Nuclear Binding, Radioactivity

Physics 102: Lecture 27

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Loomis

Hydrogen atom: Binding energy =13.6eV

Binding energy of deuteron = or 2.2Mev! That’s around 200,000 times bigger!

2.2106eV

Simplest Nucleus: Deuteron=neutron+proton

neutron proton

Very strong force

Coulomb force

electron

proton

Strong Nuclear Force

(of electron to nucleus)

ground state

2.2 MeV

Deuterium Binding Energy

Nuclei have energy level—just like atoms

12C energy levels

Note the energy scale is MeV rather than eV

energy needed to remove a proton from 12C is 16.0 MeV

energy needed to remove a neutron from 12C is 18.7 MeV

Comparing Nuclear and Atomic sizes

Hydrogen Atom: Bohr radius = 5.2910 11m

Nucleus with nucl number A: r A1/3 (1.210 15m)

Note the TREMENDOUS difference

Smaller is Bigger!

Nucleus is 104 times smaller and binding energy is 105 times larger!

13

1327Al has radius r 3.61015m

A

Z

Preflight 27.2

Where does the energy released in the nuclear reactions of the sun come from?

(1)covalent bonds between atoms 17%

(2)binding energy of electrons to the nucleus 36%

(3)binding energy of nucleons 47%

15

Binding Energy

Einstein’s famous equation E = m c2

Proton: mc2 = 938.3MeVNeutron: mc2= 939.5MeV

Deuteron: mc2 =1875.6MeV

Adding these, get 1877.8MeV

Difference is Binding energy, 2.2MeV

MDeuteron = MProton + MNeutron – |Binding Energy|

17

proton:mc2=(1.67x10-27kg)(3x108 m/s)2=1.50x10-10 J

ACT: Binding Energy

Which system “weighs” more?

1) Two balls attached by a relaxed spring.

2) Two balls attached by a stretched spring.

3) They have the same weight.

M1 = Mballs + Mspring

M2 = Mballs + Mspring + Espring/c2

M2 – M1 = Espring/c2~ 10-16 Kg

19

Iron (Fe) has most binding energy/nucleon. Lighter have too few nucleons, heavier have too many.

BIN

DIN

G E

NE

RG

Y in

MeV

/nu

cleon

92238U

10

Binding Energy Plot

Fission

Fusi

on Fusion = Combining small atoms into large

Fission = Breaking large atoms into small

21

Which element has the highest binding energy/nucleon?

Preflight 27.3

22

• Neon (Z=10) 31%

• Iron (Z=26) 22%

• Iodine (Z=53) 47%

Which of the following is most correct for the total binding energy of an Iron atom (Z=26)?

9 MeV

234 MeV

270 MeV

504 Mev

For Fe, B.E./nucleon 9MeV

2656Fe has 56 nucleons

Total B.E 56x9=504 MeV

Preflight 27.4

24

13%

28%

31%

28%

particles: nuclei 24He

particles: electrons

: photons (more energetic than x-rays) penetrate!

3 Types of Radioactivity

Easily Stopped

Stopped by metal

26

Radioactive sources

B field into screen

detector

92238U 90

234Th: example

24He recall

: example

Decay Rules

1) Nucleon Number is conserved.2) Atomic Number (charge) is conserved.3) Energy and momentum are conserved.

: example 00

* PP AZ

AZ

1) 238 = 234 + 4 Nucleon number conserved

2) 92 = 90 + 2 Charge conserved

e0111

10 pn

Needed to conserve momentum.

30

00

A nucleus undergoes decay. Which of the following is FALSE?

1. Nucleon number decreases by 4 26%

2. Neutron number decreases by 2 45%

3. Charge on nucleus increases by 2 29%

Preflight 27.6

32

decay is the emission of 24He

He42

23490

23892 ThU Ex.

Z decreases by 2(charge decreases!)

A decreases by 4

The nucleus undergoes decay. 90234Th

Which of the following is true?

1. The number of protons in the daughter nucleus increases by one.

2. The number of neutrons in the daughter nucleus increases by one.

decay is accompanied by the emission of an electron: creation of a charge -e.

In fact, inside the nucleus, and the electron and neutrino “escape.”

n p e e

Preflight 27.7

00

01

?????

23490 XTh

e 00

01

23491

23490 PaTh

e

34

ACT: Decay

Which of the following decays is NOT allowed?

HePbPo 42

21082

21484

92238U 90

234Th

40 40 0 019 20 1 0K P e

NC 147

146

1

2

3

4

238 = 234 + 4

92 = 90 + 2

214 = 210 + 4

84 = 82 + 2

14 = 14+0

6 <> 7+0

40 = 40+0+019 = 20-1+0

36

Nt

N

If the number of radioactive nuclei present is cut in half, how does the activity change?

1 It remains the same 24%

2 It is cut in half 50%

3 It doubles 26%

No. of nuclei present

decay constant

Decays per second, or “activity”

Preflight 27.8

38

ACT: Radioactivity

Nt

N

No. of nuclei present

decay constant

Decays per second, or “activity”

Start with 16 14C atoms.

After 6000 years, there are only 8 left.

How many will be left after another 6000 years?

1) 0 2) 4 3) 8Every 6000 years ½ of atoms decay

40

time

N(t)N0e t N0 2

t

T1/2

40

Decay Function

Instead of base e we can use base 2:

N(t)N0e tSurvival:

No. of nuclei present at time t

No. we started with at t=0

e t 2

tT1/2

T1/2

0.693

where

Then we can write N(t)N0e t N0 2

t

T1/2

Half life

Radioactivity Quantitatively

Nt

N

No. of nuclei present

decay constant

Decays per second, or “activity”

42

You are radioactive!

One in 8.3x1011 carbon atoms is 14C which decays with a ½ life of 5730 years. Determine # of decays/s per gram of Carbon.

Nt

N

11

2314 103.8

11002.6

mole12

0.1g

N

2/1

693.T

1-12s1083.36060243655730

693.

gatoms

106 10

decays/s 23.045

Carbon Dating

We just determined that living organisms should have a decay rate of about 0.23 events/s per gram of carbon.

The bones of an ice man are found to have a decay rate of 0.23/2 events/s per gram. We can estimate he died about 6000 years ago.

47

ACT/Preflight 27.9

The half-life for beta-decay of 14C is ~6,000 years. You test a fossil and find that only 25% of its 14C is un-decayed. How old is the fossil?

1. 3,000 years

2. 6,000 years

3. 12,000 years

At 0 years: 100% remains

At 6,000 years: 50% remains

At 12,000 years: 25% remains

49

Summary• Nuclear Reactions

– Nucleon number conserved– Charge conserved– Energy/Momentum conserved– particles = nucleii– - particles = electrons– particles = high-energy photons

• Decays– Half-Life is time for ½ of atoms to decay

50

N(t)N0e tSurvival:

T1/2

0.693

24He

See you next time!

• Take a look at Special Relativity in 14 Easy (Hyper)lessons:

http://web.hep.uiuc.edu/home/g-gollin/relativity/