Transcript A2_Unit5_Nuclear_11_Nuclear_Radius

Nuclear Radius
Nuclear Physics Lesson 11
Homework
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Research and explain how electron diffraction
can be used to determine the radius of the
nucleus (6 Marks)
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Past Paper Question on today’s material.
Complete both by Next Lesson.
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Learning Objectives
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Know how to determine a value for the index
for an equation of the form y = kxn.  EMPA!
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State and use the equation for dependence of
radius on nucleon number.
Calculate nuclear density.
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Recall the implications of the high nuclear
density compared to atomic density.
Nuclear Radius
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The nuclear radius, R, can be shown to be
related to the nucleon number, A according to:-
R  r0 A
n
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Where r0 and n are constants.
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Given R for a number of nuclei with a variety of
A, how can we determine r0 and x?
Data
Element
Boron
Neon
Argon
Nitrogen
Bromine
Antimony
Gold
Uranium
A
R/fm
11
20
40
14
80
122
197
238
2.34
2.85
3.59
2.53
4.52
5.21
6.11
6.51
Data (Suggestion)
Element
Boron
Neon
Argon
Nitrogen
Bromine
Antimony
Gold
Uranium
A
11
20
40
14
80
122
197
238
R/fm
2.34
2.85
3.59
2.53
4.52
5.21
6.11
6.51
ln A
ln R
Finding r0 and x
R  r0 A
n
Logging both sides
ln R  ln( r0 A )
n
ln R  ln r0  ln A
n
ln R  n ln A  ln r0

log (AB) = log A + log B

log (An) = n log A
This is in the form y = mx + c.
Finding r0 and x
ln R  n ln A  ln r0
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Plotting ln R against ln A should give a straight
line with gradient = n and intercept = ln r0
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Note that eln r0=r0.
Excel Plot
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From graph:-
ln R
-32.60
0.00
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gradient n = 1/3
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intercept = ln r0
intercept = -34.49
r0 = 1.05 fm
2.00
3.00
4.00
5.00
6.00
-32.80
-33.00
-33.20
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1.00
y = 0.3333x - 34.49
R² = 1
-33.40
-33.60
-33.80
ln R
Linear (ln R)
Equation
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Dependence of radius on
nucleon number:-
R  r0 A
1/ 3
[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
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Radius of a carbon nucleus ~ 3.2 × 10-15m.
Radius of a gold nucleus ~ 8.1 × 10-15m.
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Mass of a carbon nucleus ~ 2.00 × 10-26kg.
Mass of a gold nucleus ~ 3.27 × 10-25kg.
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What are the densities of the nuclei?
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Nuclear Density

Recall that density is given by:-
m

V
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You can assume that the nucleus is spherical so
that V = 4/3 πR3, so the density is given by:-
m

4 R 3
3
Nuclear Density
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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?
V  3 R  3  (r0 A )  3 r0 A
4
3
1/ 3 3
4
4
3
m  uA
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Where u is the atomic mass unit (1/12th mass of
carbon atom, close to mass of proton)
So:uA
u

4
r0 A
3
3

4
3
r0
3
Density does not depend on A!
Nuclear Density
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Nuclear density >> Atomic Density
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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.
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Example Exam Questions
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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 m3. 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 & Challenge

An often quoted random fact is that a sugar
cube of a neutron star has mass roughly equal to
the mass of all the humans on Earth.
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Making some reasonable approximations, show
whether or not this is true.
Clues
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Diameter of a neutron star ~ 25 km.
Mass of a neutron star ~ 4 ×1030 kg
Total number of humans on Earth ~ 6 billion
Average mass of humans ~ 70 kg.
Stretch & Challenge
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Assume 6 billion humans of mass 70 kg:-
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Mass of human population = 6 × 109 × 70 kg
= 4.20 × 1011 kg.
Density of neutron star =
4×1030kg/(4/3π(12,500)3) = 4.89 × 1017 kg m-3
Mass of neutron star = density × volume
= 4.89 × 1017 kg m-3 × 10-6 m3
= 4.89 × 1011 kg
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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.