (C.13)
p = p';E = E';v = v'≈ c;E ≈ pc
2
( dσ +
Z 2α 2 (!c )
⇒*
=
) dΩ , Rutherford 4 E 2 sin 4 (θ /2)
23/03/14 F. Ould-­‐Saada 11 ¡
2
EM scattering of a charged particle in the Born $ dσ '
Z 2α 2 (!c )
=
&
)
approximation à Appendix C $ '
% dΩ ( Rutherford
2
4 θ
4
E
sin
& )
§  Assume Zα=1 and use plane waves for initial and final §
§
§
§
%2(
.
$ dσ '
$ '1
2
2 θ
=&
1
−
β
sin
)
& )3
0
% dΩ ( Rutherford /
% 2 (2
states $ dσ '
Single photon exchange: à α2 &
)
% dΩ ( Mott
Rutherford: scattering of spin-­‐0 point-­‐like projectile of unit charge from fixed point-­‐like target with charge Ze β = v /c
(charge distribution of target neglected) Take into account electron-­‐spin à Mott $ dσ '
$ dσ '
$ '0
E' 2 θ
=&
)
/1+ 2τ tan & )2
€ &% dΩ )(
%
(
% 2 (1
dΩ
E
.
spin −1/ 2
Mott
Recoil of target at HE (factor E’/E) à spin-­‐1/2 formula q2
τ =−
; M target - mass
4 M 2c 2
! Mc $
! E / c$
! E '/ c $
Nucleus P = # ! & ; electron p = # !
;
p'
=
&
#! &
"p %
" p' %
"P %
4 − momentum transfer
q 2 = ( p − p')
2
2
! !
q 2 = ( p − p') = 2me2c 2 − 2(EE' /c 2 − p p' cosθ )
me ≈ 0 → pc ≈ E
⇒ q2 ≈ −
4 EE' 2 $ θ '
sin & )
%2(
c2
Q2 = −q 2
23/03/14 F. Ould-­‐Saada 12 ¡
Summary 23/03/14 ¡
F. Ould-­‐Saada 13 Final form of experimental cross-­‐
section takes into account form factor due to spatial extension of nucleus ¡
1
!
F(q 2 ) ≡
∫e
Ze
! !
Ze = ∫ f ( r )d 3 r
i
§  charge distribution within nucleus f(r) §  form factor as Fourier transform €
(magnetic interaction neglected here) ! !
q ⋅r
"
! !
f ( r )d 3 r
$ dσ '
$ dσ '
!2 2
=&
&
)
) F(q )
% dΩ ( exp t % dΩ ( Mott
€
¡
!
d 3 r = r 2 dr sin θdθdφ
Spherically symmetric charge distribution 4 π"
!
F(q 2 ) ≡
Zeq
§  integrate over angles à radial ρ(r) 23/03/14 F. Ould-­‐Saada €
∞
∫
0
' qr *
rρ (r)sin) , dr
("+
14 From Thomson 23/03/14 F. Ould-­‐Saada 15 ¡
Spherically symmetric charge distribution §  integrate over angles à radial ρ(r) !
4π "
F(q 2 ) ≡
Zeq
∞
" qr %
r
ρ
(r)sin
$ ' dr
∫
#"&
0
Simple example, hard sphere
ρ(r) = constant , r ≤ a
=0
, r>a
!
⇒ F(q 2 ) = 3[sin(b) − b(cos(b)]b −3
qa
"
!
b = tan(b) ⇒ F(q 2 ) = 0
b≡
€
è
minima in elastic cross-­‐
sections data for 58Ni and 48Ca 23/03/14 F. Ould-­‐Saada 16 ¡
dσ/dσΩ=f(θ) minima due to spatial distribution of nucleus §  In practice, ρ(r) not a hard sphere à modifications of the “zeros” §  Minimaà information about size of nucleus §
¡
Measure at fixed E and various θ (hence various q2) à Form factor extracted from cross-­‐section measurements 23/03/14 F. Ould-­‐Saada $ dσ '
$ dσ '
! 2
=&
&
)
) F(q 2 )
% dΩ ( exp t % dΩ ( Mott
€
17 ¡
Radial charge distributions of various nuclei §
a: value of radius where ρ≥ρ0/2 ρch0
ρch (r) =
1+ e(r −a )/ b
a ≈ 1.07A1/ 3 fm;b ≈ 0.54 fm
ρch0 in range 0.06 − 0.08
Charge density ~constant in nuclear interior and falls € rapidly to zero at nuclear surface ¡
23/03/14 F. Ould-­‐Saada 18 ! 3 ! 4π
1
2
4
r
f
(
r
)d
r
=
r
f (r)dr
∫
∫
¡  Mean square charge radius Ze
Ze
!!
q⋅r
§  Useful quantity stemming from i
!2
! 3!
! 3!
1
"
F(q ) ≡
e
f
(
r
)d
r
;
Ze
=
f
(
r
form factor ∫
∫ )d r
Ze
n
!
∞
1
1 % i q r cosθ ( 3 !
!2
!
Expansion : F(q ) =
∫ f (r )∑ n!'& " *) d r
¡  Derivation Ze
n =0
§  See problem 2.3 !2 ∞
4π ∞
4
π
q
!2
4
Angular integrations : F(q ) =
f (r)r 2 dr −
∫
2 ∫ f (r)r dr + ...
Ze 0
6Ze" 0
r2 ≡
!
q2 2
!2
F(q ) = 1 − 2 r + ...
6"
€
!2
dF(
q
)
r 2 = −6" 2 ! 2
dq q! 2 =0
r 2 = 0.94 A1/ 3 fm
constant from a fit range of data
5 2
r ⇒ R = 1.21A1/ 3 fm
23/03/14 3
F. Ould-­‐Saada R2 =
¡
For medium to heavy nuclei §  Nucleus often approximated to homogeneous sphere of Radius R 19 52 MeV deuterons on 54Fe ¡
Electrons not suitable for getting distribution of neutrons in nucleus Presence of neutrons taken into account by multiplying ρ by A/Z … §  à Effective nuclear matter radius Rnuclear (medium to heavy nuclei) §
ρ ch (r) → ρ ch (r) * A /Z
ρ nucl ≈ 0.17nucleons / fm 3
Rnuclear ≈ 1.2A1/ 3 fm
¡
To probe nuclear (matter) density of nuclei experimentally Hadron as projectile € energies -­‐ elastic scattering small – nucleus behaves §  At high more like absorbing sphere §
▪
λ=h/p will suffer diffractive-­‐like effects as in optics §
Nucleus as black disk of radius R §
Differential cross section has diffraction pattern with peaks and valleys ▪
▪
qR~pr θ for small θ
J1: 1st order Bessel function 23/03/14 F. Ould-­‐Saada 2
dσ $ J1 (qR) '
=&
) ; qR ≈ pRθ ;
dΩ % qR (
- 2 0 22
π0
J
(qR)
≈
sin
qR
−
/
2
[ 1 ] / πqR 2 .
1
4
.
1
π
→zeros at intervals : Δθ =
pR
20 Elastic scattering of 30.3 MeV protons: data vs optical model calculations using 2 potentials Matter density ρ(r) = f(R) 23/03/14 F. Ould-­‐Saada 21 ¡
Force binding nucleons in nuclei contribute to atom mass M(Z,A) M(Z, A) < Z(M p + me ) + N M n
ΔM(Z, A) ≡ M(Z, A) − Z(M p + me ) − N M n
§  Mass deficit: ΔM §  Binding energy: B= -­‐
ΔM c2 ¡
€
Binding energy per nucleon: B/A §  For stable or long-­‐lived nuclei, B/A peaks at 8.7 MeV for M~56 (iron) §  Excluding very light nuclei, B/A~7-­‐9 MeV 23/03/14 F. Ould-­‐Saada 22 ¡
Nuclear drop model §  a collective model of the nucleus §  describes the nuclear binding energy with a few parameters §  uses analogies with a liquid droplet §  based on the following assumptions: ▪
▪
▪
▪
23/03/14 interaction energy independent on the nucleon type Interaction attractive at a short-­‐range Interaction repulsive at large distances binding energy of the nucleus proportional number of nucleons. F. Ould-­‐Saada 23 ¡
SEMF §  Few parameters from fits to ¡
experimental data §  Some theoretical basis Atomic mass §  6 terms 5
¡
M(Z, A) = ∑ f i (Z, A)
Properties common to most nuclei, except those with very small A values i=0
f 0 (Z, A) = Z(M p + me ) + (A − Z)M n
§  (1) Interior mass densities ~equal §  (2) Total B ~proportional to masses §  f0 – mass of constituent nucleons and electrons €
¡
Analogy with classical model of liquid drop §  (1) interior densities are the same §  (2) latent heats of vaporization proportional to their masses 23/03/14 F. Ould-­‐Saada 24 ¡
¡
f0 – mass of constituent nucleons and electrons M(Z, A) = ∑ f i (Z, A)
f1 – volume term f 0 (Z, A) = Z(M p + me ) + (A − Z)M n
§  Short-­‐range attractive force; R~A1/3 à V~A ¡
f2 – surface term 5
i=0
f1 (Z, A) = −a1 A
f 2 (Z, A) = a2 A 2 / 3
§  Nucleons at surface not surrounded à correction to volume €
¡
f3 – Coulomb term §  Protons repel each other Z(Z −1)
Z2
f 3 (Z, A)
≈ a3 1/ 3
€ = a3
A1/ 3
A
(Z − A /2) 2
f 4 (Z, A) = a4
¡  f4 – asymmetry term A
§  Tendency for nuclei to €have Z=N; Pauli principle §  p from level 3 & n to level 4 à (N-­‐Z)/2àΔ
§  Transfer of (N-­‐Z)/2 nucleons à decrease of B by Δ(N-­‐Z)2/4 €
# − f (A)
§  Δ not constant but propto 1/A Z even, A − Z even
%
f 5 (Z, A) = $ 0
Z even, A − Z odd (vice - versa)
¡  f5 – pairing term: empirical %
Z odd, A − Z odd
& f (A)
§  Tendency of like nucleons in same spatial state to couple pair-­‐wise to configs with spin =0 f (A) = a5 A −1/ 2 empirical
23/03/14 F. Ould-­‐Saada 25 ¡
VSCAP av = a1, as = a2 , ac = a3 , aa = a4 , a p = a5
15.56,
17.23,
0.697,
93.14,
12.
MeV /c 2
Fit to binding energy data (solid circles) for A>20 ¡
§  Good fit for a simple formula (open circles) §  Some enhancements not reproduced ▪  Due to shell structure of nucleons within the nucleus à see section 7.3 ¡
SEMF gives correct B for some 200 stable and many more unstable nuclei §  Used to analyse stability of nuclei wrt β-­‐
decay and fission 23/03/14 F. Ould-­‐Saada 26 From Braibant Contribution to binding energy /
nucleon as function of mass number for odd-­‐A Is B/A equivalent to energy needed to remove nucleon from nucleus? ¡
¡
§  To remove a neutron – separation energy En A
Z
Y →A −1
Z X +n
E n = [ M(Z, A −1) + M n − M(Z, A)]c 2
= B(Z, A) − B(Z, A −1)
§  To remove a proton – Ep ¡
A
Z
−1
Y →ZA −1
X+p
[
]
E p = M(Z −1, A −1) + M p + me − M(Z, A) c 2
= B(Z, A) − B(Z −1, A −1) + me c 2
Ep and En are only equal to B/A in an average sense §  In practice, measurements show that Ep and En can substantially differ from average and from each other at certain values of (Z,A) ▪
23/03/14 F. Ould-­‐Saada One reason is shell structure for nucleons within nuclei – ignored in liquid drop model à chapter 7 27 Distribution of stable Nuclei. Stable and long-­‐lived occurring in nature – squares ¡
Distribution of stable nuclei – Segré plot §  Close to N=Z §  All other nuclei are unstable and decay spontaneously in various ways ▪  Isobars with large surplus of n’s: nàp (β-­‐
decays); β+: ”p”àn+e+νe ▪  (atomic) e-­‐ capture (pàn) ¡
Fe, Ni most stable nuclides §  maximum of B/A curve http://www.nndc.bnl.gov/nudat2/ §  Heavier nuclei – B/A larger due to Coulomb repulsion §  Still heavier nuclei – spontaneous decay to lighter nuclei à Q-­‐value ▪  2-­‐body:NàD1 + α (α= 4He=2p2n) ▪  Fission (spontaneous or induced): D1 and D2 ~similar mass. Z>=110 Qα = (M p − M D − Mα )c 2 = E D + Eα
§  Photon emission – EM decays 23/03/14 F. Ould-­‐Saada 28 ¡
23/03/14 http://www.nndc.bnl.gov/nudat2/ F. Ould-­‐Saada 29 Decay law ¡
dN
= λN
1Bq ≡ 1decay / s
dt
Α(t) = λ N 0 e− λt 1Ci ≡ 3.7 ×1010 decay / s
Α=−
§  Decay constant λ vs activity Α : §  Mean lifetime τ and half-­‐life t1/2 ∫ xf (x)dx
∫ f (x)dx
t dn(t)dt ∫
∫
τ≡
=
∫ dn(t)dt ∫
x≡
¡
Dating ancient specimen §
Organic specimen – radioactive 14C ▪
14C: produced in atmosphere cosmic rays on Nitrogen ▪  For constant cosmic ray activity, 14C:
12C~1:1012 in leaving organism ▪  When organism dies, ratio slowly changes with t 14Cà 14N – β-­‐decay τ=8.27x103y 23/03/14 F. Ould-­‐Saada ∞
te− λt dt
0
∞ − λt
t1/2 =
0
e dt
=
1
λ
ln 2
= τ ln 2
λ
30 ¡
Chains with decay constants λi λA
λB
λC
A →B →C →...
dN A (t)
= − λA N A ⇒ N A (t) = N A (0)e − λ A t
dt
dN B (t)
= − λB N B + λ A N A
dt
λA
N B (t) =
N A (0)[e − λ A t − e − λ B t ]
λB − λA
&
)
e − λA t
e − λB t
e − λC t
NC (t) = λA λB N A (0) (
+
+
+
(
λ
−
λ
)(
λ
−
λ
)
(
λ
−
λ
)(
λ
−
λ
)
(
λ
−
λ
)(
λ
−
λ
)
' B
A
C
A
A
B
C
B
A
C
B
C *
79
38
79
Sr→37
Rb + e + + ν e
(2.25min)
79
→36
Kr + e + + ν e
79
→35
Br + e + + ν e
23/03/14 F. Ould-­‐Saada (22.9min) ¡  λA>λB>λC – D stable (35.04hr)
¡
ΝA(t)+ΝB(t)+ΝC(t)+ΝD(t)=constant! 31 ¡
SEMF (2) §  Mass parabola: new form ▪  M(Z,A) is quadratic in Z for fixed A ▪  Minimum for Z=β/2γ §  For odd-­‐A (δ=0), SEMF is single parabola §  For even A, even-­‐even and odd-­‐
odd nuclei lie on 2 distinct vertically shifted parabolas (pairing term) §  Isobaric spectrum (same A) ¡
M(Z, A) = αA − βZ + γZ 2 +
δ
A1/ 2
as
aa
+
A1/ 3 4
β = aa + (M n − M p − me )
α = M n − av +
aa
a
+ 1/c 3
4 A
δ = ap
γ=
▪  Smallest mass stable (against β decay) €
▪  Other nuclei decay if Z not at minimum §  τ=f(Q-­‐value, Spin, …): ms à 106y 23/03/14 F. Ould-­‐Saada 32 ¡
Mass parabola – odd A §  Even-­‐N, odd-­‐Z or even-­‐Z, odd-­‐N §  Experimental mass-­‐excess from data: M(Z,A)-­‐A ▪  1a.m.u.=M(12 6 C)/12 §  Curve: theoretical SEMF prediction ▪  Minimum: 11148 Cd 111
45
111
Rh, 111
46 Pd, 47 Ag → β − decay
n → p + e− + ν e
M(Z, A) > M(Z +1, A)
23/03/14 F. Ould-­‐Saada 111
45
−
Rh → 111
46 Pd + e + ν e
(11sec)
111
46
−
Pd → 111
47 Ag + e + ν e
(22.3min)
111
47
−
Ag → 111
48 Cd + e + ν e
(7.45d)
33 ¡
Mass parabola – odd A 111
51
111
+
Sb, 111
50 Sn, 49 In → β − decay
" p"→n + e + + ν e
M(Z, A) > M(Z −1, A) + 2me
e− + p → n + ν e (e − capture)
€
23/03/14 F. Ould-­‐Saada M (Z, A) > M (Z −1, A) + ε
Excitation energy of atomic shell of daughter nucleus 111
e− + 111
Sb
→
51
50 Sn + ν e
(75sec)
111
e− + 111
50 Sn → 49 In + ν e
(35.3min)
111
e− + 111
49 In → 48 Cd + ν e
(2.8d)
34 ¡
Mass parabola – even A §  Even-­‐N, even-­‐Z or odd-­‐Z, odd-­‐N §  Nearly all stable even-­‐mass nuclei are of even-­‐even type §  Experimental mass-­‐excess data: M(Z,A)-­‐A ▪  Open circles: eve-­‐even, closed: odd-­‐odd §  Curve: theoretical SEMF prediction ▪  Lowest isobar: 102 44Ru ▪  Neighbour isobar: 102 46Pd §  Small number of even-­‐even nuclei, although beta-­‐decay energetically forbidden, (A,Z)à(A,Z+2) occurs §  Double beta decay : 2nd order weak process à Observed 2 β − decay
82
34
40
82
Se→36
Kr + 2e − + 2ν e
Ca, 76Ge,82 Se,96 Zr,100 Mo,116 Cd,128 Te,130 Te,150 Nd,
§  2e-­‐capture possible but not observed 23/03/14 F. Ould-­‐Saada €
(1019−20 yr)
238
U
2e − capture : (A,z) →(A,Z - 2)
102
46
Pd + 2e − →102
44 Ru + 2ν e
35 ¡
From Braibant 23/03/14 F. Ould-­‐Saada 36 Spontaneous fission: ¡
mass distribution of fission fragments Parent nucleus breaks into 2 daughter nuclei of ~equal masses. Equal masses unlikely. §  SEMF predicts max release energy for exactly equal masses §
▪  Isotope 254Fm -­‐ mass distribution of fission fragments Fission can also be induced by low-­‐energy neutrons or by more energetic particles ¡
n + 92 U → 56 Ba + 36 Kr
Binding energy curve à spontaneous fission energetically possible for A>100 ¡
238
92
90
U →145
57 La+ 35 Br + 3n
¡
§  154 MeV released carried by fission €
products, usually some way from stability line, and decay in steps 145
57
23/03/14 145
60
La →...→ Nd + 8.5MeV (e,ν )
F. Ould-­‐Saada Fission very rare process §  Only dominant in very heavy elements A>270 238
92
U : P( fission) ~ 3 × 10 −24 s−1 ≈ 6 × 10 −7 P(α )
254
Fm : BR( fission) ~ 0.06% ; BR(α ) ~ 99.94%
37 ¡
In SEMF, we assumed that drop/nucleus spherical §  This minimizes surface area (as) §  If surface perturbed, spherical à prolate §  as up, ac down, av constant §  Relative sizes (as, ac ) determine whether nucleus is stable or not against spontaneous fission §  Parameterize deformation: §  Change in total energy ΔE €
§  ΔE<0 à Fission 23/03/14 F. Ould-­‐Saada a = R(1+ ε );
b=
R
1+ ε
4 3 4
πR = πab 2
3
3
$ 2
'
E s = as A 2 / 3 &1+ ε 2 + ...)
% 5
(
$ 1
'
E c = ac Z 2 A −1/ 3 &1 − ε 2 + ...)
% 5
(
V=
ΔE = (E s + E c ) − (E s + E c ) semf
ε2
= (2as A 2 / 3 − ac Z 2 A −1/ 3 )
5
Z 2 2as
ΔE < 0 ⇒
≥
≈ 49
A
ac
OK for Z > 116; A ≥ 270
38 Potential energy during different stages of a fission reaction ¡
Spontaneous fission = potential barrier problem (see Appendix A) §  Activation energy determines probability for spontaneous fission ▪  ~6 MeV for heavy nuclei (provided f.e. by neutrons in induced fission) ▪  No activation needed for very heavy nuclei (dashed line) – spontaneous fission ¡  23592U – fission by thermal neutrons §  Even-­‐odd (pairing term δ=0): less tightly bound than 238 (higher in V) ¡  23892U – fission by fast neutrons §  Even-­‐even (δ<0) 23/03/14 F. Ould-­‐Saada 39 ¡
From Braibant 23/03/14 F. Ould-­‐Saada 40 ¡
In α, β decays and fission, daughter often left in excited state §  De-­‐excitation by photon emission A
Z
X * → ZA X + γ
ΔE ≈ 0.1−10MeV
§  Role of angular momentum is crucial; idem for parity (conserved in EM) §  Intrinsic parities §  Parity from angular momentum §  à more in chapter 7.8 ! ! !
Jγ = Si − S f
▪  Energy level spacing in nuclei ~some MeV (àγ-­‐rays) §
Si − S j ≤ J ≤ Si + S j
Lifetime of excited state ~10-­‐12 s (EM process ~10-­‐16 s ) M = mi − m f
P(γ ) = −1
J
Parity associated with angular à (−1)
momentum carried by the photon €
23/03/14 F. Ould-­‐Saada 41 ¡
More detailed classification of reactions in NP §  Direct reactions: assumption that projectile experiences average potential of target nucleus – reaction time 10-­‐22s ▪  Elastic scattering ▪  Inelastic scattering ▪  Pickup reaction ▪  N stripped off target, carried away by projectile ▪  Stripping reaction ▪  N stripped off projectile, transferred to target nucleus (i)
a + A →a + A
(ii) a + A →a + A*
(iii) p+16 O →d+15 O
(iv) d+16O → p+17O
€
§  Compound nucleus reactions: ▪  projectile loosely bound in nucleus ▪  reaction time much longer than just transit 23/03/14 F. Ould-­‐Saada Energy level diagram of excitation of compound nucleus and subsequent decay a + A →C * →b + B
a + A →C * →C + γ
42 ¡
Compound nucleus reactions: Total cross section §  reaction time much longer than transit §  Cross sections can show variations on a much smaller energy scale §  Density of levels high ▪  n-­‐12C scattering at En~few MEV à resonance formation in 13C with width ~tens – hundreds keV. 23/03/14 F. Ould-­‐Saada ▪  Widths of excitations decrease both with incident energy and rapidly with target nuclear mass ▪  Neutrons (neutral) have high probability of being captured by nuclei ▪  Cross sections rich in compound nucleus effects, particularly at very low energies 43 ¡
¡
Situations where particles are ejected from nucleus before full statistical equilibrium reached In collisions of heavy ions §  probability for additional mechanism: deep inelastic scattering (Section 5.8) – intermediate between direct and compound ¡
Direct and compound nucleus reactions in nuclear reactions initiated by protons ¡  Feed same final states 23/03/14 F. Ould-­‐Saada 44 ¡
Typical spectrum of energies of the nucleons emitted at fixed angle in inelastic nucleon-­‐nucleus reactions ▪  Case of incident neutron on medium mass nuclei §  N(E) of secondary particles in neutron-­‐
nucleus §  Direct reactions: cross section peaked in forward direction ▪  Falling rapidly with angle and with oscillations (see slide 16 ) §  At lowest energies, contribution from compound rather symmetric about 90o 23/03/14 F. Ould-­‐Saada 45 ¡
Many medium and large-­‐A nuclei can capture very low-­‐energy (~10-­‐100 eV) neutrons §  Neutron separation energy ~6MeV for final nucleus §  Capture leads to excitation, which often occurs in a region of high density of narrow states that show up as rich resonance structure in neutron total cross-­‐section §  Value of σ at resonance peaks (excited states of 239U) orders of magnitude greater than σ based on size of nucleus 23/03/14 F. Ould-­‐Saada 46 ¡
¡
Once formed, compound nucleus can decay to final states consistent with relevant conservation laws Neutron emission can be preferred decay §  For thermal neutrons (0.02 eV) photon emission often preferred ¡
23/03/14 Fact that radiative decay is dominant mode of compound nuclei formed by thermal neutrons is important in the use of nuclear fission to produce power in nuclear reactors F. Ould-­‐Saada 47 ¡  Pages 67-­‐69 §  2.2 , 2.4, 2.5, 2.7, §  2.10, 2.11, 2.12, 2.14 §  2.15, 2.17, 2.18 23/03/14 F. Ould-­‐Saada 48