Advanced nuclear physics [compatibility mode]

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Advanced Nuclear Physics by imran aziz

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Advanced nuclear physics [compatibility mode]

  1. 1. Advanced Nuclear Physics MOHAMMAD IMRAN AZIZ Assistant Professor PHYSICS DEPARTMENT SHIBLI NATIONAL COLLEGE, AZAMGARH (India). pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  2. 2. Nuclides The total number of protons in the nucleus of an atom is called the atomic number of the atom and is given the symbol Z. The number of electrons in an electrically-neutral atom is the same as the number of protons in the nucleus. The number of neutrons in a nucleus is known as the neutron number and is given the symbol N. The mass number of the nucleus is the total number of nucleons, that is, protons and neutrons in the nucleus. The mass number is given the symbol A and can be found by the equation Z + N = A. . pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  3. 3. why electrons cannot exist inside a nucleus: Nuclear constituents and their properties This can be explained mathematically also using Heisenberg's uncertainty principle as follows pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  4. 4. Confinement Calculation Electtron is not found in the nucleus, it means we are talking about free electron. That is free electron does not exist.n ....> p + e, here this is beta deacy and electron that becomes free is emitted out of the nucleus as free electron as can not exist due to Heisenberg uncertainty. pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  5. 5. It is common practice to represent the total angular momentum of a Nuclear Spin nucleus by the symbol I and to call it "nuclear spin". For electrons in atoms we make a clear distinction between electron spin and electron orbital angular momentum, and then combine them to give the total angular momentum. But nuclei often act as if they are a single entity with intrinsic angular momentum I. Associated with each nuclear spin is a nuclear magnetic moment which produces magnetic interactions with its environment. The nuclear spins for individual protons and neutrons parallels the treatment of electron spin, with spin 1/2 and an associated magnetic moment. The magnetic moment is much smaller than that of the electron. For the combination neutrons and protons into nuclei, the situation is more complicated. pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  6. 6. • A characteristic of the collection of protons and neutrons (which are fermions) is that a nucleus of odd mass number A will have a half-integer spin and a nucleus of even A will have integer spin. The suggestion that the angular momenta of nucleons tend to form pairs is supported by the fact that all nuclei with even Z and even N have nuclear spin I=0. The half-integer spins of the odd-A nuclides suggests that this is the nuclear spin contributed by the odd neutron pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  7. 7. Nuclear Magnetic Moments • Associated with each nuclear spin is a magnetic moment which is associated with the angular momentum of the nucleus. It is common practice to express these magnetic moments in terms of the nuclear spin in a manner parallel to the treatment of the magnetic moments of electron spin and electron orbital angular momentum. • For the electron spin and orbital cases, the magnetic moments are expressed in terms of a unit called a Bohr magneton which arises naturally in the treatment of quantized angular momentum pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  8. 8. Electric Quadrupole Moments of Nuclei The nuclear electric quadrupole moment is a parameter which describes the effective shape of the ellipsoid of nuclear charge distribution. A non-zero quadrupole moment Q indicates that the charge distribution is not spherically symmetric. By convention, the value of Q is taken to be positive if the ellipsoid is prolate and negative if it is oblate. The quantity Q0 is the classical form of the calculation represents the departure from spherical symmetry in the rest frame of the nucleus. pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  9. 9. Generally, the measured quantity is proportional to the z-component of the magnetic moment (the component along the experimentally determined direction such as the direction of an applied magnetic field, etc. ). In this treatment, the use of a "gyromagnetic ratio" or "g-factor" is introduced. The g- factor for orbital is just gL = 1, but the electron spin g-factor is approximately gS = 2 pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  10. 10. For free protons and neutrons with spin I =1/2, the magnetic moments are of the form where Proton: g = 5.5856912 +/- 0.0000022 Neutron: g = -3.8260837 +/- 0.0000018 The proton g-factor is far from the gS = 2 for the electron, and even the uncharged neutron has a sizable magnetic moment! For the neutron, this suggests that there is internal structure involving the movement of charged particles, even though the net charge of the neutron is zero. If g=2 were an expected value for the proton and g=0 were expected for the neutron, then it was noted by early researchers that the the proton g-factor is 3.6 units above its expected value and the neutron value is 3.8 units below its expected value. This approximate symmetry was used in trial models of the magnetic moment, and in retrospect is taken as an indication of the internal structure of quarks in the standard model of the proton and neutron pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  11. 11. Note that the maximum effective magnetic moment of a nucleus in nuclear magnetons will be the g-factor multiplied by the nuclear spin. For a proton with g = 5.5857 the quoted magnetic moment is m = 2.7928 nuclear magnetons. Nuclear Magnetic moment Nuclide spin I m in mN n 1/2 -1.9130418 p 1/2 +2.7928456 2H(D) 1 +0.8574376 17O 5/2 -1.89279 57Fe 1/2 +0.09062293 57Co 7/2 +4.733 93Nb 9/2 +6.1705 pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  12. 12. Nuclei • Parameters of nuclei • Strong Interaction • Binding Energy • Stable and Unstable Nuclei • Liquid-Drop Model Numerous Applications: • nuclear power • applications in medicine, biology and chemistry • evolution of stars and the Universe • nuclear weapons pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  13. 13. Size of Nuclei and Rutherford Scattering 1/3 R  R0 A - depends weakly on A (number of nucleons in the nucleus) R0  1.2 1015 m  1.2 fm Geiger, Marsden, 238 1/3 Rutherford,1910 R 92 U   1.2 fm   238   7.4 fm Scattering pattern was consistent with that expected for -particles: bare He nuclei scattering of  particles by pointlike objects having a charge of +79e (the charge of the gold nucleus). This allowed Rutherford to put an upper limit on the size of the nucleus (<310-14m for gold). Calculations were strictly classical. However, because of the Coulomb interaction between alpha-particles and nucleus, the result miraculously coincides with the exact quantum-mechanical one (recall the success of the Bohr model for atoms). To measure the size of a nucleus, one has to use more energetic  particles (or electrons, which are more R – the fitting commonly used these days) that get parameter pdfMachine close enough to get inside the nucleus. A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  14. 14. Nuclear Mass A 12 X C number of nucleons in the nucleus chemical symbol 6 Z (mass number of the nucleus) for the element number of protons in the nucleus 6 protons + 6 neutrons N  A Z (atomic number of the element) The neutron number: Atomic Mass (the mass of a neutral atom): M  Z  m p  N  mn  U nucleus / c 2  Z  me  U el .nucl . / c 2 U nucleus ,U el .nucl .  0 - attraction between nucleons in the nucleus and between electrons and the nucleus U el .nucl .  U nucleus  mp c 2 - we can neglect Uel.-nucl. and introduce a convenient mass unit: M 12 C 1u  6  1.66054 1027 kg  931.49MeV / c 2 12 Mass Unit m p  proton mass  1.007277u mn  neutron mass  1.008665u m  electron mass  0.000549u pdfMachine e A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  15. 15. Nuclear Density MA  1au  1au 3 3 3 3 3 3    R  R0 A  R0 4 4 4 1.66 1027 kg  2.4 1017 kg / m3 3 15 3   1.2 10 m  4 ! The density of neutron stars is comparable with that of nuclei. (Unstable) isotopes of tin and zinc. pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  16. 16. The Need for a “Strong Force” Which interaction controls the size of nucleons? This cannot be electromagnetic interaction: protons have the same electric charge (they would repel each other) and also there are attractive forces between protons and electrically neutral neutrons. Strong Interaction: binds protons to protons, neutrons to neutrons, and protons to neutrons with roughly the same force does not affect certain other kinds of particles (specifically electrons) is short-ranged (the range ~ 2 fm). Nucleons separated by a larger distance exert no strong forces on each other. These observations are explained by the quark model of nucleons. g Nucleons are the combination of quarks that are strong-interaction- u neutral (like an electrically-neutral atom). Two nucleons interact only if they are close enough that the distances between various pairs of d quarks are significantly different. u quarks and gluons ~ 10-15 m pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  17. 17. Structure of Matter g u El.-mag. interaction u d determines the size of quarks atoms and gluons ~ 10-15 m ~ 10-10 - 10-9 m Protons & Neutrons (nucleons) are almost an empty space (the Atom is almost an quark size is <10-18 m) empty space (the nuclear volume is ~10- 15 of the atomic volume) Strong interaction determines the size of nuclei and nucleons pdfMachine A pdf writer that produces quality PDF files with -15 ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across ~ 10 - 10 -14m nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  18. 18. Binding Energy Binding energy: Esys  EB  E parts EB  E parts  Esys -the binding energy is positive for a bound system Recall a H atom: the binding energy is 13.6 eV (the ground state energy with sign “minus”). n n p p add EB n n E parts   mi  c 2 Esys p We can compute the binding energy if we know masses of a system and its constituents: EB   mass of all particles consituting the atom    mass of the atom    c 2     Z  m1 H  N  mn  M A X   c 2 EB  M  c 2  1 Z  mass deficit pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  19. 19. Binding Energy curve Because of the short-range character of strong interaction (basically, between nearest and next-to- nearest neighbors), the interaction energy per nucleon with increasing Z saturates at the level ~ (Z/2)(# of neighbors). Binding energy per nucleon (EB/A), MeV increasing Z is caused by the long-range Coulomb B   Z 2 The decrease of the binding energy with repulsion of protons: Mass number, A The binding energy ~ 10MeV/nucleon is ~1% of the nucleon’s rest energy: we can pdfMachine consider the nucleus as a system of individual nucleons A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  20. 20. Liquid-Drop Model 2 2 A “semi-classical” model of the EB  aV A  as A2/3  ac Z  aa  A  2Z  A1/3 A nucleus: describes reasonably well the dependence EB(A): dq  r  q r  - charge density q r dq  r   4 3 2     r    4 r  dr R0 3 U  R0      0 4 0 r pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  21. 21. Liquid-Drop Model (cont’d) pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  22. 22. Limitations of Liquid-Drop Model Maria Goeppert-Mayer, J.H.D.Jensen pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  23. 23. Stable Nuclei Isotopes: all nuclei that have the same number of protons (Z) but different number of neutrons (N). Since the chemical properties of an atom are determined by the number of its electrons, isotopes of the same element have almost identical chemical properties. Example: naturally occurring isotopes of oxygen 16 17 19 8 0 8 0 8 0 Related questions: - What makes unstable nuclei unstable? - What are the mechanisms by which they transform themselves into stable nuclei? -Why do light stable nuclei tend to have N  Z? - Why do heavier nuclei tend to have more neutrons than protons? - Why are there no stable nuclei with Z>83? pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  24. 24. What makes unstable nuclei unstable? Both protons - and neutrons are energy fermions (they obey the exclusion principle). Nuclei are two- 0 r component Fermi systems. Each nuclear energy level can contain four particles: two protons (s=½) 12 and two neutrons (s=½). 6 C The potential experienced by nucleons is a 3D potential well. The ground-state configuration of the carbon-16 nucleus : If a nucleus is allowed to decrease its energy by transforming “excessive” protons (neutrons) into neutrons (protons), it will do it! protons neutrons The processes responsible for these transformations are driven by weak interaction (the fourth fundamental interaction): The weak interaction (unlike the strong interaction) affects both quarks and leptons, (unlike the el.-mag. interaction) can affect electrically neutral particles, and (unlike gravity) does not affect photons. The effective range of the weak interaction is ~ 10-18m. Some important transformation processes driven by weak interaction: n  e  p   pdfMachine p   n  e  A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in p   e  n  Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  25. 25. Why N  Z for light nuclei 13 13 8 O energy 7 N energy If the electrostatic repulsion t1/2  8.9ms of protons can be neglected (this is the case of light nuclei: recall that the positive electrostatic energy Z2), the nucleus tends to keep approximately equal numbers of protons and neutrons. protons neutrons protons neutrons 14 14 6 C energy 7 N energy Even in this case, the nucleus t1/2  5730 y can still lower its total energy: the rest energy of neutron is slightly more than the rest energy of a proton and an electron. pdfMachine A pdf writer that produces quality PDF files with ease! protons neutrons protons aziz_muhd33@yahoo.co.in neutrons Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  26. 26. Why N > Z for heavy nuclei 44 44 22 Ti energy 20 Ca energy In the heavy nuclei, the electrostatic energy cannot be neglected. As a result, the protons’ energy levels are “pushed up” with respect to the neutrons’ levels. In the “otherwise stable” 44Ti, two protons undergo the transformation into neutrons, the end product is stable 44Ca. . . . . . . . . . . protons neutrons protons neutrons The proton-neutron disbalance becomes more pronounced with increasing Z. pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  27. 27. Nuclear Masses and Sizes • Masses and binding energies – Absolute values measured with mass spectrometers. – Relative values from reactions and decays. • Nuclear Sizes – Measured with scattering experiments (leave discussion until after we have looked at Rutherford scattering). – Isotope shifts pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  28. 28. Nuclear Mass Measurements • Measure relative masses by energy released in decays or reactions. – X  Y +Z + DE – Mass difference between X and Y+Z is DE/c2. • Absolute mass by mass spectrometers (next transparency). • Mass and Binding energy: • B = [Z MH + N Mn – M(A,Z)]/c2 pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  29. 29. Mass Spectrometer • Ion Source • Velocity selector  Detector electric and magnetic forces equal and opposite Velocity – qE=qvB  v=E/B Ion Source selector • Momentum selector, circular orbit satisfies: – Mv=qBr – Measurement r gives M. pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  30. 30. Binding Energy vs A • B increases with A up to 56Fe and then slowly decreases. Why? • Lower values and not smooth at small A. pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  31. 31. Nuclear Sizes & Isotope Shift • Coulomb field modified by finite size of nucleus. • Assume a uniform charge distribution in the nucleus. Gauss’s law  r 3Ze integrate and apply boundary conditions E ( ) 2 R 4 0 r • Difference between actual potential and Coulomb Zer 2 3Ze V (r )   3 8 0 R 8 0 R  • Use 1st order perturbation theory Zer 2 3Ze Ze V ( r )   3 (r  R ) 8 0 R 8 0 R 4 0r   R Z 3/ 2 Z 2 * E   4r  ( r )[ eV ( r )] ( r )dr  ( r )  2( ) exp( Zr / a0 )  2( )3 / 2 0 a0 a0 pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  32. 32. Isotope Shifts R 2 3Ze 3 Ze Zer 2 E   4r 4( Z / a ) (  e)[ 3 ] dr 8 0R 8 0 R 4 0r   0 R 4R 5 R 4R 3 R 1 4 r 2 r 2dr  2  4 r dr  2  4 r dr  2R 2 0 5 0 3 0 r  Ze 2 4  4  3  3 E  ( 4e)( Z / a ) R [        2 ] 4 0 10  3  2  2 2 2 Ze R 3 E  ( Z / a0 ) 5 0 pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  33. 33. Isotope Shifts • Isotope shift for optical spectra • Isotope shift for X-ray spectra (bigger effect because electrons closer to nucleus) • Isotope shift for X-ray spectra for muonic atoms. Effect greatly enhanced because mm~ 207 me and a0~1/m. • All data consistent with R=R0 A1/3 with R0=1.25fm. pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  34. 34. Liquid Drop Model Nucleus • Phenomenological model to understand binding energies. • Consider a liquid drop – Ignore gravity and assume no rotation – Intermolecular force repulsive at short distances, attractive at intermediate distances and negligible at large distances  constant density. E=-an + 4pR2T B=an-bn2/3 • Analogy with nucleus – Nucleus has constant density – From nucleon nucleon scattering experiments: Nuclear force has short range repulsion and attractive at intermediate distances. – Assume charge independence of nuclear force, neutrons and protons have same strong interactions check with experiment! pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  35. 35. Mirror Nuclei • Compare binding energies of mirror nuclei (nuclei n p). Eg 73Li and 74Be. • Mass difference due to n/p mass and Coulomb R energy. Q( r ) E dQ 0 4 0 r Q( r )  Ze( r / R ) 3 dQ  3 Zer 2 / R 3 3( Ze ) 2 r 5 R ( Ze ) 2 E 6 dr  ( 3 / 5) 0 4 0 r R 4 0 R 3 e2 E c ( Z , Z  1)  [ Z ( Z  1)  ( Z  1)( Z  2)] ; Z ~ A / 2 ; R  A1 / 3 5 4 0 R EC ( Z , Z  1)  A2 / 3 pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  36. 36. Liquid Drop Model Nucleus • Phenomenological model to understand binding energies. • Consider a liquid drop – Ignore gravity and assume no rotation – Intermolecular force repulsive at short distances, attractive at intermediate distances and negligible at large distances  constant density. – n=number of molecules, T=surface tension, B=binding energy E=total energy of the drop, a,b=free constants E=-an + 4pR2T  B=an-bn2/3 surface area ~ n2/3  Analogy with nucleus  Nucleus has constant density  From nucleon-nucleon scattering experiments we know:  Nuclear force has short range repulsion and is attractive at intermediate distances.  Assume charge independence of nuclear force, neutrons and protons have same strong interactions check with experiment (Mirror Nuclei!) pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  37. 37. Coulomb Term • The nucleus is electrically charged with total charge Ze • Assume that the charge distribution is spherical and compute the reduction in binding energy due to the Coulomb interaction Ze Q(r ) ECoulomb  dQ Q (r )  Ze(r / R )3 dQ  3Zer 2 / R 3 dr 0  4 0 r to change the integral to dr ; R=outer radius of nucleus R 3( Ze) 2 r 5 ( Ze) 2 ECoulomb   dr  (3 / 5) includes self interaction of last 6 4 0 r R 4 0 R proton with itself. To correct this 0 replace Z2 with Z*(Z-1) … and remember R=R0A-1/3 in principle you could take d from this calculation Z *( Z  1) BCoulomb ( Z , A)  d but it is more accurate to take it from the overall 1/ 3 A fit of the SEMF to data (nuclei not totally pdfMachine spherical or homogeneous) A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  38. 38. Mirror Nuclei • Does the assumption of the drop model of constant binding energy for every constituent of the drop acatually hold for nuclei? • Compare binding energies of mirror nuclei (nuclei with np). Eg 73Li and 7 Be. 4 • If the assumption holds the mass difference should be due to n/p mass difference and Coulomb energy alone. • Let’s compute the Coulomb energy correction from results on previous page 3 e2 3 e2 Ecoulomb ( Z , Z  1)  [ Z ( Z  1)  ( Z  1)( Z  2)]  2( Z  1) 5 4 0 R 5 4 0 R Z ~ A / 2 ; R  R0 A1/ 3 to find that EC ( Z , Z  1)  A2 / 3  Now lets measure mirror nuclei masse, assume that the model holds and derive DECoulomb from the measurement.  This should show an A2/3 dependence  And the scaling factor should yield the correct R0 of 1.2 fm  pdfMachine if the assumptions were right A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  39. 39. “Charge symmetry” nn and pp interaction same (apart from Coulomb) pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  40. 40. More charge symmetry • Energy Levels of two mirror nuclei for a number of excited states • Corrected for n/p mass difference and Coulomb Energy DEcorrected pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  41. 41. From Charge Symmetry to Charge Independence • Mirror nuclei showed that strong interaction is the same for nn and pp. • What about np ? • Compare energy levels in “triplets” with same A, different number of n and p. e.g. 22 22 22 10 Ne 11 Na 12 Mg • If we find the same energy levels for the same spin states  Strong interaction is the same for np as nn and pp. pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  42. 42. Charge Independence DEcorrected • Same spin/parity states should have the same energy. • Yes: np=nn=pp • Note: Far more states in 22 Na. Why? 11 • Because it has more np pairs then the others • np pairs can be in any Spin- Space configuration • pp or nn pairs are excluded from the totally symmetric ones by Herr Pauli 22 10Ne • Note also that 2211Na has the 22 lowest (most bound) state, 12Mg remember for the deuteron 22 11Na on next page pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  43. 43. Charge Independence • We have shown by measurement that: – If we correct for n/p mass difference and Coulomb interaction, then energy levels in nuclei are unchanged under n  p – and we must change nothing else! I.e. spin and space wavefunctions must remain the same! • Conclusion: strong two-body interaction same for pp, pn and nn if nucleons are in the same quantum state. • Beware of the Pauli exclusion principle! eg why do we have bound state of pn but not pp or nn? – because the strong force is spin dependent and the most strongly bound spin-space configurations (deuteron) are not available to nn or pp. It’s Herr Pauli again! – Just like 2211Na on the previous triplet level schema pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  44. 44. Volume and Surface Term • We now have all we need to trust that we can apply the liquid drop model to a nucleus – constant density – same binding energy for all constituents • Volume term: BVolume ( A)   aA 2/3 • Surface term: BSurface ( A)  bA • Since we are building a phenomenological model in which the coefficients a and b will be determined by a fit to measured nuclear binding energies we must inlcude any further terms we may find with the same A dependence together with the above pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  45. 45. Asymmetry Term • Neutrons and protons are spin ½ fermions  obey Pauli exclusion principle. • If all other factors were equal nuclear ground state would have equal numbers of n & p. neutrons protons Illustration  n and p states with 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 Z-N excess protons over neutrons the extra energy is ((Z-N)/2)2 . relative to Z=N.  But how big is D ? pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  46. 46. Asymmetry Term • Assume: – p and n form two independent, non-interacting gases occupying their own square Fermi wells – kT << D – so we can neglect kT and assume T=0 – This ought to be obvious as nuclei don’t suddenly change state on a warm summers day! – Nucleons move non-relativistically (check later if this makes sense) pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  47. 47. Asymmetry Term • From stat. mech. density of states in 6d phase space = 1/h3 4 p 2 dpV dN particle  h3 here N could be the number of protons or neutrons particle • Integrate up to pf to get total number of protons Z (or Neutrons N), & Fermi Energy (all states filled up to this energy level). 3 3 4 3 Z  (4 V / 3h ) pF and V   R0 A 3 1/ 3 2 2/3 h Z 2 h Z PF  (3 / 4 ) 2/3 and E  p  EF  (3 / 4 ) 4/3 R0  A 2m 2mR02  A     EF • Change variables p  E to find avg. E E 3/ 2 dE dN / dp dN / dE   const  E1/ 2  E  0  (3 / 5) EF  EF dp / dE E1/ 2 dE 0  pdfMachine These are all standard stat. mech. results! A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  48. 48. Asymmetry Term  Compute total energy of all protons by Z*<E> call this K 2 2/3 P P 3 4/3 h Z E Total Z E  (3 / 4 ) 2 Z  5 2mR0  A   Use the above to compute total energy of Z protons and N neutrons K ETotal   Z 5/ 3  N 5/ 3  change variables from (Z,N,A) A2 / 3   to (y,A) with y=N-Z KA5/ 3 ETotal  2 / 3 5 3  (1  y / A)5/ 3  (1  y / A)5/ 3  where y/A is a small number (e) A 2  KA  ETotal  5 3 (1   )5/ 3  (1   )5/ 3  2   • Binomial expansion keep lowest term in y/A 2 3 21 35 ( N  Z ) 2 ETotal  2 KA  K note! linear terms cancel 9 A ( N  Z )2 ETotal ( Fermi  Gas )  const * This terms is only proportional to volume A comes from a fit of the (A). It has already been captured by the Volume term of the liquid pdfMachine SEMF to measurements A pdf writer that produces quality PDF files with ease! drop model aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across analytical ≈ 24 MeV nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  49. 49. Asymmetry term • From the Fermi Gas model we learn that – due to the fermionic nature of p and n we loose in binding energy if the nucleus deviates from N=Z ( N  Z )2 • The Asymmetry term: B Asymmetry ( N , Z )   c A pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  50. 50. • Observations: • Nuclei with even number of n or Pairing Term even number of p more tightly Neutron separation energy bound then with odd numbers. [MeV] in Ba isotopes See figure • Only 4 stable o-o nuclei but 153 stable e-e nuclei. • p energy levels are Coulomb 56+N 56Ba shifted wrt n  small overlap of wave functions between n and p. • Two p or two n in same energy level with opposite values of jz have AS spin state  forced into sym spatial w.f.  maximum overlap  maximum binding energy Neutron number because of short range attraction. Note: this only holds for nn and pp, not for np.  We don’t have a preference for even A pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  51. 51. Pairing Term • Measure that the Pairing effect smaller for larger A • Phenomenological*) fit to A dependence gives A-1/2 BPairing ( A)   1/ 2  A d Note: If you want to plot binding energies e-e +ive versus A it is often best to use odd A only as for these the pairing term does not e-o 0 appear o-o -ive *) For an even more insightful explanation of the A dependence read the book by Jelley pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  52. 52. Semi Empirical Mass Formula • Put everything together: 2 2 2/3 (N  Z ) Z B( N , Z )  aA  bA c  d 1/ 3  1/ 2  A A A Volume Surface Asymmetry Coulomb Pairing Term Term Term Term Term  Lets see how all of these assumptions fit reality  And find out what the constants are  Note: we went back to the simpler Z instead of Z*(Z-1) 2 pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  53. 53. Semi Empirical Mass Formula Binding Energy vs. A for beta-stable odd-A nuclei Fit parameters Iron in MeV a 15.56 Not smooth because Z b 17.23 not smooth function of A c 23.285 d 0.697 d +12 (o-o) d 0 (o-e) d -12 (e-e) pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  54. 54. Semi Empirical Mass Formula • Conclusions – Only makes sense for A≥20 – Good fit for large A (good to <1%) in most places. – Deviations are interesting  shell effects. – Coulomb term constant agrees with calculation. – Explains the valley of stability (see next lecture). – Explains energetics of radioactive decays, fission and fusion. pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  55. 55. Nuclear Shell Model • Potential between nucleons can be studied by studying bound states (pn, ppn, pnn, ppnn) or by scattering cross sections: np -> np pp -> pp nD -> nD pD -> pD • If had potential could solve Schrod. Eq. Don’t know precise form but can make general approximation • 3d Finite Well with little r-dependence (except at edge of well) • Almost spherically symmetric (fusion can be modeled as deformations but we’ll skip) • N-N interactions are limited (at high A) due to Pauli exclusion. p + n -> p’ + n’ only if state is available pdfMachine A pdf writer that produces quality PDF files with ease! Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  56. 56. Infinite Radial Well 2 d 2u  2 l ( l  1)  • Radial part of Scrod Eq  V ( r )   u  Eu  2m dr 2 2m r2    u( r )  rR ( r ) P ( r )  4u 2 • Easy to solve if l=0 2 n p2 ( k ) 2 ( hn) 2 u  sin kr k  E  2a 2m 2m 8ma 2   • For L>0, angular momentum term goes to infinity at r=0. Reduces effective wavelength, giving higher energy • Go to finite well. Wave function extends a bit outside well giving longer effective wavelength and lower energy (ala 1D square wells) pdfMachine • In nuceli, potential goes to infinity at r=0 (even with L=0) as that A pdf writer that produces quality PDF files with ease! Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the would be equivalent to nucleon “inside” other nucleon “Broadgun pdfMachine printer” and that’s it! Get yours now!
  57. 57. Angular part • If V(r) then can separate variables y(r,q,f) = R(r)Y( q,f) have spherical harmonics for angular wave function • Angular momentum then quantized like in Hydrogen (except that L>0 for n=1, etc) L2  l (l  1)  2 LZ   m l  0,1,2 m   l l n  r  quantum# • Energy doesn’t depend on m • Energy increases with increasing n (same l) • Energy increases with increasing l (same n) • If both n,l vary then use experimental observation to determine lower energy • Energy will also depend on strong magnetic coupling between pdfMachine nucleons A pdf writer that produces quality PDF files with ease! Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the • Fill up states separately for p,n “Broadgun pdfMachine printer” and that’s it! Get yours now!
  58. 58. L,S,J Coupling: Atoms vs Nuclei • ATOMS: If 2 or more electrons, Hund’s rules: • Maximise total S for lowest E (S=1 if two) • Maximise total L for lowest E (L=2 if 2 P) • Energy split by total J (J=3,2,1 for S=1,L=2) • NUCLEI: large self-coupling. Plus if 2 p (or 2 n) then will anti-align giving a state with J=0, S=0, L=0 leftover “odd” p (or n) will have two possible J = L + ½ or J = L – ½ higher J has lower energy if there are both an odd P and an odd n (which is very rare in stable) then add up Jn + Jp • Atom called LS coupling nuclei called jj pdfMachine A pdf writer that produces quality PDF files with ease! • Note that magnetic moments add differently as different g-factor Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now! for p,n
  59. 59. Spin Coupling in Nuclei • All nucleons in valence shell have same J • Strong pairing causes Jz antiparallel (3 and -3) spin wavefunction = antisymmetric space wavefunction = symmetric • This causes the N-N to be closer together and increases the attractive force between them • e-e in atoms opposite as repulsive force • Can also see in scattering of polarized particles • Even N, even Z nuclei. Total J=S=L=0 as all n,p paired off • Even N, odd Z or odd N, even Z. nuclear spin and parity determined by unpaired nucleon • Odd N, odd Z. add together unpaired n,p pdfMachine • Explains ad hoc pairing term in mass formula A pdf writer that produces quality PDF files with ease! Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  60. 60. Energy Levels in Nuclei • Levels in ascending order (both p,n) State n L degeneracy(2j+1) sum 1S1/2 1 0 2 2*** 1P3/2 1 1 4 6 1P1/2 1 1 2 8*** 1D5/2 1 2 6 14 2S1/2 2 0 2 16 1D3/2 1 2 4 20*** 1F7/2 1 3 8 28*** 2P3/2 2 1 4 32 1F5/2 1 3 6 38 2P1/2 2 1 2 40 1G9/2 1 4 10 50*** *** “magic” number is where there is a large energy gap between a pdfMachine filled shell and the next level. More tightly bound nuclei. (all filled A pdf writer that produces quality PDF files with ease! Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across subshells are slightly “magic”) nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  61. 61. Magic Numbers • Large energy gaps between some filled shells and next (unfilled) shell give larger dE/A and more made during nucleosnthesis in stars # protons #neutrons 2 He 2 He-4 6 C 6 C-12 8 O 8 O-16 20 Ca 20 28 Ni 28 Cr-52(24,28) 50 Sn 50 Ni-78 82 Pb 82 126 136 • Ni-78 (2005) doubly magic. While it is unstable, it is the much neutron rich. pdfMachine A pdf writer that produces quality PDF files with ease! • Usually more isotopes if p or n are magic. Sn has 20 isotopes, 10 Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now! of which are stable

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