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UT Arlington PHYS 3446 - Nuclear Force and Nuclear Models

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Wednesday, Sept. 27, 2006 PHYS 3446, Fall 2006Jae Yu1PHYS 3446 – Lecture #7Wednesday, Sept. 27, 2006Dr. Jae Yu1.Nature of the Nuclear Force• Shape of the Nuclear Potential• Yukawa Potential• Range of Yukawa Potential2. Nuclear Models• Liquid Drop Model• Fermi Gas Model• Shell ModelWednesday, Sept. 27, 2006 PHYS 3446, Fall 2006Jae Yu2Announcements• Workshop – 10am – 5pm, Saturday @CPB303– Did all groups purchase what you need?– Each group needs to come up with the plans for the day and discuss which group does what in which order• Some groups’ activities might interfere with others• First term exam– Date and time: 1:00 – 2:30pm, Wednesday, Oct. 4– Location: SH105– Covers: Appendix A (special relativity) + CH1 – CH3• Quiz results– Class Average: 47.4– Top score: 71– Quizzes account for 10% of the totalWednesday, Sept. 27, 2006 PHYS 3446, Fall 2006Jae Yu3• A square well nuclear potential Î provides the basis of quantum theory with discrete energy levels and corresponding bound state just like in atoms– Presence of nuclear quantum states have been confirmed through • Scattering experiments• Studies of the energies emitted in nuclear radiation• Studies of mirror nuclei and the scatterings of protons and neutrons demonstrate– Without the Coulomb effects, the forces between two neutrons, two protons or a proton and a neutron are the same • Nuclear force has nothing to do with electrical charge• Protons and neutrons behave the same under the nuclear force– Inferred as charge independence of nuclear force.Nuclear PotentialWednesday, Sept. 27, 2006 PHYS 3446, Fall 2006Jae Yu4• Strong nuclear force is independent of the electric charge carried by nucleons– Concept of strong isotopic-spin symmetry.• proton and neutron are the two different iso-spin state of the same particle called nucleon– In other words,• If Coulomb effect were turned off, protons and neutrons would be indistinguishable in their nuclear interactions• Can you give another case just like this???– This is analogues to the indistinguishability of spin up and down states in the absence of a magnetic field!!• This is called Iso-spin symmetry!!!Nuclear Potential – Iso-spin symmetryWednesday, Sept. 27, 2006 PHYS 3446, Fall 2006Jae Yu5• EM force can be understood as a result of a photon exchange– Photon propagation is described by the Maxwell’s equation– Photons propagate at the speed of light. – What does this tell you about the mass of the photon?• Massless• Coulomb potential is• What does this tell you about the range of the Coulomb force?– Long range. Why? Range of the Nuclear Force()Vr∝1rMasslessparticle exchangeWednesday, Sept. 27, 2006 PHYS 3446, Fall 2006Jae Yu6• For massive particle exchanges, the potential takes the form– What is the mass, m, in this expression? • Mass of the particle exchanged in the interaction– The force mediator mass• This form of potential is called Yukawa Potential– Formulated by Hideki Yukawa in 1934• What does Yukawa potential turn to in the limit m Æ0?– Coulomb potentialYukawa Potential()Vr∝mcre−= rWednesday, Sept. 27, 2006 PHYS 3446, Fall 2006Jae Yu7• From the form of the Yukawa potential• The range of the interaction is given by some characteristic value of r. What is this?– Compton wavelength of the mediator with mass, m: • What does this mean?– Once the mass of the mediator is known, range can be predicted– Once the range is known, the mass can be predictedRanges in Yukawa Potential()mcreVrr−∝=rer−=mc==Wednesday, Sept. 27, 2006 PHYS 3446, Fall 2006Jae Yu8• Let’s put Yukawa potential to work• What is the range of the nuclear force?– About the same as the typical size of a nucleus •1.2x10-13cm– thus the mediator mass is• This is close to the mass of a well known π meson (pion)• Thus, it was thought that π are the mediators of the nuclear forceRanges in Yukawa Potential2mc=mπ+=c≈=1971641.2MeV fmMeVfm−≈2139.6 / ;MeV c20135 /mMeVcπ=mπ−=Wednesday, Sept. 27, 2006 PHYS 3446, Fall 2006Jae Yu9• Experiments showed very different characteristics of nuclear forces than other forces• Quantification of nuclear forces and the structure of nucleus were not straightforward– Fundamentals of nuclear force were not well understood • Several phenomenological models (not theories) that describe only limited cases of experimental findings• Most the models assume central potential, just like Coulomb potentialNuclear ModelsWednesday, Sept. 27, 2006 PHYS 3446, Fall 2006Jae Yu10• An earliest phenomenological success in describing binding energy of a nucleus• Nucleus is essentially spherical with radius proportional to A1/3. – Densities are independent of the number of nucleons• Led to a model that envisions the nucleus as an incompressible liquid droplet– In this model, nucleons are equivalent to molecules• Quantum properties of individual nucleons are ignoredNuclear Models: Liquid Droplet ModelWednesday, Sept. 27, 2006 PHYS 3446, Fall 2006Jae Yu11• Nucleus is imagined to consist of – A stable central core of nucleons where nuclear force is completely saturated– A surface layer of nucleons that are not bound tightly• This weaker binding at the surface decreases the effective BE per nucleon (B/A)• Provides an attraction of the surface nucleons towards the core just as the surface tension to the liquidNuclear Models: Liquid Droplet ModelWednesday, Sept. 27, 2006 PHYS 3446, Fall 2006Jae Yu12• If a constant BE per nucleon is due to the saturation of the nuclear force, the nuclear BE can be written as: • What do you think each term does?– First term: volume energy for uniform saturated binding– Second term corrects for weaker surface tensionLiquid Droplet Model: Binding EnergyBE=• This can explain the low BE/nucleon behavior of low A nuclei– For low A nuclei, the proportion of the second term is larger.– Reflects relatively large number of surface nucleons than the core.1aA−+232aAWednesday, Sept. 27, 2006 PHYS 3446, Fall 2006Jae Yu13• Small decrease of BE for heavy nuclei can be understood as due to Coulomb repulsion– The electrostatic energies of protons have destabilizing effect• Reflecting this effect, the empirical formula for BE takes the correction term• All terms of this formula have classical origin.• This formula does not explain – Lighter nuclei with the equal number of


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UT Arlington PHYS 3446 - Nuclear Force and Nuclear Models

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