Tsoulfanidis N. Measurement and detection of radiation

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MEASUREMENT

AND DETECTlON

RADIATION

Energy

130

0.23%

(MeV)

Energy (MeV)

23%

77%

Figure

3.7

The

alpha decay scheme of

238~.

The

percentage values give the probability of decay

through the corresponding level (from Ref.

4).

Decay.

This type of decay is represented by

where

negative beta particle

electron

antineutrino

Historically, the name

particle has been given to electrons that are

emitted by nuclei undergoing beta decay. The antineutrino

(F)

is a neutral

particle with rest mass so small that it is taken equal to zero.

The energy equation of

decay is

MN(A, Z)

MN(A,Z

Tp-+

(3.48)

where MN(A,

is the

nuclear

mass and

is the electron rest mass. Using

atomic masses, Eq. 3.48 becomes (see Sec. 3.5)

The momentum equation is

Pp-+ PF

(3.50)

The

decay energy,

Qp-,

is defined as

Qp-=

M(A, Z)

M(A, Z

(3.51)

The condition for

decay to be possible is

M(A,Z)

M(A,Z

In terms of Qp-, Eq. 3.49 is rewritten in the form

Tp-+

Qp-

Equations 3.50 and 3.53 show that three particles, the nucleus, the electron,

and the antineutrino, share the energy Q,-, and their total momentum is zero.

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OF ATOMIC

AND

NUCLEAR

PHYSICS

There is an infinite number of combinations of kinetic energies and momenta

that satisfy these two equations and as a result, the energy spectrum of the betas

is continuous.

Eq.

3.53, the energy of the nucleus,

TM,

is much smaller than either T,-

because the nuclear mass is huge compared to that of the electron or the

antineutrino. For all practical purposes, TM can be neglected and

Eq.

3.53 takes

the form

in the case of

decay, the daughter nucleus may be left in an excited

state after the emission of the

particle. Then, the energy available to

become kinetic energy of the emitted particles is less. If the nucleus is left in the

ith excited state

Ei,

Eq.

3.54

takes the form

According to

Eq.

3.54, the electron and the antineutrino share the energy Qp-

(or

Emax)

and there is a certain probability that either particle may have an

energy within the limits

which means that the beta particles have a continuous energy spectrum. Let

P(T)

be the number of beta particles with kinetic energy between T and

dT.

The function B(T) has the general shape shown in Fig. 3.8. The energy

spectrum of the antineutrinos is the complement of that shown in Fig. 3.8,

consistent with

Eq.

3.57. The continuous energy spectrum of

particles

should be contrasted with the energy spectrum of internal conversion electrons

shown by Fig. 3.6.

As stated earlier, beta particles are electrons. The practical difference

between the terms

electrons

and

betas

is this: A beam of electrons of energy

consists of electrons each of which has the kinetic energy

A beam of beta

particles with energy

Emax

consists of electrons that have a continuous energy

spectrum (Fig 3.8) ranging from zero up to a maximum kinetic energy

Em,,.

Figure 3.9 shows the

P-

decay scheme of the isotope '::CS. For an example

of a

Qp-

calculation, consider the decay of '35:~s:

0.0012625(931.478 MeV)

1.1760 MeV

1.36

10-13

the ':;~a is left in the 0.6616-MeV state (which happens 93.5 percent of the

time), the available energy is

Emax

.I760

0.6616

0.5144 MeV

100

MEASUREMENT AND DETECTION OF RADIATION

Figure

3.8

typical beta energy spectrum (shows shape only; does not mean that

P-

is more

intense than

P').

For many calculations it is necessary to use the average energy of the beta

particles,

&-.

accurate equation for

has been developed: but in

practice the average energy is taken to be

Em,,

Ep-=

Decay.

The expression representing

/3+

decay is

where

positron

neutrino

Probability for

transition

Figure

3.9

The decay scheme of 137~s. The

value of the

P-

decay is

1.176

MeV (from Ref.

4).

Probability for each transition is given in percent.

REVIEW

ATOMIC

AND

NUCLEAR

PHYSICS

101

The energy equation of

P+

decay is

Using atomic masses, Eq.

3.58

becomes

The momentum equation is

The

P+

decay energy is

The condition for

P+

decay to be possible is

comparison of Eqs.

3.52

and

3.62

shows that

P-

decay is possible if the

mass of the parent is just bigger than the mass of the daughter nucleus, while

decay is possible only if the parent and daughter nuclear masses differ by at

least

2mc2

1.022

MeV.

The energy spectrum of

P+

particles is continuous, for the same reasons

the

spectrum is, and similar to that of

decay (Fig.

3.8).

The average

energy of the positrons from

decay,

&+,

is also taken to be equal to

E,,,/3

unless extremely accurate values are needed, in which case the equation given in

Ref.

should be used.

typical

P+

decay scheme is shown in Fig.

3.10.

Electron Capture.

In some cases, an atomic electron is captured by the nucleus

and a neutrino is emitted according to the equation

Figure

3.10

The decay scheme of "~a. Notice that it is

Q,,

that is plotted, not

QS+

(from Ref.

4).

102

MEASUREMENT

AND

DETECTION

RADIATION

In Eq. 3.63 all the symbols have been defined before except Be, binding energy

of the electron captured by the nucleus. This transformation is called electron

capture

(EC). In terms of atomic masses, Eq. 3.63 takes the form

The energy

QEc

released during EC is

The condition for EC to be possible is

Electron capture is an alternative to

/3+

decay. Comparison of Eqs. 3.61 and

3.66 shows that nuclei that cannot experience

/3+

decay can undergo EC, since

a smaller mass difference is required for the latter process. Of course, EC is

always possible if

/3+

decay is. For example, "~a (Fig. 3.10) decays both by

/3+

and EC.

After EC, there is a vacancy left behind that is filled by an electron falling

in from a higher orbit. Assuming that a

electron was captured, an

electron

may fill the empty state left behind. When this happens, an energy approxi-

mately equal to

B, becomes available (where B, and BL are the binding

energy of a

electron, respectively). The energy BK

BL may be emitted

as a

X-ray called fluorescent radiation, or it may be given to another atomic

electron. If this energy is given to an

electron, that particle will be emitted

with kinetic energy equal to (B,

B,)

2BL.

Atomic electrons

emitted in this way are called Auger electrons.

Whenever an atomic electron is removed and the vacancy left behind is

filled by an electron from a higher orbit, there is a competition between the

emission of Auger electrons and fluorescent radiation. The number of X-rays

emitted per vacancy in a given shell is the fluorescent yield. The fluorescent yield

increases with atomic number.

3.7.4

Particles, Antiparticles, and Electron-Positron Annihilation

Every known subatomic particle has a counterpart called the antiparticle.

charged particle and an antiparticle have the same mass, and opposite charge. If

a particle is neutral-for example, the neutron-its antiparticle is still neutral.

Then their difference is due to some other property, such as magnetic moment.

Some particles, like the photon, are identical with their own antiparticles.

antiparticle cannot exist together with the corresponding particle: when an

antiparticle meets a particle, the two react and new particles appear.

Consider the example of the electron and the "antielectron," which is the

positron. The electron and the positron are identical particles except for their

charge, which is equal to e but negative and positive, respectively. The rest mass

of either particle is equal to 0.511 MeV. A positron moving in a medium loses

REVIEW

ATOMIC

AND NUCLEAR

PHYSICS

103

energy continuously, as a result of collisions with atomic electrons (see Chap.

4).

Close to the end of its track, the positron combines with an atomic electron, the

two annihilate, and photons appear with a total energy equal to 2mc2. At least

two photons should be emitted for

conservation of energy and momentum to be

satisfied (Fig.

3.11).

Most of the time, two photons, each with energy

0.511

MeV,

are emitted.

a result, every positron emitter is also a source of 0.511-MeV

annihilation gammas.

3.7.5 Complex Decay Schemes

For many nuclei, more than one mode of decay is positive. Users of radioiso-

topic sources need information about particles emitted, energies, and probabili-

ties of emission. Many books on atomic and nuclear physics contain such

information, and the most comprehensive collection of data on this subject can

be found in the Table

Isotopes by Lederer and ~hirley.~ Figure 3.12 shows an

example of a complex decay scheme taken from that book.

3.8

THE RADIOACTIVE DECAY

LAW

Radioactive decay is spontaneous change of

nucleus. The change may result in

a new nuclide or simply change the energy of the nucleus. If there is a certain

amount of

radioisotope at hand, there is no certainty that in the next second

"so many nuclei will decay" or "none will decay." One can talk of the probabil-

ity that a nucleus will decay in a certain period of time.

The probability that a given nucleus will decay per unit time is called the

decay constant and is indicated by the letter

For a certain species,

The same for all the nuclei

2. Constant, independent of the number of nuclei present

Independent of the age of the nucleus

Consider a certain mass m of a certain radioisotope with decay constant

The number of atoms (or nuclei) in the mass m is equal to

where

6.022

loz3

Avogadro's number

atomic weight of the isotope

0.51

Figure

3.11

Electron-positron anni-

0.51

MeV

hilation.

104

MEASUREMENT

AND

DETECTION

RADIATION

Figure

3.12

complex decay scheme. For complete explanation of all the symbols and numbers see

Ref.

Half-life is given for each element's ground state, and energy of each level is given at

intermediate states.

is the neutron separation energy. Transition probabilities are indicated as

percentages (from Ref.

4).

This number of atoms decreases with time, due to the decay according to

Decrease per unit time

decay per unit time

or mathematically,

The solution of this equation is

where

N(0)

number of atoms at t

The probability that a nucleus

will

not

decay

in time t-i.e., it will survive

time t- is given by the ratio of

atoms not decaying in time t

N(0)e-At

- -

e-~t

(3.70)

atoms at

NO)

REVIEW

ATOMIC

AND

NUCLEAR

PHYSICS

105

The probability that the nucleus will decay between t and t

dt is

p(t) dt

(probability to survive to time t)(probablility to decay in dt)

ePA'

The average lifetime

of the nucleus is given by

One concept used extensively with radioisotopes in the half-life T, defined

as the time it takes for half of a certain number of nuclei to decay. Thus, using

Eq. 3.69,

N(T)

-A,

NO) 2

which then gives the relationship between

and T:

For a sample of N(t) nuclei at time

each having decay constant A, the

expected number of nuclei decaying per unit time is

where A(t)

activity of the sample at time t.

The units of activity are the Becquerel (Bq), equal to

decay/s, or the

Curie (Ci) equal to 3.7

10'' Bq. The Becquerel is the SI unit defined in 1977.

The term specific activity (SA) is used frequently. It may have one of the two

following meanings:

For solids,

activity

mass

(Bq/kg or Ci/g)

2. For gases or liquids,

activity

(Bq/m3 or Ci/cm3)

volume

Example

3.8

What is the SA of 60~o?

Answer

The SA is

AN In2

SA=-=-=-m-=

(In 2)(6.022

loz3)

m m Tm

(5.2 y)(3.16

10' s/y)(0.060 kg)

106

MEASUREMENT

AND DETECIlON

RADIATION

Example

3.9

What is the

of a liquid sample of m3 containing

kg of

32~?

Answer

The

AN In2

SA=-

=-m-=

(In 2)(1OP6 kgN6.022

loz3)

m3)(14.3 d)(864OO s/d)(O.OX kg)

There are isotopes that decay by more than one mode. Consider such an

isotope decaying by the modes 1,2,3,.

. .

(e.g., alpha, beta, gamma, etc.,

decay), and let

probability per unit time that the nucleus will decay by the ith mode

The total probability of decay (total decay constant) is

+A2

-..

+Ai

0.-

If the sample contains N(t) atoms at time t, the number of decays per unit time

by the ith mode is

The term partial

haZf-lz$e

is sometimes used to indicate a different decay mode.

is the partial half-life for the ith decay mode, using Eqs. 3.72 and 3.74, one

obtains

It should be pointed out that it is the total decay constant that is used by Eqs.

3.69 and 3.73a, and not the partial decay constants.

Example

3.10

The isotope 252~f decays by alpha decay and by spontaneous

fission. The total half-life is 2.646 years and the half-life for alpha decay is 2.731

years. What is the number of

spontaneous fissions per secondeper 10-' kg (1 g)

252

~f?

Answer

The spontaneous fission activity is

The spontaneous fission half-life is, using Eq. 3.75,

REVIEW

ATOMIC

AND

NUCLEAR

PHYSICS

107

Therefore,

Sometimes the daughter of a radioactive nucleus may also be radioactive

and decay to a third radioactive nucleus. Thus, a radioactive chain

etc.

is generated.

example of a well-known series is that of 235~, which through

combined

and

P-

decays ends up as an isotope of lead. The general equation

giving the number of atoms of the ith isotope, at time

in terms of the decay

constants of all the other isotopes in the chain was developed by

at ern an.^

&(O) is the number of atoms of the ith isotope of the series at time

0 and

then the Bateman equation takes the form

Example

3.11

Apply the Bateman equation for the second and third isotope

in a series.

Answer

3.9

NUCLEAR REACTIONS

3.9.1

General Remarks

nuclear reaction

is an interaction between two particles, a fast bombarding

particle, called the

projectile,

and a slower or stationary

target.

The products of

the reaction may be two or more particles. For the energies considered here

MeV), the products are also two particles (with the exception of fission,

which is discussed in the next section).