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This lecture:

What can we learn from scattering experiments?

Summary from last time:

- planetary model of the atom

- isotopes and mass spectroscopy

- nuclear stability due to proton and neutron pairing

- discovery of the neutron

e.g. Nuclear size – Quantified in terms of scattering cross sections ()

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Feedback from last 2 lectures

- Bigger, better red boxes!- Too much writing per slide

- Maybe too much text – more description, less reading out? more diagrams in this lecture

- 5 pm lectures: can we do anything about it?

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- Also chokka for Newman B and E : most of the “good” times taken......

- Sporadic bookings possible, but block bookings difficult.

- Some times that I can’t make (e.g. Mondays at 10)

However, some possibilities: Monday at 1pm (Newman F) or 2pm (Newman B)? Friday 9pm (Newman F)? Other days at 5pm.....

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Feedback from last 2 lectures

- Bigger, better red boxes!- Too much writing per slide

- Maybe too much text – more description, less reading out? more diagrammatic and graphical description in this lecture

- 5 pm lectures: what do we think?

- Anything else?

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Formal definition of cross section (:

Quantifies probability of collision between two particles as function of incident beam flux () and target density (n):

Rate of collision per unit volume = n

Density of target, n

Flux of incident beam ()

Flux: e.g. 60 billion neutrinos per second through your thumbnail. How many in tip of thumb at any given moment?

Flux = Number per second / Area = 60 x 109 cm-2 s-1

Density = Flux / Velocity (c) = 60 x 109 cm-2 s-1 / 3 x 1010 cm s-1

= 2 cm-2

If volume ~ 1 cm3

2 neutrinos at any given moment

- “hit” rate dependent on:a) How many targets you have to “aim” at: nb) How big the targets are: c) How often you fire at target:

Also: Flux = density (of particles in beam) x velocity

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Density of target, n

Flux of incident beam ()

=1/ (n Mean free path () – the average distance travelled before scattering:

– describes the spatial decay of beam: N (x) = N0 e – x/

– and the number of collisions experienced in distance x:

C ≡ N0 - N(x) = N0 (1- e – x/) ~ N0 x / for thin targets

probability of undergoing collision in thin target of thickness x is just n x

Sample with finite depth:

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Geometric cross section (G): Area around centre of one “target” within which centre of the “scatterer” must fall:

e.g. For hard spheres: G = (2R)2

In reality, nuclei are not hard spheres – cross section only gives an indication of size

Moreover - Collisions remove particles from incoming beam by scattering or absorption:

Three possible “collisions” and partial cross sections: (a) Elastic scattering by target, σe,

(b) Inelastic scattering by target, σi

(c) Absorption by target, σa

Total cross section is σ = σe + σi + σa

(American physicists working on the Manhattan project described the uranium nucleus as "big as a barn“:

1 barn = cross section of uranium nucleus)

Understanding cross sections

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Angle dependent scattering cross sections (dd):

However, scattering is usually function of scattering angle: So-called “differential cross section”, dσ/dΩ:

- Consider cone formed by finite element of solid angle ΔΩ at angle θ- Define Δσ: part of σ which represents probability of scattering into cone- Thus differential cross section for scattering at angle θ:

- Incident particle can scatter at any angle with respect to incident direction- Need to integrate over all solid angles to get full cross section

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In reality, geometric cross sections do not give good predictions need to introduce extension to simple geometric considerations

The Rutherford cross section (R): takes into account Coulomb scattering of charged projectiles from “static” positive nucleus

b

b determines scattering angle

derived by considering momentum conservation after impulse by Coulomb force on particle trajectory with impact parameter b(see e.g. Martin p363 for derivation)

Result:

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The Mott cross section (M):

Rutherford scattering is generally good enough to describe non-relativistic alpha scattering from small nuclei

- for high energy collisions need to take into account relativistic corrections as well as corrections due to recoil of the nucleus

- other projectiles (e.g. electrons) with non-zero spins can interact with nuclei via their magnetic moment as well as by their charge

differs significantly from Rutherford cross section for high projectile velocities ( 1) and high scattering angles ( 180°)

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1950’s: High energy electron scattering from gold nuclei saw significant deviations from Rutherford and Mott predictions:

Why??

(Father of Douglas Hofstadter)

Hofstadter’s data fell below Mott prediction- Curves separate at some critical angle- Critical angle depends on experimental parameters: nucleus being probed, energy of electron, etc.

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Electron has de Broglie wavelength

Diffraction in scattering experiments:

D

max~ / D = h / (mvD) Consider from optics the Rayleigh criterion for angular resolving power

Key Mott and Rutherford model assumption: Point-like projectile and nucleus

- Comparable in size to nucleus for high energy beams (> 100 MeV)- Only invented in 1950’s: never observed before Hofstadter

(describes diffraction limit for angular resolution of image captured through lens of size D)

In reverse:

critical angle where drops below Mott prediction yields size of nucleus, D ~ few x 10-15 m

get diffracted waves up to angle θmax ~ / D beyond this angle, scattered waves do not penetrate

- describes how waves emanate from nucleus “aperture”

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Fine structure with electron ≤ D: (i.e. electron energy > 100 MeV)

Note: minima in electron scattering data do not go to zero (non-perfect contrast) Because nucleus does not have hard edge, and is “Blurred” at edge

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Yields relationship between scattering angle θ and momentum transfer q i.e. can now write scattering cross section in terms of q

Momentum transfer in scattering experiments:

Mott cross section

Momentum transfer and scattering cross section:

(For small

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F(q) obtained from ratio of d/dexp to d/dM

Uniform central region:

Non- uniform “skin”:nuclei don’t have “hard edges”

(r) obtained from inverse Fourier transform of F(q):

Obtaining the charge distribution from electron scattering experiments:

Figure 2.4 from Martin

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Obtaining the charge distribution from electron scattering experiments:

core radius (a) a depends only on nucleon number A, while skin at surface (d) is approximately constant thickness for all

However, difficult to obtain accurate scattering measurements over wide range of angles, as scattering falls off very rapidly at large angles

In practice, parameterised charge distributions tested against data, adjusted for best fit:

Best fits to electron scattering data:

F(q) obtained from ratio of d/dexp to d/dM

(r) obtained from inverse Fourier transform of F(q):

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Measuring the mass distribution

Can be very different angle from coulomb scattering observed – need to treat scattering and absorption in nuclear mass potential

(approx. constant for of all nuclei)

Core radius (a) and depth of surface skin (d) slightly larger than for charge distribution, since usually have slightly more neutrons (mass) than protons (charge).

Difference is small: n-p attraction pulls neutrons in a bit, while p-p repulsion pushes protons out a bit, so that p and n densities not too dissimilar

After Hofstadter, experiments very quickly expanded to other scattering particles

Scattering using high energy hadrons: At high speed, α-particle overcomes Coulomb repulsion; allows interaction by strong force Neutrons: removes Coulomb term

Data still fit well to Woods-Saxon form:

Best fits to hadron scattering data:

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For next time:

Implicit assumptions so far: nuclei are spheres

What determines the shape of a nucleus?

Summary:

- formal definitions of cross section

- diffraction effects in scattering, determination of charge/mass distribution

- definitions of Rutherford and Mott cross sections for coulomb scattering