Nuclear Sizes and Shapes

The Charge distribution within a nucleus can be determined using elastic electron scattering.

If we assume a uniform distribution of protons within the volume of the nucleus, the charge distribution can give an approximation to the distribution of matter within the nucleus.

Elastic electrons scattering, is, to a first approximation, just like Rutherford Scattering. The incident charge is e instead of +2e and mass is

different.

However, we must now use quantum mechanics and relativity in the derivation of the scattering formula. We will first write down the result assuming the electron is

scattering from a point charge.

Electron Scattering If the nucleus can be considered a point charge, the electron

scattering cross section is

This is known as the Mott cross section.

Note: p = momentum of the electron.

v = speed of the electron.

= fine structure constant 1/137

This has been derived using quantum mechanics and relativity.

It includes the effects of the electron spin.

It assumes a spin-less nucleus.

The Mott cross section is exactly like the Rutherford scattering formula except for the additional part in the square brackets.

)2/(sin1

)2/(sin

1

4

2

2

2

422

2222

c

v

vp

cZ

d

d

Mott

(Not distance of

closest approach.)

Electron Scattering

But the nucleus does have a size:

The actual scattering of an electron will be the sum of the

scatterings from each piece of the nucleus.

i.e. from each volume element, dV, in the nucleus.

We need to treat the electron as a quantum mechanical

particle.

Electron Scattering

We can write the Mott cross section as

2)(

Zef

d

d

Mott

Charge of the scattering centre A function of

Scattering amplitude A()

This is like writing:

Probability Electron wave function 2

For a finite nucleus: 2

)( i

iAd

d

Amplitude for scattering from

each volume element dV.

Electron Scattering

Define: Charge density within the nucleus

We will consider scattering from a volume element at position S compared to scattering from centre C.

)(r

S can be specified by the spherical polar coordinates

(r, , )

(We use , so as not to confuse with .)

C

r

S

Electron Scattering

Charge at S: C

r

S

dddrr

dVdQ

sin)(

)(

2r

r

The scattering amplitude from S is then iedQf )(

where: ie is the phase amplitude difference compared

to scattering from the centre C.

= phase angle

d2 (can be related to and ) r

d = path length difference

= De Broglie wavelength of electron.

Electron momentum =

hp fi pp

Electron Scattering

We define the Momentum Transfer:

)( if ppq

q

ip

fp

= momentum transferred from the electron to the nucleus.

)2/sin(2 pq q

C

r

S q

Some geometry (next

page) can show that

q

rq

d2

/)()( rq

riedVf So scattering amplitude from S is

Electron Scattering

Derivation of

rq

d2

C

r

S

ip

fp

p

i rp

p

f rp

Additional path length

ppd

firprp

Therefore:

rqrqrpp

)/(

2)(22

hp

d fi

hp fi pp

Electron Scattering

Therefore the scattering amplitude from the whole nucleus is

dVefedVfAV

i

V

i

// )()()()()( rqrq

rr

2

0 0 0

2/ sin)()(r

i drddref rq

r

The total charge in the nucleus is

V

dVZe )(r

2

0 0 0

2 sin)(r

drddrr

Electron Scattering

So we can write:

Ze

dVe

ZefA V

i

/)(

)()(

rqr

Amplitude for scattering

from a point charge Ze Modification by the nuclear

charge distribution

)(FThe total differential cross section is:

2)(

A

d

d

22

)()( FZef

Mottd

dF

2)(

V

i dVeZe

F /)(

1)( rq

r

Electron Scattering

F() is more properly a function of q.

q depends on both p and .

So it is more commonly written

)2/sin(2 pq q

V

i dVeZe

qF /2 )(

1)( rq

r

)( 2qF is called a Form Factor.

It represents the modification of the point

particle scattering by the finite nuclear size.

Electron Scattering

If the Nucleus is Spherically Symmetric.

i.e.

then

Note: as

)()( r r

drqrrrZeq

qF )/sin()(4

)( 2

1)( then )0( 0 2 qFq

i.e. we have point charge like behavior.

as q increases, F(q2) decreases

Electron Scattering

F(q2) is the Form Factor.

It is essentially the Fourier transform of the charge

distribution.

If the Nucleus is Spherically Symmetric.

i.e.

then

)()( r r

drqrrrZeq

qF )/sin()(4

)( 2

V

i dVeZe

qF /2 )(

1)( rq

rMottd

dqF

d

d

22 )(

)2/(sin1

)2/(sin

1

4

2

2

2

422

2222

c

v

vp

cZ

d

d

Mott

From a point

charge Ze:

From a charge

distribution (r):

Electron Scattering To investigate the effects of nuclear

size we will take a simple “Toy Model”.

We will assume the charge distribution is spherically symmetric and has a hard edge.

i.e. (r) = constant, r < a

and (r) = 0, r > a

)(r

r

We can show that:

qaxxxx

xqF with cossin

3)(

3

2

a

Note: if x is small, 10

1)(2

2 xqF 0 as 1)( 2 qqFso

This also lets us see why we write F(q2) rather than F(,p).

Electron Scattering

qaxxxx

xqF with cossin

3)(

3

2

Actually: )(3

)( 1

2 xJx

qF

J1(x) is a Bessel function: x

J1(x)

So: 2

2 )(qF

log scale

First zero is at x = 4.5

x

Toy Model

Electron Scattering

If we take a nucleus with

The first zero is when,

fm 5.4a

Toy Model

5.4

qa

11 fm 1fm 5.4

5.4

q

Often momenta are expressed in “inverse Fermi” fm1 units

(really )

So

q

ccq /)fm 1()fm 1( 11

Then using MeV.fm 197c

MeV/c 197qFirst minimum is at

Electron Scattering

So: 2

2 )(qF

Toy Model

q MeV/c

200 400 600

fm 5.4a

1 2 3

1fm / q

Use

to relate to .

)2/sin(2 pq

20 40 60 80

If incident electron

energy = 450 MeV

Electron Scattering

In practice an experiment measures the differential cross section.

Mottd

dqF

d

d

22 )(

Calculate

Deduce

Measure

In principle it is possible to invert

V

i dVeZe

qF /2 )(

1)( rq

r

to find )(r

e.g. for 208Pb with spherical

symmetry assumed.

Electron Scattering

Usually a model for (r) is assumed and then this is used to obtain a best fit to F(q2) data.

A practical model is

This is called a Woods-Saxon shape.

The best fit to many nuclei gives,

with d 0.55 fm, for A > 40.

This is the charge radius.

Usually the nuclear radius given by

is pretty close.

d

ar

e

r

1

)( 00

a

d

r

fm 48.018.1 3/1 Aa

3/12.1 AR

2

0

Electron Scattering

More detailed fits can reveal some internal structure

in the charge distribution.

e.g.

Optical Model Electron scattering reveals the charge distribution in nuclei.

The nuclear matter distribution can be probed by scattering neutrons.

In this case the nuclear force is involved. Charged particles (e.g. protons or pions) can be used as well. This, however, introduces the extra complication of both the nuclear force and the electromagnetic interaction acting together.

The main difference between using neutrons instead of electrons as a probe, is that we do not only get elastic scattering. We also get absorption of the neutron by the nuclear matter.

A model known as the optical model is used to interpret neutron scattering experiments.

Optical Model The Optical Model:

• Describes the nucleus as a potential well with the

ability to absorb.

• This is in analogy with optics for light scattering

through a not completely transparent glass.

• The model is sometime called the “cloudy crystal

ball” model.

• The model has many parameters!

• The results for a Wood-Saxon shaped potential

well gives fm 75.0 fm, 2.1 3/1 dAa

Same as for

electron scattering.

Larger than for the charge

distribution (d 0.55 fm)