Binding Energy3.3 Binding Energy• The binding energy of a nucleus is theenergy required to separate all of theconstituent nucleons from the nucleus sothat they are all unbound and free particles.• This implies that -† MZ,AN< Z mp+ (A - Z)mn[ ]• And, of course, the mass-energy --• is the nuclear mass (no electrons)• BE defined as --• BE is always > 0.† MZ,ANc2< Z mp+ (A - Z)mn[ ]c2† BE ≡ Z mp+ (A - Z)mn[ ]c2- MZ,ANc2† MZ ,AN• To calculate the BE, we do not knownuclear masses. Therefore, use isotopicmasses --† mpc2Æ m1,1c2- mec2- EbÊ Ë Á ˆ ¯ ˜ mnc2Æ mnc2MZ,ANc2Æ MZ,Ac2- Z mec2- Ebii=1ZÂÊ Ë Á ˆ ¯ ˜ http://www.physics.valpo.edu/physLinks/atomicNuclearLinks.html• To calculate isotopic masses from D --† D = M - A Æ M = D + A1 uc2= 931.494 MeV; D26,56= -60.601MeVM26,56c2= D26,56+ A 931.494MeVuÊ Ë Á ˆ ¯ ˜ = - 60.601 MeV + 56u 931.494MeVuÊ Ë Á ˆ ¯ ˜ = 52103.065 MeVM26,56c2c2= 52103.065 MeV931.494MeVuÊ Ë Á ˆ ¯ ˜ = 55.934942u† BE = MZ ,Ac2- Z mec2- Ebii=1ZÂÊ Ë Á ˆ ¯ ˜ - Z m1,1c2- Z mec2- Eb1Ê Ë Á ˆ ¯ ˜ + A - Z( )mnc2È Î Í ˘ ˚ ˙ BE = MZ ,Ac2+ Ebii=1ZÂ- Z m1,1c2+ Z Eb1+ A - Z( )mnc2[ ]BE = MZ ,Ac2- Z m1,1c2- A - Z( )mnc2+ Ebii=1ZÂ- Z Eb1BE ª MZ ,Ac2- Z m1,1c2- A - Z( )mnc2Separation Energies & Systematic Studies• Table 3.1 - Can you see any pattern(s)?• Figure 3.16 - Describe significant features• Consider the physics that might give rise to Figure 3.16 -- can we develop a model that woulddescribe Figure 3.16?† Snª mZ,A-1+ mn( )c2- mZ,Ac2Spª mZ-1,A+ m1,1( )c2- mZ,Ac2Semi-empirical BE equation• Consistent with short-range force; nearly contactinteraction.• But nucleons on surface are less strongly bound -• Surface unbinding -† BEAª av Æ BE ª avA† BE µ -R2 IF R µ A13BE ª -asA23Semi-empirical BE equation• Coulomb force from all protons --• This effect can be calculated exactly from electrostatics -• Coulomb unbinding -† BE µ -Z (Z -1)R BE = -35e24peoZ (Z -1)RoA1/ 3Semi-empirical BE equation• Systematic studies show that the line of stability movesfrom Z = N to N > Z Why?• Coulomb force demands this -- but --• The asymmetry introduces a nuclear force unbinding --† BE = -asymA - 2Z( )2A See next slideEmpiricalSemi-empirical BE equationDEZ = Np n p nZ < NFor Z < N, there isan increased energyequal to --† Eincrease=N - Z2DEÈ Î Í ˘ ˚ ˙ N - Z2Ê Ë Á ˆ ¯ ˜ † N = A - ZEincrease=A - 2Z2Ê Ë Á ˆ ¯ ˜ 2DEEnergy jumpfor each proton# ofprotonsSemi-empirical BE equation• Systematic studies show like nucleons want to pair and inpairs are more stable (lower energy) than unpaired.• Therefore, we add (ad hoc) a pairing energy --† BE = +apA-3/4 if A & N are even BE = -apA-3/4 if A & N are odd BE = 0 if A or N is oddSemi-empirical BE equation• Combined equation for total BE is --• Systematic BE data are fit with this function giving -• Using these values of the parameters, one can thencalculate BE for any nuclide (Z,A).† BE = avA - asA2/3- acZ(Z -1)A1/3- asym(A - 2Z)2A+d † av, as, asym,d ; ac isknown.Semi-empirical mass equation• The isotopic mass of any nucleus can be calculated usingthe definition of the BE - but calculating the BE from thesemi-empirical equation:• And, at constant A, one can find the value of Z at which themass is a minimum (Zmin) - (3.30)• One can also calculate the separation energies.† MZ,ANc2= Z mp+ (A - Z)mn[ ]c2- BESemi-empirical mass equation• BE(Z) is a parabolic function of Z at constant A (isobar!)• This curve has a maximum Æ stability against decay.• The corresponding has a minimum at stability.• One curve if A is odd; two curves if A is even. (?)• Separation between the curves is -- 2d• With this semi-empirical model, one can ---– Calculate Q (energy) for decay schemes (a, b-, b+, e, p, n, fission)– Q > 0 Æ decay is possible– Q < 0 Æ decay is not possible– Put semi-empirical mass equation into Excel and calculate all ofthe masses in an isobar for a range of Z values.†
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