Nuclear Radius

Nuclear Physics Lesson 11

Homework

Research and explain how electron diffraction can be used to determine the radius of the nucleus (6 Marks)

Past Paper Question on today’s material.

Complete both by Next Lesson.

Learning Objectives

Know how to determine a value for the index for an equation of the form y = kxn. EMPA!

State and use the equation for dependence of radius on nucleon number.

Calculate nuclear density.

Recall the implications of the high nuclear density compared to atomic density.

Nuclear RadiusNuclear Radius

The nuclear radius, R, can be shown to be The nuclear radius, R, can be shown to be related to the nucleon number, A according related to the nucleon number, A according to:-to:-

Where rWhere r00 and n are constants. and n are constants.

Given R for a number of nuclei with a variety Given R for a number of nuclei with a variety of A, how can we determine rof A, how can we determine r00 and x? and x?

nArR 0

DataData

Element A R/fmBoron 11 2.34Neon 20 2.85Argon 40 3.59Nitrogen 14 2.53Bromine 80 4.52Antimony 122 5.21Gold 197 6.11Uranium 238 6.51

Data (Suggestion)Data (Suggestion)

Element A R/fm ln A ln RBoron 11 2.34Neon 20 2.85Argon 40 3.59Nitrogen 14 2.53Bromine 80 4.52Antimony 122 5.21Gold 197 6.11Uranium 238 6.51

Finding rFinding r00 and x and x

Logging both Logging both sidessides

log (AB) = log A + log log (AB) = log A + log BB

log (Alog (Ann) = n log A) = n log A

nArR 0

)ln(ln 0nArR

nArR lnlnln 0

0lnlnln rAnR This is in the form y = mx + c.

Finding rFinding r00 and x and x

Plotting ln R against ln A should give Plotting ln R against ln A should give a straight line with gradient = n and a straight line with gradient = n and intercept = ln rintercept = ln r00

Note that eNote that elnln rr00=r=r00..

0lnlnln rAnR

Excel PlotExcel Plot

From graph:-From graph:-

gradient n = 1/3gradient n = 1/3

intercept = ln rintercept = ln r00 intercept = -34.49intercept = -34.49 rr00 = 1.05 fm = 1.05 fm

Equation

Dependence of radius on nucleon number:-

3/10ArR

[The term A1/3 means the cube root of A, the nucleon number.  The term r0 is a constant with the value 1.4 × 10-15 m.  R is the nuclear radius.] What physical quantity is r0?

Rearrange in the form of A=.Try working out R for Gold (A =197 ) and Carbon (A=12)

Nuclear Density

Radius of a carbon nucleus ~ 3.2 × 10-15m. Radius of a gold nucleus ~ 8.1 × 10-15m.

Mass of a carbon nucleus ~ 2.00 × 10-26kg. Mass of a gold nucleus ~ 3.27 × 10-25kg.

What are the densities of the nuclei?

Nuclear DensityNuclear Density

Recall that density is given by:-Recall that density is given by:-

You can assume that the nucleus is You can assume that the nucleus is spherical so that V = 4/3 spherical so that V = 4/3 ππRR33, so the , so the density is given by:-density is given by:-

V

m

334 R

m

Nuclear Density Density of carbon nucleus ~ 1.46 × 1017 kg m-

3. Density of gold nucleus ~ 1.47 × 1017 kg m-3.

Very high! One teaspoon = 500 million tonnes. So pretty much the same, regardless of

element.

Ext: Work out mass of neutron star based on this density. How does it compare to solar mass?

Why the same?Why the same?

Where u is the atomic mass unit (1/12Where u is the atomic mass unit (1/12thth mass mass of carbon atom, close to mass of proton)of carbon atom, close to mass of proton)

So:-So:-

Density does not depend on A!Density does not depend on A!

ArArRV 303

433/103

4334 )(

uAm

303

4303

4 r

u

Ar

uA

Nuclear Density

Nuclear density >> Atomic Density

This implies:- Most of an atom’s mass is in its

nucleus. The nucleus is small compared to the

atom. An atom must contain a lot of empty

space.

Example Exam Questions Q1: (a)If a carbon nucleus containing 12

nucleons has a radius of 3.2 × 10-15m, what is r0?

(b) Calculate the radius of a radium nucleus containing 226 nucleons.

(c) Calculate the density of a radium nucleus if its mass is 3.75 × 10-25 kg.

Q2: A sample of pure gold has a density of 19300 kg m-3. If the density of the gold nucleus is 1.47 × 1017kg m-3 discuss what this implies about the structure of a gold atom.

Stretch & ChallengeStretch & Challenge

An often quoted random fact is that An often quoted random fact is that a sugar cube of a neutron star has a sugar cube of a neutron star has mass roughly equal to the mass of all mass roughly equal to the mass of all the humans on Earth.the humans on Earth.

Making some reasonable Making some reasonable approximations, show whether or approximations, show whether or not this is true.not this is true.

CluesClues

Diameter of a neutron star ~ 25 km.Diameter of a neutron star ~ 25 km. Mass of a neutron star ~ 4 ×10Mass of a neutron star ~ 4 ×1030 30 kgkg Total number of humans on Earth ~ Total number of humans on Earth ~

6 billion6 billion Average mass of humans ~ 70 kg.Average mass of humans ~ 70 kg.

Stretch & ChallengeStretch & Challenge

Assume 6 billion humans of mass 70 kg:-Assume 6 billion humans of mass 70 kg:-

Mass of human population = 6 × 10Mass of human population = 6 × 1099 × 70 kg × 70 kg = 4.20 × 10= 4.20 × 101111 kg. kg. Density of neutron star = Density of neutron star =

4×104×103030kg/(4/3kg/(4/3ππ(12,500)(12,500)3) 3) = 4.89 × 10= 4.89 × 101717 kg m kg m-3-3

Mass of neutron star = density × volumeMass of neutron star = density × volume = 4.89 × 10= 4.89 × 101717 kg m kg m-3-3 × 10 × 10-6-6

mm33

= 4.89 × 10= 4.89 × 101111 kg kg

Learning Objectives

Know how to determine a value for the index for an equation of the form y = kxn. EMPA!

State and use the equation for dependence of radius on nucleon number.

Calculate nuclear density.

Recall the implications of the high nuclear density compared to atomic density.