RADIOACTIVE DECAY

Principles of Nuclear Physics

NPE-503

Lecture by: Zahra Ali

Decay Equations

Decay is proportional to the # of atoms present (first order)

dN/dt = - N = AN where

N = the number of atoms of the radioactive substance present at time t

= the first order decay constant (time-1)

The number of parent atoms at any time t can be calculated as follows.

The decay equation can be rearranged and integrated over a time interval.

where No is the number of parent atoms present at time zero. Integration leads to

t

oN N e tA Ae or

Parent-Daughter Relationships

Radioactive Parent (A) Stable Daughter (B)

A B e.g. 14C 15N (stable)

Production of Daughter = Decay of Parent

, AtB

A A A A o

dNN N e

dt

A B A

2-box model Example: 14C 15N (stable)

t1/2 = 5730 years

Radioactive Parent (A)

Radioactive Daughter (B)

A B A B

source sink

BA A B B

dNN N

dt

,0( )

A BB A t t

B

B A

NN e e

,0( )

A BB A t tBB A

AA e e

A B A B

solution after assuming NB = 0 at t = 0

2-box model

mass balance for B

solution:

Secular equilibrium

time (hr)

Activity

(log scale)

daughter

t1/2

parent Total Activity

Activity of parent

and daughter at

secular equilibrium

T1/2 daughter = 0.8 hr, T1/2 parent =

Grow in of 222Rn

from 226Ra

Example:

After 5 half lives

activity of daughter =

95% of activity of parent

Assume we have a really big wind storm over the ocean so that all the inert gas 222Rn is stripped out of the surface ocean by gas exchange. The activity of the parent

of 222Rn, 226Ra, is not affected by the wind.

Then the wind stops and 222Rn starts to increase (grows in) due to decay.

How many half lives will it take for the activity of 222Rn to equal 50% (and then 95%)

of the 226Ra present?

Answer: Use the following equation:

1/ 20.693 /,0 1 t tB AA A e

Example: Rate of grow in

Radioactive Dating One application of radioactivity is the dating on

archeological and geological specimens by

measuring the concentration of radioactive isotopes.

Carbon dating: the unstable C-14 isotope, produced

during nuclear reactions in the atmosphere that

result from cosmic-ray bombardment, give a small

proportion of C-14 in the CO2 in the atmosphere.

Plants that obtain their carbon from this source

contain the same proportion of C-14 as the

atmosphere.

Radioactive Dating When a plant dies, it stops taking in carbon and its C-

14 undergoes - decay to N-14 with a half-life of 5730

years.

By measuring the proportion of C-14 in the remains,

you can determine how long ago the organism died.

Similar radioactive techniques are used with other

isotopes for dating geological specimens.

Some rocks contain the unstable K-40 isotope, a beta emitter

that decays to the stable Ar-40 nuclide with a half-life of 2.4 x

108 years.

The age of the rock can be determined by comparing the

concentrations of K-40 and Ar-40.

Radioactive Dating

Example: Before 1900 the activity per mass of

atmospheric carbon due to the presence of C-14

averaged about 0.255 Bq per gram of carbon.

a. What number of carbon atoms were C-14?In

analyzing an archeological specimen containing

500 mg of carbon, you observe 174 decays in

one hour.

b. What is the age of the specimen, assuming that

its activity per mass of carbon when it died was

that average value of the air?

Radioactive Dating

a.

nuclei10x6478.6N

s10x8359.3

Bq255.0

ANNA

s10x8359.3

s10x81.1

693.0

T

2ln

s10x81.1T

hr

s3600

da

hr24

yr

da365yr5730T

10

12

12

11

2

1

11

2

1

2

1

Radioactive Dating

b.

yr18.8019s10x5289.2t

s/10x83587.3

gBq255.0

gBq09666.0ln

t

A

Aln

ttA

Aln

1elnelntA

Aln

elnA

Alne

A

AeAA

g

Bq09666.0

g1

mg1000

mg500

Bq04833.0A

Bq04833.0s3600

hr

hr

decay174A

11

12

o

o

o

t

o

t

o

to

Radioactive Dating

Biological Effects of Radiation

As alpha particles, beta particles, neutrons, and EM radiation

such as gamma rays and x-rays, pass through matter, they lose

energy, break molecular bonds, and create ions (which is why

they are called ionizing radiation).

Excessive exposure to radiation, including sunlight, x-rays, and

all the nuclear radiations can destroy tissues.

Mild cases result in a burn, like a sunburn.

Greater exposures can cause severe illness or death by a variety of

mechanisms, including massive destruction of tissue cells, alterations

of genetic material, and destruction of the components in bone marrow

that produce red blood cells.

Calculating Radiation Doses

Radiation dosimetry is the quantitative description of the effect of radiation on living tissue.

Absorbed dose (AD) of radiation is defined as the energy delivered to the tissue per unit mass. SI unit of absorbed dose, the J/kg, is called the Gray

(Gy); 1 Gray = 1 J/kg.

The unit in more common use is the rad (radiation absorbed dose) , defined as 0.01 J/kg; 1 rad = 0.01 J/kg = 0.01 Gy.

Absorbed dose by itself is not an adequate measure of biological effect because equal energies of different kinds of radiation cause different extents of biological effect.

The variation in biological effect is described by a numerical factor called the relative biological effect (RBE), also called the quality factor (QF), of each specific radiation.

The values for RBE depend somewhat on the kind of tissue in which the radiation is absorbed and on the energy of the radiation.

Calculating Radiation Doses

X-rays with 200 keV of energy are defined to have an RBE of 1.

The biological effect is described by the product of the absorbed dose and the RBE of the radiation, this is called the biological equivalent dose (or the equivalent dose, ED). SI unit of equivalent dose is the Sievart (Sv).

1 Sv = 100 rem.

RBE units: Sv/Gy or rem/rad

1 rad = 1 rem (Rngen equivalent for man) = 0.01 J/kg.

Calculating Radiation Doses

RBE for Several Types of Radiation

Radiation RBE (Sv/Gy or

rem/rad)

X-rays and rays 1

Electrons 1 1.5

Slow neutrons 3 5

Protons 10

Particles 20

Heavy ions 20

Equations and Example Equations:

Example: During a diagnostic x-ray examination a 1.2 kg portion of a broken leg receives an equivalent dose of 0.4 mSv. a. What is the equivalent dose in mrem?

b. What is the absorbed dose in J/kg?

c. If the x-ray energy is 50 keV, how many x-ray photons are absorbed?

)()(

)()(

200)(

log

radARBEremE

GyARBESvE

raysxkeVofradindoseequalofeffect

radiationofeffecticalBioRBE

DD

DD

a.

b.

c.

mrem40rem1

mrem1000

Sv01.0

rem1

mSv1000

SvmSv4.0ED

kgJ0004.0A

rad1

kgJ01.0

rem1

rad1

mrem1000

rem1mrem40A

D

D

photons10x9925.5eV50000

photoneV10x996.2photons

eV10x996.2J10x602.1

eV1J00048.0E

J00048.0kg2.1kg

J0004.0E

1015

15

19

Example

Radiation Hazards

An ordinary chest x-ray delivers about 0.2 to 0.4 mSv to

about 5 kg of tissue.

Radiation exposure from cosmic rays and natural

radioactivity in soils, etc, is about 1 mSv (0.001 J/kg) per

year at sea level and twice that at an elevation of 5000 ft.

A whole-body dose of up to about 0.2 Sv (0.2 J/kg)

causes no immediate detectable effect.

A short-term whole-body dose of 5 Sv (5 J/kg) or more

usually causes death within a few days or weeks.

A localized dose of 100 Sv (100 J/kg) causes complete

destruction of the exposed tissues.

Radiation Hazards Long term exposure to radiation can cause various

cancers and genetic defects.

U.S. government regulations are based on maximum

yearly exposure, from all except natural resources, of 2 to

5 mSv.

Workers with occupational exposure to radiation are

permitted 50 mSv per year.

Radiation levels from nuclear power plants is not

negligible, but the health hazards from coal smoke are

serious and the natural radioactivity in the smoke from a

coal-fired power plant is believed to be 100 times as great

as that from a properly operating nuclear power plant.

Radiation Unit Basis

Roentgen (R) 1 R the quantity of x-rays or gamma rays that produces an ionization charge of

0.000258 C/kg in air.

rad (radiation

absorbed dose)

1 rad an absorbed dose of radiation of 0.01 J/kg

Gray (Gy) SI absorbed dose unit; 1 Gy = 1 J/kg = 100 rad

rem (rad

equivalent man)

Effective dose. Relative effectiveness

depends on type of radiation and is

characterized by RBE.

Sievert (Sv) SI unit of effective dose; 1 Sv = 100 rem

Effective dose (in Sv) = dose (in Gy) x RBE

Radiation Units