Download - Nucleispeciﬁedby - Universitetet i oslofolk.uio.no/farido/fys3510/NuclearPhenomenology2.pdf · 2014-03-26 · Mass’measurement’by’passing’ion’beams’ through’crossed’B,E’ﬁelds’

¡  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


>1500 unstable nuclei

¡  Mass measurement by passing ion beams through crossed B,E fields


€

! 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*


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)


€

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)


¡  Momentum & energy conservation


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


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)


€

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

23/03/14 F. Ould-‐Saada 42

▪  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.

23/03/14 F. Ould-‐Saada 43

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

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

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23/03/14 F. Ould-‐Saada 47

¡  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

23/03/14 F. Ould-‐Saada 48