Midterm obligatory exercises

FYS 3510 2011 Nuclear Physics

Try to solve as many problems as possible

but at least 50

Deadline 06052011

Nuclear Physics Exercise NP1

Explain whether the following ldquospontaneous decay processesrdquo are physically allowed or not

bull 242 + 238 496 94 x 2

242 238 496 94 2

56 + 5526 26

5626

Cm (O ) Pu (2 E = 0967 MeV) + He

M( Cm) = 242058830 u M( Pu) = 238049554 u M( He) = 4002603 u

Fe (O ) Fe(3 2

M( Fe) = 55934939 u M(

) + n

-

-bull rarr

rarr

5526

56 + 5526 26

56 55 + 226 26

238 294 x 94

+Fe) = 54938296 u M(n) = 1008665 u

Fe (O ) Fe(3 2

M( Fe) = 55934939 u M( Fe) = 54938296 u M(π ) = 1395679 MeV c

Pu (0 E = 0938 MeV)

) + π + e + ν --

+

bull

bull

rarr

rarr38

23894

23 2310 11

23 2310 11

+

Pu (0 ) + γ

M( Pu) = 238049554 u

Ne (5 2 ) Na(3 2

M( Ne) = 22994465 u M( Na) = 22989768 u

+) + e + ν -

+

bull rarr

Note the superscript refers to an excited nucleus

Nuclear Physics Exercise NP2 (a) formfactors

The form factor for a spherically symmetric charge distribution is given as

a) Write down the expression for the form factor for elastic electron scattering from a spherical charge distribution with uniform charge density and a sharp surface (radius R)Determine the location of the minima in the elastic scattering cross section for an electron beam of 200 MeV energy impinging on a target consisting of 197Au atoms What is the ratio between the differential cross sections at scattering angles of 180o and 0o Similarly for an α-particle beam with the same de Broglie wavelength as the electrons locate the minima in the cross section (if any) and find the ratio of the differetial cross sections at 180o and 0o Throughout the exercise assume the target recoil to be negligible

infin

2 2

0

sin(|q|r )F(q ) = 4π r drρ(r )

(|q|r )

where the charge density ρ(r) is normalized to

infin

2

04π r drρ(r ) = 1

2F(q )

Nuclear Physics Exercise NP2 (b)b) Elastic electron scattering on a charge distribution reveals the following form factor

where with being the transferred momentum and R some constant radius Propose a charge distribution which can produce such a form factor Determine the mean square radius for the distribution in terms of R

The nucleus can be approximately described as an α-particle with 4 loosely bound neutrons moving in a diffuse halo about the core Explain qualitatively how the electron scattering form factor measured for would differ from that of

2 -31 3F(q ) = + x (sinx - xcosx)

2 2x =|q|R q

8

2 He

8

2 He 4

2 He

Nuclear Physics Exercise NP3Fission

bull Assume a uranium nucleus 23692U breaks up spontaneously

into two roughly equal parts Estimate the reduction in electrostatic energy of the nuclei What is the relationship of this to total change of the energy

(assume uniform charge distiribution with radius R=12x 10-13

A013 cm)

bull BR Martin 210

bull The half-life of 239Pu has been determined by immersing a sphere of 239Pu of mass 1201gm in liquid nitrogen of a volume enough to stop all α-particles and measure the rate of evaporation of the liquid The evaporation rate corresponded to a power of 0231W Calculate to the nearest hundred years the half-life of 239Pu given that the energy of its decay alpha particles is 5144 MeV (Tak into account the recoil energy of the product nucleus

1MeV=160206x10-13 joule 1 atomic mass unit = 166x10-24 gm

bull BR Martin 28 211 212 213 (look through)

Nuclear Physics Exercise NP4Alpha- decay

bull An element of low atomic number Z can undergo allowed positron β-decay Let p0 is the maximum possible momentum of the positron supposing p0ltltmc (m-positron mass) Γβ is the beta-decay rate An alternative process is K-electron capture the nucleus capturing a K-shell electron and undergoing the same nuclear transition with emission of neutrino Let ΓK is the decay rate of this process Compute ratio ΓKΓβ You can treat the wave function of the K-shell electron as hydrogenic and can ignore the electron binding energy

bull BRM 215 79

Nuclear Physics Exercise NP4Beta- decays

Nuclear Physics Exercise NP5Nuclear reactions

bull BRM 216 (radioactive decay)

bull BRM 78 (gamma ndash decay)

bull BRM 710 711 (beta-decay)

bull Shell model BRM 71 72 73 74

bull Deformed nuclei BRM 75 76

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull At CERN Large Hadron Collider there were several poton-proton runs at energies

radics = 900 236 and 7000 GeV Calculate the main characteristics of these reactions

Ecm Elab β γ maximal rapidity ycm available for protons with mass mp=0938 GeV

and for pions with mπ=0139 GeV Draw y(β ) - dependence

bull Consider the production of the K+ in the reaction γprarrΚ+ +Λ- Give the minimal γ-ray energy in the laboratory system where p is at rest

when this reaction can take place - What will be the minimal γ- energy if the proton is not free but is bound in

the nucleus Take into account the Fermi motion with pF=250 MeVc

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

bull What is the smallest Elab energy of pp interactions for production K-meson with mass

mK=0494 GeV Write also the possible reactions

bull Proton photon and electron (me=0511 MeV) have the same wave length λ = 10-11 m

Which time will it take for each of them to fly the distance L=10 m

bull The Λ decays in flight into a proton and a πminus meson Λ rarrpπ- If the Λ has a velocity of 08c - what is the maximal momentum that the πminus can have in the laboratory system- What is the maximum component of laboratory momentum perpendicular

to the Λ direction

bull Nucleus of 10B transits from excited state with energy E= 072 MeVto the ground state with emission of a photon with half-life time t12=67x10-10 secWhat is energy uncertainty ΔE of the emitted γ

bull α-particles with kinetic energy T=65 MeV scatter on the 197Au nucleiDefine a) impact parameter b (at scattering angle θ=90o)

b) minimal distance rmin between α-particles and 197Au nucleic-d) kinetic T and potential E energy of α-particle at this point

Nuclear Physics Exercise NP6 Kinematics and Rutherford scattering

bull At CERN Large Hadron Collider there were several poton-proton runs at energies

radics = 900 2360 and 7000 GeV Calculate the main characteristics of these reactions

Ecm Elab β γ maximal rapidity ycm available for protons with mass mp=0938 GeV

and for pions with mπ=0139 GeV Draw y(β ) ndash dependence

s=(E 1+E2 )2 ndash (p1 +p2 )2= (2Ecm )2 if m1=m2 because for cms p1 = - p2

=(Elab +m2)2 - plab2= m1

2+ m22+ 2 Elab m2 = 2m2+ 2mElab

Ecm = radics2 Elab = s(2m) - m p=(E2-m2)12

β = pcmEcm = plab(Elab+m) γ = Ecmm

ymax= 05 ln (E+pl)(E-pl)= 05 ln (E+pl)2(E2-pl

2)= ln (E+pl)mt = ln 2Em

90000 236000 7000004317687848Sqrt s 90000 236000 700000

Ecm 45000 118000 350000

Pcm 449999022 1179999627 3499999874

Elab 4317687848 2968868998 2611940205

Plab 4317687848 2968868998 2611940205

Beta 09999978275 09999996841 09999999641

Betal 09999978275 09999996841 09999999641

Gamma 479744 1257995 3731343

ymax 68663 78304 89176

ymaxpi 87756 973969 1082694

β

lab

cms

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Consider the production of the K+ in the reaction γprarrΚ+ +Λ

- Give the minimal γ-ray energy in the laboratory system where proton is at rest when this reaction can take place

- What will be the minimal γ- energy if the proton is not free but is bound in the nucleus Take into account the Fermi motion with pF=250 MeVc

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull What is the smallest Elab energy of pp interactions for production K-meson with mass

mK=0494 GeV Write also the possible reactions

The reaction p+p-gt p+p+K++K- is possible but the minimal energy is demanded for

p+p-gtp+Λ+K+ where both baryon number and strangeness are conserved

radics gt Σi mi - condition for energy threshold where mi are masses of the produced particles

s=(Elab +m2)2 - plab2= m1

2+ m22+ 2 Elab m2 = 2m2+ 2mElab

So threshold energy

Elab = 12m2 ( (Σi mi )2- m1

2- m22) = ( (Σi mi )

2- 2m2)(2m)= (0939 + 0494 +1116)2- 209392 GeV

= 4734 GeV

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Proton photon and electron (me=0511 MeV) have the same wave length λ = 10-11 m

Which time will it take for each of them to fly the distance L=10 m

λ = (ħc)pc =10-11 m =gt p= (ħc) λc = (200 MeV Fm)(10-11m 10-15 mFm)c=2 106 MeVc

p=mvγ=mvradic(1-v2c2) =gt

v=1 radic(m2p2+1c2) = ħc c ((mc2)2 λ2 +(ħc)2)12

t=Lv= L ((mc2)2 λ2 +(ħc)2)12 ħc c

For proton t = 10 m ((9383 MeV)2 (104 Fm)2 +(197 MeV Fm)2)12(197 MeV Fm x 3 108 msec)

=16 10-3 sec

For electron t = 10 m ((0511 MeV)2 (104 Fm)2 +(197 MeV Fm)2)12(197 MeV Fm x 3 108 msec)=

= 9 10-7 sec

For photon t = Lc= 10 m 3 108 msec =33 10-8 sec

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

The Λ decays in flight into a proton and a πminus meson Λ rarrpπ-

If the Λ has a velocity of 08c - what is the maximal momentum that the πminus can have in the laboratory system- What is the maximum component of laboratory momentum perpendicular to the Λ direction

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Nucleus of 10B transits from excited state with energy E= 072 MeV

to the ground state with emission of a photon with half-life time t12=67x10-10 sec

What is energy uncertainty ΔE of the emitted γ

kinematics of γ-decay is energy and momentum conservation law

recoil of the nucleus pγ=pA (1)

E=Eγ +EA0 +EAkin (2)

E=ΔE= E - EA0

(1) is connected with kinetic energy of the nucleus in the ground state

EAkin =pA22M = pγ

22M and Eγ = pγ

from here Eγ =M( (1+2EM)12 -1) = M(1+EM-1)=E

recoil can be neglected

EAkin =Eγ22M =E 22M =(072 MeV)2(2x1081x931 MeV) =26x10-5 MeV

EAkin Eγ = 26x10-5 MeV072 MeV=36x10-5

ΔEτ= ΔEtln2 lt ħ =gt ΔE lt ħ ln2 t = 0693x658x10-22 MeV x sec 67x10-10 sec=68x10-7ev

bull α-particles with kinetic energy T=65 MeV scatter on the 197Au nucleiDefine a) impact parameter b (at scattering angle θ=90o)

b) (drdt=0) minimal distance rmin between α-particles and 197Au nucleic-d) kinetic T and potential E energy of α-particle at this point

r = rmin производная = 0 = 0

) кинетическая энергия T = T - E = 65 МeV -54 MeV=11 MeV

54 МэВ = 11 МэВ

Nuclear Physics Exercise NP1

Explain whether the following ldquospontaneous decay processesrdquo are physically allowed or not

bull 242 + 238 496 94 x 2

242 238 496 94 2

56 + 5526 26

5626

Cm (O ) Pu (2 E = 0967 MeV) + He

M( Cm) = 242058830 u M( Pu) = 238049554 u M( He) = 4002603 u

Fe (O ) Fe(3 2

M( Fe) = 55934939 u M(

) + n

-

-bull rarr

rarr

5526

56 + 5526 26

56 55 + 226 26

238 294 x 94

+Fe) = 54938296 u M(n) = 1008665 u

Fe (O ) Fe(3 2

M( Fe) = 55934939 u M( Fe) = 54938296 u M(π ) = 1395679 MeV c

Pu (0 E = 0938 MeV)

) + π + e + ν --

+

bull

bull

rarr

rarr38

23894

23 2310 11

23 2310 11

+

Pu (0 ) + γ

M( Pu) = 238049554 u

Ne (5 2 ) Na(3 2

M( Ne) = 22994465 u M( Na) = 22989768 u

+) + e + ν -

+

bull rarr

Note the superscript refers to an excited nucleus

Nuclear Physics Exercise NP2 (a) formfactors

The form factor for a spherically symmetric charge distribution is given as

a) Write down the expression for the form factor for elastic electron scattering from a spherical charge distribution with uniform charge density and a sharp surface (radius R)Determine the location of the minima in the elastic scattering cross section for an electron beam of 200 MeV energy impinging on a target consisting of 197Au atoms What is the ratio between the differential cross sections at scattering angles of 180o and 0o Similarly for an α-particle beam with the same de Broglie wavelength as the electrons locate the minima in the cross section (if any) and find the ratio of the differetial cross sections at 180o and 0o Throughout the exercise assume the target recoil to be negligible

infin

2 2

0

sin(|q|r )F(q ) = 4π r drρ(r )

(|q|r )

where the charge density ρ(r) is normalized to

infin

2

04π r drρ(r ) = 1

2F(q )

Nuclear Physics Exercise NP2 (b)b) Elastic electron scattering on a charge distribution reveals the following form factor

where with being the transferred momentum and R some constant radius Propose a charge distribution which can produce such a form factor Determine the mean square radius for the distribution in terms of R

The nucleus can be approximately described as an α-particle with 4 loosely bound neutrons moving in a diffuse halo about the core Explain qualitatively how the electron scattering form factor measured for would differ from that of

2 -31 3F(q ) = + x (sinx - xcosx)

2 2x =|q|R q

8

2 He

8

2 He 4

2 He

Nuclear Physics Exercise NP3Fission

bull Assume a uranium nucleus 23692U breaks up spontaneously

into two roughly equal parts Estimate the reduction in electrostatic energy of the nuclei What is the relationship of this to total change of the energy

(assume uniform charge distiribution with radius R=12x 10-13

A013 cm)

bull BR Martin 210

bull The half-life of 239Pu has been determined by immersing a sphere of 239Pu of mass 1201gm in liquid nitrogen of a volume enough to stop all α-particles and measure the rate of evaporation of the liquid The evaporation rate corresponded to a power of 0231W Calculate to the nearest hundred years the half-life of 239Pu given that the energy of its decay alpha particles is 5144 MeV (Tak into account the recoil energy of the product nucleus

1MeV=160206x10-13 joule 1 atomic mass unit = 166x10-24 gm

bull BR Martin 28 211 212 213 (look through)

Nuclear Physics Exercise NP4Alpha- decay

bull An element of low atomic number Z can undergo allowed positron β-decay Let p0 is the maximum possible momentum of the positron supposing p0ltltmc (m-positron mass) Γβ is the beta-decay rate An alternative process is K-electron capture the nucleus capturing a K-shell electron and undergoing the same nuclear transition with emission of neutrino Let ΓK is the decay rate of this process Compute ratio ΓKΓβ You can treat the wave function of the K-shell electron as hydrogenic and can ignore the electron binding energy

bull BRM 215 79

Nuclear Physics Exercise NP4Beta- decays

Nuclear Physics Exercise NP5Nuclear reactions

bull BRM 216 (radioactive decay)

bull BRM 78 (gamma ndash decay)

bull BRM 710 711 (beta-decay)

bull Shell model BRM 71 72 73 74

bull Deformed nuclei BRM 75 76

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull At CERN Large Hadron Collider there were several poton-proton runs at energies

radics = 900 236 and 7000 GeV Calculate the main characteristics of these reactions

Ecm Elab β γ maximal rapidity ycm available for protons with mass mp=0938 GeV

and for pions with mπ=0139 GeV Draw y(β ) - dependence

bull Consider the production of the K+ in the reaction γprarrΚ+ +Λ- Give the minimal γ-ray energy in the laboratory system where p is at rest

when this reaction can take place - What will be the minimal γ- energy if the proton is not free but is bound in

the nucleus Take into account the Fermi motion with pF=250 MeVc

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

bull What is the smallest Elab energy of pp interactions for production K-meson with mass

mK=0494 GeV Write also the possible reactions

bull Proton photon and electron (me=0511 MeV) have the same wave length λ = 10-11 m

Which time will it take for each of them to fly the distance L=10 m

bull The Λ decays in flight into a proton and a πminus meson Λ rarrpπ- If the Λ has a velocity of 08c - what is the maximal momentum that the πminus can have in the laboratory system- What is the maximum component of laboratory momentum perpendicular

to the Λ direction

bull Nucleus of 10B transits from excited state with energy E= 072 MeVto the ground state with emission of a photon with half-life time t12=67x10-10 secWhat is energy uncertainty ΔE of the emitted γ

bull α-particles with kinetic energy T=65 MeV scatter on the 197Au nucleiDefine a) impact parameter b (at scattering angle θ=90o)

b) minimal distance rmin between α-particles and 197Au nucleic-d) kinetic T and potential E energy of α-particle at this point

Nuclear Physics Exercise NP6 Kinematics and Rutherford scattering

bull At CERN Large Hadron Collider there were several poton-proton runs at energies

radics = 900 2360 and 7000 GeV Calculate the main characteristics of these reactions

Ecm Elab β γ maximal rapidity ycm available for protons with mass mp=0938 GeV

and for pions with mπ=0139 GeV Draw y(β ) ndash dependence

s=(E 1+E2 )2 ndash (p1 +p2 )2= (2Ecm )2 if m1=m2 because for cms p1 = - p2

=(Elab +m2)2 - plab2= m1

2+ m22+ 2 Elab m2 = 2m2+ 2mElab

Ecm = radics2 Elab = s(2m) - m p=(E2-m2)12

β = pcmEcm = plab(Elab+m) γ = Ecmm

ymax= 05 ln (E+pl)(E-pl)= 05 ln (E+pl)2(E2-pl

2)= ln (E+pl)mt = ln 2Em

90000 236000 7000004317687848Sqrt s 90000 236000 700000

Ecm 45000 118000 350000

Pcm 449999022 1179999627 3499999874

Elab 4317687848 2968868998 2611940205

Plab 4317687848 2968868998 2611940205

Beta 09999978275 09999996841 09999999641

Betal 09999978275 09999996841 09999999641

Gamma 479744 1257995 3731343

ymax 68663 78304 89176

ymaxpi 87756 973969 1082694

β

lab

cms

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Consider the production of the K+ in the reaction γprarrΚ+ +Λ

- Give the minimal γ-ray energy in the laboratory system where proton is at rest when this reaction can take place

- What will be the minimal γ- energy if the proton is not free but is bound in the nucleus Take into account the Fermi motion with pF=250 MeVc

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull What is the smallest Elab energy of pp interactions for production K-meson with mass

mK=0494 GeV Write also the possible reactions

The reaction p+p-gt p+p+K++K- is possible but the minimal energy is demanded for

p+p-gtp+Λ+K+ where both baryon number and strangeness are conserved

radics gt Σi mi - condition for energy threshold where mi are masses of the produced particles

s=(Elab +m2)2 - plab2= m1

2+ m22+ 2 Elab m2 = 2m2+ 2mElab

So threshold energy

Elab = 12m2 ( (Σi mi )2- m1

2- m22) = ( (Σi mi )

2- 2m2)(2m)= (0939 + 0494 +1116)2- 209392 GeV

= 4734 GeV

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Proton photon and electron (me=0511 MeV) have the same wave length λ = 10-11 m

Which time will it take for each of them to fly the distance L=10 m

λ = (ħc)pc =10-11 m =gt p= (ħc) λc = (200 MeV Fm)(10-11m 10-15 mFm)c=2 106 MeVc

p=mvγ=mvradic(1-v2c2) =gt

v=1 radic(m2p2+1c2) = ħc c ((mc2)2 λ2 +(ħc)2)12

t=Lv= L ((mc2)2 λ2 +(ħc)2)12 ħc c

For proton t = 10 m ((9383 MeV)2 (104 Fm)2 +(197 MeV Fm)2)12(197 MeV Fm x 3 108 msec)

=16 10-3 sec

For electron t = 10 m ((0511 MeV)2 (104 Fm)2 +(197 MeV Fm)2)12(197 MeV Fm x 3 108 msec)=

= 9 10-7 sec

For photon t = Lc= 10 m 3 108 msec =33 10-8 sec

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

The Λ decays in flight into a proton and a πminus meson Λ rarrpπ-

If the Λ has a velocity of 08c - what is the maximal momentum that the πminus can have in the laboratory system- What is the maximum component of laboratory momentum perpendicular to the Λ direction

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Nucleus of 10B transits from excited state with energy E= 072 MeV

to the ground state with emission of a photon with half-life time t12=67x10-10 sec

What is energy uncertainty ΔE of the emitted γ

kinematics of γ-decay is energy and momentum conservation law

recoil of the nucleus pγ=pA (1)

E=Eγ +EA0 +EAkin (2)

E=ΔE= E - EA0

(1) is connected with kinetic energy of the nucleus in the ground state

EAkin =pA22M = pγ

22M and Eγ = pγ

from here Eγ =M( (1+2EM)12 -1) = M(1+EM-1)=E

recoil can be neglected

EAkin =Eγ22M =E 22M =(072 MeV)2(2x1081x931 MeV) =26x10-5 MeV

EAkin Eγ = 26x10-5 MeV072 MeV=36x10-5

ΔEτ= ΔEtln2 lt ħ =gt ΔE lt ħ ln2 t = 0693x658x10-22 MeV x sec 67x10-10 sec=68x10-7ev

bull α-particles with kinetic energy T=65 MeV scatter on the 197Au nucleiDefine a) impact parameter b (at scattering angle θ=90o)

b) (drdt=0) minimal distance rmin between α-particles and 197Au nucleic-d) kinetic T and potential E energy of α-particle at this point

r = rmin производная = 0 = 0

) кинетическая энергия T = T - E = 65 МeV -54 MeV=11 MeV

54 МэВ = 11 МэВ

Nuclear Physics Exercise NP2 (a) formfactors

The form factor for a spherically symmetric charge distribution is given as

a) Write down the expression for the form factor for elastic electron scattering from a spherical charge distribution with uniform charge density and a sharp surface (radius R)Determine the location of the minima in the elastic scattering cross section for an electron beam of 200 MeV energy impinging on a target consisting of 197Au atoms What is the ratio between the differential cross sections at scattering angles of 180o and 0o Similarly for an α-particle beam with the same de Broglie wavelength as the electrons locate the minima in the cross section (if any) and find the ratio of the differetial cross sections at 180o and 0o Throughout the exercise assume the target recoil to be negligible

infin

2 2

0

sin(|q|r )F(q ) = 4π r drρ(r )

(|q|r )

where the charge density ρ(r) is normalized to

infin

2

04π r drρ(r ) = 1

2F(q )

Nuclear Physics Exercise NP2 (b)b) Elastic electron scattering on a charge distribution reveals the following form factor

where with being the transferred momentum and R some constant radius Propose a charge distribution which can produce such a form factor Determine the mean square radius for the distribution in terms of R

The nucleus can be approximately described as an α-particle with 4 loosely bound neutrons moving in a diffuse halo about the core Explain qualitatively how the electron scattering form factor measured for would differ from that of

2 -31 3F(q ) = + x (sinx - xcosx)

2 2x =|q|R q

8

2 He

8

2 He 4

2 He

Nuclear Physics Exercise NP3Fission

bull Assume a uranium nucleus 23692U breaks up spontaneously

into two roughly equal parts Estimate the reduction in electrostatic energy of the nuclei What is the relationship of this to total change of the energy

(assume uniform charge distiribution with radius R=12x 10-13

A013 cm)

bull BR Martin 210

bull The half-life of 239Pu has been determined by immersing a sphere of 239Pu of mass 1201gm in liquid nitrogen of a volume enough to stop all α-particles and measure the rate of evaporation of the liquid The evaporation rate corresponded to a power of 0231W Calculate to the nearest hundred years the half-life of 239Pu given that the energy of its decay alpha particles is 5144 MeV (Tak into account the recoil energy of the product nucleus

1MeV=160206x10-13 joule 1 atomic mass unit = 166x10-24 gm

bull BR Martin 28 211 212 213 (look through)

Nuclear Physics Exercise NP4Alpha- decay

bull An element of low atomic number Z can undergo allowed positron β-decay Let p0 is the maximum possible momentum of the positron supposing p0ltltmc (m-positron mass) Γβ is the beta-decay rate An alternative process is K-electron capture the nucleus capturing a K-shell electron and undergoing the same nuclear transition with emission of neutrino Let ΓK is the decay rate of this process Compute ratio ΓKΓβ You can treat the wave function of the K-shell electron as hydrogenic and can ignore the electron binding energy

bull BRM 215 79

Nuclear Physics Exercise NP4Beta- decays

Nuclear Physics Exercise NP5Nuclear reactions

bull BRM 216 (radioactive decay)

bull BRM 78 (gamma ndash decay)

bull BRM 710 711 (beta-decay)

bull Shell model BRM 71 72 73 74

bull Deformed nuclei BRM 75 76

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull At CERN Large Hadron Collider there were several poton-proton runs at energies

radics = 900 236 and 7000 GeV Calculate the main characteristics of these reactions

Ecm Elab β γ maximal rapidity ycm available for protons with mass mp=0938 GeV

and for pions with mπ=0139 GeV Draw y(β ) - dependence

bull Consider the production of the K+ in the reaction γprarrΚ+ +Λ- Give the minimal γ-ray energy in the laboratory system where p is at rest

when this reaction can take place - What will be the minimal γ- energy if the proton is not free but is bound in

the nucleus Take into account the Fermi motion with pF=250 MeVc

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

bull What is the smallest Elab energy of pp interactions for production K-meson with mass

mK=0494 GeV Write also the possible reactions

bull Proton photon and electron (me=0511 MeV) have the same wave length λ = 10-11 m

Which time will it take for each of them to fly the distance L=10 m

bull The Λ decays in flight into a proton and a πminus meson Λ rarrpπ- If the Λ has a velocity of 08c - what is the maximal momentum that the πminus can have in the laboratory system- What is the maximum component of laboratory momentum perpendicular

to the Λ direction

bull Nucleus of 10B transits from excited state with energy E= 072 MeVto the ground state with emission of a photon with half-life time t12=67x10-10 secWhat is energy uncertainty ΔE of the emitted γ

bull α-particles with kinetic energy T=65 MeV scatter on the 197Au nucleiDefine a) impact parameter b (at scattering angle θ=90o)

b) minimal distance rmin between α-particles and 197Au nucleic-d) kinetic T and potential E energy of α-particle at this point

Nuclear Physics Exercise NP6 Kinematics and Rutherford scattering

bull At CERN Large Hadron Collider there were several poton-proton runs at energies

radics = 900 2360 and 7000 GeV Calculate the main characteristics of these reactions

Ecm Elab β γ maximal rapidity ycm available for protons with mass mp=0938 GeV

and for pions with mπ=0139 GeV Draw y(β ) ndash dependence

s=(E 1+E2 )2 ndash (p1 +p2 )2= (2Ecm )2 if m1=m2 because for cms p1 = - p2

=(Elab +m2)2 - plab2= m1

2+ m22+ 2 Elab m2 = 2m2+ 2mElab

Ecm = radics2 Elab = s(2m) - m p=(E2-m2)12

β = pcmEcm = plab(Elab+m) γ = Ecmm

ymax= 05 ln (E+pl)(E-pl)= 05 ln (E+pl)2(E2-pl

2)= ln (E+pl)mt = ln 2Em

90000 236000 7000004317687848Sqrt s 90000 236000 700000

Ecm 45000 118000 350000

Pcm 449999022 1179999627 3499999874

Elab 4317687848 2968868998 2611940205

Plab 4317687848 2968868998 2611940205

Beta 09999978275 09999996841 09999999641

Betal 09999978275 09999996841 09999999641

Gamma 479744 1257995 3731343

ymax 68663 78304 89176

ymaxpi 87756 973969 1082694

β

lab

cms

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Consider the production of the K+ in the reaction γprarrΚ+ +Λ

- Give the minimal γ-ray energy in the laboratory system where proton is at rest when this reaction can take place

- What will be the minimal γ- energy if the proton is not free but is bound in the nucleus Take into account the Fermi motion with pF=250 MeVc

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull What is the smallest Elab energy of pp interactions for production K-meson with mass

mK=0494 GeV Write also the possible reactions

The reaction p+p-gt p+p+K++K- is possible but the minimal energy is demanded for

p+p-gtp+Λ+K+ where both baryon number and strangeness are conserved

radics gt Σi mi - condition for energy threshold where mi are masses of the produced particles

s=(Elab +m2)2 - plab2= m1

2+ m22+ 2 Elab m2 = 2m2+ 2mElab

So threshold energy

Elab = 12m2 ( (Σi mi )2- m1

2- m22) = ( (Σi mi )

2- 2m2)(2m)= (0939 + 0494 +1116)2- 209392 GeV

= 4734 GeV

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Proton photon and electron (me=0511 MeV) have the same wave length λ = 10-11 m

Which time will it take for each of them to fly the distance L=10 m

λ = (ħc)pc =10-11 m =gt p= (ħc) λc = (200 MeV Fm)(10-11m 10-15 mFm)c=2 106 MeVc

p=mvγ=mvradic(1-v2c2) =gt

v=1 radic(m2p2+1c2) = ħc c ((mc2)2 λ2 +(ħc)2)12

t=Lv= L ((mc2)2 λ2 +(ħc)2)12 ħc c

For proton t = 10 m ((9383 MeV)2 (104 Fm)2 +(197 MeV Fm)2)12(197 MeV Fm x 3 108 msec)

=16 10-3 sec

For electron t = 10 m ((0511 MeV)2 (104 Fm)2 +(197 MeV Fm)2)12(197 MeV Fm x 3 108 msec)=

= 9 10-7 sec

For photon t = Lc= 10 m 3 108 msec =33 10-8 sec

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

The Λ decays in flight into a proton and a πminus meson Λ rarrpπ-

If the Λ has a velocity of 08c - what is the maximal momentum that the πminus can have in the laboratory system- What is the maximum component of laboratory momentum perpendicular to the Λ direction

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Nucleus of 10B transits from excited state with energy E= 072 MeV

to the ground state with emission of a photon with half-life time t12=67x10-10 sec

What is energy uncertainty ΔE of the emitted γ

kinematics of γ-decay is energy and momentum conservation law

recoil of the nucleus pγ=pA (1)

E=Eγ +EA0 +EAkin (2)

E=ΔE= E - EA0

(1) is connected with kinetic energy of the nucleus in the ground state

EAkin =pA22M = pγ

22M and Eγ = pγ

from here Eγ =M( (1+2EM)12 -1) = M(1+EM-1)=E

recoil can be neglected

EAkin =Eγ22M =E 22M =(072 MeV)2(2x1081x931 MeV) =26x10-5 MeV

EAkin Eγ = 26x10-5 MeV072 MeV=36x10-5

ΔEτ= ΔEtln2 lt ħ =gt ΔE lt ħ ln2 t = 0693x658x10-22 MeV x sec 67x10-10 sec=68x10-7ev

bull α-particles with kinetic energy T=65 MeV scatter on the 197Au nucleiDefine a) impact parameter b (at scattering angle θ=90o)

b) (drdt=0) minimal distance rmin between α-particles and 197Au nucleic-d) kinetic T and potential E energy of α-particle at this point

r = rmin производная = 0 = 0

) кинетическая энергия T = T - E = 65 МeV -54 MeV=11 MeV

54 МэВ = 11 МэВ

Nuclear Physics Exercise NP2 (b)b) Elastic electron scattering on a charge distribution reveals the following form factor

where with being the transferred momentum and R some constant radius Propose a charge distribution which can produce such a form factor Determine the mean square radius for the distribution in terms of R

The nucleus can be approximately described as an α-particle with 4 loosely bound neutrons moving in a diffuse halo about the core Explain qualitatively how the electron scattering form factor measured for would differ from that of

2 -31 3F(q ) = + x (sinx - xcosx)

2 2x =|q|R q

8

2 He

8

2 He 4

2 He

Nuclear Physics Exercise NP3Fission

bull Assume a uranium nucleus 23692U breaks up spontaneously

into two roughly equal parts Estimate the reduction in electrostatic energy of the nuclei What is the relationship of this to total change of the energy

(assume uniform charge distiribution with radius R=12x 10-13

A013 cm)

bull BR Martin 210

bull The half-life of 239Pu has been determined by immersing a sphere of 239Pu of mass 1201gm in liquid nitrogen of a volume enough to stop all α-particles and measure the rate of evaporation of the liquid The evaporation rate corresponded to a power of 0231W Calculate to the nearest hundred years the half-life of 239Pu given that the energy of its decay alpha particles is 5144 MeV (Tak into account the recoil energy of the product nucleus

1MeV=160206x10-13 joule 1 atomic mass unit = 166x10-24 gm

bull BR Martin 28 211 212 213 (look through)

Nuclear Physics Exercise NP4Alpha- decay

bull An element of low atomic number Z can undergo allowed positron β-decay Let p0 is the maximum possible momentum of the positron supposing p0ltltmc (m-positron mass) Γβ is the beta-decay rate An alternative process is K-electron capture the nucleus capturing a K-shell electron and undergoing the same nuclear transition with emission of neutrino Let ΓK is the decay rate of this process Compute ratio ΓKΓβ You can treat the wave function of the K-shell electron as hydrogenic and can ignore the electron binding energy

bull BRM 215 79

Nuclear Physics Exercise NP4Beta- decays

Nuclear Physics Exercise NP5Nuclear reactions

bull BRM 216 (radioactive decay)

bull BRM 78 (gamma ndash decay)

bull BRM 710 711 (beta-decay)

bull Shell model BRM 71 72 73 74

bull Deformed nuclei BRM 75 76

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull At CERN Large Hadron Collider there were several poton-proton runs at energies

radics = 900 236 and 7000 GeV Calculate the main characteristics of these reactions

Ecm Elab β γ maximal rapidity ycm available for protons with mass mp=0938 GeV

and for pions with mπ=0139 GeV Draw y(β ) - dependence

bull Consider the production of the K+ in the reaction γprarrΚ+ +Λ- Give the minimal γ-ray energy in the laboratory system where p is at rest

when this reaction can take place - What will be the minimal γ- energy if the proton is not free but is bound in

the nucleus Take into account the Fermi motion with pF=250 MeVc

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

bull What is the smallest Elab energy of pp interactions for production K-meson with mass

mK=0494 GeV Write also the possible reactions

bull Proton photon and electron (me=0511 MeV) have the same wave length λ = 10-11 m

Which time will it take for each of them to fly the distance L=10 m

bull The Λ decays in flight into a proton and a πminus meson Λ rarrpπ- If the Λ has a velocity of 08c - what is the maximal momentum that the πminus can have in the laboratory system- What is the maximum component of laboratory momentum perpendicular

to the Λ direction

bull Nucleus of 10B transits from excited state with energy E= 072 MeVto the ground state with emission of a photon with half-life time t12=67x10-10 secWhat is energy uncertainty ΔE of the emitted γ

bull α-particles with kinetic energy T=65 MeV scatter on the 197Au nucleiDefine a) impact parameter b (at scattering angle θ=90o)

b) minimal distance rmin between α-particles and 197Au nucleic-d) kinetic T and potential E energy of α-particle at this point

Nuclear Physics Exercise NP6 Kinematics and Rutherford scattering

bull At CERN Large Hadron Collider there were several poton-proton runs at energies

radics = 900 2360 and 7000 GeV Calculate the main characteristics of these reactions

Ecm Elab β γ maximal rapidity ycm available for protons with mass mp=0938 GeV

and for pions with mπ=0139 GeV Draw y(β ) ndash dependence

s=(E 1+E2 )2 ndash (p1 +p2 )2= (2Ecm )2 if m1=m2 because for cms p1 = - p2

=(Elab +m2)2 - plab2= m1

2+ m22+ 2 Elab m2 = 2m2+ 2mElab

Ecm = radics2 Elab = s(2m) - m p=(E2-m2)12

β = pcmEcm = plab(Elab+m) γ = Ecmm

ymax= 05 ln (E+pl)(E-pl)= 05 ln (E+pl)2(E2-pl

2)= ln (E+pl)mt = ln 2Em

90000 236000 7000004317687848Sqrt s 90000 236000 700000

Ecm 45000 118000 350000

Pcm 449999022 1179999627 3499999874

Elab 4317687848 2968868998 2611940205

Plab 4317687848 2968868998 2611940205

Beta 09999978275 09999996841 09999999641

Betal 09999978275 09999996841 09999999641

Gamma 479744 1257995 3731343

ymax 68663 78304 89176

ymaxpi 87756 973969 1082694

β

lab

cms

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Consider the production of the K+ in the reaction γprarrΚ+ +Λ

- Give the minimal γ-ray energy in the laboratory system where proton is at rest when this reaction can take place

- What will be the minimal γ- energy if the proton is not free but is bound in the nucleus Take into account the Fermi motion with pF=250 MeVc

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull What is the smallest Elab energy of pp interactions for production K-meson with mass

mK=0494 GeV Write also the possible reactions

The reaction p+p-gt p+p+K++K- is possible but the minimal energy is demanded for

p+p-gtp+Λ+K+ where both baryon number and strangeness are conserved

radics gt Σi mi - condition for energy threshold where mi are masses of the produced particles

s=(Elab +m2)2 - plab2= m1

2+ m22+ 2 Elab m2 = 2m2+ 2mElab

So threshold energy

Elab = 12m2 ( (Σi mi )2- m1

2- m22) = ( (Σi mi )

2- 2m2)(2m)= (0939 + 0494 +1116)2- 209392 GeV

= 4734 GeV

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Proton photon and electron (me=0511 MeV) have the same wave length λ = 10-11 m

Which time will it take for each of them to fly the distance L=10 m

λ = (ħc)pc =10-11 m =gt p= (ħc) λc = (200 MeV Fm)(10-11m 10-15 mFm)c=2 106 MeVc

p=mvγ=mvradic(1-v2c2) =gt

v=1 radic(m2p2+1c2) = ħc c ((mc2)2 λ2 +(ħc)2)12

t=Lv= L ((mc2)2 λ2 +(ħc)2)12 ħc c

For proton t = 10 m ((9383 MeV)2 (104 Fm)2 +(197 MeV Fm)2)12(197 MeV Fm x 3 108 msec)

=16 10-3 sec

For electron t = 10 m ((0511 MeV)2 (104 Fm)2 +(197 MeV Fm)2)12(197 MeV Fm x 3 108 msec)=

= 9 10-7 sec

For photon t = Lc= 10 m 3 108 msec =33 10-8 sec

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

The Λ decays in flight into a proton and a πminus meson Λ rarrpπ-

If the Λ has a velocity of 08c - what is the maximal momentum that the πminus can have in the laboratory system- What is the maximum component of laboratory momentum perpendicular to the Λ direction

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Nucleus of 10B transits from excited state with energy E= 072 MeV

to the ground state with emission of a photon with half-life time t12=67x10-10 sec

What is energy uncertainty ΔE of the emitted γ

kinematics of γ-decay is energy and momentum conservation law

recoil of the nucleus pγ=pA (1)

E=Eγ +EA0 +EAkin (2)

E=ΔE= E - EA0

(1) is connected with kinetic energy of the nucleus in the ground state

EAkin =pA22M = pγ

22M and Eγ = pγ

from here Eγ =M( (1+2EM)12 -1) = M(1+EM-1)=E

recoil can be neglected

EAkin =Eγ22M =E 22M =(072 MeV)2(2x1081x931 MeV) =26x10-5 MeV

EAkin Eγ = 26x10-5 MeV072 MeV=36x10-5

ΔEτ= ΔEtln2 lt ħ =gt ΔE lt ħ ln2 t = 0693x658x10-22 MeV x sec 67x10-10 sec=68x10-7ev

bull α-particles with kinetic energy T=65 MeV scatter on the 197Au nucleiDefine a) impact parameter b (at scattering angle θ=90o)

b) (drdt=0) minimal distance rmin between α-particles and 197Au nucleic-d) kinetic T and potential E energy of α-particle at this point

r = rmin производная = 0 = 0

) кинетическая энергия T = T - E = 65 МeV -54 MeV=11 MeV

54 МэВ = 11 МэВ

Nuclear Physics Exercise NP3Fission

bull Assume a uranium nucleus 23692U breaks up spontaneously

into two roughly equal parts Estimate the reduction in electrostatic energy of the nuclei What is the relationship of this to total change of the energy

(assume uniform charge distiribution with radius R=12x 10-13

A013 cm)

bull BR Martin 210

bull The half-life of 239Pu has been determined by immersing a sphere of 239Pu of mass 1201gm in liquid nitrogen of a volume enough to stop all α-particles and measure the rate of evaporation of the liquid The evaporation rate corresponded to a power of 0231W Calculate to the nearest hundred years the half-life of 239Pu given that the energy of its decay alpha particles is 5144 MeV (Tak into account the recoil energy of the product nucleus

1MeV=160206x10-13 joule 1 atomic mass unit = 166x10-24 gm

bull BR Martin 28 211 212 213 (look through)

Nuclear Physics Exercise NP4Alpha- decay

bull An element of low atomic number Z can undergo allowed positron β-decay Let p0 is the maximum possible momentum of the positron supposing p0ltltmc (m-positron mass) Γβ is the beta-decay rate An alternative process is K-electron capture the nucleus capturing a K-shell electron and undergoing the same nuclear transition with emission of neutrino Let ΓK is the decay rate of this process Compute ratio ΓKΓβ You can treat the wave function of the K-shell electron as hydrogenic and can ignore the electron binding energy

bull BRM 215 79

Nuclear Physics Exercise NP4Beta- decays

Nuclear Physics Exercise NP5Nuclear reactions

bull BRM 216 (radioactive decay)

bull BRM 78 (gamma ndash decay)

bull BRM 710 711 (beta-decay)

bull Shell model BRM 71 72 73 74

bull Deformed nuclei BRM 75 76

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull At CERN Large Hadron Collider there were several poton-proton runs at energies

radics = 900 236 and 7000 GeV Calculate the main characteristics of these reactions

Ecm Elab β γ maximal rapidity ycm available for protons with mass mp=0938 GeV

and for pions with mπ=0139 GeV Draw y(β ) - dependence

bull Consider the production of the K+ in the reaction γprarrΚ+ +Λ- Give the minimal γ-ray energy in the laboratory system where p is at rest

when this reaction can take place - What will be the minimal γ- energy if the proton is not free but is bound in

the nucleus Take into account the Fermi motion with pF=250 MeVc

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

bull What is the smallest Elab energy of pp interactions for production K-meson with mass

mK=0494 GeV Write also the possible reactions

bull Proton photon and electron (me=0511 MeV) have the same wave length λ = 10-11 m

Which time will it take for each of them to fly the distance L=10 m

bull The Λ decays in flight into a proton and a πminus meson Λ rarrpπ- If the Λ has a velocity of 08c - what is the maximal momentum that the πminus can have in the laboratory system- What is the maximum component of laboratory momentum perpendicular

to the Λ direction

bull Nucleus of 10B transits from excited state with energy E= 072 MeVto the ground state with emission of a photon with half-life time t12=67x10-10 secWhat is energy uncertainty ΔE of the emitted γ

bull α-particles with kinetic energy T=65 MeV scatter on the 197Au nucleiDefine a) impact parameter b (at scattering angle θ=90o)

b) minimal distance rmin between α-particles and 197Au nucleic-d) kinetic T and potential E energy of α-particle at this point

Nuclear Physics Exercise NP6 Kinematics and Rutherford scattering

bull At CERN Large Hadron Collider there were several poton-proton runs at energies

radics = 900 2360 and 7000 GeV Calculate the main characteristics of these reactions

Ecm Elab β γ maximal rapidity ycm available for protons with mass mp=0938 GeV

and for pions with mπ=0139 GeV Draw y(β ) ndash dependence

s=(E 1+E2 )2 ndash (p1 +p2 )2= (2Ecm )2 if m1=m2 because for cms p1 = - p2

=(Elab +m2)2 - plab2= m1

2+ m22+ 2 Elab m2 = 2m2+ 2mElab

Ecm = radics2 Elab = s(2m) - m p=(E2-m2)12

β = pcmEcm = plab(Elab+m) γ = Ecmm

ymax= 05 ln (E+pl)(E-pl)= 05 ln (E+pl)2(E2-pl

2)= ln (E+pl)mt = ln 2Em

90000 236000 7000004317687848Sqrt s 90000 236000 700000

Ecm 45000 118000 350000

Pcm 449999022 1179999627 3499999874

Elab 4317687848 2968868998 2611940205

Plab 4317687848 2968868998 2611940205

Beta 09999978275 09999996841 09999999641

Betal 09999978275 09999996841 09999999641

Gamma 479744 1257995 3731343

ymax 68663 78304 89176

ymaxpi 87756 973969 1082694

β

lab

cms

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Consider the production of the K+ in the reaction γprarrΚ+ +Λ

- Give the minimal γ-ray energy in the laboratory system where proton is at rest when this reaction can take place

- What will be the minimal γ- energy if the proton is not free but is bound in the nucleus Take into account the Fermi motion with pF=250 MeVc

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull What is the smallest Elab energy of pp interactions for production K-meson with mass

mK=0494 GeV Write also the possible reactions

The reaction p+p-gt p+p+K++K- is possible but the minimal energy is demanded for

p+p-gtp+Λ+K+ where both baryon number and strangeness are conserved

radics gt Σi mi - condition for energy threshold where mi are masses of the produced particles

s=(Elab +m2)2 - plab2= m1

2+ m22+ 2 Elab m2 = 2m2+ 2mElab

So threshold energy

Elab = 12m2 ( (Σi mi )2- m1

2- m22) = ( (Σi mi )

2- 2m2)(2m)= (0939 + 0494 +1116)2- 209392 GeV

= 4734 GeV

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Proton photon and electron (me=0511 MeV) have the same wave length λ = 10-11 m

Which time will it take for each of them to fly the distance L=10 m

λ = (ħc)pc =10-11 m =gt p= (ħc) λc = (200 MeV Fm)(10-11m 10-15 mFm)c=2 106 MeVc

p=mvγ=mvradic(1-v2c2) =gt

v=1 radic(m2p2+1c2) = ħc c ((mc2)2 λ2 +(ħc)2)12

t=Lv= L ((mc2)2 λ2 +(ħc)2)12 ħc c

For proton t = 10 m ((9383 MeV)2 (104 Fm)2 +(197 MeV Fm)2)12(197 MeV Fm x 3 108 msec)

=16 10-3 sec

For electron t = 10 m ((0511 MeV)2 (104 Fm)2 +(197 MeV Fm)2)12(197 MeV Fm x 3 108 msec)=

= 9 10-7 sec

For photon t = Lc= 10 m 3 108 msec =33 10-8 sec

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

The Λ decays in flight into a proton and a πminus meson Λ rarrpπ-

If the Λ has a velocity of 08c - what is the maximal momentum that the πminus can have in the laboratory system- What is the maximum component of laboratory momentum perpendicular to the Λ direction

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Nucleus of 10B transits from excited state with energy E= 072 MeV

to the ground state with emission of a photon with half-life time t12=67x10-10 sec

What is energy uncertainty ΔE of the emitted γ

kinematics of γ-decay is energy and momentum conservation law

recoil of the nucleus pγ=pA (1)

E=Eγ +EA0 +EAkin (2)

E=ΔE= E - EA0

(1) is connected with kinetic energy of the nucleus in the ground state

EAkin =pA22M = pγ

22M and Eγ = pγ

from here Eγ =M( (1+2EM)12 -1) = M(1+EM-1)=E

recoil can be neglected

EAkin =Eγ22M =E 22M =(072 MeV)2(2x1081x931 MeV) =26x10-5 MeV

EAkin Eγ = 26x10-5 MeV072 MeV=36x10-5

ΔEτ= ΔEtln2 lt ħ =gt ΔE lt ħ ln2 t = 0693x658x10-22 MeV x sec 67x10-10 sec=68x10-7ev

bull α-particles with kinetic energy T=65 MeV scatter on the 197Au nucleiDefine a) impact parameter b (at scattering angle θ=90o)

b) (drdt=0) minimal distance rmin between α-particles and 197Au nucleic-d) kinetic T and potential E energy of α-particle at this point

r = rmin производная = 0 = 0

) кинетическая энергия T = T - E = 65 МeV -54 MeV=11 MeV

54 МэВ = 11 МэВ

bull The half-life of 239Pu has been determined by immersing a sphere of 239Pu of mass 1201gm in liquid nitrogen of a volume enough to stop all α-particles and measure the rate of evaporation of the liquid The evaporation rate corresponded to a power of 0231W Calculate to the nearest hundred years the half-life of 239Pu given that the energy of its decay alpha particles is 5144 MeV (Tak into account the recoil energy of the product nucleus

1MeV=160206x10-13 joule 1 atomic mass unit = 166x10-24 gm

bull BR Martin 28 211 212 213 (look through)

Nuclear Physics Exercise NP4Alpha- decay

bull An element of low atomic number Z can undergo allowed positron β-decay Let p0 is the maximum possible momentum of the positron supposing p0ltltmc (m-positron mass) Γβ is the beta-decay rate An alternative process is K-electron capture the nucleus capturing a K-shell electron and undergoing the same nuclear transition with emission of neutrino Let ΓK is the decay rate of this process Compute ratio ΓKΓβ You can treat the wave function of the K-shell electron as hydrogenic and can ignore the electron binding energy

bull BRM 215 79

Nuclear Physics Exercise NP4Beta- decays

Nuclear Physics Exercise NP5Nuclear reactions

bull BRM 216 (radioactive decay)

bull BRM 78 (gamma ndash decay)

bull BRM 710 711 (beta-decay)

bull Shell model BRM 71 72 73 74

bull Deformed nuclei BRM 75 76

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull At CERN Large Hadron Collider there were several poton-proton runs at energies

radics = 900 236 and 7000 GeV Calculate the main characteristics of these reactions

Ecm Elab β γ maximal rapidity ycm available for protons with mass mp=0938 GeV

and for pions with mπ=0139 GeV Draw y(β ) - dependence

bull Consider the production of the K+ in the reaction γprarrΚ+ +Λ- Give the minimal γ-ray energy in the laboratory system where p is at rest

when this reaction can take place - What will be the minimal γ- energy if the proton is not free but is bound in

the nucleus Take into account the Fermi motion with pF=250 MeVc

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

bull What is the smallest Elab energy of pp interactions for production K-meson with mass

mK=0494 GeV Write also the possible reactions

bull Proton photon and electron (me=0511 MeV) have the same wave length λ = 10-11 m

Which time will it take for each of them to fly the distance L=10 m

bull The Λ decays in flight into a proton and a πminus meson Λ rarrpπ- If the Λ has a velocity of 08c - what is the maximal momentum that the πminus can have in the laboratory system- What is the maximum component of laboratory momentum perpendicular

to the Λ direction

bull Nucleus of 10B transits from excited state with energy E= 072 MeVto the ground state with emission of a photon with half-life time t12=67x10-10 secWhat is energy uncertainty ΔE of the emitted γ

bull α-particles with kinetic energy T=65 MeV scatter on the 197Au nucleiDefine a) impact parameter b (at scattering angle θ=90o)

b) minimal distance rmin between α-particles and 197Au nucleic-d) kinetic T and potential E energy of α-particle at this point

Nuclear Physics Exercise NP6 Kinematics and Rutherford scattering

bull At CERN Large Hadron Collider there were several poton-proton runs at energies

radics = 900 2360 and 7000 GeV Calculate the main characteristics of these reactions

Ecm Elab β γ maximal rapidity ycm available for protons with mass mp=0938 GeV

and for pions with mπ=0139 GeV Draw y(β ) ndash dependence

s=(E 1+E2 )2 ndash (p1 +p2 )2= (2Ecm )2 if m1=m2 because for cms p1 = - p2

=(Elab +m2)2 - plab2= m1

2+ m22+ 2 Elab m2 = 2m2+ 2mElab

Ecm = radics2 Elab = s(2m) - m p=(E2-m2)12

β = pcmEcm = plab(Elab+m) γ = Ecmm

ymax= 05 ln (E+pl)(E-pl)= 05 ln (E+pl)2(E2-pl

2)= ln (E+pl)mt = ln 2Em

90000 236000 7000004317687848Sqrt s 90000 236000 700000

Ecm 45000 118000 350000

Pcm 449999022 1179999627 3499999874

Elab 4317687848 2968868998 2611940205

Plab 4317687848 2968868998 2611940205

Beta 09999978275 09999996841 09999999641

Betal 09999978275 09999996841 09999999641

Gamma 479744 1257995 3731343

ymax 68663 78304 89176

ymaxpi 87756 973969 1082694

β

lab

cms

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Consider the production of the K+ in the reaction γprarrΚ+ +Λ

- Give the minimal γ-ray energy in the laboratory system where proton is at rest when this reaction can take place

- What will be the minimal γ- energy if the proton is not free but is bound in the nucleus Take into account the Fermi motion with pF=250 MeVc

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull What is the smallest Elab energy of pp interactions for production K-meson with mass

mK=0494 GeV Write also the possible reactions

The reaction p+p-gt p+p+K++K- is possible but the minimal energy is demanded for

p+p-gtp+Λ+K+ where both baryon number and strangeness are conserved

radics gt Σi mi - condition for energy threshold where mi are masses of the produced particles

s=(Elab +m2)2 - plab2= m1

2+ m22+ 2 Elab m2 = 2m2+ 2mElab

So threshold energy

Elab = 12m2 ( (Σi mi )2- m1

2- m22) = ( (Σi mi )

2- 2m2)(2m)= (0939 + 0494 +1116)2- 209392 GeV

= 4734 GeV

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Proton photon and electron (me=0511 MeV) have the same wave length λ = 10-11 m

Which time will it take for each of them to fly the distance L=10 m

λ = (ħc)pc =10-11 m =gt p= (ħc) λc = (200 MeV Fm)(10-11m 10-15 mFm)c=2 106 MeVc

p=mvγ=mvradic(1-v2c2) =gt

v=1 radic(m2p2+1c2) = ħc c ((mc2)2 λ2 +(ħc)2)12

t=Lv= L ((mc2)2 λ2 +(ħc)2)12 ħc c

For proton t = 10 m ((9383 MeV)2 (104 Fm)2 +(197 MeV Fm)2)12(197 MeV Fm x 3 108 msec)

=16 10-3 sec

For electron t = 10 m ((0511 MeV)2 (104 Fm)2 +(197 MeV Fm)2)12(197 MeV Fm x 3 108 msec)=

= 9 10-7 sec

For photon t = Lc= 10 m 3 108 msec =33 10-8 sec

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

The Λ decays in flight into a proton and a πminus meson Λ rarrpπ-

If the Λ has a velocity of 08c - what is the maximal momentum that the πminus can have in the laboratory system- What is the maximum component of laboratory momentum perpendicular to the Λ direction

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Nucleus of 10B transits from excited state with energy E= 072 MeV

to the ground state with emission of a photon with half-life time t12=67x10-10 sec

What is energy uncertainty ΔE of the emitted γ

kinematics of γ-decay is energy and momentum conservation law

recoil of the nucleus pγ=pA (1)

E=Eγ +EA0 +EAkin (2)

E=ΔE= E - EA0

(1) is connected with kinetic energy of the nucleus in the ground state

EAkin =pA22M = pγ

22M and Eγ = pγ

from here Eγ =M( (1+2EM)12 -1) = M(1+EM-1)=E

recoil can be neglected

EAkin =Eγ22M =E 22M =(072 MeV)2(2x1081x931 MeV) =26x10-5 MeV

EAkin Eγ = 26x10-5 MeV072 MeV=36x10-5

ΔEτ= ΔEtln2 lt ħ =gt ΔE lt ħ ln2 t = 0693x658x10-22 MeV x sec 67x10-10 sec=68x10-7ev

bull α-particles with kinetic energy T=65 MeV scatter on the 197Au nucleiDefine a) impact parameter b (at scattering angle θ=90o)

b) (drdt=0) minimal distance rmin between α-particles and 197Au nucleic-d) kinetic T and potential E energy of α-particle at this point

r = rmin производная = 0 = 0

) кинетическая энергия T = T - E = 65 МeV -54 MeV=11 MeV

54 МэВ = 11 МэВ

bull An element of low atomic number Z can undergo allowed positron β-decay Let p0 is the maximum possible momentum of the positron supposing p0ltltmc (m-positron mass) Γβ is the beta-decay rate An alternative process is K-electron capture the nucleus capturing a K-shell electron and undergoing the same nuclear transition with emission of neutrino Let ΓK is the decay rate of this process Compute ratio ΓKΓβ You can treat the wave function of the K-shell electron as hydrogenic and can ignore the electron binding energy

bull BRM 215 79

Nuclear Physics Exercise NP4Beta- decays

Nuclear Physics Exercise NP5Nuclear reactions

bull BRM 216 (radioactive decay)

bull BRM 78 (gamma ndash decay)

bull BRM 710 711 (beta-decay)

bull Shell model BRM 71 72 73 74

bull Deformed nuclei BRM 75 76

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull At CERN Large Hadron Collider there were several poton-proton runs at energies

radics = 900 236 and 7000 GeV Calculate the main characteristics of these reactions

Ecm Elab β γ maximal rapidity ycm available for protons with mass mp=0938 GeV

and for pions with mπ=0139 GeV Draw y(β ) - dependence

bull Consider the production of the K+ in the reaction γprarrΚ+ +Λ- Give the minimal γ-ray energy in the laboratory system where p is at rest

when this reaction can take place - What will be the minimal γ- energy if the proton is not free but is bound in

the nucleus Take into account the Fermi motion with pF=250 MeVc

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

bull What is the smallest Elab energy of pp interactions for production K-meson with mass

mK=0494 GeV Write also the possible reactions

bull Proton photon and electron (me=0511 MeV) have the same wave length λ = 10-11 m

Which time will it take for each of them to fly the distance L=10 m

bull The Λ decays in flight into a proton and a πminus meson Λ rarrpπ- If the Λ has a velocity of 08c - what is the maximal momentum that the πminus can have in the laboratory system- What is the maximum component of laboratory momentum perpendicular

to the Λ direction

bull Nucleus of 10B transits from excited state with energy E= 072 MeVto the ground state with emission of a photon with half-life time t12=67x10-10 secWhat is energy uncertainty ΔE of the emitted γ

bull α-particles with kinetic energy T=65 MeV scatter on the 197Au nucleiDefine a) impact parameter b (at scattering angle θ=90o)

b) minimal distance rmin between α-particles and 197Au nucleic-d) kinetic T and potential E energy of α-particle at this point

Nuclear Physics Exercise NP6 Kinematics and Rutherford scattering

bull At CERN Large Hadron Collider there were several poton-proton runs at energies

radics = 900 2360 and 7000 GeV Calculate the main characteristics of these reactions

Ecm Elab β γ maximal rapidity ycm available for protons with mass mp=0938 GeV

and for pions with mπ=0139 GeV Draw y(β ) ndash dependence

s=(E 1+E2 )2 ndash (p1 +p2 )2= (2Ecm )2 if m1=m2 because for cms p1 = - p2

=(Elab +m2)2 - plab2= m1

2+ m22+ 2 Elab m2 = 2m2+ 2mElab

Ecm = radics2 Elab = s(2m) - m p=(E2-m2)12

β = pcmEcm = plab(Elab+m) γ = Ecmm

ymax= 05 ln (E+pl)(E-pl)= 05 ln (E+pl)2(E2-pl

2)= ln (E+pl)mt = ln 2Em

90000 236000 7000004317687848Sqrt s 90000 236000 700000

Ecm 45000 118000 350000

Pcm 449999022 1179999627 3499999874

Elab 4317687848 2968868998 2611940205

Plab 4317687848 2968868998 2611940205

Beta 09999978275 09999996841 09999999641

Betal 09999978275 09999996841 09999999641

Gamma 479744 1257995 3731343

ymax 68663 78304 89176

ymaxpi 87756 973969 1082694

β

lab

cms

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Consider the production of the K+ in the reaction γprarrΚ+ +Λ

- Give the minimal γ-ray energy in the laboratory system where proton is at rest when this reaction can take place

- What will be the minimal γ- energy if the proton is not free but is bound in the nucleus Take into account the Fermi motion with pF=250 MeVc

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull What is the smallest Elab energy of pp interactions for production K-meson with mass

mK=0494 GeV Write also the possible reactions

The reaction p+p-gt p+p+K++K- is possible but the minimal energy is demanded for

p+p-gtp+Λ+K+ where both baryon number and strangeness are conserved

radics gt Σi mi - condition for energy threshold where mi are masses of the produced particles

s=(Elab +m2)2 - plab2= m1

2+ m22+ 2 Elab m2 = 2m2+ 2mElab

So threshold energy

Elab = 12m2 ( (Σi mi )2- m1

2- m22) = ( (Σi mi )

2- 2m2)(2m)= (0939 + 0494 +1116)2- 209392 GeV

= 4734 GeV

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Proton photon and electron (me=0511 MeV) have the same wave length λ = 10-11 m

Which time will it take for each of them to fly the distance L=10 m

λ = (ħc)pc =10-11 m =gt p= (ħc) λc = (200 MeV Fm)(10-11m 10-15 mFm)c=2 106 MeVc

p=mvγ=mvradic(1-v2c2) =gt

v=1 radic(m2p2+1c2) = ħc c ((mc2)2 λ2 +(ħc)2)12

t=Lv= L ((mc2)2 λ2 +(ħc)2)12 ħc c

For proton t = 10 m ((9383 MeV)2 (104 Fm)2 +(197 MeV Fm)2)12(197 MeV Fm x 3 108 msec)

=16 10-3 sec

For electron t = 10 m ((0511 MeV)2 (104 Fm)2 +(197 MeV Fm)2)12(197 MeV Fm x 3 108 msec)=

= 9 10-7 sec

For photon t = Lc= 10 m 3 108 msec =33 10-8 sec

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

The Λ decays in flight into a proton and a πminus meson Λ rarrpπ-

If the Λ has a velocity of 08c - what is the maximal momentum that the πminus can have in the laboratory system- What is the maximum component of laboratory momentum perpendicular to the Λ direction

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Nucleus of 10B transits from excited state with energy E= 072 MeV

to the ground state with emission of a photon with half-life time t12=67x10-10 sec

What is energy uncertainty ΔE of the emitted γ

kinematics of γ-decay is energy and momentum conservation law

recoil of the nucleus pγ=pA (1)

E=Eγ +EA0 +EAkin (2)

E=ΔE= E - EA0

(1) is connected with kinetic energy of the nucleus in the ground state

EAkin =pA22M = pγ

22M and Eγ = pγ

from here Eγ =M( (1+2EM)12 -1) = M(1+EM-1)=E

recoil can be neglected

EAkin =Eγ22M =E 22M =(072 MeV)2(2x1081x931 MeV) =26x10-5 MeV

EAkin Eγ = 26x10-5 MeV072 MeV=36x10-5

ΔEτ= ΔEtln2 lt ħ =gt ΔE lt ħ ln2 t = 0693x658x10-22 MeV x sec 67x10-10 sec=68x10-7ev

bull α-particles with kinetic energy T=65 MeV scatter on the 197Au nucleiDefine a) impact parameter b (at scattering angle θ=90o)

b) (drdt=0) minimal distance rmin between α-particles and 197Au nucleic-d) kinetic T and potential E energy of α-particle at this point

r = rmin производная = 0 = 0

) кинетическая энергия T = T - E = 65 МeV -54 MeV=11 MeV

54 МэВ = 11 МэВ

Nuclear Physics Exercise NP5Nuclear reactions

bull BRM 216 (radioactive decay)

bull BRM 78 (gamma ndash decay)

bull BRM 710 711 (beta-decay)

bull Shell model BRM 71 72 73 74

bull Deformed nuclei BRM 75 76

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull At CERN Large Hadron Collider there were several poton-proton runs at energies

radics = 900 236 and 7000 GeV Calculate the main characteristics of these reactions

Ecm Elab β γ maximal rapidity ycm available for protons with mass mp=0938 GeV

and for pions with mπ=0139 GeV Draw y(β ) - dependence

bull Consider the production of the K+ in the reaction γprarrΚ+ +Λ- Give the minimal γ-ray energy in the laboratory system where p is at rest

when this reaction can take place - What will be the minimal γ- energy if the proton is not free but is bound in

the nucleus Take into account the Fermi motion with pF=250 MeVc

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

bull What is the smallest Elab energy of pp interactions for production K-meson with mass

mK=0494 GeV Write also the possible reactions

bull Proton photon and electron (me=0511 MeV) have the same wave length λ = 10-11 m

Which time will it take for each of them to fly the distance L=10 m

bull The Λ decays in flight into a proton and a πminus meson Λ rarrpπ- If the Λ has a velocity of 08c - what is the maximal momentum that the πminus can have in the laboratory system- What is the maximum component of laboratory momentum perpendicular

to the Λ direction

bull Nucleus of 10B transits from excited state with energy E= 072 MeVto the ground state with emission of a photon with half-life time t12=67x10-10 secWhat is energy uncertainty ΔE of the emitted γ

bull α-particles with kinetic energy T=65 MeV scatter on the 197Au nucleiDefine a) impact parameter b (at scattering angle θ=90o)

b) minimal distance rmin between α-particles and 197Au nucleic-d) kinetic T and potential E energy of α-particle at this point

Nuclear Physics Exercise NP6 Kinematics and Rutherford scattering

bull At CERN Large Hadron Collider there were several poton-proton runs at energies

radics = 900 2360 and 7000 GeV Calculate the main characteristics of these reactions

Ecm Elab β γ maximal rapidity ycm available for protons with mass mp=0938 GeV

and for pions with mπ=0139 GeV Draw y(β ) ndash dependence

s=(E 1+E2 )2 ndash (p1 +p2 )2= (2Ecm )2 if m1=m2 because for cms p1 = - p2

=(Elab +m2)2 - plab2= m1

2+ m22+ 2 Elab m2 = 2m2+ 2mElab

Ecm = radics2 Elab = s(2m) - m p=(E2-m2)12

β = pcmEcm = plab(Elab+m) γ = Ecmm

ymax= 05 ln (E+pl)(E-pl)= 05 ln (E+pl)2(E2-pl

2)= ln (E+pl)mt = ln 2Em

90000 236000 7000004317687848Sqrt s 90000 236000 700000

Ecm 45000 118000 350000

Pcm 449999022 1179999627 3499999874

Elab 4317687848 2968868998 2611940205

Plab 4317687848 2968868998 2611940205

Beta 09999978275 09999996841 09999999641

Betal 09999978275 09999996841 09999999641

Gamma 479744 1257995 3731343

ymax 68663 78304 89176

ymaxpi 87756 973969 1082694

β

lab

cms

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Consider the production of the K+ in the reaction γprarrΚ+ +Λ

- Give the minimal γ-ray energy in the laboratory system where proton is at rest when this reaction can take place

- What will be the minimal γ- energy if the proton is not free but is bound in the nucleus Take into account the Fermi motion with pF=250 MeVc

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull What is the smallest Elab energy of pp interactions for production K-meson with mass

mK=0494 GeV Write also the possible reactions

The reaction p+p-gt p+p+K++K- is possible but the minimal energy is demanded for

p+p-gtp+Λ+K+ where both baryon number and strangeness are conserved

radics gt Σi mi - condition for energy threshold where mi are masses of the produced particles

s=(Elab +m2)2 - plab2= m1

2+ m22+ 2 Elab m2 = 2m2+ 2mElab

So threshold energy

Elab = 12m2 ( (Σi mi )2- m1

2- m22) = ( (Σi mi )

2- 2m2)(2m)= (0939 + 0494 +1116)2- 209392 GeV

= 4734 GeV

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Proton photon and electron (me=0511 MeV) have the same wave length λ = 10-11 m

Which time will it take for each of them to fly the distance L=10 m

λ = (ħc)pc =10-11 m =gt p= (ħc) λc = (200 MeV Fm)(10-11m 10-15 mFm)c=2 106 MeVc

p=mvγ=mvradic(1-v2c2) =gt

v=1 radic(m2p2+1c2) = ħc c ((mc2)2 λ2 +(ħc)2)12

t=Lv= L ((mc2)2 λ2 +(ħc)2)12 ħc c

For proton t = 10 m ((9383 MeV)2 (104 Fm)2 +(197 MeV Fm)2)12(197 MeV Fm x 3 108 msec)

=16 10-3 sec

For electron t = 10 m ((0511 MeV)2 (104 Fm)2 +(197 MeV Fm)2)12(197 MeV Fm x 3 108 msec)=

= 9 10-7 sec

For photon t = Lc= 10 m 3 108 msec =33 10-8 sec

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

The Λ decays in flight into a proton and a πminus meson Λ rarrpπ-

If the Λ has a velocity of 08c - what is the maximal momentum that the πminus can have in the laboratory system- What is the maximum component of laboratory momentum perpendicular to the Λ direction

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Nucleus of 10B transits from excited state with energy E= 072 MeV

to the ground state with emission of a photon with half-life time t12=67x10-10 sec

What is energy uncertainty ΔE of the emitted γ

kinematics of γ-decay is energy and momentum conservation law

recoil of the nucleus pγ=pA (1)

E=Eγ +EA0 +EAkin (2)

E=ΔE= E - EA0

(1) is connected with kinetic energy of the nucleus in the ground state

EAkin =pA22M = pγ

22M and Eγ = pγ

from here Eγ =M( (1+2EM)12 -1) = M(1+EM-1)=E

recoil can be neglected

EAkin =Eγ22M =E 22M =(072 MeV)2(2x1081x931 MeV) =26x10-5 MeV

EAkin Eγ = 26x10-5 MeV072 MeV=36x10-5

ΔEτ= ΔEtln2 lt ħ =gt ΔE lt ħ ln2 t = 0693x658x10-22 MeV x sec 67x10-10 sec=68x10-7ev

bull α-particles with kinetic energy T=65 MeV scatter on the 197Au nucleiDefine a) impact parameter b (at scattering angle θ=90o)

b) (drdt=0) minimal distance rmin between α-particles and 197Au nucleic-d) kinetic T and potential E energy of α-particle at this point

r = rmin производная = 0 = 0

) кинетическая энергия T = T - E = 65 МeV -54 MeV=11 MeV

54 МэВ = 11 МэВ

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull At CERN Large Hadron Collider there were several poton-proton runs at energies

radics = 900 236 and 7000 GeV Calculate the main characteristics of these reactions

Ecm Elab β γ maximal rapidity ycm available for protons with mass mp=0938 GeV

and for pions with mπ=0139 GeV Draw y(β ) - dependence

bull Consider the production of the K+ in the reaction γprarrΚ+ +Λ- Give the minimal γ-ray energy in the laboratory system where p is at rest

when this reaction can take place - What will be the minimal γ- energy if the proton is not free but is bound in

the nucleus Take into account the Fermi motion with pF=250 MeVc

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

bull What is the smallest Elab energy of pp interactions for production K-meson with mass

mK=0494 GeV Write also the possible reactions

bull Proton photon and electron (me=0511 MeV) have the same wave length λ = 10-11 m

Which time will it take for each of them to fly the distance L=10 m

bull The Λ decays in flight into a proton and a πminus meson Λ rarrpπ- If the Λ has a velocity of 08c - what is the maximal momentum that the πminus can have in the laboratory system- What is the maximum component of laboratory momentum perpendicular

to the Λ direction

bull Nucleus of 10B transits from excited state with energy E= 072 MeVto the ground state with emission of a photon with half-life time t12=67x10-10 secWhat is energy uncertainty ΔE of the emitted γ

bull α-particles with kinetic energy T=65 MeV scatter on the 197Au nucleiDefine a) impact parameter b (at scattering angle θ=90o)

b) minimal distance rmin between α-particles and 197Au nucleic-d) kinetic T and potential E energy of α-particle at this point

Nuclear Physics Exercise NP6 Kinematics and Rutherford scattering

bull At CERN Large Hadron Collider there were several poton-proton runs at energies

radics = 900 2360 and 7000 GeV Calculate the main characteristics of these reactions

Ecm Elab β γ maximal rapidity ycm available for protons with mass mp=0938 GeV

and for pions with mπ=0139 GeV Draw y(β ) ndash dependence

s=(E 1+E2 )2 ndash (p1 +p2 )2= (2Ecm )2 if m1=m2 because for cms p1 = - p2

=(Elab +m2)2 - plab2= m1

2+ m22+ 2 Elab m2 = 2m2+ 2mElab

Ecm = radics2 Elab = s(2m) - m p=(E2-m2)12

β = pcmEcm = plab(Elab+m) γ = Ecmm

ymax= 05 ln (E+pl)(E-pl)= 05 ln (E+pl)2(E2-pl

2)= ln (E+pl)mt = ln 2Em

90000 236000 7000004317687848Sqrt s 90000 236000 700000

Ecm 45000 118000 350000

Pcm 449999022 1179999627 3499999874

Elab 4317687848 2968868998 2611940205

Plab 4317687848 2968868998 2611940205

Beta 09999978275 09999996841 09999999641

Betal 09999978275 09999996841 09999999641

Gamma 479744 1257995 3731343

ymax 68663 78304 89176

ymaxpi 87756 973969 1082694

β

lab

cms

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Consider the production of the K+ in the reaction γprarrΚ+ +Λ

- Give the minimal γ-ray energy in the laboratory system where proton is at rest when this reaction can take place

- What will be the minimal γ- energy if the proton is not free but is bound in the nucleus Take into account the Fermi motion with pF=250 MeVc

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull What is the smallest Elab energy of pp interactions for production K-meson with mass

mK=0494 GeV Write also the possible reactions

The reaction p+p-gt p+p+K++K- is possible but the minimal energy is demanded for

p+p-gtp+Λ+K+ where both baryon number and strangeness are conserved

radics gt Σi mi - condition for energy threshold where mi are masses of the produced particles

s=(Elab +m2)2 - plab2= m1

2+ m22+ 2 Elab m2 = 2m2+ 2mElab

So threshold energy

Elab = 12m2 ( (Σi mi )2- m1

2- m22) = ( (Σi mi )

2- 2m2)(2m)= (0939 + 0494 +1116)2- 209392 GeV

= 4734 GeV

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Proton photon and electron (me=0511 MeV) have the same wave length λ = 10-11 m

Which time will it take for each of them to fly the distance L=10 m

λ = (ħc)pc =10-11 m =gt p= (ħc) λc = (200 MeV Fm)(10-11m 10-15 mFm)c=2 106 MeVc

p=mvγ=mvradic(1-v2c2) =gt

v=1 radic(m2p2+1c2) = ħc c ((mc2)2 λ2 +(ħc)2)12

t=Lv= L ((mc2)2 λ2 +(ħc)2)12 ħc c

For proton t = 10 m ((9383 MeV)2 (104 Fm)2 +(197 MeV Fm)2)12(197 MeV Fm x 3 108 msec)

=16 10-3 sec

For electron t = 10 m ((0511 MeV)2 (104 Fm)2 +(197 MeV Fm)2)12(197 MeV Fm x 3 108 msec)=

= 9 10-7 sec

For photon t = Lc= 10 m 3 108 msec =33 10-8 sec

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

The Λ decays in flight into a proton and a πminus meson Λ rarrpπ-

If the Λ has a velocity of 08c - what is the maximal momentum that the πminus can have in the laboratory system- What is the maximum component of laboratory momentum perpendicular to the Λ direction

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Nucleus of 10B transits from excited state with energy E= 072 MeV

to the ground state with emission of a photon with half-life time t12=67x10-10 sec

What is energy uncertainty ΔE of the emitted γ

kinematics of γ-decay is energy and momentum conservation law

recoil of the nucleus pγ=pA (1)

E=Eγ +EA0 +EAkin (2)

E=ΔE= E - EA0

(1) is connected with kinetic energy of the nucleus in the ground state

EAkin =pA22M = pγ

22M and Eγ = pγ

from here Eγ =M( (1+2EM)12 -1) = M(1+EM-1)=E

recoil can be neglected

EAkin =Eγ22M =E 22M =(072 MeV)2(2x1081x931 MeV) =26x10-5 MeV

EAkin Eγ = 26x10-5 MeV072 MeV=36x10-5

ΔEτ= ΔEtln2 lt ħ =gt ΔE lt ħ ln2 t = 0693x658x10-22 MeV x sec 67x10-10 sec=68x10-7ev

bull α-particles with kinetic energy T=65 MeV scatter on the 197Au nucleiDefine a) impact parameter b (at scattering angle θ=90o)

b) (drdt=0) minimal distance rmin between α-particles and 197Au nucleic-d) kinetic T and potential E energy of α-particle at this point

r = rmin производная = 0 = 0

) кинетическая энергия T = T - E = 65 МeV -54 MeV=11 MeV

54 МэВ = 11 МэВ

bull The Λ decays in flight into a proton and a πminus meson Λ rarrpπ- If the Λ has a velocity of 08c - what is the maximal momentum that the πminus can have in the laboratory system- What is the maximum component of laboratory momentum perpendicular

to the Λ direction

bull Nucleus of 10B transits from excited state with energy E= 072 MeVto the ground state with emission of a photon with half-life time t12=67x10-10 secWhat is energy uncertainty ΔE of the emitted γ

bull α-particles with kinetic energy T=65 MeV scatter on the 197Au nucleiDefine a) impact parameter b (at scattering angle θ=90o)

b) minimal distance rmin between α-particles and 197Au nucleic-d) kinetic T and potential E energy of α-particle at this point

Nuclear Physics Exercise NP6 Kinematics and Rutherford scattering

bull At CERN Large Hadron Collider there were several poton-proton runs at energies

radics = 900 2360 and 7000 GeV Calculate the main characteristics of these reactions

Ecm Elab β γ maximal rapidity ycm available for protons with mass mp=0938 GeV

and for pions with mπ=0139 GeV Draw y(β ) ndash dependence

s=(E 1+E2 )2 ndash (p1 +p2 )2= (2Ecm )2 if m1=m2 because for cms p1 = - p2

=(Elab +m2)2 - plab2= m1

2+ m22+ 2 Elab m2 = 2m2+ 2mElab

Ecm = radics2 Elab = s(2m) - m p=(E2-m2)12

β = pcmEcm = plab(Elab+m) γ = Ecmm

ymax= 05 ln (E+pl)(E-pl)= 05 ln (E+pl)2(E2-pl

2)= ln (E+pl)mt = ln 2Em

90000 236000 7000004317687848Sqrt s 90000 236000 700000

Ecm 45000 118000 350000

Pcm 449999022 1179999627 3499999874

Elab 4317687848 2968868998 2611940205

Plab 4317687848 2968868998 2611940205

Beta 09999978275 09999996841 09999999641

Betal 09999978275 09999996841 09999999641

Gamma 479744 1257995 3731343

ymax 68663 78304 89176

ymaxpi 87756 973969 1082694

β

lab

cms

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Consider the production of the K+ in the reaction γprarrΚ+ +Λ

- Give the minimal γ-ray energy in the laboratory system where proton is at rest when this reaction can take place

- What will be the minimal γ- energy if the proton is not free but is bound in the nucleus Take into account the Fermi motion with pF=250 MeVc

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull What is the smallest Elab energy of pp interactions for production K-meson with mass

mK=0494 GeV Write also the possible reactions

The reaction p+p-gt p+p+K++K- is possible but the minimal energy is demanded for

p+p-gtp+Λ+K+ where both baryon number and strangeness are conserved

radics gt Σi mi - condition for energy threshold where mi are masses of the produced particles

s=(Elab +m2)2 - plab2= m1

2+ m22+ 2 Elab m2 = 2m2+ 2mElab

So threshold energy

Elab = 12m2 ( (Σi mi )2- m1

2- m22) = ( (Σi mi )

2- 2m2)(2m)= (0939 + 0494 +1116)2- 209392 GeV

= 4734 GeV

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Proton photon and electron (me=0511 MeV) have the same wave length λ = 10-11 m

Which time will it take for each of them to fly the distance L=10 m

λ = (ħc)pc =10-11 m =gt p= (ħc) λc = (200 MeV Fm)(10-11m 10-15 mFm)c=2 106 MeVc

p=mvγ=mvradic(1-v2c2) =gt

v=1 radic(m2p2+1c2) = ħc c ((mc2)2 λ2 +(ħc)2)12

t=Lv= L ((mc2)2 λ2 +(ħc)2)12 ħc c

For proton t = 10 m ((9383 MeV)2 (104 Fm)2 +(197 MeV Fm)2)12(197 MeV Fm x 3 108 msec)

=16 10-3 sec

For electron t = 10 m ((0511 MeV)2 (104 Fm)2 +(197 MeV Fm)2)12(197 MeV Fm x 3 108 msec)=

= 9 10-7 sec

For photon t = Lc= 10 m 3 108 msec =33 10-8 sec

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

The Λ decays in flight into a proton and a πminus meson Λ rarrpπ-

If the Λ has a velocity of 08c - what is the maximal momentum that the πminus can have in the laboratory system- What is the maximum component of laboratory momentum perpendicular to the Λ direction

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Nucleus of 10B transits from excited state with energy E= 072 MeV

to the ground state with emission of a photon with half-life time t12=67x10-10 sec

What is energy uncertainty ΔE of the emitted γ

kinematics of γ-decay is energy and momentum conservation law

recoil of the nucleus pγ=pA (1)

E=Eγ +EA0 +EAkin (2)

E=ΔE= E - EA0

(1) is connected with kinetic energy of the nucleus in the ground state

EAkin =pA22M = pγ

22M and Eγ = pγ

from here Eγ =M( (1+2EM)12 -1) = M(1+EM-1)=E

recoil can be neglected

EAkin =Eγ22M =E 22M =(072 MeV)2(2x1081x931 MeV) =26x10-5 MeV

EAkin Eγ = 26x10-5 MeV072 MeV=36x10-5

ΔEτ= ΔEtln2 lt ħ =gt ΔE lt ħ ln2 t = 0693x658x10-22 MeV x sec 67x10-10 sec=68x10-7ev

bull α-particles with kinetic energy T=65 MeV scatter on the 197Au nucleiDefine a) impact parameter b (at scattering angle θ=90o)

b) (drdt=0) minimal distance rmin between α-particles and 197Au nucleic-d) kinetic T and potential E energy of α-particle at this point

r = rmin производная = 0 = 0

) кинетическая энергия T = T - E = 65 МeV -54 MeV=11 MeV

54 МэВ = 11 МэВ

Nuclear Physics Exercise NP6 Kinematics and Rutherford scattering

bull At CERN Large Hadron Collider there were several poton-proton runs at energies

radics = 900 2360 and 7000 GeV Calculate the main characteristics of these reactions

Ecm Elab β γ maximal rapidity ycm available for protons with mass mp=0938 GeV

and for pions with mπ=0139 GeV Draw y(β ) ndash dependence

s=(E 1+E2 )2 ndash (p1 +p2 )2= (2Ecm )2 if m1=m2 because for cms p1 = - p2

=(Elab +m2)2 - plab2= m1

2+ m22+ 2 Elab m2 = 2m2+ 2mElab

Ecm = radics2 Elab = s(2m) - m p=(E2-m2)12

β = pcmEcm = plab(Elab+m) γ = Ecmm

ymax= 05 ln (E+pl)(E-pl)= 05 ln (E+pl)2(E2-pl

2)= ln (E+pl)mt = ln 2Em

90000 236000 7000004317687848Sqrt s 90000 236000 700000

Ecm 45000 118000 350000

Pcm 449999022 1179999627 3499999874

Elab 4317687848 2968868998 2611940205

Plab 4317687848 2968868998 2611940205

Beta 09999978275 09999996841 09999999641

Betal 09999978275 09999996841 09999999641

Gamma 479744 1257995 3731343

ymax 68663 78304 89176

ymaxpi 87756 973969 1082694

β

lab

cms

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Consider the production of the K+ in the reaction γprarrΚ+ +Λ

- Give the minimal γ-ray energy in the laboratory system where proton is at rest when this reaction can take place

- What will be the minimal γ- energy if the proton is not free but is bound in the nucleus Take into account the Fermi motion with pF=250 MeVc

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull What is the smallest Elab energy of pp interactions for production K-meson with mass

mK=0494 GeV Write also the possible reactions

The reaction p+p-gt p+p+K++K- is possible but the minimal energy is demanded for

p+p-gtp+Λ+K+ where both baryon number and strangeness are conserved

radics gt Σi mi - condition for energy threshold where mi are masses of the produced particles

s=(Elab +m2)2 - plab2= m1

2+ m22+ 2 Elab m2 = 2m2+ 2mElab

So threshold energy

Elab = 12m2 ( (Σi mi )2- m1

2- m22) = ( (Σi mi )

2- 2m2)(2m)= (0939 + 0494 +1116)2- 209392 GeV

= 4734 GeV

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Proton photon and electron (me=0511 MeV) have the same wave length λ = 10-11 m

Which time will it take for each of them to fly the distance L=10 m

λ = (ħc)pc =10-11 m =gt p= (ħc) λc = (200 MeV Fm)(10-11m 10-15 mFm)c=2 106 MeVc

p=mvγ=mvradic(1-v2c2) =gt

v=1 radic(m2p2+1c2) = ħc c ((mc2)2 λ2 +(ħc)2)12

t=Lv= L ((mc2)2 λ2 +(ħc)2)12 ħc c

For proton t = 10 m ((9383 MeV)2 (104 Fm)2 +(197 MeV Fm)2)12(197 MeV Fm x 3 108 msec)

=16 10-3 sec

For electron t = 10 m ((0511 MeV)2 (104 Fm)2 +(197 MeV Fm)2)12(197 MeV Fm x 3 108 msec)=

= 9 10-7 sec

For photon t = Lc= 10 m 3 108 msec =33 10-8 sec

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

The Λ decays in flight into a proton and a πminus meson Λ rarrpπ-

If the Λ has a velocity of 08c - what is the maximal momentum that the πminus can have in the laboratory system- What is the maximum component of laboratory momentum perpendicular to the Λ direction

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Nucleus of 10B transits from excited state with energy E= 072 MeV

to the ground state with emission of a photon with half-life time t12=67x10-10 sec

What is energy uncertainty ΔE of the emitted γ

kinematics of γ-decay is energy and momentum conservation law

recoil of the nucleus pγ=pA (1)

E=Eγ +EA0 +EAkin (2)

E=ΔE= E - EA0

(1) is connected with kinetic energy of the nucleus in the ground state

EAkin =pA22M = pγ

22M and Eγ = pγ

from here Eγ =M( (1+2EM)12 -1) = M(1+EM-1)=E

recoil can be neglected

EAkin =Eγ22M =E 22M =(072 MeV)2(2x1081x931 MeV) =26x10-5 MeV

EAkin Eγ = 26x10-5 MeV072 MeV=36x10-5

ΔEτ= ΔEtln2 lt ħ =gt ΔE lt ħ ln2 t = 0693x658x10-22 MeV x sec 67x10-10 sec=68x10-7ev

bull α-particles with kinetic energy T=65 MeV scatter on the 197Au nucleiDefine a) impact parameter b (at scattering angle θ=90o)

b) (drdt=0) minimal distance rmin between α-particles and 197Au nucleic-d) kinetic T and potential E energy of α-particle at this point

r = rmin производная = 0 = 0

) кинетическая энергия T = T - E = 65 МeV -54 MeV=11 MeV

54 МэВ = 11 МэВ

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Consider the production of the K+ in the reaction γprarrΚ+ +Λ

- Give the minimal γ-ray energy in the laboratory system where proton is at rest when this reaction can take place

- What will be the minimal γ- energy if the proton is not free but is bound in the nucleus Take into account the Fermi motion with pF=250 MeVc

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull What is the smallest Elab energy of pp interactions for production K-meson with mass

mK=0494 GeV Write also the possible reactions

The reaction p+p-gt p+p+K++K- is possible but the minimal energy is demanded for

p+p-gtp+Λ+K+ where both baryon number and strangeness are conserved

radics gt Σi mi - condition for energy threshold where mi are masses of the produced particles

s=(Elab +m2)2 - plab2= m1

2+ m22+ 2 Elab m2 = 2m2+ 2mElab

So threshold energy

Elab = 12m2 ( (Σi mi )2- m1

2- m22) = ( (Σi mi )

2- 2m2)(2m)= (0939 + 0494 +1116)2- 209392 GeV

= 4734 GeV

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Proton photon and electron (me=0511 MeV) have the same wave length λ = 10-11 m

Which time will it take for each of them to fly the distance L=10 m

λ = (ħc)pc =10-11 m =gt p= (ħc) λc = (200 MeV Fm)(10-11m 10-15 mFm)c=2 106 MeVc

p=mvγ=mvradic(1-v2c2) =gt

v=1 radic(m2p2+1c2) = ħc c ((mc2)2 λ2 +(ħc)2)12

t=Lv= L ((mc2)2 λ2 +(ħc)2)12 ħc c

For proton t = 10 m ((9383 MeV)2 (104 Fm)2 +(197 MeV Fm)2)12(197 MeV Fm x 3 108 msec)

=16 10-3 sec

For electron t = 10 m ((0511 MeV)2 (104 Fm)2 +(197 MeV Fm)2)12(197 MeV Fm x 3 108 msec)=

= 9 10-7 sec

For photon t = Lc= 10 m 3 108 msec =33 10-8 sec

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

The Λ decays in flight into a proton and a πminus meson Λ rarrpπ-

If the Λ has a velocity of 08c - what is the maximal momentum that the πminus can have in the laboratory system- What is the maximum component of laboratory momentum perpendicular to the Λ direction

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Nucleus of 10B transits from excited state with energy E= 072 MeV

to the ground state with emission of a photon with half-life time t12=67x10-10 sec

What is energy uncertainty ΔE of the emitted γ

kinematics of γ-decay is energy and momentum conservation law

recoil of the nucleus pγ=pA (1)

E=Eγ +EA0 +EAkin (2)

E=ΔE= E - EA0

(1) is connected with kinetic energy of the nucleus in the ground state

EAkin =pA22M = pγ

22M and Eγ = pγ

from here Eγ =M( (1+2EM)12 -1) = M(1+EM-1)=E

recoil can be neglected

EAkin =Eγ22M =E 22M =(072 MeV)2(2x1081x931 MeV) =26x10-5 MeV

EAkin Eγ = 26x10-5 MeV072 MeV=36x10-5

ΔEτ= ΔEtln2 lt ħ =gt ΔE lt ħ ln2 t = 0693x658x10-22 MeV x sec 67x10-10 sec=68x10-7ev

bull α-particles with kinetic energy T=65 MeV scatter on the 197Au nucleiDefine a) impact parameter b (at scattering angle θ=90o)

b) (drdt=0) minimal distance rmin between α-particles and 197Au nucleic-d) kinetic T and potential E energy of α-particle at this point

r = rmin производная = 0 = 0

) кинетическая энергия T = T - E = 65 МeV -54 MeV=11 MeV

54 МэВ = 11 МэВ

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull What is the smallest Elab energy of pp interactions for production K-meson with mass

mK=0494 GeV Write also the possible reactions

The reaction p+p-gt p+p+K++K- is possible but the minimal energy is demanded for

p+p-gtp+Λ+K+ where both baryon number and strangeness are conserved

radics gt Σi mi - condition for energy threshold where mi are masses of the produced particles

s=(Elab +m2)2 - plab2= m1

2+ m22+ 2 Elab m2 = 2m2+ 2mElab

So threshold energy

Elab = 12m2 ( (Σi mi )2- m1

2- m22) = ( (Σi mi )

2- 2m2)(2m)= (0939 + 0494 +1116)2- 209392 GeV

= 4734 GeV

mp= 0939 GeVc2 mπ= 0139 GeVc2 mK= 0494 GeVc2 mΛ= 1116 GeVc2

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Proton photon and electron (me=0511 MeV) have the same wave length λ = 10-11 m

Which time will it take for each of them to fly the distance L=10 m

λ = (ħc)pc =10-11 m =gt p= (ħc) λc = (200 MeV Fm)(10-11m 10-15 mFm)c=2 106 MeVc

p=mvγ=mvradic(1-v2c2) =gt

v=1 radic(m2p2+1c2) = ħc c ((mc2)2 λ2 +(ħc)2)12

t=Lv= L ((mc2)2 λ2 +(ħc)2)12 ħc c

For proton t = 10 m ((9383 MeV)2 (104 Fm)2 +(197 MeV Fm)2)12(197 MeV Fm x 3 108 msec)

=16 10-3 sec

For electron t = 10 m ((0511 MeV)2 (104 Fm)2 +(197 MeV Fm)2)12(197 MeV Fm x 3 108 msec)=

= 9 10-7 sec

For photon t = Lc= 10 m 3 108 msec =33 10-8 sec

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

The Λ decays in flight into a proton and a πminus meson Λ rarrpπ-

If the Λ has a velocity of 08c - what is the maximal momentum that the πminus can have in the laboratory system- What is the maximum component of laboratory momentum perpendicular to the Λ direction

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Nucleus of 10B transits from excited state with energy E= 072 MeV

to the ground state with emission of a photon with half-life time t12=67x10-10 sec

What is energy uncertainty ΔE of the emitted γ

kinematics of γ-decay is energy and momentum conservation law

recoil of the nucleus pγ=pA (1)

E=Eγ +EA0 +EAkin (2)

E=ΔE= E - EA0

(1) is connected with kinetic energy of the nucleus in the ground state

EAkin =pA22M = pγ

22M and Eγ = pγ

from here Eγ =M( (1+2EM)12 -1) = M(1+EM-1)=E

recoil can be neglected

EAkin =Eγ22M =E 22M =(072 MeV)2(2x1081x931 MeV) =26x10-5 MeV

EAkin Eγ = 26x10-5 MeV072 MeV=36x10-5

ΔEτ= ΔEtln2 lt ħ =gt ΔE lt ħ ln2 t = 0693x658x10-22 MeV x sec 67x10-10 sec=68x10-7ev

bull α-particles with kinetic energy T=65 MeV scatter on the 197Au nucleiDefine a) impact parameter b (at scattering angle θ=90o)

b) (drdt=0) minimal distance rmin between α-particles and 197Au nucleic-d) kinetic T and potential E energy of α-particle at this point

r = rmin производная = 0 = 0

) кинетическая энергия T = T - E = 65 МeV -54 MeV=11 MeV

54 МэВ = 11 МэВ

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Proton photon and electron (me=0511 MeV) have the same wave length λ = 10-11 m

Which time will it take for each of them to fly the distance L=10 m

λ = (ħc)pc =10-11 m =gt p= (ħc) λc = (200 MeV Fm)(10-11m 10-15 mFm)c=2 106 MeVc

p=mvγ=mvradic(1-v2c2) =gt

v=1 radic(m2p2+1c2) = ħc c ((mc2)2 λ2 +(ħc)2)12

t=Lv= L ((mc2)2 λ2 +(ħc)2)12 ħc c

For proton t = 10 m ((9383 MeV)2 (104 Fm)2 +(197 MeV Fm)2)12(197 MeV Fm x 3 108 msec)

=16 10-3 sec

For electron t = 10 m ((0511 MeV)2 (104 Fm)2 +(197 MeV Fm)2)12(197 MeV Fm x 3 108 msec)=

= 9 10-7 sec

For photon t = Lc= 10 m 3 108 msec =33 10-8 sec

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

The Λ decays in flight into a proton and a πminus meson Λ rarrpπ-

If the Λ has a velocity of 08c - what is the maximal momentum that the πminus can have in the laboratory system- What is the maximum component of laboratory momentum perpendicular to the Λ direction

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Nucleus of 10B transits from excited state with energy E= 072 MeV

to the ground state with emission of a photon with half-life time t12=67x10-10 sec

What is energy uncertainty ΔE of the emitted γ

kinematics of γ-decay is energy and momentum conservation law

recoil of the nucleus pγ=pA (1)

E=Eγ +EA0 +EAkin (2)

E=ΔE= E - EA0

(1) is connected with kinetic energy of the nucleus in the ground state

EAkin =pA22M = pγ

22M and Eγ = pγ

from here Eγ =M( (1+2EM)12 -1) = M(1+EM-1)=E

recoil can be neglected

EAkin =Eγ22M =E 22M =(072 MeV)2(2x1081x931 MeV) =26x10-5 MeV

EAkin Eγ = 26x10-5 MeV072 MeV=36x10-5

ΔEτ= ΔEtln2 lt ħ =gt ΔE lt ħ ln2 t = 0693x658x10-22 MeV x sec 67x10-10 sec=68x10-7ev

bull α-particles with kinetic energy T=65 MeV scatter on the 197Au nucleiDefine a) impact parameter b (at scattering angle θ=90o)

b) (drdt=0) minimal distance rmin between α-particles and 197Au nucleic-d) kinetic T and potential E energy of α-particle at this point

r = rmin производная = 0 = 0

) кинетическая энергия T = T - E = 65 МeV -54 MeV=11 MeV

54 МэВ = 11 МэВ

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

The Λ decays in flight into a proton and a πminus meson Λ rarrpπ-

If the Λ has a velocity of 08c - what is the maximal momentum that the πminus can have in the laboratory system- What is the maximum component of laboratory momentum perpendicular to the Λ direction

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Nucleus of 10B transits from excited state with energy E= 072 MeV

to the ground state with emission of a photon with half-life time t12=67x10-10 sec

What is energy uncertainty ΔE of the emitted γ

kinematics of γ-decay is energy and momentum conservation law

recoil of the nucleus pγ=pA (1)

E=Eγ +EA0 +EAkin (2)

E=ΔE= E - EA0

(1) is connected with kinetic energy of the nucleus in the ground state

EAkin =pA22M = pγ

22M and Eγ = pγ

from here Eγ =M( (1+2EM)12 -1) = M(1+EM-1)=E

recoil can be neglected

EAkin =Eγ22M =E 22M =(072 MeV)2(2x1081x931 MeV) =26x10-5 MeV

EAkin Eγ = 26x10-5 MeV072 MeV=36x10-5

ΔEτ= ΔEtln2 lt ħ =gt ΔE lt ħ ln2 t = 0693x658x10-22 MeV x sec 67x10-10 sec=68x10-7ev

bull α-particles with kinetic energy T=65 MeV scatter on the 197Au nucleiDefine a) impact parameter b (at scattering angle θ=90o)

b) (drdt=0) minimal distance rmin between α-particles and 197Au nucleic-d) kinetic T and potential E energy of α-particle at this point

r = rmin производная = 0 = 0

) кинетическая энергия T = T - E = 65 МeV -54 MeV=11 MeV

54 МэВ = 11 МэВ

Nuclear Physics Exercise NP6Kinematics and Rutherford scattering

bull Nucleus of 10B transits from excited state with energy E= 072 MeV

to the ground state with emission of a photon with half-life time t12=67x10-10 sec

What is energy uncertainty ΔE of the emitted γ

kinematics of γ-decay is energy and momentum conservation law

recoil of the nucleus pγ=pA (1)

E=Eγ +EA0 +EAkin (2)

E=ΔE= E - EA0

(1) is connected with kinetic energy of the nucleus in the ground state

EAkin =pA22M = pγ

22M and Eγ = pγ

from here Eγ =M( (1+2EM)12 -1) = M(1+EM-1)=E

recoil can be neglected

EAkin =Eγ22M =E 22M =(072 MeV)2(2x1081x931 MeV) =26x10-5 MeV

EAkin Eγ = 26x10-5 MeV072 MeV=36x10-5

ΔEτ= ΔEtln2 lt ħ =gt ΔE lt ħ ln2 t = 0693x658x10-22 MeV x sec 67x10-10 sec=68x10-7ev

bull α-particles with kinetic energy T=65 MeV scatter on the 197Au nucleiDefine a) impact parameter b (at scattering angle θ=90o)

b) (drdt=0) minimal distance rmin between α-particles and 197Au nucleic-d) kinetic T and potential E energy of α-particle at this point

r = rmin производная = 0 = 0

) кинетическая энергия T = T - E = 65 МeV -54 MeV=11 MeV

54 МэВ = 11 МэВ

bull α-particles with kinetic energy T=65 MeV scatter on the 197Au nucleiDefine a) impact parameter b (at scattering angle θ=90o)

b) (drdt=0) minimal distance rmin between α-particles and 197Au nucleic-d) kinetic T and potential E energy of α-particle at this point

r = rmin производная = 0 = 0

) кинетическая энергия T = T - E = 65 МeV -54 MeV=11 MeV

54 МэВ = 11 МэВ