Preamble Masses and Binding Energy Neutron seperation

NuclearPhysics &

Open Source

Sandeep SGhugre

PreambleSoftware

Masses andBindingEnergy

Neutronseperationenergy

RadioactiveDecay

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PreambleMasses and Binding EnergyNeutron seperation energy

Radioactive Decay

Sandeep S Ghugre Nuclear Physics & Open Source

NuclearPhysics &

Open Source

Sandeep SGhugre

PreambleSoftware



RadioactiveDecay

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Radioactive Decay

Nuclear Physics & Open Source

Sandeep S Ghugre

UGC DAE CSR KC

www.iuc.res.in

April 5, 2021


NuclearPhysics &

Open Source

Sandeep SGhugre

PreambleSoftware



RadioactiveDecay

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Radioactive Decay

Software

1 PreambleSoftware

2 Masses and Binding Energy

3 Neutron seperation energy

4 Radioactive Decay


NuclearPhysics &

Open Source

Sandeep SGhugre

PreambleSoftware



RadioactiveDecay

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Radioactive Decay

Software

This lecture series attempts to use the Open Source toolsin elucidating some of the fundamental aspects (concepts)in Nuclear Physics.The presentation may be distributed outside this audiencefollowing specific permission.An excellent repository can be found at Link , whichcontains teaching material for the Nuclear Structurecourse at Michigan State University.


https://github.com/NuclearStructure/

NuclearPhysics &

Open Source

Sandeep SGhugre

PreambleSoftware



RadioactiveDecay

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Radioactive Decay

Software

The Python software Link can be used for the numericalexercises.

Use of Spreadsheet is also possible for some of the exercises.


https://www.python.org/

NuclearPhysics &

Open Source

Sandeep SGhugre

PreambleSoftware



RadioactiveDecay

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Radioactive Decay

1 PreambleSoftware



4 Radioactive Decay


NuclearPhysics &

Open Source

Sandeep SGhugre

PreambleSoftware



RadioactiveDecay

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Radioactive Decay

Nucleus provides us with a unique playground to explore thetwo extreme, degrees of freedom. viz.

the single particle andthe collective

The dimensions of our playground is around 10−15 fm, with anextremely high density viz. ρnucl ∼ 1017 kg/m3.


NuclearPhysics &

Open Source

Sandeep SGhugre

PreambleSoftware



RadioactiveDecay

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Radioactive Decay

The operational system of unit used for nucleonic mass is theAtomic Mass Unit, (denoted by u) .

We know that 1u is given by

1u = M(12C)/12 = 931.49386MeV /c2

Accordingly, the nucleonic masses are given by

mp = 938.28MeV /c2 = 1.007276 umn = 939.257MeV /c2 = 1.008664 u


NuclearPhysics &

Open Source

Sandeep SGhugre

PreambleSoftware



RadioactiveDecay

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Radioactive Decay

Atomic masses are tabulated in terms of mass excess, wherethe mass excess is defined as

∆M(N,Z ) = M(N,Z )− uA

Conventionally, the mass table (compilations) are available andduring this course we shall limit ourself to the compilation byAudi, Wapstra and Co-workers.

The mass tables in Excel format has been made available in thepublic domain by Dr. Gordon Gilmore of Nuclear TrainingServices Ltd.


NuclearPhysics &

Open Source

Sandeep SGhugre

PreambleSoftware



RadioactiveDecay

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Radioactive Decay

The energy spent (locked up) in assembling the system ofnucleons appears as deficit in the mass of the resultant nucleus.

This is referred to as Mass Defect, denoted by (δm), which inthe unit of u is defined as

δm = (mass of constituents) − (mass of the nucleus)

= (Z ∗ mp) + (A − Z ) ∗ mn − Mnucleus


NuclearPhysics &

Open Source

Sandeep SGhugre

PreambleSoftware



RadioactiveDecay

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Radioactive Decay

Mass Defect

This mass defect, is related to the energy released when thenucleus, is split into it’s constituent independent nucleons, andis referred to as Binding Energy, (BE ), which is related to δmas

B.EMeV = δmu ⊗ 931.49

Figure: G.Audi and A.H.Wapstra, Nuclear Physics A595 vol. 4p.409-480, December 25, 1995


NuclearPhysics &

Open Source

Sandeep SGhugre

PreambleSoftware



RadioactiveDecay

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Radioactive Decay

In terms of mass excess (∆M), the BE is obtained as

BE (Z ,N) = Z ∗∆Hc2 + N ∗∆nc2 −∆(N,Z )c2 (1)

where ∆Hc2 = 7.2890 MeV and ∗∆nc2 = 8.0713 MeV.

Calculate the BE for alpha particle, from the data provided inthe mass table.SolutionFor the alpha particle (N = Z = 2), from the mass table wefind that

∆M = 2.4249 MeVBE = [(2 ∗ 7.2890) + (2 ∗ 8.0713)]− 2.4249

= 28.29 MeV


NuclearPhysics &

Open Source

Sandeep SGhugre

PreambleSoftware



RadioactiveDecay

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Radioactive Decay

We know that the plot of Binding Energy per nucleon as afunction of A holds the entire story of the amazing nuclear zoo.

saturation of nuclear forces.presence of shell structure.dominance of pairing.

Using the Excel sheet generate the plot of Binding Energy pernucleon as a function of A.


NuclearPhysics &

Open Source

Sandeep SGhugre

PreambleSoftware



RadioactiveDecay

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Radioactive Decay

Using the given Excel sheet (which has been modified from theoriginal sheet available in the public domain) , plug in thefollowing formulae for the respective columns

Z×mp → F4∗1.007825

N×mn → E4∗1.008665

Consti Mass → (T4+U4)

B.EMeV → (V 4−R4)∗931.49BEA MeV

A

→ W 4B4


NuclearPhysics &

Open Source

Sandeep SGhugre

PreambleSoftware



RadioactiveDecay

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Radioactive Decay

Plot


NuclearPhysics &

Open Source

Sandeep SGhugre

PreambleSoftware



RadioactiveDecay

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Radioactive Decay

1 PreambleSoftware



4 Radioactive Decay


NuclearPhysics &

Open Source

Sandeep SGhugre

PreambleSoftware



RadioactiveDecay

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Radioactive Decay

Now we can represent the reaction which essentially removesone nucleon, say a neutron from the parent nucleus viz.

AZ X N → A−1

Z X N−1 + n

20982Pb 127 → 208

82X (126) + n

Neutron seperation energy , is the amount of energy needed toremove a neutron from the nucleus and is denoted by Sn ,

Sn = BE (209Pb)− BE (208Pb)


NuclearPhysics &

Open Source

Sandeep SGhugre

PreambleSoftware



RadioactiveDecay

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Radioactive Decay

Use the mass table and tabulate the binding energy for Pbisotopes.Using an excel file we obtain the difference betweenA & (A − 1) isotopes.

Figure: Calculation of separation energy using Spreadsheet.


NuclearPhysics &

Open Source

Sandeep SGhugre

PreambleSoftware



RadioactiveDecay

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Radioactive Decay

Figure: Neutron separation energy for Pb isotopes as a function ofneutron number.

A pronounced discontinuity is observed at N = 126 , theneutron magic number


NuclearPhysics &

Open Source

Sandeep SGhugre

PreambleSoftware



RadioactiveDecay

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Radioactive Decay

1 PreambleSoftware



4 Radioactive Decay


NuclearPhysics &

Open Source

Sandeep SGhugre

PreambleSoftware



RadioactiveDecay

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Radioactive Decay

Let us attempt to solve the radioactive decay equationdndt = −λdt.

The solution would be attempted numerically using Euler’smethod and we shall compare the results with the analyticalsolution N(t) = N0 · e−λt


NuclearPhysics &

Open Source

Sandeep SGhugre

PreambleSoftware



RadioactiveDecay

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Radioactive Decay

Let us start with a very simple first order ODE, say

dydx = f (x , y)

y(x0) = y0

Thus starting with x0, y0&f0 we can generate the subsequentvalues for a given h,

dydx

= x + y

y(0) = 1[a, b

]=

[0, 1

]Let us choose h = 0.5

Hence N = b−ah = 2

Now x = 0, 0.5, 1.0

yn+1 = yn + hf (xn, yn)

y0 = 1 (x0 = 0)y1 = y0 + hf (x0, y0)

= y0 + h(x0 + y0)

= 1 + 0.5(0 + 1)= 1.5 (x1 = 0.5)

y2 = y1 + hf (x1, y1)

= y1 + h(x1 + y1)

= 1.5 + 0.5(0.5 + 1.5)= 2.5


NuclearPhysics &

Open Source

Sandeep SGhugre

PreambleSoftware



RadioactiveDecay

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Radioactive Decay

1 Create a column with values of t2 In the Column D2 enter the value of dt3 The first entry for time should be 04 The first entry for N should be N0 = 5.5 The first entry for dN/dt should be 0.3*C5.6 The first entry for N(New) should be the initial value of

N0

Figure: Euler Method & Spreadsheet for Radioactive Decay.


NuclearPhysics &

Open Source

Sandeep SGhugre

PreambleSoftware



RadioactiveDecay

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Radioactive Decay

1 Now create the enteries for time2 In the row corresponding to the time enter the following

B5+D$2

where B5 corresponds to the previous cell3 Continue till you attain the desired time range

Figure: Euler Method & Spreadsheet for Radioactive Decay.Sandeep S Ghugre Nuclear Physics & Open Source

NuclearPhysics &

Open Source

Sandeep SGhugre

PreambleSoftware



RadioactiveDecay

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Radioactive Decay

1 Now create the entry for N2 In the row corresponding to the N enter the following E5

where E5 corresponds to the earlier value of N



NuclearPhysics &

Open Source

Sandeep SGhugre

PreambleSoftware



RadioactiveDecay

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Radioactive Decay

1 Now create the entry for dn/dt2 In the row corresponding to the dN/dt enter the following

(-0.3)*C6

where C6 corresponds to the earlier N(new)



NuclearPhysics &

Open Source

Sandeep SGhugre

PreambleSoftware



RadioactiveDecay

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Radioactive Decay

1 Now create the entry for N(new)2 In the row corresponding to the N(new) enter the

following C6+D6*D$2



Documents

Preamble Masses and Binding Energy Neutron seperation