Nuclei radii and masses

Book page 492 - 496

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Geiger ndash Marsden Gold foil experiment bull Proof for the nuclear model of the atom

bull Estimate nuclear diameter of hydrogen atom

bull Extend the theory to any atom use ldquoZrdquo

bull Let Z be the number of positive charges in the nucleus

bull eZ = charge of nucleus

bull e = elementary charge of electron

bull Rutherfords Gold Foil Experiment

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Energy of electron moving between energy levels

bull 119879119864119899 = minus2120587211989811989011989621198904

1198992ℎ2 for hydrogen

bull Extend for any atom

bull 119879119864119899 = minus2120587211989811989011989621198904

ℎ2 ∙1198852

1198992

bull This is the total energy for an electron in the nth level of any atom with proton number Z

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Distance of closest approach bull Assume an alpha particle is on collision curse with

a gold nucleus

bull Ignore the recoil of the gold nucleus

bull 119870119864120572 far away from nucleus = KE

bull As it comes closer to the nucleus due to the Coulomb force KE changes into electrostatic PE

bull At a specific instant the alpha particle is at rest

Distance of closes approach

Path of reflected 120572 119901119886119903119905119894119888119897119890

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Closest approach d

bull TE = KE + PE

= 0 + 1198961198902

119877119899

bull TE = 1

41205871205760∙

119885119890times2119890

119877119899

bull 119889 =1198851198902

21205871205760119879119864

bull TE of 120572 119901119886119903119905119894119888119897119890 = 40119872119890119881 = 40 times 106 times 16 times 10minus19 = 64 times 10minus13119869

bull 119889 =79times 16times10minus19 2

2120587times885times10minus12times64times10minus13 = 568 times 10minus14119898

119877119899 = 119889 1199021 = 119885119890 = gold atom 1199022 = 2119890 = 120572 119901119886119903119905119894119888119897119890

Z = of positive charges 119885119860119906 = 79

119896 =1

41205871205760

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Distance of closest approach bull This distance will depend on original KE of alpha particle

bull If KE is increased a point is reached where Coulomb scattering does not take place

bull This is an estimate

bull At a separation of about 10minus15119898 Coulomb force is overtaken by strong nuclear force

bull Nuclear forces are attractive and short ranged

size of force between nucleons depends only on their nucleon separation

copycgrahamphysicscom 2016

Conclusion bull Nucleons are very close

together they repel

bull 1 fermi apart = 10minus15119898 they experience the strong nuclear force

bull 4fm apart the strong force is neglible they experience electrostatic force of repulsion

bull The nucleus becomes unstable

copycgrahamphysicscom 2016

Wave properties bull If the electrons are accelerated to very high energies

the wave like properties become more important

bull De Broglie wavelength λ =ℎ

119901 and 119864 =

ℎ119888

λ

λ = ℎ

119901=

ℎ119888

119864

bull Assume the electron has an energy of 300MeV

bull λ =ℎ119888

119864=

663times10minus34times3times108

300times106times16times10minus19 = 415 times 10minus15119898

bull From diffraction sin 120579 =122λ

119863

bull D = nuclear diameter of target atom copycgrahamphysicscom 2016

1st minimum when 120579 = 750

bull 119863 =122times415times10minus15

sin 75= 524 times 10minus15119898

bull This is the nuclear diameter of the target atom

bull Experimental it can be shown that the maximum radius R of the nucleus is given by

119877 = 119877011986013

bull 1198770 = constant = Fermiradius = 120 times 10minus15119898 = 12119891119898

bull A = nucleon mass number

copycgrahamphysicscom 2016

Example

bull Determine the radius of an uranium ndash 238 nucleus

Solution

bull 119877 = 11987701198601

3

bull 119877 = 12119891119898 times 2381

3 = 7437119891119898

copycgrahamphysicscom 2016

Nuclear density bull Assume a nucleus is spherical its volume can be calculated

bull 119881 =4

31205871198773 =

4

31205871198601198770

3

bull Density of nucleus

120588 =119898

119881=

119860119906

43

12058711986011987703

=3119906

412058711987703

bull u = uniform atomic mass

bull Au = total mass of nucleus

bull A = nucleon number

copycgrahamphysicscom 2016

Meaning of calculation

bull 120588 =3119906

412058711987703

bull Each part in the equation is a constant

bull The density of any nucleus is independent of the number of nucleons in the nucleus

bull Densities are about the same for all nuclei

bull Only macroscopic objects with same density as nuclei are neutron stars

copycgrahamphysicscom 2016

Distance of closest approach bull Assume an alpha particle is on collision curse with

a gold nucleus

bull Ignore the recoil of the gold nucleus

bull 119870119864120572 far away from nucleus = KE

bull As it comes closer to the nucleus due to the Coulomb force KE changes into electrostatic PE

bull At a specific instant the alpha particle is at rest

Distance of closes approach

Path of reflected 120572 119901119886119903119905119894119888119897119890

copycgrahamphysicscom 2016

Closest approach d

bull TE = KE + PE

= 0 + 1198961198902

119877119899

bull TE = 1

41205871205760∙

119885119890times2119890

119877119899

bull 119889 =1198851198902

21205871205760119879119864

bull TE of 120572 119901119886119903119905119894119888119897119890 = 40119872119890119881 = 40 times 106 times 16 times 10minus19 = 64 times 10minus13119869

bull 119889 =79times 16times10minus19 2

2120587times885times10minus12times64times10minus13 = 568 times 10minus14119898

119877119899 = 119889 1199021 = 119885119890 = gold atom 1199022 = 2119890 = 120572 119901119886119903119905119894119888119897119890

Z = of positive charges 119885119860119906 = 79

119896 =1

41205871205760

copycgrahamphysicscom 2016

Distance of closest approach bull This distance will depend on original KE of alpha particle

bull If KE is increased a point is reached where Coulomb scattering does not take place

bull This is an estimate

bull At a separation of about 10minus15119898 Coulomb force is overtaken by strong nuclear force

bull Nuclear forces are attractive and short ranged

size of force between nucleons depends only on their nucleon separation

copycgrahamphysicscom 2016

Conclusion bull Nucleons are very close

together they repel

bull 1 fermi apart = 10minus15119898 they experience the strong nuclear force

bull 4fm apart the strong force is neglible they experience electrostatic force of repulsion

bull The nucleus becomes unstable

copycgrahamphysicscom 2016

Wave properties bull If the electrons are accelerated to very high energies

the wave like properties become more important

bull De Broglie wavelength λ =ℎ

119901 and 119864 =

ℎ119888

λ

λ = ℎ

119901=

ℎ119888

119864

bull Assume the electron has an energy of 300MeV

bull λ =ℎ119888

119864=

663times10minus34times3times108

300times106times16times10minus19 = 415 times 10minus15119898

bull From diffraction sin 120579 =122λ

119863

bull D = nuclear diameter of target atom copycgrahamphysicscom 2016

1st minimum when 120579 = 750

bull 119863 =122times415times10minus15

sin 75= 524 times 10minus15119898

bull This is the nuclear diameter of the target atom

bull Experimental it can be shown that the maximum radius R of the nucleus is given by

119877 = 119877011986013

bull 1198770 = constant = Fermiradius = 120 times 10minus15119898 = 12119891119898

bull A = nucleon mass number

copycgrahamphysicscom 2016

Example

bull Determine the radius of an uranium ndash 238 nucleus

Solution

bull 119877 = 11987701198601

3

bull 119877 = 12119891119898 times 2381

3 = 7437119891119898

copycgrahamphysicscom 2016

Nuclear density bull Assume a nucleus is spherical its volume can be calculated

bull 119881 =4

31205871198773 =

4

31205871198601198770

3

bull Density of nucleus

120588 =119898

119881=

119860119906

43

12058711986011987703

=3119906

412058711987703

bull u = uniform atomic mass

bull Au = total mass of nucleus

bull A = nucleon number

copycgrahamphysicscom 2016

Meaning of calculation

bull 120588 =3119906

412058711987703

bull Each part in the equation is a constant

bull The density of any nucleus is independent of the number of nucleons in the nucleus

bull Densities are about the same for all nuclei

bull Only macroscopic objects with same density as nuclei are neutron stars

copycgrahamphysicscom 2016

Closest approach d

bull TE = KE + PE

= 0 + 1198961198902

119877119899

bull TE = 1

41205871205760∙

119885119890times2119890

119877119899

bull 119889 =1198851198902

21205871205760119879119864

bull TE of 120572 119901119886119903119905119894119888119897119890 = 40119872119890119881 = 40 times 106 times 16 times 10minus19 = 64 times 10minus13119869

bull 119889 =79times 16times10minus19 2

2120587times885times10minus12times64times10minus13 = 568 times 10minus14119898

119877119899 = 119889 1199021 = 119885119890 = gold atom 1199022 = 2119890 = 120572 119901119886119903119905119894119888119897119890

Z = of positive charges 119885119860119906 = 79

119896 =1

41205871205760

copycgrahamphysicscom 2016

Distance of closest approach bull This distance will depend on original KE of alpha particle

bull If KE is increased a point is reached where Coulomb scattering does not take place

bull This is an estimate

bull At a separation of about 10minus15119898 Coulomb force is overtaken by strong nuclear force

bull Nuclear forces are attractive and short ranged

size of force between nucleons depends only on their nucleon separation

copycgrahamphysicscom 2016

Conclusion bull Nucleons are very close

together they repel

bull 1 fermi apart = 10minus15119898 they experience the strong nuclear force

bull 4fm apart the strong force is neglible they experience electrostatic force of repulsion

bull The nucleus becomes unstable

copycgrahamphysicscom 2016

Wave properties bull If the electrons are accelerated to very high energies

the wave like properties become more important

bull De Broglie wavelength λ =ℎ

119901 and 119864 =

ℎ119888

λ

λ = ℎ

119901=

ℎ119888

119864

bull Assume the electron has an energy of 300MeV

bull λ =ℎ119888

119864=

663times10minus34times3times108

300times106times16times10minus19 = 415 times 10minus15119898

bull From diffraction sin 120579 =122λ

119863

bull D = nuclear diameter of target atom copycgrahamphysicscom 2016

1st minimum when 120579 = 750

bull 119863 =122times415times10minus15

sin 75= 524 times 10minus15119898

bull This is the nuclear diameter of the target atom

bull Experimental it can be shown that the maximum radius R of the nucleus is given by

119877 = 119877011986013

bull 1198770 = constant = Fermiradius = 120 times 10minus15119898 = 12119891119898

bull A = nucleon mass number

copycgrahamphysicscom 2016

Example

bull Determine the radius of an uranium ndash 238 nucleus

Solution

bull 119877 = 11987701198601

3

bull 119877 = 12119891119898 times 2381

3 = 7437119891119898

copycgrahamphysicscom 2016

Nuclear density bull Assume a nucleus is spherical its volume can be calculated

bull 119881 =4

31205871198773 =

4

31205871198601198770

3

bull Density of nucleus

120588 =119898

119881=

119860119906

43

12058711986011987703

=3119906

412058711987703

bull u = uniform atomic mass

bull Au = total mass of nucleus

bull A = nucleon number

copycgrahamphysicscom 2016

Meaning of calculation

bull 120588 =3119906

412058711987703

bull Each part in the equation is a constant

bull The density of any nucleus is independent of the number of nucleons in the nucleus

bull Densities are about the same for all nuclei

bull Only macroscopic objects with same density as nuclei are neutron stars

copycgrahamphysicscom 2016

Distance of closest approach bull This distance will depend on original KE of alpha particle

bull If KE is increased a point is reached where Coulomb scattering does not take place

bull This is an estimate

bull At a separation of about 10minus15119898 Coulomb force is overtaken by strong nuclear force

bull Nuclear forces are attractive and short ranged

size of force between nucleons depends only on their nucleon separation

copycgrahamphysicscom 2016

Conclusion bull Nucleons are very close

together they repel

bull 1 fermi apart = 10minus15119898 they experience the strong nuclear force

bull 4fm apart the strong force is neglible they experience electrostatic force of repulsion

bull The nucleus becomes unstable

copycgrahamphysicscom 2016

Wave properties bull If the electrons are accelerated to very high energies

the wave like properties become more important

bull De Broglie wavelength λ =ℎ

119901 and 119864 =

ℎ119888

λ

λ = ℎ

119901=

ℎ119888

119864

bull Assume the electron has an energy of 300MeV

bull λ =ℎ119888

119864=

663times10minus34times3times108

300times106times16times10minus19 = 415 times 10minus15119898

bull From diffraction sin 120579 =122λ

119863

bull D = nuclear diameter of target atom copycgrahamphysicscom 2016

1st minimum when 120579 = 750

bull 119863 =122times415times10minus15

sin 75= 524 times 10minus15119898

bull This is the nuclear diameter of the target atom

bull Experimental it can be shown that the maximum radius R of the nucleus is given by

119877 = 119877011986013

bull 1198770 = constant = Fermiradius = 120 times 10minus15119898 = 12119891119898

bull A = nucleon mass number

copycgrahamphysicscom 2016

Example

bull Determine the radius of an uranium ndash 238 nucleus

Solution

bull 119877 = 11987701198601

3

bull 119877 = 12119891119898 times 2381

3 = 7437119891119898

copycgrahamphysicscom 2016

Nuclear density bull Assume a nucleus is spherical its volume can be calculated

bull 119881 =4

31205871198773 =

4

31205871198601198770

3

bull Density of nucleus

120588 =119898

119881=

119860119906

43

12058711986011987703

=3119906

412058711987703

bull u = uniform atomic mass

bull Au = total mass of nucleus

bull A = nucleon number

copycgrahamphysicscom 2016

Meaning of calculation

bull 120588 =3119906

412058711987703

bull Each part in the equation is a constant

bull The density of any nucleus is independent of the number of nucleons in the nucleus

bull Densities are about the same for all nuclei

bull Only macroscopic objects with same density as nuclei are neutron stars

copycgrahamphysicscom 2016

Conclusion bull Nucleons are very close

together they repel

bull 1 fermi apart = 10minus15119898 they experience the strong nuclear force

bull 4fm apart the strong force is neglible they experience electrostatic force of repulsion

bull The nucleus becomes unstable

copycgrahamphysicscom 2016

Wave properties bull If the electrons are accelerated to very high energies

the wave like properties become more important

bull De Broglie wavelength λ =ℎ

119901 and 119864 =

ℎ119888

λ

λ = ℎ

119901=

ℎ119888

119864

bull Assume the electron has an energy of 300MeV

bull λ =ℎ119888

119864=

663times10minus34times3times108

300times106times16times10minus19 = 415 times 10minus15119898

bull From diffraction sin 120579 =122λ

119863

bull D = nuclear diameter of target atom copycgrahamphysicscom 2016

1st minimum when 120579 = 750

bull 119863 =122times415times10minus15

sin 75= 524 times 10minus15119898

bull This is the nuclear diameter of the target atom

bull Experimental it can be shown that the maximum radius R of the nucleus is given by

119877 = 119877011986013

bull 1198770 = constant = Fermiradius = 120 times 10minus15119898 = 12119891119898

bull A = nucleon mass number

copycgrahamphysicscom 2016

Example

bull Determine the radius of an uranium ndash 238 nucleus

Solution

bull 119877 = 11987701198601

3

bull 119877 = 12119891119898 times 2381

3 = 7437119891119898

copycgrahamphysicscom 2016

Nuclear density bull Assume a nucleus is spherical its volume can be calculated

bull 119881 =4

31205871198773 =

4

31205871198601198770

3

bull Density of nucleus

120588 =119898

119881=

119860119906

43

12058711986011987703

=3119906

412058711987703

bull u = uniform atomic mass

bull Au = total mass of nucleus

bull A = nucleon number

copycgrahamphysicscom 2016

Meaning of calculation

bull 120588 =3119906

412058711987703

bull Each part in the equation is a constant

bull The density of any nucleus is independent of the number of nucleons in the nucleus

bull Densities are about the same for all nuclei

bull Only macroscopic objects with same density as nuclei are neutron stars

copycgrahamphysicscom 2016

Wave properties bull If the electrons are accelerated to very high energies

the wave like properties become more important

bull De Broglie wavelength λ =ℎ

119901 and 119864 =

ℎ119888

λ

λ = ℎ

119901=

ℎ119888

119864

bull Assume the electron has an energy of 300MeV

bull λ =ℎ119888

119864=

663times10minus34times3times108

300times106times16times10minus19 = 415 times 10minus15119898

bull From diffraction sin 120579 =122λ

119863

bull D = nuclear diameter of target atom copycgrahamphysicscom 2016

1st minimum when 120579 = 750

bull 119863 =122times415times10minus15

sin 75= 524 times 10minus15119898

bull This is the nuclear diameter of the target atom

bull Experimental it can be shown that the maximum radius R of the nucleus is given by

119877 = 119877011986013

bull 1198770 = constant = Fermiradius = 120 times 10minus15119898 = 12119891119898

bull A = nucleon mass number

copycgrahamphysicscom 2016

Example

bull Determine the radius of an uranium ndash 238 nucleus

Solution

bull 119877 = 11987701198601

3

bull 119877 = 12119891119898 times 2381

3 = 7437119891119898

copycgrahamphysicscom 2016

Nuclear density bull Assume a nucleus is spherical its volume can be calculated

bull 119881 =4

31205871198773 =

4

31205871198601198770

3

bull Density of nucleus

120588 =119898

119881=

119860119906

43

12058711986011987703

=3119906

412058711987703

bull u = uniform atomic mass

bull Au = total mass of nucleus

bull A = nucleon number

copycgrahamphysicscom 2016

Meaning of calculation

bull 120588 =3119906

412058711987703

bull Each part in the equation is a constant

bull The density of any nucleus is independent of the number of nucleons in the nucleus

bull Densities are about the same for all nuclei

bull Only macroscopic objects with same density as nuclei are neutron stars

copycgrahamphysicscom 2016

1st minimum when 120579 = 750

bull 119863 =122times415times10minus15

sin 75= 524 times 10minus15119898

bull This is the nuclear diameter of the target atom

bull Experimental it can be shown that the maximum radius R of the nucleus is given by

119877 = 119877011986013

bull 1198770 = constant = Fermiradius = 120 times 10minus15119898 = 12119891119898

bull A = nucleon mass number

copycgrahamphysicscom 2016

Example

bull Determine the radius of an uranium ndash 238 nucleus

Solution

bull 119877 = 11987701198601

3

bull 119877 = 12119891119898 times 2381

3 = 7437119891119898

copycgrahamphysicscom 2016

Nuclear density bull Assume a nucleus is spherical its volume can be calculated

bull 119881 =4

31205871198773 =

4

31205871198601198770

3

bull Density of nucleus

120588 =119898

119881=

119860119906

43

12058711986011987703

=3119906

412058711987703

bull u = uniform atomic mass

bull Au = total mass of nucleus

bull A = nucleon number

copycgrahamphysicscom 2016

Meaning of calculation

bull 120588 =3119906

412058711987703

bull Each part in the equation is a constant

bull The density of any nucleus is independent of the number of nucleons in the nucleus

bull Densities are about the same for all nuclei

bull Only macroscopic objects with same density as nuclei are neutron stars

copycgrahamphysicscom 2016

Example

bull Determine the radius of an uranium ndash 238 nucleus

Solution

bull 119877 = 11987701198601

3

bull 119877 = 12119891119898 times 2381

3 = 7437119891119898

copycgrahamphysicscom 2016

Nuclear density bull Assume a nucleus is spherical its volume can be calculated

bull 119881 =4

31205871198773 =

4

31205871198601198770

3

bull Density of nucleus

120588 =119898

119881=

119860119906

43

12058711986011987703

=3119906

412058711987703

bull u = uniform atomic mass

bull Au = total mass of nucleus

bull A = nucleon number

copycgrahamphysicscom 2016

Meaning of calculation

bull 120588 =3119906

412058711987703

bull Each part in the equation is a constant

bull The density of any nucleus is independent of the number of nucleons in the nucleus

bull Densities are about the same for all nuclei

bull Only macroscopic objects with same density as nuclei are neutron stars

copycgrahamphysicscom 2016

Nuclear density bull Assume a nucleus is spherical its volume can be calculated

bull 119881 =4

31205871198773 =

4

31205871198601198770

3

bull Density of nucleus

120588 =119898

119881=

119860119906

43

12058711986011987703

=3119906

412058711987703

bull u = uniform atomic mass

bull Au = total mass of nucleus

bull A = nucleon number

copycgrahamphysicscom 2016

Meaning of calculation

bull 120588 =3119906

412058711987703

bull Each part in the equation is a constant

bull The density of any nucleus is independent of the number of nucleons in the nucleus

bull Densities are about the same for all nuclei

bull Only macroscopic objects with same density as nuclei are neutron stars

copycgrahamphysicscom 2016

Meaning of calculation

bull 120588 =3119906

412058711987703

bull Each part in the equation is a constant

bull The density of any nucleus is independent of the number of nucleons in the nucleus

bull Densities are about the same for all nuclei

bull Only macroscopic objects with same density as nuclei are neutron stars

copycgrahamphysicscom 2016