nuclear masses and binding energy - oregon state...
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Nuclear Masses and Binding Energy Lesson 3
Nuclear Masses
• Nuclear masses and atomic masses
mnuclc2 = Matomicc
2 ! [Zmelectronc2 + Belectron (Z)]
Belectron (Z) =15.73Z 7 / 3eV
Because Belectron(Z)is so small, it is neglected in most situations.
Mass Changes in Beta Decay
• β- decay
14C!14N + "# + $ e
Energy = [(m(14C) + 6melectron ) # (m(14N) + 6melectron ) #m("
#)]c 2
Energy = [M(14C) #M(14N)]c 2
• β+ decay
64Cu!64Ni" + # + + $ e
Energy = [(m(64Cu) + 29melectron ) " (m(64Ni) + 28melectron ) "melectron "m(#
+)]c 2
Energy = [M(64Cu) "M(64Ni) " 2melectron ]c2
Mass Changes in Beta Decay
• EC decay
207Bi+ + e!"207Pb+ # e
Energy = [(m(207Bi) + 83melectron ) ! (m(207Pb) + 82melectron )]c
2
Energy = [M(207Bi) !M(207Pb)]c 2
Conclusion: All calculations can be done with atomic masses
Nomenclature
• Sign convention: Q=(massesreactants-massesproducts)c2
Q has the opposite sign as ΔH Q=+ exothermic Q=- endothermic
Nomenclature
• Total binding energy, Btot(A,Z) Btot(A,Z)=[Z(M(1H))+(A-Z)M(n)-M(A,Z)]c2 • Binding energy per nucleon Bave(A,Z)= Btot(A,Z)/A • Mass excess (Δ) M(A,Z)-A See appendix of book for mass tables
Nomenclature
• Packing fraction (M-A)/A
• Separation energy, S Sn=[M(A-1,Z)+M(n)-M(A,Z)]c2
Sp=[M(A-1,Z-1)+M(1H)-M(A,Z)]c2
Binding energy per nucleon
Separation energy systematics
Abundances
Semi-empirical mass equation
Btot (A,Z) = avA ! asA2 / 3 ! ac
Z 2
A1/ 3! aa
(A ! 2Z)2
A± "
Terms
• Volume avA • Surface -asA2/3
• Coulomb -acZ2/A1/3
ECoulomb = 35Z 2e2
RR =1.2A1/ 3
ECoulomb = 0.72 Z 2
A1/ 3
Asymmetry term
!aa(A ! 2Z)2
A= !aa
(N ! Z)2
A
To make AZ from Z=N=A/2, need to move q protons qΔ in energy, thus the work involved is q2Δ=(N-Z)2Δ/4. If we add that Δ=1/A, we are done.
Pairing term A Z N # stable e e e 201
o e o 69
o o e 61
e o o 4
! = +11A"1/ 2…ee! = 0…oe,eo! = "11A"1/ 2…oo
Relative importance of terms
Values of coefficients
av =15.56MeVas =17.23MeVac = 0.7MeVaa = 23.285MeV
Modern version of semi-empirical mass equation (Myers
and Swiatecki)
Btot (A,Z) = c1A 1! kN ! ZA
" # $
% & ' 2(
) *
+
, - ! c2A
2 / 3 1! k N ! ZA
" # $
% & ' 2(
) *
+
, - ! c3
Z 2
A1/ 3+ c4
Z 2
A+ .
c1 =15.677MeVc2 =18.56MeVc3 = 0.717MeVc4 =1.211MeVk =1.79! =11A"1/ 2
Mass parabolas and Valley of beta stability
M (Z,A) = Z •M (1H )c2 + (A − Z )M (n)c2 − Btot (Z,A)
Btot (Z,A) = avA − asA2/3 − ac
Z 2
A1/3− aa
(A − 2Z )2
A
aa(A − 2Z )2
A= aa
A2 − 4AZ + 4Z 2
A= aa A − 4Z + 4Z
2
A⎛⎝⎜
⎞⎠⎟
M = A M (n)c2 − av +asA1/3
+ aa⎡⎣⎢
⎤⎦⎥+ Z M (1H )c2 −M (n)c2 − 4aa⎡⎣ ⎤⎦ + Z
2 acA1/3
+ 4aaA
⎛⎝⎜
⎞⎠⎟
This is the equation of a parabola, a+bZ+cZ2
Where is the minimum of the parabolas?
!M!Z
" # $
% & ' A
= 0 = b + 2cZA
ZA = (b2c
= M(1H) (M(n) ( 4aa
2 acA1/ 3
+ 4aaA
" # $
% & '
ZA
A) 12
8180 + 0.6A2 / 3
Valley of Beta Stability
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