Decay Rates

10/25/10 
Radioactive Decay

Decay (disintegration) is the process by which a radionuclide changes its number of neutrons and from an unstable combination to a more stable combination. Some people prefer to use the term transformation rather than decay. E l i l d b t d it d l Examples inclu

de 

b

e

ta 

decay, positron 

decay, elec

capture and alpha decay. In radioactive decay, mass is always lost. This maconverted to energy and released. The released ecarried off by any charged particles and/or photonemitted (i.e., radiation).
Activity (A)

Activity is a quantity given the symbol A. It is the dof a specified radionuclide in a sample, i.e., the nudecays per unit time of that radionuclide. The International Commission on Radiation Units Measurements (ICRU) definition (ICRU Report 60"The activity, A, of an amount of radioactive nuclidparticular energy state at a given time is the quotieby dt, where dN is the expectation value of the nuspontaneous nuclear transformations from that enstate in the time interval dt.“
Activity (A)

The term “spontaneous” indicates that the transforare an inherent characteristic of the material and ainduced by external forces (such as neutron bombardment). Because decay is a random process, it is impossibpreci l h t th ti it f l i ill b 

isely what the activity of a sample is, or will be

only specify what the activity was. As such, the ICemploys the term "expectation value", i.e. the activdecay rate we expect to have.
Units of Activity
Units of Activity - the curie (Ci)

Originally, the curie was defined by the InternationRadiological Congress in 1910 as the quantity of remanation (radon gas) in equilibrium with one graradium. Radium itself was quantified by weighing.Later , the curie was defined as the activity of one pure 

Ra-226. 

Today, the curie (Ci) is defined as 3.7 x 1010 dps. According to Ernest Rutherford, the unit was namehonor of both Pierre Curie (the latter had been run a coach) and Marie.
Units of Activity - the curie (Ci)

Multiples of the curie

1 curie (Ci) = 3.7 x 1010 dps (2.22 x 1

1 millicurie (mCi) = 3.7 x 10

7 dps (2.22 x 1

1 i i ( Ci) 3 7 10

4 d (2 22 1

1 microcurie 

(

uCi) = 

3.7 x 10 

4

dps (2.22 x 1

1 nanocurie (nCi) = 3.7 x 101 dps (2.22 x 1

1 picocurie (pCi) = 3.7 x 10-2 dps (2.22 dpm
Units of Activity - the becquerel (B

The becquerel (Bq), named after Henri Becquerel discovered radioactivity, is the basic unit of activity Systeme International (the SI system). Multiples of the becquerel

1 bec

q ( q) uerel 

(

Bq) = 1 d

p

s

1 kilobecquerel (kBq) = 10

3 dps

1 megabecquerel (MBq) = 10

6 dps

1 gigabecquerel (GBq) = 10

9 dps

1 terabecquerel (TBq) = 1012 dps
Units of Activity

In real scientific literature, the units of activity woube expressed as dps or dpm . This is an admissiothis handout is not real scientific literature.The “s”, as in “dps”, is a true unit. The second is a amount of time used as a standard of measure. However, the “d” in “dps” is not a unit. A disintegraevent, or a thing. It is not a specific amount of anyThe scientific literature would employ “s-1” instead 

s-1 means per second
Decay Constant
Decay Constant (8)

The decay constant, symbolized by the Greek lettelambda, can be described (not quite correctly) as tfraction of the atoms (or activity) of a radionuclide expected to decay per unit time. e.g., a decay constant of 0.25 s-1 implies that 25% radionuclide radionuclide s' atoms (or activity) decay per second atoms (or activity) decay per second

It can also be thought of as the proportionality conbetween the number of atoms of a radionuclide anactivity:
Decay Constant (8)

Unfortunately, decay constants aren't found by loothem up in a table or list. They must be calculated the following equation:

where T is the half-life of the radionuclide

Each radionuclide has its own unique decay const
Decay Constant (8) - examples

a. Calculate the decay constant of I-131 in days-1:

Since the half-life of iodine-131 is 8.04 days

This implies (incorrectly) that 8.62% of the activity per day. 
Decay Constant (8) - examples

b. Calculate the decay constant of I-131 in hr-1:

This implies (incorrectly) that 0 359% decays peThis implies (incorrectly) that 0.359% decays pe

c. Calculate the decay constant of I-131 in y-1:

This implies (incorrectly) that 3150% decays pe
Decay Constant (8)

These three decay constants are all correct.What is incorrect is the idea that the decay constafraction of the atoms or activity of a nuclide that deunit time.More correctly the decay constant is the instantanMore correctly, the decay constant is the instantan

decay fraction – the fraction of the activity that decinfinitesimally small amount of time.
The Decay Constant is Immutable 

In general, the decay constant (or half-life) of a radis not affected by time, space, pressure, temperatuelectromagnetic fields, etc. However, there are exceptions: the chemical form sample can affect the decay constant of radionuclundergo electron capture and/or internal conversioundergo electron capture and/or internal conversio

This happens because the chemical bonds formed atom affect the availability of the electrons for capture/conversion. For example, the decay constBe-7 shows a 0.1% difference between beryllium BeF

2. Tc-99m, which undergoes internal conversio

nearly 100% of its decays (transitions), shows a 0

difference in decay constants between KTcO

4 and 
The Mean Life

The mean (average) life of a radionuclide is calculfollows:J is the mean (average) life of the radionuclide (e 

is the mean (average) life of the radionuclide (e.

8Is the decay constant of the radionuclide (e.g., hr

T is the half life of the radionuclide (e.g., hr)
Decay Equation
The Decay Equation

The following equation calculates the activity (A) fospecified number of atoms of a radionuclide (N)

A = 

8

N

The number of decays per unit time equals the frathe atoms decaying per unit time multiplied by the the atoms decaying per unit time multiplied by the 

of atoms. Unfortunately, this equation is rarely of practical vabecause the number of atoms is continually decre
The Decay Equation

However, the next equations are useful because tus the number of atoms or activity of the radionuclfunction of time:Nt is the number of atoms of a radionuclide at time N

0 is the number of atoms of a radionuclide at time 

At is the activity of a radionuclide at time tA

0 is the activity of a radionuclide at time zero

e is the base of the natural log system, e = 2.7182
The Decay Equation - example

Calculate the activity of a 100 Ci I-131 source aftedecayed for 1.5 days. Before using an equation, it’s a good idea to make guesstimate as to the answer.1 5 days is about one 1.5 days is about one

-fifth of iodine fifth of iodine s' 8 day half 8 day half-life 

life

days reduces the activity to 50 percent of what it w

days should only reduce it to ninety percent or so original activity. Therefore, our best guess is that tactivity of the iodine will be about 90 Ci. Now we are ready to use the decay equation.
The Decay Equation - example

Since this answer is right in line with our estimate, wprobably did the calculation correctly. 
The Meaning of e-8t

The term e-8t is the fraction that remains.

As such, the equation A = A0 e-8t means that the a

that you have is the activity you started with times fraction that remains!

Si il l th ti N N 8t Similarly, the equation N = N0 e th t th 

-8t means that the 

of atoms of a radionuclide that you have is the numstarted with times the fraction that remains!
Back Calculating Activities

In some situations, it is necessary to back calculatactivity of a source to some earlier date:At is the activity at some earlier specified time t A

0 is the activity at time zero

Note, At is greater than A

0
Calculating the Decay Time

Should it be necessary to calculate how much time required for a radionuclide to decay from an initial (A

0) to a specified final activity (At), the following e

can be used:

For what it is worth, the value At/A

0 is the fraction o

original activity remaining.
Calculating the Decay Time

The number of half lives required to reduce its origactivity (A

0) to a desired value (At) is calculated as 

A h it t k As suc

h it

t

a

kes: 

3.32 half lives for the activity to decay to 10% 

original activity6.65 half lives for the activity to decay to 1% ooriginal activity9.97 half lives for the activity to decay to 0.1% original activity 
Graphing Activity as 

a Function of Time a Function of Time
tivity

100

50

Linear Plot

A0

Act

Number of Half-lives

50

0

0 1 2 3 4 5 6

25

12.56.25

A = A0 e-8t
Problem with a Linear Plot

The problem with a linear plot is that a curved line hard to draw (as if anyone actually does that).It is hard to interpolate between data points.It is hard to extra

py p olate be

yond data 

points.
A semilog plot such as the one on the following sli

solve all our problems!On the semilog plot, time is indicated with a linear while activity (or number of atoms) is indicated witlo

garithmic scale.

Semilog Plot
100

10

Semilog Plot

5025

Logarithmic 12.5

A0

A = A0 e-8t

Linear scale

10

1 0 1 2 3 4 5 6 7 

6.253.121.56

g

scale
tivity

100

10

The straight line represents aexponential decrease of acti

After seveli th 

Semilog Plot

5025

12.5

Act

Number of Half-lives

10

1

lives, the a

less than 1what it sta

0 1 2 3 4 5 6 7 

6.253.121.56
Logarithmic 

scale

128

6432

256512

1024

100

1000

16

4

32

21

10

8

1 

A log scale can begin with any number except zero
The specific activity of a radionuclide is its activity 

mass (of the pure material). The "old" and SI units and Bq/kg respectively. As might be expected from the original definition ocurie, the specific activity of Ra-226 is close to 1 CSaying that something has a high specific activity Specific ActivitySaying that something has

a high specific activity 

same as saying that it has a short half-life. To calculate the specific activity (SpA) in Bq/g, we following formula:

SpA = N

g

8

Ng is the number of atoms of the nuclide in one g

pure material and 

8 is the decay constant in units 
Example: calculate the specific activity of U-235. 

First, calculate the number of U-235 atoms (N

g) in 

Since there are 235 grams of U-235 per mole, in othere will be 1/235 moles, i.e, 4.255 x 10-3 moles. 

Specific Activity – example calculatSince there are 6.023 x 1023 atoms per mole (Ava

number), there are the following number of atoms in one gram of U-235:N

g = 4.255 x 10-3 moles/gram x 6.023 x 1023 a

= 2.563 x 1021 atoms/gram
Next, calculate the decay constant (

λ) for U-235, 

(T) of which is 7.04 x 10

8 a (years). 

Because we want to calculate an activity in dps, thof U-235 should be expressed in seconds.

λ = 0 693/

T

Specific Activity – example calculat

=

0.693/

T

= 0.693/(7.04 x 10

8 a x 365.25 d/a x 24 h/d x 3

= 0.693/2.222 x 1016

s

= 3.119 x 10-17

s-1
Finally, multiply λ times Ng to calculate the specific 

SpA = Ng λ

= 2.563 x 1021 atoms/g x 3.119 x 

= 7 995 x 104 s-1 g-1 (dps per gram)

Specific Activity – example calculat

= 7.995 x 104 s 1 g 1 (dps per gram)= 7.995 x 104 Bq g-1 (Bq per gram)

= 2.16 x 10-6 Ci g-1 (Ci per gram)
Air Kerma Strength

Air kerma strength is the quantity used to describe strength of brachytherapy sources. It can be thoua substitute for activity. More specifically, it is the arate at some specified distance from the source. Tequation for the air kerma strength is:

S

k

= K l

2

S

k

K l

S

K is the air kerma strength (e.g., uGy m

2/h

K is the air kerma rate in free space (e.g., ul is the reference distance (m)

The units of air kerma strength are often given the U where 1 U = 1 uGy m

2/h
Air Kerma Strength

Air kerma strength is an empirical measured quanrather than a calculated one. It is inappropriate to calculate the air kerma rate obrachytherapy source based on its activity (e.g., mSimilarly it is inappropriate to calculate the activity Similarly, it is inappropriate to calculate the activity 

source based on its air kerma strength.
Air Kerma Strength

Despite the previous comments, conversion factor“apparent” activity to air kerma strength are requir

real world. The following are from Med Phys 18 (3448.

Au-198 seeds 1 mCi 

≈ 2.06 U

Pd-103 seeds 1 mCi 

≈ 1.29 U

I-125 seeds 1 mCi 

≈ 1.27 U

See Code of practice for brachytherapy physics: Rthe AAPM Radiation Therapy Committee Task Gro56 (Medical Physics 24 (10); 1557-1598,Oct. 199
Appendix A

The Meaning of (1 8t – e- )
The Meaning of (1 - e

-

8

t

)

The term (1 - e-8t ) is the fraction that decayed.

If e-8t = 0.25, then 25% of the original number of 

remain.

And 

(1 

– 0.25

) g means that 0.75 or 75% of the ori

g

number of atoms have decayed over the time periThe number of decays of a radionuclide during tim

N

d = N

0 (1 - e-8t )

N

d is the number of decays during time t

N

0 is the number of atoms of the radionuclide at t 
The Meaning of (1 - e

-

8

t

)

When a radioactive sample is counted with a detenumber of counts is:

C = N

0 E (1 - e-8t )

C is the number of counts

N

0 is the number of atoms of the radionuclid

start of the count (t = 0)E is the detector efficiency (counts/decay)
The Meaning of (1 - e

-

8

t

)

Rewriting the previous equation gives:Multiply both sides by 

8 gives:

Hence:
Appendix B

Some Obsolete Units of Activ
Some Obsolete Units of Activity

While the following units are no longer used, they be encountered in early literature and/or on old radsources.1. Rutherford The Rutherford (rd) was defined in 1930 by the International Radium Standard Commission as 10

Such a unit was needed because the Commission to using the curie to describe the activities of radiooutside the radium family (radium or its daughters
Some Obsolete Units of Activity

2. Strength The strength was a poorly defined unit primarily embetween 1904 and 1920. It was related to the intethe gamma rays emitted by uranium. If a radium swas said to have a stren

g ,, th of 500,000, it meant th

a gamma intensity 500,000 times greater than an mass of uranium. The terms intensity and activity wsometimes used to mean the same thing.
Some Obsolete Units of Activity

3. UranieThe uranie was defined in 1905 by W. D. Butcherpresident of the Rontgen Society, as the quantity oradiation emitted by one gram of uranium. One graradium was e

qual to one million uranies. 
Some Obsolete Units of Activity

4. Mache This unit was defined in 1904 and used exclusivelradon. In general, it was used to express the activradon in one liter of water (occasionally air). One mwas equal to 364 pCi. If it wasn't stated, it could usinferred that the activity was per liter inferred that the activity was per liter. 

The Mache Unit was applied in different ways and difficult to equate it to a radium activity. However, arough approximation, a 2700 Mache Units usually a radium-226 activity of at least 1 uCi.
Some Obsolete Units of Activity

5. EmanOne eman was equal to 100 pCi of radon. In other respects, the eman was similar to the mache unit.
Some Obsolete Units of Activity

6. Special Curie Unit for Natural Uranium Natural uranium, whether in ore, or in chemically pform, is a mix of three isotopes: U-234, U-235 and In one gram of pure natural uranium, these isotopehave the following activities: 7.36 x 10

5 dpm U-234

10

4 dpm

U 235; and 7 36 x 10

5 10 dpm

U 238

4 dpm

U

-235; and

7.36

x 10

5 dpm

U

-238. 

Originally, limits on natural uranium concentrations water etc. were expressed in milligrams. By massuranium is almost entirely U-238. Later, when it becommon to express those same limits in units of ahad to be clarified whether the limit expressed the activity (summed activities of U-234, U-235 and U

just the activity of the U-238. 
Some Obsolete Units of Activity

6. Special Unit for Natural Uranium - continueTo reduce the confusion, the ICRP (1955) stated " of natural uranium is considered to correspond to dis/sec from U-238, 3.7 x 1010 dis/sec from U-234 

10

8 dis/sec from U-235". Later, the value for U-235 

corrected to 1.7 x 10

9 dis

/sec. In other words, the 

natural uranium was 7.57 x 1010 dps (4.542 x 1012 

Therefore, the activity of one gram of pure uranium 

be 3.3 x 10-7 curies.