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ICRU REPORT 60.

Fundamental Quantities

and Units for lonizing

Radiation

lssued: 30 December 1998

IN.TERNATIONAL COMMISSION ON RADIATION

UNITS ANO MEASUREMENTS

791O WOODMONT AVEN UEBETHESDA, MARYLAND 20814

U.S.A.

Stralingsbeschermingsdienst TU/e

Leergang 3, 2000-2001


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Fundamental Quantities and

Units for Ionizing RadiationIntroduction

This report supersedes PartA of ICRU Report 33 (ICRU, 1980), dealing with quantities and units forgeneral use. Part B of ICRU Report 33, coveringquantities and units for use in radiation protection,has already been replaced by ICRU Report 51 (ICRU,

1993a) entitled Quantities and Units in Radiation

Protection Dosimetry.The present report deals with the fundamental

- ; ..- {Uantities and units for ionizing radiation. Drafts ofits main sections, namely radiometry, interactioncoefficients and dosimetry, have been published forcomment in the ICRU News. The ICRU appreciatesthe assistance rendered by scientific bodies andindividuals who submitted comments, and hopesthat this process will facilitate the acceptance of thereport.

The report is structured in five majar sections,

each of which is followed by tables summarizing, foreach quantity, its symbol, unit and the relation used

in its definition.

Section 1 deals with terms and mathematicalconventions used throughout the report.

Section 2, entitled Radiometry, presents quantities required for the specification of radiation fields.Two classes of quantities are used referring either tothe number of particles or to the energy transportedy them. Accordingly, the definitions of radiometric

' - ,..J.uantities are grouped into pairs. Both scalar andvectorial quantities are defined.

Interaction coe:fficients and related quantities arecovered in Section 3. The fundamental interaction

coefficient is the cross section. All other coefficientsdefined in this section can be expressed in terms ofcross section or differential cross section. The defin-

tion of the linear energy transfer (LET) given in thepresent report differs from that given previously(ICRU, 1980) by the inclusion ofthe binding energyfor all collisions.

Section 4 deals with dosimetric quantities whichdescribe the processes by which particle energy is

converted and finally deposited in matter. Accordingly, the definitions of dosimetric quantities arepresented in two parts entitled Conversion ofEnergyand Deposition ofEnergy, respectively. The first partincludes a new quantity, cerna (converted energy perunit mass) for charged particles, paralleling kerma

(kinetic energy released per unit mass) for uncharged particles. Cerna differs from kerma in thatcerna involves the energy lost in electronic collisionsby the incoming charged particles while kerma involves the energy imparted to outgoing chargedparticles . In the second part on deposition of energy,a new quantity termed energy deposit is introduced.

Energy deposit, i.e., the energy deposited in a singleinteraction, is the fundamental quantity in terms ofwhich all other quantities presented in the sectioncan be defined. These are the traditional stochasticquantities, energy imparted, lineal energy and specific energy, the latter leading to the non-stochasticquantity absorbed dose.

Quantities related to radioactivity are defined inSection 5.

Much work has been devoted to the current docu ment to ensure it is scientifically rigorous and asconsistent as possible with similar publications used

in other fields of physics. It is hoped that this reportrepresents a modest step towards a universal scientific language.

1


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l. General Considerations

This section deals with terms and mathematical

conventions used throughout the report.

1.1 Quantities and Units

Quantities, when used for the quantitative description of physical phenomena or objects, are generally

called physical quantities. A unit is a selected reference sample of a quantity with which other quanti

ties of the same kind are compared. Every quantity

may be expressed as the product of a numerical valueand a unit. As a quantity remains unchanged when

the unit in which it is expressed changes, its numeri

cal value is modified accordingly.Quantities can be multiplied or divided by one

another resulting in other quantities. Thus, all quan

tities can be derived from a set of base quantities.The resulting quantities are called derived quantities.

Asystem of units is obtained in the same way byfirst defining units for the base quantities, the baseunits, and then forming derived units. A system issaid to be coherent if no numerical factors other thanthe number 1 occur in the expressions of derived

units.

The ICRU recommends the use of the International System of U its (SI) (BIPM, 1998). In thissystem, the base units are metre, kilogram, second,

ampere, kelvin, mole, and candela, for the base

quantities length, mass, time, electric current, ther

modynamic temperature, amount of substance, and

luminous intensity, respectively.

Sorne derived SI units are given special names,such as coulomb for ampere second. Other derived

units are given special names only when they areused with certain derived quantities. Special names

currently in use in this restricted category are

becquerel (equal to reciproca! second for activityof a radionuclide) and gray (equal to joule perkilogram for absorbed dose, kerma, cerna and spe

cific energy). Sorne examples of SI units are given in

Table 1.1.

There are also a few units outside of the intemational system that may be used with SI. For sorne of

these, their values in terms of SI units are obtained

experimentally. Two of these are used in current

ICRU documents-electron volt (symbol eV) and

(unified) atomic mass unit (symbol u). Others, such

as day, hour and minute, are not coherent with thesystem but, because of long usage, are permitted to

be used with SI (see Table 1.2).

Decimal multiples and submultiples of SI units2 can be formed using the SI prefixes (see Table 1.3).

TABLE 1.1-SIunits used in this report

Category ofunits Quantity N ame Symbol

SI base units length metre m

mass kilogram kg

time second S

amount of substance mole mol

SI derived units with electric charge coulomb especial names energy joule J(general use) solid angle steradian sr

power watt wSI derived units with activity becquerel Bq

special names absorbed dose, gray Gy

(restricted use) kerma, cerna, spe-

cific energy

1.2 Ionizing Radiation

Ionization produced by particles is the process bywhich one or more electrons are liberated in colli sions of the particles with atoms or molecules. Thismay be distinguished from excitation, which is atransfer of electrons to higher energy levels in atomsor molecules and generally requires less energy.

When charged particles have slowed down suffi

ciently, ionization becomes less likely or impossibleand the particles increasingly dissipate their remain-

TABLE 1.2-Some unts used with the SI

Category ofunits Quantity Name Symbol

Units widely used time minute min

hour hday d

Units whose values in energy electron volta eVSI are obtained mass (unified) atomie mass u

experimentally unita

a 1eV= 1.602 177 33(49) 10-19J. 1u= 1.660 540 2(10) 10-27kg. The digits in parentheses are the one-standard-deviation

uncertainty in the last digits ofthe given value (CODATA, 1986).

TABLE 1.3-Sl prefixesa

Factor Pref ix Symbol Factor Prefix Symbol

1024 yotta y w-1 deci d1021 zetta z w-2 centi e1018 exa E w-3 milli m1015 peta p w-s micro J.l1012 tera T w-9 nano n109 giga G 10-12 pico p1Q6 mega M 10-15 femto f103 kilo k 10-18 atto a

102

hecto h 10-21

zepto z

101 deca da lQ-24 yocto y,....---......_....,

a The prefix symbol attached to the unit symbol constitutes anew symbol, e.g., 1fm2 = no-15 m)2 = w-30 m2.


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I

----Ing energy in other processes such as excitation orelastic scattering. Thus, near the end oftheir range,charged particles that were ionizing become nonionizing.

The term ionizing radiation refers to charged

particles (e.g., electrons or protons) and unchargedparticles (e.g.,photons or neutrons) that can produceionizations in a medium. In the condensed phase, thedifference between ionization and excitation can

become blurred. A pragmatic approach for dealingwith this ambiguity is to adopt a threshold for theenergy that can be transferred to the medium at thelocations called energy transfer points (see Section4.2.1). This implies cut-off energies below whichcharged particles are not considered to be ionizing.Below such energies, their ranges are minute. Hence,

the choice of the cut-off energies does not materially

affect the spatial distribution of energy depositionxcept at the smallest distances that may be of----concern in microdosimetry. The choice of the thresh

old value depends on the application; for example, a

-value of 10eV- may be appropriate for radiobiology.

1.3 Stochastic and Non-Stochastic Quantities

Differences between results from repeated observations are common in physics. These can arise fromimperfect measurement systems, or from the factthat many physical phenomena are subject to inherent fiuctuations. Thus, one distinguishes between anon-stochastic quantity with its unique value and a

stochastic quantity, the values of which follow aprobability distribution. In many instances, thisdistinction is not significant because the probabilitydistribution is very narrow. For example, the measurement of an electric current commonly involveso many electrons that fiuctuations contribute negli-

-":.tibly to inaccuracy in the measurement. Althoughsimilar considerations often apply to radiation, fluctuations can play a significant role, and may need to

be considered explicitly.Certain stochastic processes follow a Poisson distri

bution, a distribution uniquely determined by itsmean value. A typical example of such a process isradioactive decay. However, more complex distributions are involved in energy deposition. In thisreport, because of their relevance, four stochasticquantities are defined explicitly, namely energy deposit, e (see Section 4.2.1), energy imparted, e (seeSection 4.2.2), lineal energy, y (see Section 4.2.3), andspecific energy, z (see Section 4.2.4).

For example, the specific energy,z, is defined asthe quotient ofthe energy imparted, e, and the mass,m. Repeated measurements would provide an est

mate of the probability distribution of z and of itsfirst moment, z,which approaches the absorbed dose,.J, (see Section 4.2.5) as the mass becomes small.Knowledge of the distribution of z may not be

required for the determination of the absorbed dose,

D. However, knowledge of the distribution of zcorresponding to a known D can be important because in the irradiated mass element, m, the effectsof radiation are more closely related to z than to D,

and the values ofz can differ greatly fromD for smallvalues of m (e.g.,biological cells).

1.4 Mathematical Conventions

To permit characterization of a radiation field andits interactions with matter, many of the quantitiesdefined in this report are considered as functions ofother quantities. For simplicity in presentation, thearguments on which a quantity depends often willnot be stated explicitly. In sorne instances, the distribution of a quantity with respect to another quantitycan be defined. The distribution function of a discretequantity, such as the particle number N, (see Section2.1.1) will be treated as if it were continuous, since Nis usually a very large number. Distributions withrespect to energy are Jrequently required. For example, the distribution offluence (see Section 2.1.3)with respect to particle energy is given by (see Eq.2.1.6a)


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2. Radiometry

Radiation measurements and investigations of

adiation effects require various degrees of specifica- - on of the radiation field at the point of interest.

- . adiation fields consisting of various types of par-

eles, such as photons, electrons, neutrons, or pro

-. 1ns, are characterized by radiometric quantities

- hich apply in free space and in matter.

Two classes of quantities are used in the character

_ation of a radiation field, referring either to the

Limber of particles or to the energy transported by

- 1em.Accordingly, most ofthe definitions ofradiomet

ic quantities given in this report can be grouped into

. airs.

Both scalar and vectorial quantities are used in

radiometry and here they are treated separately.

Formal definitions of quantities deemed to be of

particular relevance are presented in boxes. Equiva

lent definitions which are used in particular applica

tions are given in the text. Distributions of sorne

radiometric quantities with respect to energy are given when they will be required later in the report.

An extended set of quantities relevant to radiometry

is presented in Tables 2.1 and 2.2.

2.1 Scalar Radiometric QuantitiesConsideration of radiometric quantities begins

with the definition of the most general quantities

associated with the radiation field, namely, theparticle number, N, and the radiant energy, R (seeSection 2.1.1). The full description of the radiation

field, however, requires information on the type and

the energy of the partid es as well as on their spatial,

directional and temporal distributions. In the pre

sent report, the specification of the radiation field is

achieved with increasing detail, by defining radiomet

ric quantities through successive differentiations of

N andR with respect to time, area, volume, direction or energy. Thus, these quantities relate to a particu

lar value of each variable of differentiation. This

procedure provides the simplest definitions of quan

tities such as fluence and energy fluence (see Section

2.1.3), often used in the common situation where

radiation interactions are independent of the direc

tion and temporal distribution of the incoming par

ticles.

The scalar radiometric quantities defined in this

reportare used also for fields of optical and ultravio

let radiations, sometimes under different names.

The equivalence between the various terminologies is noted in connection with the relevant defini

tions.

4

2.1.1 :particle Number, Radiant Energy

The particle number, N, is the number of particles that are emitted, transferred, or received.

Unit: 1

The radiant energy,R, is the energy (excludingrest energy) of the particles that are emitted,

transferred or received.

Unit: J

For particles of energyE (excluding rest energy),

the radiant energy,R, is equal to the productNE.The distributions, NE and RE, of the particlenumber and the radiant energy with respect toenergy are given by

(2.1.1a)

and

RE= dR/dE, (2.1.1b)

where dN is the number of particles with energy

betweenE andE + dE and dR is their radiant energy.The two distributions are related by

(2.1.2)

The volumic particle number, n, is given by

n = d.NidV, (2.1.3)

where dN is the number of particles inthe volume dV.nis also termed number density ofparticles (ISO, 1993).

2.1.2 Flux, Energy Flux

The flux,N, is the quotient of dN by dt, where dNis the increment of the particle number in thetime interval dt, thus

. dNN=-

dt

Unit: s-1

The energy flux,R, is the quotient of dR by dt,.where dR is the increment of radiant energy inthe time interval dt, thus

.dRR=dt

Unit: W

-----.....\


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,__ These quantities frequently refer to a limited

spatial region, e.g., the flux of particles emergingfrom a collimator. For source emission, the flux in all

directions is generally considered.

For visible light and related electromagnetic radia

tions, the energy flux is defined as power emitted,transmitted, or received in the form of radiation and

termed radiant flux or radiant power (CIE, 1987).

The term flux has been employed for the quantity

termed fluence rate in the present report (see Section

2.1.4). This usage is discouraged because of the

possible confusion with the above definition offiux.

2.1.3 Fluence, Energy Fluence

The fiuence, 4>, is the quotient of dNby da, wheredN is the number of particles incident on a sphere

The distributions, 4>E and '/!E, of the fiuence andenergy fiuence with respect to energy are given by

cpE = dcJJ/dE (2.1.6a)

and

(2.1.6b)

where d4> is the fiuence of particles of energy between E and E + dE and d1JI' is their energy fiuence.

The relationship between the two distributions is

given by

(2.1.7)

The energy fiuence is related to the quantity

radiant exposure defined, for fields ofvisible light, as

the quotient of the radiant energy incident on a

surface element by the area of that element (CIE,

..-- of cross-sectional area da, thus

dN4>=-

da

Unit: m-2

The energy fiuence, P, is the quotient of dR byda, where dR is the radiant energy incident on asphere of cross-sectional area da, thus

dR

1Jt=-da

Unit: Jm-2

The use of a sphere of cross sectional area da expresses in the simplest manner the fact that one

considers an area da perpendicular to the direction

uf each particle. The quantities fluence and energy

fiuence are applicable in the common situation in which

radiation interactions are independent of the directionof the incoming particles. Incertain situations, quantities (defined below) involving the differential solid angle,

d!2, in a specified direction are required.

In dosimetric calculations, fiuence is frequently

expressed in terms of the lengths of the particle trajecto

ries. It can be shown that the fiuence, cJJ, is given by

4> = dl/dV, (2.1.4)

where dl is the sum ofthe particle trajectory lengthsin the volume dV.

For a radiation field that does not vary over the

time interval, t, and which is composed of particleswith velocity u, the fluence, 4>, is given by

1987). When a parallel beam is incident atan angle ewith the normal direction to a given surface element,the radiant exposure is equal to 1JI' cos e.

2.1.4 Fluence Rate, Energy Fluence Rate

The fiuence rate, , is the quotient of d4>by dt,where d4> is the increment of the fiuence in thetime interval dt, thus

d

= nut,

where n is the volumic particle number.

(2.1.5)


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2.1.5 Particle Radiance, Energy Radiance

. .The particle radiance, 27 is the quotient of dn.E and 1.P'n,E of the vectorialparticle radiance and the vectorial energy radiance,with respect to energy, are given by

. .

lPn.E = D4>n,E (2.2.1a)and

(2.2.lb)


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'------w' here


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tl>n m-2 s-1 sr-1 fl n Sect. 2.2.1

'l'n wm-2 sr-1 fltJtn Sect. 2.2.1

vo um c par c enumber n

volumic radiant

m-3 dN/dV Eq. 2.1.3 tion of vecto-

rial energy

energy w Jm-3

dR/dVradiance

n.Eenergy distribu- vector a u-tion of ence rate < volumic par-

ticle number nE m-3 J-1vectorial energy

dnldE fluence rate '1'

energy distribu"' energy str u-tion of tion of vecto-

volumic rial fluence

radiant rate


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1

3. Interaction Coefficients and Related Quantities

Interaction processes occur between radiation and

matter. In an interaction, the energy or the direction(or both) of the incident partele is altered or the particleis absorbed. The interaction may be followed by theemission of one or several secondary particles. Thelikelihood of such interactions is characterized by interaction coefficients. They refer to a specific interactionprocess, type and energy ofradiation, target or material.

The fundamental interaction coefficient is thecross section (see Section 3.1). All other interactioncoefficients defined in this report can be expressed interms of cross sections or differential cross sections.

Interaction coefficients and related quantities discussed in this section are listed in Table 3.1.

3.1 Cross Section

The cross section, u, of a target entity, for aparticular interaction produced by incidentcharged or uncharged particles, is the quotient ofPby 4>, where P is the probability of that interaction for a single target entity when subjected tothe particle fluence, 4>, thus

p

u= cp

Unit: m2

A special unit often used for the cross section is the barn, b, defined by

1b = 10-28 m2 = 100 fm2

A full description of an interaction process requires, inter alia, the knowledge of the distributionsof cross sections in terms of energy and direction of

3.2 Mass Attenuation Coefficient

The mass attenuation coefficient, J.LIp, of amaterial, for uncharged particles, is the quotientof dN/Nbyp dl, where dN/N is the fraction of

particles that experience interactions in traversing a distance dl in the material ofdensityp, thus

J.L 1 dN-=--p pdlN'

Unit: m2 kg- 1

J.L is the linear attenuation coefficient. The probability that at normal incidence a particle undergoes aninteraction in a ma:teriallayer ofthickness dl is J.L dl.

The reciproca! of J.L is called the mean free path ofan uncharged particle.

The linear attenuation coefficient, J.L, depends onthe density, p, of the absorber. This dependence is

largely removed by using the mass attenuation

coefficient,J.LIp.The mass attenuation coefficient can be expressed

in terms of the total cross section, cr. The mass

attenuation coefficient is the product of u and N.JM,

where NA is the Avogadro constant and M is themolar mass of the target material, thus

(3.2.1)

where aJ is the component cross section relating to

interaction of typeJ.Relation 3.2.1 can be written as

J.L nt

1 -

all emergent particles resulting from the interaction.

Such distributions, sometimes called differential cross

-=-a

p p '(3.2.2)

sections, are obtained by differentiations of awithrespect to energy and solid angle (see Eq. 3.4.2).

If incident particles of a given type and energy canundergo different and independent types of interaction in a target entity, the resulting cross section,

sometimes called the total cross section, u, is expressed by the sum of the component cross sections,a:, hence

where nt is the volumic number oftarget entities, i.e.,the number of target entities in a volume element

divided by its volume.

The mass attenuation coefficient of a compound

material is usually treated as if the latter consisted

of independent atoms. Thus,

u=_ aJ = -::PJ,. J

and UJ is the component crosssection relating to an interaction of typeJ.

of type L, aL the total cross section for an entity L,and uL,J the cross section of an interaction of typ'eJfor a single target entity of type L. Relation 3.2.3,

which ignores the changes in the molecular, chemi- 9


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di/el

cal or crystalline environment of an atom, is justi:fiedin most cases, but can occasionally lead to errors, forexample in the interaction of low-energy photonswith molecules (Hubbell, 1969).

3.3 Mass Energy Transfer Coefficient

The mass energy transfer coefficient, 11-tr/p, ofa material, for uncharged particles, is the quotient of dRtr/Rbyp dl, where dRu:IR is the fractionof incident radiant energy that is transferred to kinetic energy of charged particles by interactions, in traversing a distance dl in the material ofdensity p, thus

Unit: m2 kg- 1

In calculations relating to photons, the binding energy is usually included in the mass energy transfer coefficient. In materials consisting of elements ofmodest atomic number, this is generally signi:ficant

below photon energies of 1keV.If incident uncharged particles of a given type and

energy can produce several types of independentinteractions in a target entity, the mass energy

transfer coefficient can be expressed in terms of thecomponent cross sections, CIJ,by the relation

(3.3.1)

wherefJ is the average fraction of the incident particleenergy that is transferred to kinetic energy of chargedpartides in an interaction of typeJ,NA is the Avogadroconstant and Mis the molar mass ofthe target material.

The mass energy transfer coefficient is related to themass attenuation coefficient, p.,/p,by the equation

11-tr J.L

where (nt)L and crL,J have the same meaning as in Eq.3.2.3 and fL.J is the average fraction of the incident

particle energy that is transferred to kinetic energyof charged particles in an interaction oftype J with a

target entity of type L. Relation 3.3.3 implies thesame approximations as relation 3.2.3.The product of JLtriP for a material and (1 - g),

where g is the fraction of the energy of liberatedcharged particles that is lost in radiative processes inthe material, is called the mass energy absorption coefficient,J.Lerlp,of the material for uncharged partid es.

The mass energy absorption coefficient of a compound material depends on the stopping power (seeSection 3.4) of the material. Thus its evaluationcannot, in principie, be reduced to a simple summation of the mass energy absorption coefficient of theatomic constituents (Seltzer, 1993). Such a summa

tion can provide an adequate approximation whenthe value ofg is sufficiently small.

3.4 Mass Stopping Power

The mass stopping power, SIp, of a material, forcharged particles, is the quotient of dE by p dl,where dE is the energy lost by a charged particlein traversing a distance dl in the material ofdensity p, thus

S 1dE

p p dl.

Unit: Jm2 kg-1

E may be expressed in eV and, hence, SIp may beexpressed in eV m2 kg-1 or sorne convenient multiples or submultiples, such as MeV cm2 g-1

S =dE/dl denotes thelinear stopping power.The mass stopping power can be expressed as a

sum of independent components by

where

o:.0...,;>

ao.

.::::

where

-=- f,p p

(3.3.2)

( = s., isthe mass electronic (or collision)P P

stopping power due to collisions with

electrons,

The mass energy transfer coefficient of a com

pound materi'al is usually treated as if the latterconsisted of independent atoms. Thus,

(:) = Srad is themass radiative stopping

P rad Ppower dueto emission ofbremsstrahlung

-11-tr -1""' (nt)L "" fL,J (J'L,J, (3.3.3)in the electric fields of atomic nuclei

0 p p L J or atomic electrons,


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1-1(dEl) =-Snuc is the mass nuclear stopping The linear energy transfer, LJ., can also be ex P nuc P

power 1 dueto elastic Coulomb collisions

in which recoil energy is imparted to

atoms.

pressed by

LJ.d.Eke.J.

=Se- , (3.5.1)

In addition, one can consider energy losses due to

inelastic nuclear processes.The separate mass stopping power components

can be expressed in terms of cross sections. Forexample, the mass electronic (or collision) stopping

power for an atom can be expressed as

(3.4.2)

vhere NA is the Avogadro constant, M the molar- mass of the atom, Z its atomic number, da/dw the

differential cross section (per atomic electron) forcollisions and w is the energy loss.

Forming the quotient SetfP greatly reduces, butdoes not eliminate, the dependence on the density ofthe material (see ICRU, 1984, 1993b, where thedensity effect and the stopping powers for com

pounds are discussed).

3.5 Linear Energy Transfer (LET)

The linear energy transfer or restricted linear electronic stopping power, L,1, of a material, for charged particles, is the quotient of dE.1

by dl, where dE.1 is the energy lost by a chargedparticle due to electronic collisions in traversing a

distance dl, minus the sum ofthe kinetic energiesof all the electrons released with kinetic energies

in excess of Ll, thus

where Se1 is the linear electronic stopping power and

d.Eke.J. is the sum ofthe kinetic energies, greater than' of all the electrons released by the charged

particle traversing a distance dl.The definition expresses the following energy

balance: energy lost by the primary charged par

ticle in collisions with electrons, along a track

segment dl, minus energy carried away by secon

dary electrons having kinetic energies greater than Ll,

equals energy considered as 'locally transferred',

although the definition specifies an energy cutoff, .:1,

and not a range cutoff.

This definition differs from that previously given

(ICRU, 1980) in two respects. First,L.1 now includesthe binding energy for all collisions. As a conse

quence, L0 refers to the energy lost that does not

reappear as kinetic energy of released electrons.

Second, the threshold of the kinetic energy of the

released electrons is now Ll as opposed to Ll minus thebinding energy.

In order to simplify notation, .:1 may be ex

pressed in eV. Then L100 is understood to be thelinear energy transfer for an energy cutoff of 100 eV.Lx, which is equal to Seb may be replaced by L andis sometimes termed unrestricted linear e rzergytransfer.

3.6 Radiation Chemical Yield

The radiation chemical yield, G(x), of an en tity, x, is the quotient ofn(x) by E, where n(x) is themean amount of substance of that entity pro

duced, destroyed, or changed in a system by theenergy imparted, E, to the matter of that system,thus

Unit: Jm-1

E .1 may be expressed in eV and henceL .1 may beexpressed in eV m-1,or sorne convenient multiples or

Unit: mol J-1

G(x)n(x)=-.

E

submultiples, such as keV .um-1.

1The established term "mass nuclear stopping power" is amisnomer because it does not pertain to nuclear interactions.

The mole is the amount of substance of a system which contains as many elementary entities as thereare atoms in 0.012 kg of carbon-12. The elementary

entities must be specified and may be atoms, molecules, ions, electrons, other particles or specitledgroups of such particles (BIPM, 1998).

A related quantity, called G value, has been de- 11


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fined as the mean number of entities produced,destroyed or changed by an energy imparted of100 eV. The unit in which the G value is expressed is(100 eV)- 1.AG value of1 (100 eV)- 1corresponds toaradiation chemical yield of0.104 ,umolJ-1.

In salid state theory, a concept similar to W is theaverage energy required for the formation of ahole-electron pair.

T.-\.BLE 3.1-Interaction coefficients and related quantities

3.7 Mean Energy Expended in a Gas per IonPairFormed

The mean energy expended in a gas per ionpair form.ed, W,is the quotient ofE byN, whereN is the mean number of ion pairs formed whenthe initial kinetic energyE of a charged particle iscompletely dissipated in the gas, thus

E

W=N.

Unit: J

:Name

cross sectionmass attenua

tion coeffi

cient

linear attenua

tion coeffi

cient

mean free path

mass energy

transfer coef

ficient

mass energy

absorption

coefficient

mass stopping

Symbol Unit

\Vhere it appearsDefinition inthe report

PfcJ> Sect. 3.1

d.Nip dlN Sect. 3.2

d.NIN dl Sect. 3.2N dlld.N Sect. 3.2

dRtriP dl R Sect. 3.3

(.Ltrlp) (1-g) Sect. 3.3

W may also be expressed in eV.It follows from the definition of W that the ions

produced by bremsstrahlung or other secondary

power

linear stopping

power

linear energy

Slp J m2 kg- 1 dE!p dl

dE/dl

Sect. 3.4

Sect. 3.4

radiation emitted by the charged particles are includedinN.

In certain cases, it may be necessary to focus

transferradiation

chemical

dEjdl Sect.3.5

attention on the variation in the mean energy ex pended per ion pair along the path of the particle; then the concept of a differential W is required, as

yield

meanenergy

expended in agas perion

G(x) mol J-1 n(x)/e Sect. 3.6

defined in ICRU Report 31 (ICRU, 1979). pairformed w J EIN Sect. 3.7

2


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4. Dosimetry

The effects of radiation on matter depend on theradiation field, as specified by the radiometric quantities defined in Sections 2.1 and 2.2, and on theinteractions between radiation and matter, as characterized by the interaction quantities defined inSections 3.1 to 3.5. Dosimetric quantities, which aredevised to provide a physical measure to correlatewith actual or potential effects, are, in essence,

products of radiometric quantities and interactioncoefficients. In calculations, the values of the relevant quantities of each type must be known, while measurements often do not require this information.

Radiation interacts with matter in a series ofprocesses in which particle energy is converted and

finally deposited in matter. The dosimetric quanti,ies which describe these processes are presented

--below in two sections dealing with the conversionand with the deposition of energy.

The quantity dEtr includes the _kinetic energy of

the Auger electrons.For a fiuence, e[>, of uncharged particles of energy

E, the kerma,K, in a specified material is given by

K = c[>EJ1t!p, (4.1.1)

where J.Lt!P is the mass energy transfer coefficient ofthe material for these particles.

The kerma per unit :fiuence, K/4>, is termed thekerma coefficient for uncharged particles of energyEin a specified material. The term kerma coefficient isused in preference to the term kerma factor previously used as the word coefficient implies a physicaldimension whereas the word factor does not.

In dosimetric calculations, the kerma, K, is usually expressed in terms of the distribution, c[>E, of theuncharged particle fiuence with respect to energy(see Eq. 2.1.6a). The kerma,K, is then given by

4.1 Conversion ofEnergy

The term conversion of energy refers to the trans

K = f c[>EE J.Ltr dE, p

(4.1.2)

fer of energy from ionizing particles to secondaryionizing particles. The quantity kerma relates to thekinetic energy of the charged particles liberated byuncharged particles; the energy expended againstthe binding energies, usually a relatively small

component, is, by definition, not included. In addition to kenna, a quantity called cema is defined whichrelates to the energy lost by charged particles (e.g.,electrons, protons, alpha particles) in collisions withatomic electrons. By definition, the binding energies areincluded. Cerna differs from kerma in that cerna involves the energy lost in electronic collisions by theincoming charged particles while kerma involves the

,, nergy imparted to outgoing charged particles.

4.1.1 Kerma2

The kerma, K, is the quotient of dEtr by dm,where dEtr is the sum of the initial kinetic energies of all the charged particles liberated by

uncharged particles in a mass dm of material,thus

where J.Lt/p is the mass energy transfer coefficient ofthe material for uncharged particles of energy E.

The expression of kenna in terms of :fiuence impliesthat one can refer to a value ofkerma or kerma rate for aspecified material at a point in free space, or inside a

different material. Thus, one can speak., for example, ofthe air kerma ata point inside a water phantom.

Although kerma is a quantity which concernsinitial transfer of energy to matter, it is sometimes

used as an approximation to absorbed dose. Equality of absorbed dose and kerma is approached to thedegree that charged particle equilibrium exists, thatradiative losses are negligible, and that the energy of

the uncharged particles is large compared to thebinding energy of the released charged particles.Charged particle equilibrium exists at a point if thedistribution of the charged particle radiance withrespect to energy (see Eq. 2.1.9a) is constant withindistances equal to the maximum charged particle range.

4.1.2 Kerma Rate

Unit: Jkg-1

dEtrK= dm.

The kerma rate, k, is the quotient of dK by dt,where dK is the increment of kerma in the time interval dt, thus

The special name for the unit of kerma is gray(Gy).

2 netic nergy eleased per unit mass.

Unit: J kg-1

s-1

Ifthe special name gray is used, the unit ofkermarate is gray per second (Gy s-1).

13


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cpE-

4.1.3 Exposure

The exposure, X, is the quotient of dQ by dm,where dQ is the absolute value ofthe total chargeofthe ions of one sign produced in air when all theelectrons and positrons liberated or created by

photons in air ofmass dm are completely stoppedin air, thus

Unit: ekg-1

The ionization produced by Auger electrons isincluded in dQ. The ionization due to photons emit

ted by radiative processes (i.e.,bremsstrahlung andfluorescence photons) is not to be included in dQ.Except for this difference, significant at high energies, the exposure, as defined above, is the ionizationanalogue of the air kerma. Exposure can be ex

pressed in terms of the distribution, ct>E, of thefluence with respect to the photon energy, E, and themass energy transfer coefficient, 1-Lt!p, for air and forthat energy as follows

4.1.5 Cema3

The cema, e, is the quotient of dECby dm, wheredEe is the energy lost by charged particles, exceptsecondary electrons, in electronic collisions in amass dm of a material, thus

dEC

e= dm

Unit: Jkg-1

The special name ofthe unit of cerna is gray (Gy).

The energy lost by charged particles in electroniccollisions includes the energy expended against binding energy and any kinetic energy of the liberated

electrons, referred to as secondary electrons. Thus,energy subsequently lost by all secondary electronsis excluded from dEc.

The cerna, e, can be expressed in terms of thedistribution ct>E, ofthe charged particle fluence, withrespect to energy (see Eq. 2.1.6a). According to thedefinition of cerna, the distribution cpE does notinclude the contribution of secondary electrons to the

fluence. The cerna, e, is thus given by

(4.1.3) e= fcpESe

-1

dE=f L:o

dE,p p

(4.1.4)

where e is the elementary charge, W is the meanenergy expended in air per ion pair formed, g is thefraction of the energy of the electrons liberated byphotons that is lost in radiative processes in air.

For photon energies of the order of 1MeV or below,where the value of g is small, Eq. 4.1.3 may beapproximated byX = e1W K(1 -g), whereK is theair kerma for primary photons andg is the meanvalue ofg averaged over the distribution of the airkerma with respect to the electron energy.

where Selpis the mass electronic stopping power of aspecified material for charged particles of energy E,and Loo is the corresponding unrestricted linearenergy transfer.

For charged particles of high energies, it may beundesirable to disregard energy transport by secondary electrons of all energies. A modified concept,

restricted cema, e;1, (Kellerer et al., 1992) is thendefined as the integral

As in the case of kerma, it may be convenient torefer to a value of exposure or of exposure rate in free

eJ.=fL!l

4>E:- dE.p

(4.1.5)

space or at a point inside a material different from air; one can speak, for example, of the exposure at a

point inside a water phantom.

4.1.4 Exposure Rate

The exposure rate,X, is the quotient of dXby dt,where dX is the increment of exposure in the time interval dt, thus

Unit: ekg-1s-114

This di:ffers from the integral in Eq. 4.1.4 in thatLxis replaced by L!l and that the distribution ct>E: nowincludes secondary electrons with kinetic energiesgreater than .1. For .1 = oo, restricted cerna isidentical to cerna.

The expression of cerna and restricted cerna interms of fluence implies that one can refer to theirvalues for a specified material at a point in freespace, or inside a different material. Thus, one can

speak, for example, of tissue cerna in air (Kellerer etal., 1992).

The quantities called cerna and restricted cernacan be used as approxirnations to absorbed dose fromcharged particles. Equality of absorbed dose and

3 -.onverted nergy per unit mass.


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\._ .;ema is approached to the degree that secondaryelectron equilibrium exists and that radiative lossesand those due to elastic nuclear collisions are negligible. Such an equilibrium is achieved at a point ifthe fluence of secondary electrons is constant within

distances equal to their maximum range. For restricted cerna, only partial secondary electron equilibrium, up to kinetic energy Ll, is required.

4.1.6 Cema Rate

The cerna rate, , is the quotient of dC by dt, where dC is the increment of cerna in the time interval dt, thus

- Unit: Jkg-1s-1

If the special name gray is used, the unit of cernarate is gray per second (Gy s-1).

4.2 Deposition of Energy

In this section, certain stochastic quantities areintroduced.Energy deposit is the fundamental quantity in terms of which all other quantities presented

here can be defined.

4.2.1 Energy Deposit

The energy deposit, Ei, is the energy deposited ina single interaction, i, thus

where Ein is the energy of the incident ionizingparticle (excluding rest energy), Eout is the sum ofthe energies of all ionizing particles leaving the

interaction (excludingrest energy), Q is the changein the rest energies of the nucleus and of all

particles involved in the interaction (Q > 0: decrease of rest energy; Q < 0: increase of restenergy).

Unit: J

E may also be expressed in eV.Ei may be considered as the energy deposited at the

point of interaction, which is called the transferpoint, i.e., the location where an ionizing particleloses kinetic energy. The quantum-mechanical uncertainties ofthis location are ignored.

The energy deposits and transfer points, without- -rurther details of the interactions that cause them,

are sufficient for a description ofthe spatial distribution of energy deposition by ionizing particles.

4.2.2 Energy Imparted

The energy imparted, E, to the matter in a givenvolume is the sum of all energy deposits in thevolume, thus

where the summation is performed over all en

ergy deposits, Ei, in that volume.

Unit: J

E may also be expressed in eV.

The energy deposits over which the summation isperformed may belong to one or more (energy deposition) events; for example, they may belong to one orseveral statistically independent partid e tracks. Theterm event denotes the imparting of energy to matter by statistically correlated particles. Examplesinclude a proton and its secondary electrons, anelectron-positron pair or the primary and secondary

particles in nuclear reactions.

If the energy imparted to the matter in a given volume is dueto a single event, it is equal to the sumof the energy deposits in the volume associated with

the event.If

the energy imparted to the matter in agiven volume is dueto several events, it is equal tothe sum of the individual energies imparted to thematter in the volume due to each event.

The mean energy imparted, , to the matter in a

given volume equals the radiant energy, Rim of allthose charged and uncharged ionizing particles whichenter the volume minus the radiant energy, Rouh ofall those charged and uncharged ionizing particleswhich leave the volume, plus the sum, Q, of allchanges of the rest energy of nuclei and elementaryparticles which occur in the volume (Q > 0: decreaseofrest energy; Q < 0: increase ofrest energy), thus

(4.2.1)

4.2.3 Lineal Energy

The lineal energy, y, is the quotient of Esby l,where Es is the energy imparted to the matter in a given volume by a single (energy deposition)

event and l is the mean chord length of thatvolume, thus

Unit: Jm-115


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Es is the sum of the energy deposits E in a volume

from a single event and may be expressed in eV.

Hence y may be expressed in multiples and submul

tiples ofeV and m, e.g., in keV .um-1.The mean chord length of a volume is the mean

length of randomly oriented chords (uniform isotropic randomness) through that volume. For a convexbody, it can be shown that. the mean chord length, l,equals 4V/A, where V is the volume and A is thesurface area (Cauchy, 1850; Kellerer, 1980).

It is useful to consider the probability distribution

ofy. The value of the distribution function, F(y), isthe probability that the lineal energy due to a single

(energy deposition) event is equal to or less than y.

The probability density, f(y), is the derivative ofF(y), thus

dF(y)

probability that a specific energy less than or equal

to z is deposited if one event has occurred Theprobability density, {1(z), is the derivative of F1(z),

thus

(4.2.4)

For convex volumes, y and the increment, z, ofspecific energy due to a single (energy deposition)

event are related by

pA

y=--z, (4.2.5)

where A is the surface area of the volume, and p isthe density of matter in the volume.

f(y) =d.Y. (4.2.2) 4.2.5 Absorbed DoseF(y) andf(y) are independent of absorbed dose andabsorbed dose rate.

4.2.4 Specific Energy

The absorbed dose, D, is the quotient of dbydm, where d is the mean energy imparted to

matter ofmass dm, thus

The specific energy (imparted), z, is the quotient of E by m, where E is the energy imparted tomatter ofmass m, thus

E

Unit: J kg-1

dD= dm.

Unit: Jkg-1

z =-.m

The special name for the unit of absorbed dose is gray(Gy).

The special name for the unit of specific energy is gray (Gy).

The specific energy may be due to one or more(energy deposition) events. The distribution function,F(z), is the probability that the specific energyis equal to or less thanz. The probability density,

f(z), is the derivative ofF(z), thus

dF(z)f(z) = . (4.2.3)

F(z) andf(z) depend on absorbed dose. The probabil

ity density f(z) includes a discrete component (a

Dirac delta function) atz = O for the probability of noenergy deposition.

The distribution function of the specific energy

deposited in a single event, F1(z), is the conditional

16

In the limit of a small domain, the mean specific

energyz is equal to the absorbed doseD.

4.2.6 Absorbed Dose Rate

The absorbed dose rate,D, is the quotient of dDby dt, where dD is the increment of absorbed dosein the time interval dt, thus

. dD

D =dt.

Unit: J kg-1s-1

If the special name gray is used, the unit ofabsorbed dose rate is gray per second (Gy s-1).


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coefficient J rn2 kg-1 Gyrn2 K/4> Sect. 4.1.1kerma rate k J kg-1 s-1 Gy s-1 dKJdt Sect. 4.1.2

irnparted E

lineal energy yJJrn-1

E

Esfl

Sect. 4.2.2

Sect. 4.2.3

specific

energy z g- Gy elm ect. . . absorbed

dose D J kg-1 Gy de/dm Sect. 4.2.5absorbed

dose rate iJ J kg-1s-1 Gy s-1 dD/dt Sect. 4.2.6

o

J-.TABLE 4.1-Dosimetric quantities- Conversion of energy TABLE 4.2-Dosimetric quantities-Deposition ofenergy >,

"" Where it appears '\'here it EName Symbol Unit Definition in the report appears a;

in the

kerma K J kg-1 Gy d.Eu-fdm Sect. 4.1.1 N ame Symbol Unit Definition reponkerma

energy

deposit E J En- Eout + Q Sect. 4.2.1exposure X e kg- 1 dQ/dm Sect. 4.1.3 energyexposure rate x e kg- 1 s-1 cL"G'dt Sect. 4.1.4cerna e J kg-1 Gy d.Ecldm Sect. 4.1.5restricted cerna c.l J kg-1 Gy Sect. 4.1.5cerna rate e J kg-1 s-1 Gy s-1 dC/dt Sect.4.1.6

17


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5. Radioactivity

The term radioactivity refers to those spontaneoustransformations that involve changes ofthe nuclei of

atoms. The energy released in such transformationsis emitted as photons and/or other radiations.

Radioactivity is a stochastic process. The wholeatom is involved inthis process because nuclear transformations also can affect the atomic shell structure andcause emission of electrons, photons or both.

Atoms are subdivided into nuclides. A nuclide is aspecies of atoms having a specified number of protons and neutrons in its nucleus. Unstable nuclides,that transform to stable or unstable progeny, arecalled radionuclides. The transformation results inanother nuclide orina transition toa lower energy

state ofthe same nuclide.

5.1 Decay Constant

The decay constant, A, of a radionuclide in aparticular energy state is the quotient of dP by dt,where dP is the probability that a given nucleus undergoes a spontaneous nuclear transformationfrom that energy state in the time interval dt,thus

Unit: s-1

The quantity (ln 2)/A, commonly called the halflife, T112, of a radionuclide, is the mean time taken forthe radionuclides in the particular energy state todecrease to one half of their initial number.

5.2 Activity

The activity,A, ofan amount ofa radionuclide ina particular energy state at a given time, is thequotient of dN by dt, where dN is the number ofspontaneous nuclear transformations from thatenergy state in the time interval dt, thus

Unit: s-1

The activity,A, of an amount of a radionuclide in aparticular energy state is equal to the product of thedecay constant, A, for that state, and the number 1V ofnuclei in that state, thus

A=AN. (5.2.1)

5.3 Air Kerma-Rate Constant

The air kerma-rate constant, Ta, of a radionuclide emitting photons is the quotient of l 2KabyA,whereKa is the air kerma rate due to photons ofenergy greater than o, at a distance l in vacuo

from a pont source of this nuclide having anactivityA, thus

Unit: m2 J kg-1

If the special names gray (Gy) and becquerel (Bq)are used, the unit of air kerma-rate constant ism2 Gy Bq-1s-1

The photons referred to in the definition include

gamma rays, characteristic x rays, and interna!bremsstrahlung. The air kerma-rate constant, a characteristic of a

radionuclide, is defined in terms of an ideal pointsource. In a source of finite size, attenuation and scattering occur, and annihilation radiation and externa! bremsstrahlung may be produced. In sorne cases,these processes require significant corrections.

Any medium intervening between the source and

the point of measurement will give rise to absorptionand scattering for which corrections are needed.

The selection of the value of o depends upon theapplication. To simpli:fy notation and ensure unifor

mity, it is recommended that obe expressed in keV.For example, T5 is understood to be the air kermarate constant with a photon energy cutoff of 5 keV.

TABLE 5.1-Quantities related to radioactivity

Whereit

appears

in the

Name Symbol Unit Definition report

decay

The special name for the unit of activity is bec querel (Bq).

The "particular energy state" is the ground state of

constant A.

halflife Tv2

activity Aairkerma-rate con-

s-1 d.P/dt Sect. 5.1

S Cln 2)/A. Sect. 5.1s-1

Bq dN/dt Sect. 5.2

8 the radionuclide unless otherwise specified. stant T6 m2 J kg-1 m2 Gy Bq-1s-1 12K6/A Sect. 5.3


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ICRU (1993b). International Commission on Radia

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