Like the alchemist’s dreamof chemical transformation

Nuclear physics has found the meansof transmutation

Nuclear Reactions

Besides his famous scattering of particles off gold and lead foil, Rutherford observed the transmutation:

OHHeN 17

8

1

1

4

2

14

7

OpN 17

8

14

7 or, if you prefer

Though a more compact form is often used:

OpN 17

8

14

7 ),(

Whenever energetic particles(from a nuclear reactor or an accelerator)

irradiate matter there is the possibility of a nuclear reaction

Target (Projectile, Detected Particle) Residual Nucleus

the projectile and emitted particle are enclosed in brackets between the target and daughter nuclei

OHHeN 17

8

1

1

4

2

14

7

OpN 17

8

14

7 or, if you prefer

Though a more compact form is often used:

OpN 17

8

14

7 ),(

Target (Projectile, Detected Particle) Residual Nucleus

Convenient as the bracketed part can be used by itself to refer to a particular class of reactions like ( , p) or (n, ).

•compound nucleus reactions and •direct reactions.

Classification of Nuclear Reactions

divided roughly into the two groups

Classification of Nuclear Reactions

direct reactions are of the types

•scattering reaction projectile and scattered (detected) particle are the same

•elastic scattering residual nucleus left in ground state

•inelastic scattering residual nucleus left in excited state

Even inelastic proton scattering is strongly peaked in the forward direction with the cross-section only gradually changing with higher energies.

Ei , pi

Ef , pf

EN , pN

recoilNfiEEE

,

recoilNfippp

,

The simple 2-body kinematics of scattering fixes the energy of particles scattered through .

Num

ber

of p

arti

cles

wit

h E

f

8.0 8.5 9.0 9.5 10.0Energy Ef of scattered particle (MeV)

The predicted elastic scattering calculatedfrom the kinematics of a 2-body collision.

Measuring Ef at a fixed angle

Only Ef should be observed at .

Classification of Nuclear ReactionsAddition details on direct reactions

• pickup reactionsincident projectile collects additional nucleons from the target O + d O + H (d, 3H)

Ca + He Ca + (3He,)

•inelastic scatteringindividual collisions between the incoming projectile and a single target nucleon; the incident particle emerges with reduced energy

2311

2412

Na + He Mg + d

16 8

15 8

31

4120

32

4020

32

9040

9140

Zr + d Zr + p (d,p)(3He,d)

•stripping reactionsincident projectile leaves one or more nucleons behind in the target

•charge exchange reactionsA proton (neutron) enters the nucleus, but emerges as a neutron (proton)

“exchanging charge” with one of the nucleons

Direct reactions are often described as “surface” reactions• involving interactions with individual nucleons within the target• occurring very rapidly (on the order of 10-22 seconds)

Restricted to highest energy projectiles (>20-MeV)(from accelerators)

There is evidence for an altogether different mechanismtaking place at more moderate energies

• slower process – taking 10-16 to 10-18 seconds involving an intermediate, unstable, short-lived state outliving direct contact of projectile & target• final products ejected isotropically compared to the “forward-peaked” cross-section we described for the direct processes

• for incident energies <20-MeV governs all “natural” processes where the projectiles are natural decay products

The most obvious evidence for long lived intermediate states in nuclear reactions is the strongly resonant nature of nuclear cross-sections.

This is illustrated in the figure at right which shows the indium total cross section for neutrons.

The energy of these long lived states is defined to a few eV and if we apply the uncertainty relation dEdt ~ h we see this implies a lifetime of ~ 10-16 s which is very long compared to the time it takes a nucleon to traverse a nucleus ~ 10-22 s.

Recall:Direct reactions do not involve the formation of some intermediate state; their characteristic time of interaction is more like 10-22 s. Variations in their cross-sections, as a function of energy, are spread over a few MeV.

sec10sec)/m103/(meters) 102(/)4(2 22814

0

3/1

0

3/1 crAr

Consider the time for a relativistic to cross/pass through a medium mass (A=125u) nucleus

Let’s compare de Broglie wavelength for

1 MeV (“low” energy) proton

5-10 MeV ’s (typical for nuclear reaction studies)

100-MeV

What could possibly distinguish low from high energy collisions?

mEseVph 2/1013570.4/ 15

)1)(/938(2/1013570.4 215 MeVcMeVseV

fmm

MeVcMeVseV

41054.4

)10)(/3727(2/1013570.415

215

)100)(/3727(2/1013570.4 215 MeVcMeVseV

fm4.1

fmmsms 301086.2)/103)(1054844.9( 14823

0

3/1 rA

0r

Low energy projectiles cannot see anything finer than the whole target nucleus

The resultant collisions are with the entire nucleus,and the energy is shared across it.

High energy projectiles are better focused on individual target nucleons

Their collisions are billiard ball-likeand result in

Direct Reactions.

The intermediate state (itself never directly observed) is referred to as the

Compound Nucleus.

nXXN

A

ZN

A

Z

1

3

2

The above figure shows the contributing cross-sections for the reactions

nXXN

A

ZN

A

Z2

1

2

2

nXXN

A

ZN

A

Z3

1

1

2

nXXN

A

ZN

A

Z4

12 and so on…

Alpha particle energy

Cro

ss S

ectio

n

,n

,2n

,3n,4n

Total

With increasing energy • more and more neutrons are likely to be knocked free• it becomes less and less likely for an individual neutron to be freed

This is NOT explicable as an -nucleon collision, the shares itsenergy across the nucleus, elevating it to some excited state.

Alpha particle energy

Cro

ss S

ect

ion

,n

,2n

,3n,4n

Total

V = V0

V = 0

R

E

Consider a neutron (or -particle) with energy E

within a square well potential.

At r = R there is a large confining force

which as we saw in Problem Set #1 can produce reflections.

Let’s consider the simple ℓ=0 case, where we can substitute:r

u

02

2

in

in Kudr

ud for )(

202

2 VEm

K

inside the well.

with u satisfying:

02

2

in

in Kudr

ud )(

202

2 VEm

K

inside the well:

r

u

Since uinside must vanish at the origin: KrCuinin

sin

02

2

out

out udr

ud k 2

2 2

mEkoutside the well:

For scattering off this potential, we want to consider final states that are traveling outwards outside the well:

ri

outouteCu k for r > R

Continuity requiresdr

du

udr

du

uout

out

in

in

11 at r = R

or kiKRK cot

The energy of the inside states must be expressed as a complex number!

In scattering experiments we are not dealing with simple“stationary states” with real-valued energy eigenvalues.

/)2/(/ 0),,(),,(),,,(tiEiiEt erertr

but with E = E0 i/2

2//0),,( ttiE

eer

te 2

2//0),,(),,,( ttiE

eertr

A stationary state of energy E0

that decays!

because the probability of finding the particle in that state:

These are “virtually bound” states or resonances.

For a short range or abrupt-sided potential there exist quasi-bound or virtual single particle states

of positive energy.

Long range potentials (like the coulomb potential) have no such states.

The projectile can be momentarily trapped in one of these excitedstates, sharing its energy through interactions with the nucleons inside the nucleus: raising some of them into excited states, itself

dropping into lower states.

This is the formation of the many particle excited state

which is the compound nucleus.

At this stage all memory of the original mode of excitation is lost.

At a later time when a decay occurs, the energy is once more

concentrated in a single particle.

To quantum mechanically describe a particle being absorbed, we resort to the use of a complex potential in what is called the optical model.

Consider a traveling wave moving in a potential V then this plane wavefunction is written

The Optical Model

ikxeψ 22 /)(8 VEmk

If the potential V is replaced by V + iW then k also becomes complex and the wavefunction can be written

where

xkxikeeψ 21

and now here

21ikkk

A decreasing amplitude means that the transmitted particle

is being absorbed.

xkxikeeψ 21

xkxikkixikki eee 22121 2)()(*

now describes a traveling wave of decreasing amplitude:

In most cases V >> W.

To make an estimate of the mean free path (the “attenuation length”) we will assume that condition:

)/(1/)(8

/)(822

22

VEiWVEm

iWVEmk

)](2/[1/)(8 22 VEiWVEm

replacing the kinetic energy term (E - V) by the expression mv2/2 gives

)}/(1){/2(

)/(1/42

22222

mviWhmv

mviWhvmk

21

/2/2

ikk

hvWihmvk

2p/h = 2/

where is the de Broglie wavelength of incident projectile

xkxikkixikki eee 22121 2)()(* the mean free path - the distance over which the intensity ( * )

is attenuated to 1/eth its initial value – is given by

W

h

k

42

1

2

the mean free time - the time by which the intensity ( * ) is attenuated to 1/eth its initial value

W

ht

4

since this gives a (mean) distance where tvW

h

4

To describe both diffraction and scattering phenomena with the optical model

requires an imaginary potential of a few MeV.

W

ht

4xikxik eeψ 21

The SHELL MODEL we relied on earlier has a potential depth ~40 MeV.

If the attenuation distance is of the same order as the nuclear radius (3-6 fm) then W 6-8 MeV.

NOTE: this is entirely consistent with the lifetime of the virtual single particle state before absorption of about 1022 sec.

)64sec/(1013570.4)4/( 15 MeVeVWht 23105.5 t ~ 10-22

bYCXa *Writing the compound nucleus reaction as

the cross-section (which remember expresses a probability) can be expressed as

b

aba

the widthfor decay to b

the totaldecay width

b is the “partial width”   is the total width (ħ times the total decay probability).

the cross-sectionfor absorbing a

and forming the compound nucleus

)()( bY to decaying ofy probabilitC* forming ofy probabilit

The behavior of the cross-section with energy depends on the relative sizes of & the spacing between energy levels.

For low excitation of a nucleus the energy levels are relatively well spaced and the cross-section exhibits resonance while at higher energies of excitation the width will overlap several energy levels and the cross-section varies much more slowly with energy.

This is the continuum.

The energy at which the resonance continuum transitionoccurs depends upon A.

For A ~ 20 ~10 MeV while for A ~ 200 ~100 keV.

For well separated levels, an individual state will decay as exp(- t/ ħ)RECALL: we write such a wave function as

)2/(/0)( ttiEeer

the 1st exponential gives the normal oscillatory time dependence of a wavefunction with E0 (here the energy above ground state).

The second exponential gives the decay of the state.

dEeE iEt /)( AThis convolution you’ll recognize as the Fourier transform!

Since it is decaying, such a state is not a solution of Schrödinger’s equation with a static potential (not a stationary state).

It can,however,be considered a superposition of such states

]4/)[(4

)()(

22

0

2

2

2

EE

rEA

The function A(E) can be obtained by Fourier transform which yields

Giving the cross-section the form

22

0)2/()(

EE

Ca

The constant C depends upon the phase space available to the incident particle

2

2

4

)12(

p

hC a

where p is the momentum of the incident particle, ℓ its orbital angular momentum, and a the partial width for the decay back to the initial state.

22

0

2

)2/()(

)12(

4

1

EE ba

ba

All together

ph /where

This is the Breit-Wigner formula for a single level reaction cross-section.For example - in the case of elastic scattering (a = b)

at the maximum of the resonance (E = E0) the cross-section is

2

22 )12(

a

elastic

And if no other processes to compete with (a=)

/)12(2 elastic

the maximum possible elastic cross-section.

2

2 )12)((

aa

inelastic

4

)12(2 inelastic

For the inelastic cross-section (b = a) at the maximum of its

resonance (E = E0)

with a maximum value of

when a = /2.

When the separation of energy levels is much smaller than the total width there are many levels contributing to a given process.

This mixture of levels

described as a compound nucleus.

Furthermore

The subsequent decay of this intermediate state is largely independent of its mode of formation. A given compound nucleus may be formed by any of several reactions but the probability of a certain type of final state is only dependent upon the amount of excitation energy.

In a compound nucleus all sense of direction of the incident particle is lost the produced particles are ejected in an essentially isotropic distribution

in the center of mass frame.

The projectile is captured by the target forming an intermediate state

the compound nucleus

BB

BeC

LiN

O

HeO

pF

10

5

10

5

8

4

12

6

6

3

14

7

16

8

3

2

17

8

19

9

BB

BB

BeC

BeC

LiN

LiN

O

HeO

HF

dF

Ne

nNe

pF

11

5

9

5

10

5

10

5

9

4

11

6

8

4

12

6

7

3

13

7

6

3

14

7

16

8

3

2

17

8

3

1

17

9

18

9

20

10

19

10

19

9

2010[ Ne]*

Here are plotted the yields of

decay products from the compound nucleus

64Zn formed by

2 different routes

Note: the relative cross-sectionsfor the different processes

63Cu(p,n)63Zn and

60Ni(,n)63Zn remain ~constant

across theplotted energies.