applications of nuclear physics

Tony Weidberg Nuclear Physics Lectures 1

Applications of Nuclear Physics

• Fusion– How the sun works– Fusion reactor

• Radioactive dating– C dating– Rb/Sr age of the Earth


Fusion in the Sun

• Where nuclear physics meets astrophysics and has a big surprise for particle physics.

• Neutrinos• Heavier Elements

– Up to Fe– Beyond Fe

• Sun by Numbers:L=3.86 1026 WM=1.99 1030 kgR=6.96 108 m


How to power the sun• Try gravity

• Too short!• By elimination must be nuclear fusion.• Energy per particle (nuclei/electron)

• Gives plasma, ionised H and He.

MYrLUt

JR

GMU

3~/

108.3 412

keVM

MUE

S

P 1~)(~


PP Chain

• Very long range weather forecast very cold• But only ~ 10% H atoms converted to He

MeV49.5HeHp)2( 32

21

MeV42.0eHpp)1( e21

MeV86.12ppHeHeHe)3( 42

32

32

MeV02.12ee)4(

MeV55.6)H(E

MeV26.0E


Physics of Nuclear Fusion• All reactions at low energy are suppressed by

Coulomb barrier (cf decay). • Reaction rate: convolution of MB distribution

and barrier penetration (EG= Gamow Energy)

• Problem:) too small to measure! Extrapolated from higher energy or from n scattering.

2

0

2212

42

)exp()0(~)(

c

eZZmcE

E

EE

G

G


Example C


Reaction Rates & Coulomb Barrier• From definition of

• Main contribution around min

)v(vNNR ba

2/3B

1/3G03/2

1/2G

BT)(kEE0

2E

E

Tk

1

dE

dφ

)Tk2

mvexp()

Tk

m()

2()v(P

B

22/3

B

2/1

mvdvdEmv2

1E;dv)v(P)v(v)v(v 2

0

E/ETk/E)E(;dE)]E(exp[)E()v(v GB0


Cross Sections and W.I.• Consider first reaction pp chain

• Cross section small even above Coulomb barrier because this is a weak interaction

• Order of magnitude estimate

• At 1 MeV s=36b; tnuclear~10-23s; tdecay~900s

~10-25b• This reaction is the bottleneck explains long time

scales for nuclear fusion to consume all the H in the core of the sun.

MeV42.0eHpp e21

decay

nuclearS t

t~


Heavier Elements• He to Si:

• 8Be unstable! Resonance in C12 enhances rate.• Heavier elements up to Fe

– Photo-disintegration n,p and . These can be absorbed by other nuclei to build up heavier nuclei up to Fe.

• Fe most stable nucleus, how do we make heavier nuclei?

HeSiOO

OCHe

CBeHe

BeHeHe

4281616

16124

1284

844


Fusion Reactors

• Use deuterium + tritium:

– Large energy release– Large cross-section at low energy– Deuterium abundant (0.015% of H).– Breed Tritium in Lithium blanket– .

MeV62.17nHeHH 42

31

21

MeV8.4HeHLin

nHeHMeV46.2Lin42

31

63

42

31

73


Fusion Reactors

• Energy out > Energy in

• Lawson criteria (assume kBT=20 keV).– number density D ions : – Cross-section: – Confinement time for plasma: tc

– Energy released per fusion: Efusion

cfusion2

out tEvE

TkE Bin c1319

inout t)sm10(~E/E


Magnetic Confinement

• Confine plasma with magnetic fields.– Toroidal field: ions spiral around field

lines.– Poloidal fields: focus ions away from

walls.

• Heating:– RF power accelerates electrons– Current pulse causes further heating.


Jet


Magnetic Confinement Fusion

• JET passed break-even (ie achieved Lawson criteria).


Inertial Confinement Fusion

Very Big Laser

Mirrors

D-T Pellet


Inertial Confinement Fusion


Radioactive Dating

• C14/C12 for organic matter age of dead trees etc.

• Rb/Sr in rocks age of earth.


Carbon Dating

• C14 produced by Cosmic rays (mainly neutrons) at the top of the atmosphere.

• C14 mixes in atmosphere and absorbed by plants/trees constant ratio C14 / C12 . Ratio decreases when plant dies. t1/2=5700 years.

• Either– Rate of C14 radioactive decays– Count C14 atoms in sample by Accelerator Mass

Spectrometer.

• Which is better?• Why won’t this work in the future?


Carbon Dating Calibration


How Old Is The Earth?

• Rb87 Sr87: decay t1/2=4.8 1010 yr

• Assume no initial daughter nuclei get age from ratio of daughter/parent now.

)t(N)t(N)t(N 0p1P1D

)tt(exp()t(N)t(N 010p1P

)t(N

)t(Nln

1t

1p

0p

)t(N

)t(N1ln

1t

1p

1D


Improved Calculation• Allow for initial daughters to be present.• Need another isotope of the daughter D’ which is stable

and not a product of a radioactive decay chain. • Plot vs straight line fit age and initial ratio.

)t(N)t(N)t(N)t(N 0p0D1P1D

)t(N

)t(N

1D

1D

)t(N

)t(N

1D

1P

)t(N

)t(N)t(N

)t(N

)t(N)t(N

0D

0p0D

1D

1P1D

)t(N

)t(N]1)t[exp(

)t(N

)t(N

)t(N

)t(N

0D

0D

1D

1P

1D

1D


Age of Earth

• Rb/Sr method• Stable isotope of

daughter is Sr86

• Fit gives age of earth=4.53 109 years. S

r87/

Sr8

6

Rb87/Sr86

1.0 4.0


Cross-Sections

• Why concept is important– Learn about dynamics of interaction and/or

constituents (cf Feynman’s watches).– Needed for practical calculations.

• Experimental Definition• How to calculate

– Fermi Golden Rule– Breit-Wigner Resonances– QM calculation of Rutherford Scattering


Definition of • a+bx

• Effective area or reaction to occur is

Beam a

dx

Na

Na(0) particles type a/unit time hit target b

Nb atoms b/unit volume

Number /unit area= Nb dx

Probability interaction = Nbdx

dNa=-Na Nb dx

Na(x)=Na(0) exp(-x/) ; =1/(Nb )


Reaction Rates• Na beam particles/unit volume, speed v

• Flux F= Na v

• Rate/target b atom R=F• Thin target x<<: R=(Na

T) F Total

• This is total cross section. Can also define differential cross sections, as a function of reaction product, energy, transverse momentum, angle etc.

• dR(a+bc+d)/dE=(NaT) F d(a+bc+d) /dE


Cross Section Calculations

• Use NRQM to calculate cross sections:

• Calculation (blackboard) gives Breit-Wigner resonance for decay of excited state

nn0nn )/tiEexp()t(a)t(;H

dti

)EE(PH2

)t(a

4/)EE(

H)t(a

nm2

mn2

n

22nm

2mn2

n

4)EE(

1

2)EE(P

22nm

nm


Breit-Wigner Resonance

• Important in atomic, nuclear and particle physics.

• Uncertainty relationship

• Determine lifetimes of states from width.

• t=1/=FWHM;

~tE


Fermi Golden Rule• Decays to a channel i (range of states n).

Density of states ni(E). Assume narrow resonance

dE)EE(P)E(nH2

P 0i2

0ii

)E(nH2

P 0i2

0ii

TotaliiTotali

i RPR;R;P

)E(nH2

R 0i2

0ii

i


Cross Section

• Breit Wigner cross section.

• Definition of and flux F:

v

k4

)2(

V)E(n;v

dk

dE;

m2

)k(E

k4)2(

V)k(n

vVF

)r.kiexp(V

FR

2

3

2

23

1

2/1


Breit-Wigner Cross Section

• Combine rate, flux & density states

4/)EE(

E

)E(n

1

2

1R

)E(nH2)E(

4/)EE(

H)t(aR

2201

f1i

210i

f22

01

201f2

o

4/)EE(

E

2

1

k4V

v)2(

v

V22

01

f1i2

3


Breit-Wigner Cross Section

4/)EE(k 2201

fi2

n + 16O 17O


Low Energy Resonances

• n + Cd total cross section.

• Cross section scales ~ 1/E1/2 at low E.

• B-W: 1/k2 and ~n(E)~k


Rutherford Scattering 1

cosddrrr

)cosiqrexp(2ZZVH

rdr

)r.qiexp(ZZVH

kkq

rd)r.kiexp(r

ZZ)r.kiexp(VH

)r.kiexp(V;)r.kiexp(V

1c;c4

e;

r

ZZ)r(V

221

1fi

321

1fi

fi

3f

21i

1fi

f2/1

fi2/1

i

0

221


Rutherford Scattering 2

2211

fi

22211

fi

211fi

211fi

2221

1fi

q

4ZZVH

q)a/1(

iq2

iq

2ZZVH

iqa/1

1

iqa/1

1

iq

2ZZVH

dr)iqa/1exp(r)iqa/1exp(iq

2ZZVH

a)a/rexp();r(xV

drriqr

)iqrexp()iqrexp(2ZZVH


Rutherford Scattering 3• Use Fermi Golden Rule:

f

2fi dE

dnH

2R

)2/(sinp4)cos1(p2)pp(q

qv)4(

)ZZ(p4

d

d

v

V

)2(v

Vp

Vq

4ZZ2

d

d

vVF;F/R

d)2(v

Vp)E(n

v/1dE

dp;

dE

dp

dp

dn

dE

dn;

4

d

h

Vp4

dp

dn

2222fi

2

422

221

2

3

22

221

1

3

2

32


Low Energy Experiment• Scattering of on Au & Ag agree with calculation

assuming point nucleus

Sin4(/2)

dN

/dco

s


Higher Energy

• Deviation from Rutherford scattering at higher energy determine charge distribution in the nucleus.

• Form factors is F.T. of charge distribution.

applications of nuclear physics

Documents

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