# nuclear masses and binding energy - oregon state u ?· nuclear masses • nuclear masses and...     Post on 17-Sep-2018

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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) =15.73Z7 / 3eV

Because Belectron(Z)is so small, it is neglected in most situations.

• Mass Changes in Beta Decay

- decay

14C!14N + "# + \$ eEnergy = [(m(14C) + 6melectron ) # (m(

14N) + 6melectron ) #m("#)]c 2

Energy = [M(14C) #M(14N)]c 2

+ decay

64Cu!64Ni" + # + + \$ eEnergy = [(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+ # eEnergy = [(m(207Bi) + 83melectron ) ! (m(

207Pb) + 82melectron )]c2

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 H 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 () 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 = 0.72Z 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 q in energy, thus the work involved is q2=(N-Z)2/4. If we add that =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/ 2ee! = 0oe,eo! = "11A"1/ 2oo

• 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

" # \$

% & '

ZAA

) 12

8180 + 0.6A2 / 3

• Valley of Beta Stability

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