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Nuclear Physics
An introduction


  Brief history
  Binding energy
  Semi empirical mass formula or the Liquid drop
  model
  Radioactivity
  Nuclear energy & some applications
Why Study Nuclear Physics?
   To understand origin of different nuclei
    ◦ Big bang: H, He and Li
    ◦ Stars: elements up to Fe
    ◦ Supernova: heavy elements
 We are all made of stardust
 Applications are plenty
    ◦   Energy (Fission, fusion, transmutation)
    ◦   Medicine (Radiotherapy, MRI)
    ◦   Instrumentation (e.g. spectroscopy)
    ◦   Devices (e.g. Smoke detector)
    ◦   Radioactive dating
Brief history
   1896 Becquerel - radioactivity
   1897 Thomson - electron
   1898 Curies – radium
   1911 Rutherford – nucleus
   1932 Chadwick - neutron
Dimensions
Basics
   The number of protons inside the nucleus is designated by
    Z and is known as the Atomic Number
   The number of neutrons inside the nucleus is designated
    by N and is known as the Neutron Number
   The mass number, A, is the sum of the atomic number and
    the neutron number A = Z + N
   The mass number is an integer and is only approximately
    equal to the atomic weight of a element
   A nuclide is a single nuclear species having a specific Z
                                                       A
                                                       Z EN
    and N. The notation that is used to designate the nuclides
    is
   Nuclei with same Z, but differing N  Isotopes
   Nuclei with same N, but differing Z  Isotones
   Nuclei with same A  Isobars
Basic properties
   Size
    ◦ Most nuclei are nearly spherical, with the radius being
      given by1/3 fm
      R 1.2 A


   Density
    ◦ The nucleus has approximately constant density ~ 1017
      kg/m3
   Binding energy
    ◦ When you measure the mass of an atom you find that it
      is less than the sum of its parts
              BE Z M H N M N M ( A, Z ) c2
    ◦ The difference is known as the binding energy and is
      given by
Nuclear binding energy
Models of the nucleus

   No fundamental theory that can explain all observed
    properties of the nucleus exists
   Several models developed to explain some of the
    observed properties
   Liquid Drop Model–Nucleons are treated as
    molecules in a liquid
   Shell Model–Similar to central field approximation in
    atomic structure
Liquid drop model
Bethe-Wiezsacker mass formula (1935)
Assumptions
  Each nucleon in interacting solely with its nearest neighbours
       Equivalent to atoms in a solid or molecules in liquid which
  move         freely while maintaining fixed intermolecular
  distance
       Vibrations in solid would be too high for stability
   Nucleus ~ charged liquid drop

We may consider different effects term-wise

Volume term
  Bulk binding energy    volume      EV    aV A
    Ev     R3
         = (r0 A1/3)3
Surface term
   Surface area = 4 r 2         4 (r0 A 1/ 3 )2        4 r02 A 2/ 3

    Surface energy       aS A 2/ 3


Coulomb term
 The work done to bring together Z protons from infinity
            e
    V
          4 0r
 For Z ( Z 1) / 2 pairs of protons
          Z ( Z 1)      Z ( Z 1)e2 1
    EC             V
               2            8 0    r              AV


             1/ 3             Z ( Z 1)
     r   A          EC     aC
                                 A1/ 3
Asymmetry term
                                Neutron and proton states with
 Neutrons          Protons     same spacing .

                                Crosses represent initially
                               occupied states in ground state.

                                If three protons were turned into
                               neutrons the extra energy required
                               would be 3 3 .

                                In general if there are N Z
                               excess protons over neutrons the
                               extra energy is [(N Z)/2]2 .
                               relative to Z = N.
              (N        Z )2
E Asym   aa
                    A                1/A
Pairing term
   Like Cooper pair formation, the nucleons also can pair
   Some energy is spent in binding the pairs
   BE(Nucleus with paired nucleons)
       > BE(Nucleus with unpaired nucleons)
                            BE (even-Z , odd-N )
    BE (even-Z , even-N )                              BE (odd-Z , odd-N )
                            BE (odd-Z , even-N )

= +ve                              0                     -ve

   Its observed that this effect smaller for larger A
   Phenomenological fit to A dependence  EPair               1/A1/3


                              E Pair   ap       1/ 3
                                            A
e=even    o=odd
                                                               + 33.5 MeV (e-e)
                                                           ap= 0 MeV (o-e or e-o)
   av=14.1 MeV       ac=0.595 MeV                               - 33.5 MeV (o-o)

                                            2                2
                      2
                           3
                                        Z           (N Z )
EBind   av A as A                  ac       1
                                                 aa              ap    1/ 3
                                        A    3        A               A
          as=13.0 MeV                       aa=19.0 MeV



          BE ( N , Z )
                                        Constraint for most stable isotope
              N        Z       Const.
#include<stdio.h>
#include<math.h>
#include<string.h>
FILE *fout1;
main()
{
int iA,iZ;
float A,Z,del;
float VEP,SEP,CEP,AEP,PEP,BEP;
float av=14.1,as=13.0,ac=0.595,aa=19.0,ap=33.5;
fout1=fopen("BEP.OUT","w");
fprintf(fout1,"       Z        A        VEP        SEP      CEP       AEP       PEP       BEP");
for (iA=1;iA<=300;iA++)
 {A=(float)(iA);
  Z=0.5*A/(1.0+pow(A,2.0/3.0)*ac/(4.0*aa));
  iZ=(int)(Z);
  Z=(float)(iZ);
  printf("n%f    %f",Z,A);
  VEP=av;
  SEP=-as/pow(A,1.0/3.0);
  CEP=-ac*Z*Z/pow(A,4.0/3.0);
  AEP=-aa*pow((A-2*Z)/A,2);
  if(iA%2 != 0)
    del=0;
  else
  { if(iZ%2 != 0)
       del=-1;
    else
       del=1;
  }
  PEP=ap*del/pow(A,4.0/3.0);
  BEP=VEP+SEP+CEP+AEP+PEP;

fprintf(fout1,"n%10.4f%10.4f%10.4f%10.4f%10.4f%10.4f%10.4f%10.4f",Z,A,VEP,SEP,CEP,AEP,PEP,BEP);
 }
}
15




             10



                            Volume          Surface
BE/A (MeV)




              5
                            Coulomb         Asymmetry
                            Pairing         Total
              0




              -5




             -10
                   0   50   100       150       200     250

                                  A
 nuclear physics
200

          180

          160

          140

          120
Protons




                            N=Z
          100

           80

           60
                                    beta stability

           40

           20

            0
                0   50        100             150

                         Neutrons
SHE – discovery in nuclear labs
The Chart of Nuclides
Present scenario




          2900 nuclei till year 2000
          3090 till August 2008
          3000 more to be discovered
Classification of Decays
                                        -decay:
                                    •   emission of Helium nucleus
                                    •   ZZ-2
Protons




                                    •   NN-2
                                    •   AA-4
                 EC                     --decay
                                    •   emission of e- and
                                    •   ZZ+1
                                    •   NN-1
                                    •   A=const
                                        +-decay
                                    • emission of e+ and
                                    • ZZ-1
                                    • NN+1
                  Neutrons          • A=const
                                    Electron Capture (EC)
                                    • absorbtion of e- and emiss
                 -decay             • ZZ-1
                • emission of       • NN+1
                • Z,N,A all const   • A=const         21
Spin                        1 1                        3
       S    s( s 1)            1                       
                            2 2                       2
             1
       ms
             2

Magnetic Moment
                                           e                        27
   Nuclear magneton          N                      5.051 10              J/T
                                          2m p
                                                                               
  Proton      pz        2.793     N                    pz has same direction as S
                                                                        
   Neutron         nz    1.913       N                nz is opposite to S

  Magnetic energy U m                      z   B,    E       2   z   B

                          Nuclear Zeeman effect
Practical Applications

   Nuclear fission for energy generation.
    ◦ No greenhouse gasses
    ◦ Safety and storage of radioactive material.

   Nuclear fusion
    ◦ No safety issue (not a bomb)
    ◦ Less radioactive material but still some technical
      difficulties.

   Nuclear transmutation of radioactive waste
    with neutrons.
    ◦ Turn long lived isotopes  stable or short lived.
Medical Applications

   Radiotherapy for cancer
    ◦ Kill cancer cells.
    ◦ Used for 100 years but can be improved by better
      delivery and dosimetery
    ◦ Heavy ion beams can give more localised energy
      deposition.


   Medical Imaging
    ◦ MRI (Nuclear magnetic resonance)
    ◦ X-rays (better detectors  lower doses)
    ◦ Many others…
Other Applications

   Radioactive Dating
    ◦ C14/C12 gives ages for dead
      plants/animals/people.
    ◦ Rb/Sr gives age of earth as 4.5 Gyr.

   Element analysis
    ◦ Forenesic (eg date As in hair).
    ◦ Biology (eg elements in blood cells)
    ◦ Archaeology (eg provenance via isotope
      ratios).
Carbon Dating
   C14 produced by Cosmic rays (mainly
    neutrons) at the top of the atmosphere.
    ◦ n N14  p C14
 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?

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nuclear physics

  • 1. Nuclear Physics An introduction Brief history Binding energy Semi empirical mass formula or the Liquid drop model Radioactivity Nuclear energy & some applications
  • 2. Why Study Nuclear Physics?  To understand origin of different nuclei ◦ Big bang: H, He and Li ◦ Stars: elements up to Fe ◦ Supernova: heavy elements  We are all made of stardust  Applications are plenty ◦ Energy (Fission, fusion, transmutation) ◦ Medicine (Radiotherapy, MRI) ◦ Instrumentation (e.g. spectroscopy) ◦ Devices (e.g. Smoke detector) ◦ Radioactive dating
  • 3. Brief history  1896 Becquerel - radioactivity  1897 Thomson - electron  1898 Curies – radium  1911 Rutherford – nucleus  1932 Chadwick - neutron
  • 5. Basics  The number of protons inside the nucleus is designated by Z and is known as the Atomic Number  The number of neutrons inside the nucleus is designated by N and is known as the Neutron Number  The mass number, A, is the sum of the atomic number and the neutron number A = Z + N  The mass number is an integer and is only approximately equal to the atomic weight of a element  A nuclide is a single nuclear species having a specific Z A Z EN and N. The notation that is used to designate the nuclides is  Nuclei with same Z, but differing N  Isotopes  Nuclei with same N, but differing Z  Isotones  Nuclei with same A  Isobars
  • 6. Basic properties  Size ◦ Most nuclei are nearly spherical, with the radius being given by1/3 fm R 1.2 A  Density ◦ The nucleus has approximately constant density ~ 1017 kg/m3  Binding energy ◦ When you measure the mass of an atom you find that it is less than the sum of its parts BE Z M H N M N M ( A, Z ) c2 ◦ The difference is known as the binding energy and is given by
  • 8. Models of the nucleus  No fundamental theory that can explain all observed properties of the nucleus exists  Several models developed to explain some of the observed properties  Liquid Drop Model–Nucleons are treated as molecules in a liquid  Shell Model–Similar to central field approximation in atomic structure
  • 9. Liquid drop model Bethe-Wiezsacker mass formula (1935) Assumptions Each nucleon in interacting solely with its nearest neighbours Equivalent to atoms in a solid or molecules in liquid which move freely while maintaining fixed intermolecular distance Vibrations in solid would be too high for stability  Nucleus ~ charged liquid drop We may consider different effects term-wise Volume term Bulk binding energy volume EV aV A Ev R3 = (r0 A1/3)3
  • 10. Surface term Surface area = 4 r 2 4 (r0 A 1/ 3 )2 4 r02 A 2/ 3 Surface energy aS A 2/ 3 Coulomb term The work done to bring together Z protons from infinity e V 4 0r For Z ( Z 1) / 2 pairs of protons Z ( Z 1) Z ( Z 1)e2 1 EC V 2 8 0 r AV 1/ 3 Z ( Z 1) r A EC aC A1/ 3
  • 11. Asymmetry term  Neutron and proton states with Neutrons Protons same spacing .  Crosses represent initially occupied states in ground state.  If three protons were turned into neutrons the extra energy required would be 3 3 .  In general if there are N Z excess protons over neutrons the extra energy is [(N Z)/2]2 . relative to Z = N. (N Z )2 E Asym aa A  1/A
  • 12. Pairing term  Like Cooper pair formation, the nucleons also can pair  Some energy is spent in binding the pairs  BE(Nucleus with paired nucleons) > BE(Nucleus with unpaired nucleons) BE (even-Z , odd-N ) BE (even-Z , even-N ) BE (odd-Z , odd-N ) BE (odd-Z , even-N ) = +ve 0 -ve  Its observed that this effect smaller for larger A  Phenomenological fit to A dependence  EPair 1/A1/3 E Pair ap 1/ 3 A
  • 13. e=even o=odd + 33.5 MeV (e-e) ap= 0 MeV (o-e or e-o) av=14.1 MeV ac=0.595 MeV - 33.5 MeV (o-o) 2 2 2 3 Z (N Z ) EBind av A as A ac 1 aa ap 1/ 3 A 3 A A as=13.0 MeV aa=19.0 MeV BE ( N , Z ) Constraint for most stable isotope N Z Const.
  • 14. #include<stdio.h> #include<math.h> #include<string.h> FILE *fout1; main() { int iA,iZ; float A,Z,del; float VEP,SEP,CEP,AEP,PEP,BEP; float av=14.1,as=13.0,ac=0.595,aa=19.0,ap=33.5; fout1=fopen("BEP.OUT","w"); fprintf(fout1," Z A VEP SEP CEP AEP PEP BEP"); for (iA=1;iA<=300;iA++) {A=(float)(iA); Z=0.5*A/(1.0+pow(A,2.0/3.0)*ac/(4.0*aa)); iZ=(int)(Z); Z=(float)(iZ); printf("n%f %f",Z,A); VEP=av; SEP=-as/pow(A,1.0/3.0); CEP=-ac*Z*Z/pow(A,4.0/3.0); AEP=-aa*pow((A-2*Z)/A,2); if(iA%2 != 0) del=0; else { if(iZ%2 != 0) del=-1; else del=1; } PEP=ap*del/pow(A,4.0/3.0); BEP=VEP+SEP+CEP+AEP+PEP; fprintf(fout1,"n%10.4f%10.4f%10.4f%10.4f%10.4f%10.4f%10.4f%10.4f",Z,A,VEP,SEP,CEP,AEP,PEP,BEP); } }
  • 15. 15 10 Volume Surface BE/A (MeV) 5 Coulomb Asymmetry Pairing Total 0 -5 -10 0 50 100 150 200 250 A
  • 17. 200 180 160 140 120 Protons N=Z 100 80 60 beta stability 40 20 0 0 50 100 150 Neutrons
  • 18. SHE – discovery in nuclear labs
  • 19. The Chart of Nuclides
  • 20. Present scenario 2900 nuclei till year 2000 3090 till August 2008 3000 more to be discovered
  • 21. Classification of Decays -decay: • emission of Helium nucleus • ZZ-2 Protons • NN-2 • AA-4 EC --decay • emission of e- and • ZZ+1 • NN-1 • A=const +-decay • emission of e+ and • ZZ-1 • NN+1 Neutrons • A=const Electron Capture (EC) • absorbtion of e- and emiss -decay • ZZ-1 • emission of • NN+1 • Z,N,A all const • A=const 21
  • 22. Spin 1 1 3 S s( s 1)  1   2 2 2 1 ms 2 Magnetic Moment e 27 Nuclear magneton N 5.051 10 J/T 2m p   Proton pz 2.793 N pz has same direction as S   Neutron nz  1.913 N nz is opposite to S Magnetic energy U m z B, E 2 z B Nuclear Zeeman effect
  • 23. Practical Applications  Nuclear fission for energy generation. ◦ No greenhouse gasses ◦ Safety and storage of radioactive material.  Nuclear fusion ◦ No safety issue (not a bomb) ◦ Less radioactive material but still some technical difficulties.  Nuclear transmutation of radioactive waste with neutrons. ◦ Turn long lived isotopes  stable or short lived.
  • 24. Medical Applications  Radiotherapy for cancer ◦ Kill cancer cells. ◦ Used for 100 years but can be improved by better delivery and dosimetery ◦ Heavy ion beams can give more localised energy deposition.  Medical Imaging ◦ MRI (Nuclear magnetic resonance) ◦ X-rays (better detectors  lower doses) ◦ Many others…
  • 25. Other Applications  Radioactive Dating ◦ C14/C12 gives ages for dead plants/animals/people. ◦ Rb/Sr gives age of earth as 4.5 Gyr.  Element analysis ◦ Forenesic (eg date As in hair). ◦ Biology (eg elements in blood cells) ◦ Archaeology (eg provenance via isotope ratios).
  • 26. Carbon Dating  C14 produced by Cosmic rays (mainly neutrons) at the top of the atmosphere. ◦ n N14  p C14  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?