nucleispecifiedby - universitetet i oslofolk.uio.no/farido/fys3510/nuclearphenomenology2.pdf ·...
TRANSCRIPT
¡ 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 1
¡ 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
¡ Nucleus mass: § Fundamental measurable
quantity uniquely defining nuclide
§ As test of nuclear models and models of short-‐lived exotic nuclei
¡ Measure of mass § Deflection spectrometers § Kinematic analysis § Penning Trap
measurements
23/03/14 F. Ould-‐Saada 2
>1500 unstable nuclei
¡ Mass measurement by passing ion beams through crossed B,E fields
23/03/14 F. Ould-‐Saada 3
€
! F = q! v ×
! B 1 + q
! E
! E ⊥! B 1⇒ F = qvB1 − qE
F = 0⇒ v =EB1
§ Isotopes separated and focused onto a detector (photographic plate)
§ In practice, to achieve higher accuracy, measure mass differences
§ ΔM/M ~ 10-‐6
€
mv 2
ρ= qvB2
qm
=E
B1B2ρ=
EB2ρ
¡ Masses from kinematics of nuclear reactions § Inelastic reaction
A(a,a)A* , short-‐lived nucleus
§ Non-‐relativistic kinematics à mass difference ΔE
¡ ΔE iteratively from formula from measurements of kinetic energies Ei and Ef è mass of A*
23/03/14 F. Ould-‐Saada 4
a(Ei,!pi )+ A(mAc
2,!0)→ a(Ef ,
!pf )+ A*( !E, !"p)
Etot (initial) = Ei +mac2 +mAc
2
Etot ( final) = Ef + !E +mac2 + !mc2
ΔE ≡ ( !m−mA )c2 = Ei −Ef − !E =pi
2
2ma
−pf
2
2ma
−!p2
2 !mp− conservation→ pi = pf cosθ + !px; 0 = pf sinθ − !py
ΔE = Ei 1− ma
!m%
&'
(
)*−Ef 1+ ma
!m%
&'
(
)*+
2ma
!mEiEf cosθ
¡ General reaction A(a,b)B à mass difference with Q kinetic energy released in reaction
¡ Kinetic energies in formula measured in Laboratory frame ¡ In centre-‐of-‐mass (See appendix B for CM vs Lab)
23/03/14 F. Ould-‐Saada 5
€
ΔE ≡ ( ˜ m −mA )c 2 = Ei 1−ma
˜ m %
& '
(
) * − E f 1+
ma
˜ m %
& '
(
) * +
2ma
˜ m EiE f cosθ
€
ΔE = Ei 1−ma
mB
$
% &
'
( ) − E f 1+
mb
mB
$
% &
'
( ) +
2mB
mambEiE f cosθ +Q€
ECM = Elab 1+ma
mA
"
# $
%
& '
−1
¡ 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
¡ Momentum & energy conservation
23/03/14 F. Ould-‐Saada 7
t = e⇒ mt =me <<mα ⇒ no large angle scattering (las)t = Au⇒ mt >>mα ⇒ LAS
Coulomb scattering neglected
mα
!vi =mα
!vf +mt!vt
mαvi2 =mαvf
2 +mtvt2
!"#
$#⇒
mαvi2 =mαvf
2 +mt2
mα
vt2 + 2mt
!vf ⋅!vt( )
vt2 1− mt
mα
(
)*
+
,-= 2!vf ⋅!vt
23/03/14 F. Ould-‐Saada 8
Coulomb scattering: M>>m
b: impact parameter py due to Coulomb force: F=dp/dt[ ]
⇒Δp = zZe2
4πε0r2 cosφ
−∞
+∞
∫ dt
(5) ⇒ 2mvsin(θ / 2) = zZe2
4πε0
1bv"
#$
%
&' cosφ−(π−θ )/2
+(π−θ )/2
∫ dφ Impact parameter: b = zZe2
8πε0
⋅1Ekin
cot θ2"
#$%
&'
€
Angular momentum conservation : mvb = mr2 dφdt
(5)
Scattering symmetric about y - axis →along y pi = −mv sin(θ /2) = −pf = p ⇒ Δp = 2mv sin(θ /2)
23/03/14 F. Ould-‐Saada 9
€
b =zZe2
8πε 0
⋅1Ekin
cot θ2&
' ( )
* +
⇒ dσdΩ&
' (
)
* + Rutherford
=zZe2
16πε 0Ekin
&
' (
)
* +
2
cosec4 θ2&
' ( )
* + =
zZe2
16πε 0Ekin
&
' (
)
* +
21
sin4 θ2&
' ( )
* +
€
initial flux of particles : Jintensity between b and b + db : 2πbJ dbequal to rate of scattered particles into dΩ = 2π sinθ dθdW = 2πbJ db
€
dW = 2πbJ dbsingle target particle : (see 1.60)
dW = J dσdΩ
dΩ = 2πJ sinθ dθ dσdΩ
dσdΩ
=b
sinθ⋅dbdθ
d(cot x) = −(sin x)−2dx
¡ 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 10
€
dσdΩ
=1
4π 2!4p'2
vv'M(" q 2)
2 " q = " p − " p '
M(" q ) = V (" r )ei" q ⋅" r !∫ d3" r
V (" r ) = VC (" r ) = −
αZ(!c)r
€
MC (! q ) = limλ→ 0
−αZ("c)e−λr
r&
' (
)
* + e
i! q ⋅! r "∫ d3! r
! q along z − axis : ! q ⋅ ! r = qrcosθ
MC (! q ) = −4π ("c)αZ"
qlimλ→ 0
e−λr sin(qr"
)dr0
∞
∫
MC (! q ) = −4π ("c)αZ"2
q2
¡ Rutherford formula § Scattering angle θ small
23/03/14 F. Ould-‐Saada 11
dσdΩ
=1
4π 2!4p '2
vv 'M ("q2 )
2MC (!q) = − 4π ("c)αZ"
2
q2
€
⇒dσdΩ
= 4Z 2α 2(!c)2 p'2
vv'q4
p2 = p'2 = 2mEkin ; v = v '= 2Ekin /m ; q = 2psin(θ /2)⇒ previous Rutherford formula (C.13)p = p';E = E ';v = v'≈ c;E ≈ pc
⇒dσdΩ(
) *
+
, - Rutherford
=Z 2α 2 !c( )2
4E 2 sin4 θ /2( )
¡ EM scattering of a charged particle in the Born approximation à Appendix C § Assume Zα=1 and use plane waves for initial and final
states § Single photon exchange: à α2 § Rutherford: scattering of spin-‐0 point-‐like projectile of
unit charge from fixed point-‐like target with charge Ze (charge distribution of target neglected)
§ Take into account electron-‐spin à Mott
§ Recoil of target at HE (factor E’/E) à spin-‐1/2 formula
23/03/14 F. Ould-‐Saada 12
€
dσdΩ$
% &
'
( )
spin−1/ 2
=dσdΩ$
% &
'
( )
Mott
E 'E
1+ 2τ tan2 θ2$
% & '
( )
-
. /
0
1 2
τ = −q2
4M 2c 2 ; M target - mass
q2 = p − p'( )2= 2me
2c 2 − 2(EE ' /c 2 −! p ! p ' cosθ )
me ≈ 0 → pc ≈ E ⇒ q2 ≈ −4EE '
c 2 sin2 θ2$
% & '
( )
Q2 = −q2
€
dσdΩ$
% &
'
( ) Rutherford
=Z 2α 2 !c( )2
4E 2 sin4 θ2$
% & '
( )
dσdΩ$
% &
'
( ) Mott
=dσdΩ$
% &
'
( ) Rutherford
1− β2 sin2 θ2$
% & '
( )
.
/ 0
1
2 3
β = v /c
Nucleus P =Mc!P
!
"#
$
%& ; electron p =
E / c!p!
"#
$
%& ; p ' =
E '/ c!p '!
"#
$
%&
4−momentum transfer q2 = p− p '( )2
¡ Summary ¡
23/03/14 F. Ould-‐Saada 13
¡ Spherically symmetric charge distribution § integrate over angles à radial ρ(r)
¡ Final form of experimental cross-‐section takes into account form factor due to spatial extension of nucleus § charge distribution within nucleus f(r) § form factor as Fourier transform (magnetic interaction neglected here)
23/03/14 F. Ould-‐Saada 14
€
F(! q 2) ≡ 1Ze
ei! q ⋅! r " f (! r )d3! r ∫
Ze = f (! r )d3! r ∫
€
dσdΩ$
% &
'
( ) exp t
=dσdΩ$
% &
'
( )
Mott
F(! q 2)2
€
d3! r = r2drsinθdθdφ
F(! q 2) ≡ 4π"Zeq
rρ(r)sin qr"
'
( )
*
+ , dr
0
∞
∫
23/03/14 F. Ould-‐Saada 15
From Thomson
¡ Spherically symmetric charge distribution § integrate over angles à radial ρ(r)
è minima in elastic cross-‐sections
23/03/14 F. Ould-‐Saada 16
€
Simple example, hard sphereρ(r) = constant , r ≤ a = 0 , r > a⇒ F(! q 2) = 3 sin(b) − b(cos(b)[ ]b−3
b ≡ qa"
b = tan(b)⇒ F(! q 2) = 0
F(!q2 ) ≡ 4π"Zeq
rρ(r)sin qr"
"
#$
%
&'dr
0
∞
∫
data for 58Ni and 48Ca
¡ 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 17
€
dσdΩ$
% &
'
( ) exp t
=dσdΩ$
% &
'
( )
Mott
F(! q 2)2
¡ Radial charge distributions of various nuclei § a: value of radius where ρ≥ρ0/2
23/03/14 F. Ould-‐Saada 18
€
ρch (r) =ρch
0
1+ e(r−a ) / b
a ≈1.07A1/ 3 fm;b ≈ 0.54 fmρch
0 in range 0.06 − 0.08
¡ Charge density ~constant in nuclear interior and falls rapidly to zero at nuclear surface
¡ Mean square charge radius § Useful quantity stemming from
form factor
23/03/14 F. Ould-‐Saada 19
r2 ≡1Ze
r2∫ f (!r )d3!r = 4πZe
r4∫ f (r)dr
F(!q2 ) ≡ 1Ze
ei!q⋅!r" f (!r )d3!r∫ ; Ze = f (!r )d3!r∫
€
Expansion : F(! q 2) =1Ze
f (! r ) 1n!n =0
∞
∑ i ! q rcosθ"
%
& '
(
) *
n
d3! r ∫
Angular integrations : F(! q 2) =4πZe
f (r)r2dr0
∞
∫ −4π! q 2
6Ze"2 f (r)r4dr + ...0
∞
∫
¡ Derivation § See problem 2.3
€
F(! q 2) =1−! q 2
6"2 r2 + ...
r2 = −6"2 dF(! q 2)d! q 2 !
q 2 =0
r2 = 0.94A1/ 3 fm constant from a fit range of data
R2 =53
r2 ⇒ R =1.21A1/ 3 fm
¡ For medium to heavy nuclei § Nucleus often
approximated to homogeneous sphere of Radius R
¡ 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)
23/03/14 F. Ould-‐Saada 20
€
ρch (r)→ρch (r) * A /Zρnucl ≈ 0.17nucleons / fm
3
Rnuclear ≈1.2A1/ 3 fm
52 MeV deuterons on 54Fe
§ Differential cross section has diffraction pattern with peaks and valleys ▪ qR~pr θ for small θ ▪ J1: 1st order Bessel function
€
dσdΩ
=J1(qR)qR
$
% &
'
( )
2
; qR ≈ pRθ;
J1(qR)[ ]2≈
2πqR-
. /
0
1 2 sin2 qR − π
4-
. /
0
1 2
→zeros at intervals :Δθ =πpR
¡ To probe nuclear (matter) density of nuclei experimentally § Hadron as projectile § At high energies -‐ elastic scattering small – nucleus behaves
more like absorbing sphere ▪ λ=h/p will suffer diffractive-‐like effects as in optics
§ Nucleus as black disk of radius R
23/03/14 F. Ould-‐Saada 21
Elastic scattering of 30.3 MeV protons: data vs optical model calculations using 2 potentials
Matter density ρ(r) = f(R)
¡ Force binding nucleons in nuclei contribute to atom mass M(Z,A) § Mass deficit: ΔM § Binding energy: B= -‐ΔM c2
23/03/14 F. Ould-‐Saada 22
€
M(Z,A) < Z(Mp +me ) + N Mn
ΔM(Z,A) ≡ M(Z,A) − Z(Mp +me ) − N Mn
¡ 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
¡ 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: ▪ 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.
23/03/14 F. Ould-‐Saada 23
¡ Atomic mass § 6 terms
§ f0 – mass of constituent nucleons and electrons
23/03/14 F. Ould-‐Saada 24
€
M(Z,A) = fi(Z,A)i=0
5
∑
f0(Z,A) = Z(Mp +me ) + (A − Z)Mn
¡ SEMF § Few parameters from fits to
experimental data § Some theoretical basis
¡ Properties common to most nuclei, except those with very small A values § (1) Interior mass densities ~equal § (2) Total B ~proportional to masses
¡ Analogy with classical model of liquid drop § (1) interior densities are the same § (2) latent heats of vaporization
proportional to their masses
€
f5(Z,A) =
− f (A) Z even, A − Z even0 Z even, A − Z odd (vice - versa)f (A) Z odd, A − Z odd
#
$ %
& %
f (A) = a5A−1/ 2 empirical
¡ f0 – mass of constituent nucleons and electrons
23/03/14 F. Ould-‐Saada 25
€
f3(Z,A) = a3Z(Z −1)A1/ 3
≈ a3Z 2
A1/ 3
€
M(Z,A) = fi(Z,A)i=0
5
∑
f0(Z,A) = Z(Mp +me ) + (A − Z)Mn¡ f1 – volume term § Short-‐range attractive force; R~A1/3 à V~A
¡ f2 – surface term § Nucleons at surface not surrounded à correction to volume
¡ f3 – Coulomb term § Protons repel each other
¡ f4 – asymmetry term § Tendency for nuclei to have Z=N; Pauli principle § p from level 3 & n to level 4 à (N-‐Z)/2àΔ
§ Transfer of (N-‐Z)/2 nucleons à decrease of B by Δ(N-‐Z)2/4 § Δ not constant but propto 1/A
¡ f5 – pairing term: empirical § Tendency of like nucleons in same spatial state
to couple pair-‐wise to configs with spin =0
€
f1(Z,A) = −a1A f2(Z,A) = a2A2 / 3
€
f4 (Z,A) = a4(Z − A /2)2
A
¡ VSCAP
€
av = a1, as = a2, ac = a3, aa = a4 , ap = a5
15.56, 17.23, 0.697, 93.14, 12. MeV /c 2
23/03/14 F. Ould-‐Saada 26
¡ 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
¡ 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?
¡ 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) ▪ One reason is shell structure for nucleons within nuclei –
ignored in liquid drop model à chapter 7 23/03/14 F. Ould-‐Saada 27
§ To remove a neutron – separation energy En
§ To remove a proton – Ep
€
ZAY→ Z
A −1X + n
En = M(Z,A −1) + Mn −M(Z,A)[ ]c 2
= B(Z,A) − B(Z,A −1)
ZAY→Z −1
A −1X + p
Ep = M(Z −1,A −1) + Mp +me −M(Z,A)[ ]c 2
= B(Z,A) − B(Z −1,A −1) +mec2
From Braibant
¡ 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: nàp (β-‐
decays); β+: ”p”àn+e+νe ▪ (atomic) e-‐ capture (pàn)
¡ Fe, Ni most stable nuclides § maximum of B/A curve § Heavier nuclei – B/A larger due to Coulomb
repulsion § Still heavier nuclei – spontaneous decay to
lighter nuclei à Q-‐value ▪ 2-‐body:NàD1 + α (α= 4He=2p2n) ▪ Fission (spontaneous or induced): D1 and D2
~similar mass. Z>=110
§ Photon emission – EM decays 23/03/14 F. Ould-‐Saada 28
Distribution of stable Nuclei. Stable and long-‐lived occurring in nature – squares
€
Qα = (Mp −MD −Mα )c2 = ED + Eα
http://www.nndc.bnl.gov/nudat2/
¡
23/03/14 F. Ould-‐Saada 29 http://www.nndc.bnl.gov/nudat2/
¡ Decay law § Decay constant λ vs activity Α : § Mean lifetime τ and half-‐life t1/2
Α = −dNdt
= λN 1Bq ≡1decay / s
Α(t) = λN0e−λt 1Ci ≡ 3.7×1010decay / s
x ≡xf (x)dx∫f (x)dx∫
τ ≡t dn(t)dt∫dn(t)dt∫
=te−λt
0
∞
∫ dt
e−λt0
∞
∫ dt=1λ
t1/2 =ln2λ
= τ ln2
23/03/14 F. Ould-‐Saada 30
¡ 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
€
A→λAB→
λBC→
λC...
dNA (t)dt
= −λANA ⇒ NA (t) = NA (0)e−λA t
dNB (t)dt
= −λBNB + λANA
NB (t) =λA
λB − λANA (0) e
−λA t − e−λB t[ ]
NC (t) = λAλBNA (0)e−λA t
(λB − λA )(λC − λA )+
e−λB t
(λA − λB )(λC − λB )+
e−λC t
(λA − λC )(λB − λC )&
' (
)
* +
€
3879Sr→37
79Rb + e+ +ν e (2.25min) →36
79Kr + e+ +ν e (22.9min) →35
79Br + e+ +ν e (35.04hr)
¡ λA>λB>λC – D stable ¡ ΝA(t)+ΝB(t)+ΝC(t)+ΝD(t)=constant!
¡ Chains with decay constants λi
23/03/14 F. Ould-‐Saada 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
¡
23/03/14 F. Ould-‐Saada 32
€
M(Z,A) = αA − βZ + γZ 2 +δA1/ 2
α = Mn − av +asA1/ 3
+aa4
β = aa + (Mn −Mp −me )
γ =aa4
+acA1/ 3
δ = ap
§ For even A, even-‐even and odd-‐odd nuclei lie on 2 distinct vertically shifted parabolas (pairing term)
§ Isobaric spectrum (same A) ▪ Smallest mass stable (against β decay) ▪ Other nuclei decay if Z not at
minimum
§ τ=f(Q-‐value, Spin, …): ms à 106y
¡ 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
23/03/14 F. Ould-‐Saada 33
45111Rh→ 46
111Pd + e− +νe (11sec)
46111Pd→ 47
111Ag+ e− +νe (22.3min)
47111Ag→ 48
111Cd + e− +νe (7.45d)
€
45111Rh, 46
111Pd, 47111Ag →β − decay
n→ p + e− +ν eM(Z,A) > M(Z +1,A)
¡ Mass parabola – odd A
23/03/14 F. Ould-‐Saada 34
€
51111Sb, 50
111Sn, 49111In →β+ − decay
" p"→n + e+ +ν e
M(Z,A) > M(Z −1,A) + 2me
e− + p→ n+νe (e− capture)M (Z,A)>M (Z −1,A)+εe− + 51
111Sb→ 50111Sn+νe (75sec)
e− + 50111Sn→ 49
111In+νe (35.3min)e− + 49
111In→ 48111Cd +νe (2.8d)
Excitation energy of atomic shell of daughter nucleus
23/03/14 F. Ould-‐Saada 35
¡ 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
€
2β − decay (1019−20 yr)
3482Se→36
82Kr + 2e− + 2ν e40Ca, 76Ge,82Se,96Zr,100Mo,116Cd,128Te,130Te,150Nd, 238U
§ 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
€
2e − capture : (A,z) →(A,Z - 2)
46102Pd + 2e−→44
102Ru + 2νe§ 2e-‐capture possible but not observed
¡
23/03/14 F. Ould-‐Saada 36
From Braibant
¡ Binding energy curve à spontaneous fission energetically possible for A>100
§ 154 MeV released carried by fission
products, usually some way from stability line, and decay in steps
¡ Spontaneous fission: § 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
23/03/14 F. Ould-‐Saada 37
€
92238U→57
145La+3590Br + 3n
€
92238U :P( fission) ~ 3 ×10−24 s−1 ≈ 6 ×10−7P(α)254Fm : BR( fission) ~ 0.06% ; BR(α) ~ 99.94%
mass distribution of fission fragments
¡ Fission very rare process § Only dominant in very heavy
elements A>270
€
57145La→...→60
145Nd + 8.5MeV (e,ν)
n+ 92U→ 56Ba+ 36Kr
€
ΔE = (Es + Ec ) − (Es + Ec )semf =ε 2
52asA
2 / 3 − acZ2A−1/ 3( )
ΔE < 0 ⇒ Z 2
A≥
2asac
≈ 49
OK for Z >116; A ≥ 270
§ Parameterize deformation: § Change in total energy ΔE § ΔE<0 à Fission
23/03/14 F. Ould-‐Saada 38
€
a = R(1+ε); b =R
1+ε
V =43πR3 =
43πab2
Es = asA2 / 3 1+
25ε 2 + ...
$
% &
'
( )
Ec = acZ2A−1/ 3 1− 1
5ε 2 + ...
$
% &
'
( )
¡ 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
¡ Spontaneous fission = potential barrier problem (see Appendix A)
¡ 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)
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Potential energy during different stages of a fission reaction
§ 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
¡
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From Braibant
¡ In α, β decays and fission, daughter often left in excited state § De-‐excitation by photon
emission
▪ Energy level spacing in nuclei ~some MeV (àγ-‐rays)
§ Lifetime of excited state ~10-‐12
s (EM process ~10-‐16 s )
ZAX*→ Z
AX +γΔE ≈ 0.1−10MeV
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§ Role of angular momentum is crucial; idem for parity (conserved in EM) § Intrinsic parities § Parity from angular
momentum § à more in chapter 7.8
Parity associated with angular à momentum carried by the photon
€
! J γ =! S i −! S f
Si − S j ≤ J ≤ Si + S j
M = mi −m f
P(γ) = −1(−1)J
§ Compound nucleus reactions: ▪ projectile loosely bound in nucleus ▪ reaction time much longer than just transit
¡ More detailed classification of reactions in NP § Direct reactions: assumption that projectile experiences average
potential of target nucleus – reaction time 10-‐22s
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▪ Elastic scattering ▪ Inelastic scattering ▪ Pickup reaction ▪ N stripped off target, carried away by projectile
▪ Stripping reaction ▪ N stripped off projectile, transferred to target nucleus
Energy level diagram of excitation of compound nucleus and subsequent decay €
(i) a + A→a + A(ii) a + A→a + A*
(iii) p+16O→d+15O
(iv) d+16O→ p+17O
€
a + A→C* →b + Ba + A→C* →C +γ
¡ Compound nucleus reactions: § 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.
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Total cross section
▪ 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
¡ 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
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¡ Direct and compound nucleus reactions in nuclear reactions initiated by protons ¡ Feed same final states
¡ 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
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¡ 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
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¡ 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
¡ 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
¡ 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
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