1. The Size and Shape of Nuclei
Chapter 3
The Size and Shape of Nuclei
◎ The size of nuclei
● The scattering of electron by nuclei
◎ The nuclear electric charge distribution
● The nuclear electric form-factor
◎ Nuclear scattering and nuclear size
◎ The shape of nuclei

2. “7And I will establish My covenant between Me and you and your seed after you throughout their generations for an everlasting covenant, to be God to you and to your seed after you. 8And I will give you and to your seed after you the land of your sojournings, all the land of Canaan, for an everlasting possession; and I will be their God.” (Genesis 17:7 ~ 8)

3. Borders of the land of Canaan

4. Nuclei are not always spherical!
§ 3.1 The size of nuclei
What we are interested here is
the size of nuclei
how the nuclear matter is distributed inside a sphere, if the nucleus
is spherical.
In fact
Nuclei are not always spherical!
A prolate deformed nucleus

5. What we mean by the size of a nucleus ?
Rough visualization of an electron "wave function". (Art by Blake Stacey.)
The electron probability density for the first few hydrogen atom electron orbitals shown as cross-sections. These orbitals form an orthonormal basis for the wave function of the electron.
In atomic physics, the boundary of an atom is not sharp since the wave function of the outer electrons decreases monotonically.
We have a similar situation in the nuclear physics!

6. These deviations occur at short distances of approach.
the finite charge density distributions overlap.
2 strong nuclear forces between the α particle and the target nucleus comes into play.
Rutherford’s formula breaks down when the kinetic energy of the incoming α particle is too high and it comes too close to the target nucleus.

7. Size - Rutherford’s result
measurements by Rutherford et al
nuclei size of about m
Nuclear radii, R, are seen to increase with the total number of nucleons, A,
R = r0A1/ 3
(1)
To reveal more detail needs smaller de Broglie wavelength for the probing particle

8. To reveal more detail needs smaller de Broglie wavelength for the probing particle
(2)
We need to use proper incident particles with proper kinetic energies for probing nuclear structures.

9. Charge The charge of the nucleus is just the sum of the proton charges
(2) The hydrogen atom is precisely neutral
(3) The neutron is precisely neutral - but has a magnetic moment
- an internal charge structure - and so a composite object
net hydrogen charge is less than 1 x e
Hydrogen n=4 level
Neutron charge is less than 2 x e
However the neutron does have a magnetic moment
moving charges
a composite object - not fundamental

10. 1 The nucleus has mass density, ρ = mass divided by its volume.
There are two different kinds of distributions which we are considering.
Nuclear matter distribution
Nuclear charge distribution
1 The nucleus has mass density, ρ = mass divided by its volume.
mass of the neutron
(3)
neutrons and
protons have the same
mass to 0.14%
= about 1.82 x 1017 kg/m3
so nuclear density does not depend on the atomic mass number

11. The density distributions for some nuclei

12. It is roughly a constant.
2 Nuclear charge distribution ρc = total charge divided by its volume.
(4)
Since N ~ Z ~ A/2
(5)
It is roughly a constant.

13. = 1.2 F for charge distribution
We may use electrons to probe the charge distribution in a nucleus because electrons do not experience the nuclear force.
Accelerated neutrons can be used as a probe for nuclear matter distribution since neutrons are electrically neutral and do not experience Coulomb force.
R = r0A1/ 3
r0 = 1.4 F for nuclear
matter distribution
= 1.2 F for charge distribution
Rough description
Nuclear charge distribution
Nuclear matter distribution

14. § 3.2 The scattering of electrons by nuclei
When a gold (
) foil is bombarded by energetic α particles:
(1) R > m Coulomb scattering
(2) R < m Nuclear potential
a-1 is an estimation of the short nuclear interaction range
a-1 ≈ 1 ~ 2 F

15. We may use electron beams as probes to study nuclear charge distribution.
― electron scattering
Early problems with electron beams:
Beam energy is not mono-energetic.
Electron’s linear momentum too small and easily
experiences large angle scattering as well as multiple
scatterings
3 Small incident kinetic energy
A typical electron energy from beta decay.
wave length too long !!
de Broglie’s wave length

16. In early years there were not comprehensible results through electron scattering data due to insufficient electron’s kinetic energy.
If we are to acquire sensible results out of electron scatterings the de Broglie wavelength must be of the order of 10 F.
The corresponding linear momentum (pe) for an electron is
And the kinetic energy of an electron must be around 124 MeV.
The scattering of energetic electrons (Te > 100 MeV) is a very important tool in the investigation of the nuclear size.

17. This is the Mott formula.
The formula for the differential cross-section for the elastic scattering of relativistic electrons was derived by Mott using relativistic quantum mechanics. Mott formula is analogous to Rutherford’s for α particle scattering.
The Rutherford formula for the differential angular cross-section for the elastic scattering of a non-relativistic, spin-less particles of unit charge, momentum P, and velocity v, at a fixed (no recoil) target nucleus of atomic number Z, is
(6)
If the incident particle is an electron which is unpolarized and can be relativistic (v→c), the formula becomes
(7)
This is the Mott formula.

18. Assumption in the Mott formula:
1. Relativistic quantum mechanical treatment for the electron;
2. First-order perturbation theory is adequate to calculate the scattering
cross section;
There is no nuclear recoil;
The nuclear electric charge is point-like;
The nuclear spin is zero.
(7)

19. § 3.3 The electric charge distribution
Nuclear charge distribution
There are two models which are to be used in the study of nuclear charge distributions.
Model I is a sharp-edged charge distribution which is very unlikely but can be tested.
Model II softens the hard edges by assuming a charge distribution with a mathematical form normally associated with the Fermi-Dirac statistics but which, applied to nuclei, is called the Saxon-Woods form.

20. Model I
Model II a t ρ r
(8)
Saxon-Woods form

21. ρ(r) = ρ0, r < a ρ(r) = 0, r > a Model I
Since the total charge in the nucleus is Ze. Therefore the following equation must be satisfied.
(9)

22. For a spherical nucleus
Model II a t ρ r
(8)
(10)
Total Charge
In a nucleus
For a spherical nucleus

23. “19But God said, No, but Sarah your wife will bear you a son, and you shall call his name Isaac; and I will establish My covenant with him for an everlasting covenant for his seed after him. 20And as for Ishmael, I have heard you; indeed, I have blessed him and will make him fruitful and will multiply him exceedingly. Twelve princes will he beget, and I will make him a great nation.” (Genesis 17:19 ~ 20)

24. Scattering experiments
incident particle is a plane wave
Scattered wave fronts are spherical
intensity variations on wave-front
due to diffraction
diffraction pattern observed from
plane wave striking a target
analogy with optics: like optical
diffraction
but nucleus has blurred edges

25. “Fraunhofer diffraction pattern”.
The optical description of the Fraunhofer diffraction pattern
In optics if the obstacle (lens) size D is much larger than the wavelength λ of an incoming wave (D >>λ) there would appear diffraction pattern. The observed pattern on a screen which is located behind the obstacle is called the
“Fraunhofer diffraction pattern”.
Mathematically we perform a Fourier transform on the obstacle and is able to describe diffraction patterns seen on the screen. In other words diffraction pattern is no less than the Fourier transformed image of the obstacle.
For a disk lens of diameter D the first minimum appearing on the screen satisfies the following relation.
(11)
First minimum of the diffraction pattern

26. The Fraunhofer diffraction pattern
First minimum of the diffraction pattern

27. In the electron scattering experiment we may treat the incoming electrons as quantum mechanical waves. The wavelength of incident electrons is inversely proportional to the electron’s linear momentum.
The nuclear charge distribution is regarded as the “optical obstacle” which stays in the path of incoming “electronic wave”.
The angular distribution of differential cross sections measured on various scattering angles can be regarded as the diffraction patterns after the “electronic wave” passing through the nuclear charge distribution.

28. The first maximum occurs at
θ ≈ 24°
D = 2a = 2  4.1 = 8.2 F
Te = 450 MeV →  = 2.76 F
The first maximum occurs at

29. Angular differential cross section
Te = 502 MeV
λ=2.47 F
Electron scattering
Proton scattering
Tp = 1000 MeV
λ= 0.73 F
λ=2.47 F
First minimum
Angular differential cross section
Optical diffraction pattern

30. § 3.4 The nuclear electric form-factor
For electron scattering we take the nucleus to have charge Ze where e is the charge on the proton.
If the nucleus is point-like the measured differential cross section (dσ/dΩ) is dependent on the scattering amplitude Zef(θ) at large distance at an angle θ, so that
(12)
For a point-like nucleus

31. (12)
where
(13)
is dependent on various scattering details such as the wavelength of the incident wave, energy, or momentum etc.
In general A(θ) is a complex function which is called the scattering amplitude.

32. In the case of point Coulomb scattering
and the scattering amplitude for point Coulomb scattering is
(14)

33. In the case of point Mott scattering
and the scattering amplitude for the point Mott scattering is
(15)

34. Form factor F(θ)
While the nuclear charge is no longer a point we need a form factor F(θ) to modify our scattering formula. Namely,
(16)
(17)

35. Scattering amplitude A(θ)
(17)
The form factor F(θ) is connected to the internal charge distribution of a nucleus. It is actually the Fourier transform of the nuclear charge density ρ(r).
Form factor F(θ)
Scattering amplitude A(θ)
Differential cross section [dσ/dΩ(θ)]

36. The incident and scattered electron have momentum p and p’ with
If that charge is spread out then an element of charge d(Ze) at a point r will give rise to a contribution to the amplitude of eiδf(θ)d(Ze) where δ is the extra ‘optical’ phase introduced by wave scattering by the element of charge at the point r compared to zero phase for scattering at r = 0.
The incident and scattered electron have momentum p and p’ with
p =∣p∣=∣p’∣. p p’
The momentum transfer q [q = 2psin(θ/2)] is along OZ, O being the nuclear center.

37. The ‘optical ray’ P1OP1’ (path a) is taken to have zero relative path length.
The ray P2SP2’ (path b) has equal angles of incidence and reflection with the ray P1OP1’ at the plane AXA’ which is perpendicular to OZ.
Due to different lengths of two paths there will be a phase difference δ for waves coming from two paths. la lb
(path a)
(18)
(path b)

38. r
The contribution to the scattering amplitude for a ‘point charge’ with volume element dV and located at the position r is
(19)

39. r And the scattering amplitude is the volume integral of the eq. (19).
(20)

40. P0
P0’
The line OZ bisect the angle ∠(P1OP1’) and intersect with the line AA’ at the point X.
Every path that is reflected from the plane AX’A (⊥line OZ) is of the same ‘optical path length’ . Therefore all paths reflected from the plane AX’A have the same ‘optical path length’ .
The optical length difference between P2SP2’ and P1OP1’
= The optical length difference between P0XP0’ and P1OP1’ d

41. P0
P0’ θ P1
P1’ X O
d/2
(21)

42. It is evident from the figure that (OX) = (OS) cos α = r cos αla lb
It is evident from the figure that
(OX) = (OS) cos α = r cos α
(22)

43. The phase difference δcan then be expressed as
Finally
(23)
Here λ = p/h, and p is the linear momentum of the incident electron.

44. With λ = p/h the equation (23) can also be written as
(24)
Since the magnitude of the momentum transfer (q) for the electron elastic scattering on a nucleus is [2p sin(θ/2)] it can be shown geometrically that
(25)

45. (20)
Substitute the expression in the equation (25) into the equation (20) we get
(26)
More specifically
(27)

46. Now we have for the total charge
So we see that we can write
(28)

47. F(θ) is called a form factor.
Thus the Mott (or Rutherford) scattering amplitude Zef(θ) is changed by a factor F(θ) and the scattering formula becomes
(29)
F(θ) is called a form factor.
Somewhat formally it could be written
(30)

48. (30)
Since the form factor is more properly a function of q than of θ it has become useful to write F(q2) rather than F(θ).
It is immediately recognized that the form factor F(q2) [or F(θ)] is actually the Fourier transform of the nuclear charge distribution.

49. 1. When q → 0 (without momentum transfer), F(q2) → 1 and the
The physical meaning of the form factor F(q2).
1. When q → 0 (without momentum transfer), F(q2) → 1 and the
scattering is no different from that for a point-like nucleus. As q increases the oscillatory nature of the exponential in equation (30) for an extended nucleus reduces from 1 and the scattering is reduced.
(30)
An extended electric charge has greater difficulty in taking up the momentum transfer than does the point-like arrangement of the same total charge.

50. If the nuclear charge distribution is of spherical symmetry
please show that
(a).
(31)
(b).
(32)

51. If the nuclear charge distribution is like a step function shownρ0
please show that
(33) a
where

52. 2. If a nucleus is seen as a point charge without any extensive
distribution then we have the form factor simply as 1.
The point charge is located at the origin.
For the form factor value of other than 1 there is some sort of extended charge distribution being detected.

53. is a rapidly changing periodic function.
3. High-q transfer measurement shows characteristics on the nuclear surface.
(32)
is a rapidly changing periodic function.
The value of integration in the equation (32) would be very small if the charge distribution function ρ(r) is almost constant.
It is only when the distribution function ρ(r) itself changing rapidly can we expect a notable value from the integral.
The foregoing argument points out that data collected from
high-q transfer measurement actually reveal
characteristics on the nuclear surface on which charge distribution changes rapidly with coordinate locations

54. The mean square average of the nuclear charge extension
4. Low-q transfer measurement shows the mean square average
of the nuclear charge extension.
The mean square average of the nuclear charge extension

55. In the mathematical description the form factor F(q2) is the Fourier transform of the nuclear charge distribution function.
In principle we should be able to know all the details of nuclear charge distribution if we can acquire all necessary information out of the momentum transfer measurement.
In reality it is impossible to make a thorough measurement for our needful purpose. We are in a situation of fragmental information as far as the nuclear charge distribution is concerned.
With this limitation in mind we can only fit our data taken from the electron scattering to the Wood-Saxon function at this stage.

56. a t ρ r
By varying two parameters, a and d, in the function until data are reasonably fit to the shape of the presumed distribution can we more or less tell how charges are distributed in a nucleus.

57. The measured differential scattering cross-section dσ/dΩ for the scattering of 450 MeV electrons by Ni-58. The positions of the diffraction minima should be matched against those of the figure below.

58. The first maximum occurs at
θ ≈ 24°
D = 2a = 2  4.1 = 8.2 F
Te = 450 MeV →  = 2.76 F
The first maximum occurs at

59. Here shows results from some nuclei with momentum transfer q = 800 MeV/c. It is apparent that our Woods-Saxon model is only an approximation.

60. If we neglect the finer details, our Woods-Saxon model gives a modestly reasonable description of nuclear size.
An analysis of the data made by Hofstadter and Collard (1967) gave for the half-point radius.
Where the ± indicates the range of values found in nuclei with A > 40. Below A = 40 there are marked changes in thickness with A.

61. Problems (I)
1. Show that, for a spherically symmetric charge distribution,
where〈r2〉is the mean square of the electric charge distribution.
2. Show that the form factor for the charge distribution of model I is

62. Problems (II) 3. Find the form factor for a charge distribution
4. An electron of momentum 330 MeV/c is scattered at an angle
of 10° by a calcium nucleus. Assuming no recoil, find the momentum transfer and its reduced de Broglie wavelength. Also calculate the Mott differential cross-section (point-like nucleus), and by what factor it is reduced if the calcium nucleus (A = 40) can be assumed to be represented by model I with a = 1.2A1/3 F.

63. § 3.5 Nuclear scattering and nuclear size
Electron scattering
the distribution of electric charge in a nucleus
Neutron scattering
the distribution of matter in a nucleus
Other particles which can be employed in measurements
protons
and even the long-lived elementary particles….
α-particles
deuterons
tritons
helions

64. protons
and even the long-lived elementary particles….
α-particles
deuterons
tritons
helions
These particles are charged so that scattering is caused by the combined effect of both the nuclear and Coulomb forces. Considerable task is required to take away effect from Coulomb force.

65. Elastic scattering of 14 MeV neutrons by nickel
The observed diffraction pattern with peaks and valleys is the characteristic of scattered waves from an absorbing object with a moderately well-defined boundary.
The diffraction pattern can be interpreted by the optical model.

66. The Optical Model Potential
The Coulomb potential for proton only
The nuclear potential well
The imaginary potential representing absorption of the incident nucleon
The spin-orbit interaction
1. V, W and VLS are expected to and do vary with energy.
2. The functions f1(r), f2(r), fLS(r) are usually
taken to have the familiar Woods-Saxon form

67. Nuclear matter distribution Nuclear charge distribution
For f1(r), f2(r), and fLS(r)
Roughly We have the following results
a = 1.2A1/3 F
d = 0.75 F
Nuclear matter distribution
a = 1.18A1/3 ± 0.48 F
d = 0.55 ± 0.07 F
Nuclear charge distribution

68. Possible shapes of nuclei apart from spherical
1. If the charge distribution is not
spherically symmetric the nucleus can have electric moments other than monopole. This will manifest by its effect on the optical spectroscopy of the atom.
2. A non-spherical nucleus will have rotational states of motion and are identifiable in the spectrum of excited states.

69. ~ The End ~