ICRU 85a revised 2011-04-QuantitiesUnits(revised).pdf

Volume 11 No 1 2011 ISSN 1473-6691 (print)
ISSN 1742-3422 (online)
Journal of the ICRU
OXFORD UNIVERSITY PRESS INTERNATIONAL COMMISSION ON
RADIATION UNITS AND
MEASUREMENTS
ICRU REPORT 85a
Fundamental Quantities and Units
for Ionizing Radiation (Revised)
October 2011

FUNDAMENTAL QUANTITIES AND UNITS
FOR IONIZING RADIATION (Revised)
THE INTERNATIONAL COMMISSION ON
RADIATION UNITS AND
MEASUREMENTS
OCTOBER 2011
Published by Oxford University Press
ICRU REPORT No. 85
Journal of the ICRU Volume 11 No 1 2011
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Editor’s Note
The original version of ICRU Report 85 published in the JICRU, Vol. 11, No. 1, 2011, had a number of unfor-
tunate typographical errors. The ultimate responsibility for the failure to catch these errors on page proofs
rests, of course, with the Editor. The basic nature of this Report and the errors in rather important
summary tables of SI prefixes, quantities, and units, suggested strongly that a completely corrected Report
be made available. Thus, this version (Report 85a) corrects those and other errors, and it is hoped that this
corrected version is typographically error-free. If further errors are indeed detected, please contact the
ICRU so that these can be corrected in a future update.
Stephen M. Seltzer
Scientific Editor, JICRU
Gaithersburg, Maryland
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FUNDAMENTAL QUANTITIES AND UNITS
FOR IONIZING RADIATION (Revised)
Report Committee
S. M. Seltzer (Chairman), National Institute of Standards and Technology, Gaithersburg, Maryland USA
D. T. Bartlett, Abingdon, United Kingdom
D. T. Burns, Bureau International des Poids et Mesures, Sèvres, France
G. Dietze, Braunschweig, Germany
H.-G. Menzel, European Organization for Nuclear Research (CERN), Geneva, Switzerland
H. G. Paretzke, Institute for Radiation Protection, Neuherberg, Germany
A. Wambersie, Catholic University of Louvain, Brussels, Belgium
Consultants to the Report Committee
J. Tada, Riken Yokohama Institute, Yokohama, Japan
The Commission wishes to express its appreciation to the individuals involved in the preparation of this
Report for the time and efforts that they devoted to this task and to express its appreciation to the
organizations with which they are affiliated.
All rights reserved. No part of this book may be reproduced, stored in retrieval systems or transmitted in
any form by any means, electronic, electrostatic, magnetic, mechanical photocopying, recording or
otherwise, without the permission in writing from the publishers.
British Library Cataloguing in Publication Data. A Catalogue record of this book is available at the British
Library.
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The International Commission on Radiation Units and
Measurements
Introduction
The International Commission on Radiation Units
and Measurements (ICRU), since its inception in
1925, has had as its principal objective the develop-
ment of internationally acceptable recommendations
regarding:
(1) quantities and units of radiation and radioactivity,
(2) procedures suitable for the measurement
and application of these quantities in clinical
radiology and radiobiology, and
(3) physical data needed in the application of these
procedures, the use of which tends to assure
uniformity in reporting.
The Commission also considers and makes similar
types of recommendations for the radiation protec-
tion field. In this connection, its work is carried
out in close cooperation with the International
Commission on Radiological Protection (ICRP).
Policy
The ICRU endeavors to collect and evaluate
the latest data and information pertinent to the
problems of radiation measurement and dosimetry
and to recommend the most acceptable values and
techniques for current use.
The Commission’s recommendations are kept
under continual review in order to keep abreast of
the rapidly expanding uses of radiation.
The ICRU feels that it is the responsibility of
national organizations to introduce their own
detailed technical procedures for the development
and maintenance of standards. However, it urges
that all countries adhere as closely as possible to
the internationally recommended basic concepts of
radiation quantities and units.
The Commission feels that its responsibility lies
in developing a system of quantities and units having
the widest possible range of applicability. Situations
can arise from time to time for which an expedient
solution of a current problem might seem advisable.
Generally speaking, however, the Commission feels
that action based on expediency is inadvisable
from a long-term viewpoint; it endeavors to base
its decisions on the long-range advantages to be
expected.
The ICRU invites and welcomes constructive
comments and suggestions regarding its rec-
ommendations and reports. These may be trans-
mitted to the Chairman.
Current Program
The Commission recognizes its obligation to
provide guidance and recommendations in the areas
of radiation therapy, radiation protection, and the
compilation of data important to these fields, and to
scientific research and industrial applications of
radiation. Increasingly, the Commission is focusing
on the problems of protection of the patient and
evaluation of image quality in diagnostic radiology.
These activities do not diminish the ICRU’s commit-
ment to the provision of a rigorously defined set of
quantities and units useful in a very broad range of
scientific endeavors.
The Commission is currently engaged in the
formulation of ICRU Reports treating the following
subjects:
Alternatives to Absorbed Dose for Quantification
and Reporting of Low Doses and Other
Heterogeneous Exposures
Bioeffect Modeling and Biologically Equivalent
Dose Concepts in Radiation Therapy
Concepts and Terms for Recording and Reporting
Gynecologic Brachytherapy
Harmonization of Prescribing, Recording, and
Reporting Radiotherapy
Image Quality and Patient Dose in Computed
Tomography
Key Data for Measurement Standards in the
Dosimetry of Ionizing Radiation
Measurement and Reporting of Radon Exposure
Operational Radiation Protection Quantities for
External Radiation
Prescribing, Recording, and Reporting Ion-Beam
Therapy
doi:10.1093/jicru/nd
Oxford University Press
# International Commission on Radiation Units and Measurements 2011
Journal of the ICRU Vol 11 No 1 (2011) Report 85
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Small-Field Photon Dosimetry and Applications in
Radiotherapy
The Commission continually reviews radiation
science with the aim of identifying areas in which
the development of guidance and recommendations
can make an important contribution.
The ICRU’s Relationship with Other
Organizations
In addition to its close relationship with the ICRP,
the ICRU has developed relationships with national
and international agencies and organizations. In
these relationships, the ICRU is looked to for
primary guidance in matters relating to quantities,
units, and measurements for ionizing radiation, and
their applications in the radiological sciences. In
1960, through a special liaison agreement, the
ICRU entered into consultative status with the
International Atomic Energy Agency (IAEA).
The Commission has a formal relationship with the
United Nations Scientific Committee on the Effects
of Atomic Radiation (UNSCEAR), whereby ICRU
observers are invited to attend annual UNSCEAR
meetings. The Commission and the International
Organization for Standardization (ISO) informally
exchange notifications of meetings, and the ICRU is
formally designated for liaison with two of the ISO
technical committees. The ICRU also enjoys a
strong relationship with its sister organization, the
National Council on Radiation Protection and
Measurements (NCRP). In essence, these organiz-
ations were founded concurrently by the same indi-
viduals. Presently, this long-standing relationship is
formally acknowledged by a special liaison agree-
ment. The ICRU also exchanges reports with the
following organizations:
Bureau International de Métrologie Légale
Bureau International des Poids et Mesures
European Commission
Council for International Organizations of Medical
Sciences
Food and Agriculture Organization of the United
Nations
International Council for Science
International Electrotechnical Commission
International Labour Office
International Organization for Medical Physics
International Radiation Protection Association
International Union of Pure and Applied Physics
United Nations Educational, Scientific and Cultural
Organization
The Commission has found its relationship with
all of these organizations fruitful and of substantial
benefit to the ICRU program.
Operating Funds
In recent years, principal financial support has
been provided by the European Commission, the
National Cancer Institute of the US Department of
Health and Human Services, and the International
Atomic Energy Agency. In addition, during the last
10 years, financial support has been received from
the following organizations:
American Association of Physicists in Medicine
Belgian Nuclear Research Centre
Canadian Nuclear Safety Commission
Electricité de France
Helmholtz Zentrum München
Hitachi, Ltd.
International Radiation Protection Association
International Society of Radiology
Ion Beam Applications, S.A.
Japanese Society of Radiological Technology
MDS Nordion
National Institute of Standards and Technology
Nederlandse Vereniging voor Radiologie
Philips Medical Systems, Incorporated
Radiological Society of North America
Siemens Medical Solutions
US Department of Energy
Varian Medical Systems
In addition to the direct monetary support pro-
vided by these organizations, many organizations
provide indirect support for the Commission’s
program. This support is provided in many forms,
including, among others, subsidies for (1) the time
of individuals participating in ICRU activities,
(2) travel costs involved in ICRU meetings, and
(3) meeting facilities and services.
In recognition of the fact that its work is made
possible by the generous support provided by all
of the organizations supporting its program, the
Commission expresses its deep appreciation.
Hans-Georg Menzel
Chairman, ICRU
Geneva, Switzerland
FUNDAMENTAL QUANTITIES AND UNITS FOR IONIZING RADIATION
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Fundamental Quantities and Units for Ionizing Radiation
Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2. General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Quantities and Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Ionizing Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Stochastic and Non-Stochastic Quantities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Mathematical Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3. Radiometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1 Scalar Radiometric Quantities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1.1 Particle Number, Radiant Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1.2 Flux, Energy Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.1.3 Fluence, Energy Fluence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.1.4 Fluence Rate, Energy-Fluence Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1.5 Particle Radiance, Energy Radiance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Vector Radiometric Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2.1 Vector Particle Radiance, Vector Energy Radiance . . . . . . . . . . . . . . . . . . . . . . . 14
3.2.2 Vector Fluence Rate, Vector Energy-Fluence Rate . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2.3 Vector Fluence, Vector Energy Fluence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4. Interaction Coefficients and Related Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.1 Cross Section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.2 Mass Attenuation Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.3 Mass Energy-Transfer Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.4 Mass Stopping Power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.5 Linear Energy Transfer (LET) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.6 Radiation Chemical Yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.7 Ionization Yield in a Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.8 Mean Energy Expended in a Gas per Ion Pair Formed . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5. Dosimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.1 Conversion of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.1.1 Kerma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.1.2 Kerma Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.1.3 Exposure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
doi:10.1093/jicru/ndr012
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5.1.4 Exposure Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.1.5 Cema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.1.6 Cema Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.2 Deposition of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.2.1 Energy Deposit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.2.2 Energy Imparted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.2.3 Lineal Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.2.4 Specific Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.2.5 Absorbed Dose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.2.6 Absorbed-Dose Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6. Radioactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6.1 Decay Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6.2 Activity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6.3 Air-Kerma-Rate Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
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Preface
The International Commission on Radiation Units
and Measurements (ICRU) was originally conceived
at the First International Congress of Radiology in
1925 as the International X-Ray Unit Committee,
later to become the International Commission on
Radiological Units before adopting its current name.
Its primary objective was to provide an internation-
ally agreed upon unit for measurement of radiation
as applied in medicine. The ICRU established the
first internationally acceptable unit for exposure,
the röntgen, in 1928. In 1950, the ICRU expanded
its role to consider all concepts, quantities, and units
for ionizing radiation, encompassing not only
medical applications but also those in industry, radi-
ation protection, and nuclear energy. The ICRU has
continued to recommend new quantities and units
as the need arose, for example, absorbed dose (1950),
the rad (1953), fluence (1962), kerma (1968), and
cema (1998).
Thus, quantities and units for ionizing radiation
represent the most basic element of the core
mission of the ICRU. This is evidenced by the
series of Reports on fundamental quantities and
units: Report 10a, Radiation Quantities and Units
(1962); Report 11, Radiation Quantities and Units
(1980); and Report 60, Fundamental Quantities
and Units for Ionizing Radiation (1998). This list
excludes those Reports that primarily include rec-
ommendations for quantities and units specifically
intended for radiation protection, which are devel-
oped in liaison and collaboration with the
International Commission on Radiation Protection
(ICRP).
The development of the present Report was
prompted by a few criticisms of Report 60, and
while basically introducing no new quantities it
does strive for more-precisely worded definitions
and clarity.
Stephen M. Seltzer
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Abstract
Definitions of fundamental quantities, and their
units, for ionizing radiation are given, which rep-
resent the recommendation of the International
Commission on Radiation Units and Measurements
(ICRU).
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1. Introduction
There has been a gradual evolution in the estab-
lishment and definitions of fundamental quantities
for ionizing radiation since the formation of the
ICRU (see, e.g., the series of separate Reports on
quantities and units: ICRU, 1962; 1968; 1971;
1980; 1998). The goal is a set of quantities (and
their units) of use in the measurement of and in
transport calculations for ionizing radiation in
practical applications. The definitions of these
quantities should be precise and logically consist-
ent, and possess the utmost scientific validity and
mathematical rigor. Although much progress has
been made toward this goal, the effort has no doubt
fallen short of perfection due to compromises
among the unavoidable ambiguities inherent in the
real natural world and the need nonetheless for a
basic set of useful quantities.
This Report supersedes ICRU Report 60,
Fundamental Quantities and Units for Ionizing
Radiation (1998). Definitions are not radically
changed: rather, the refinements presented in this
Report mainly correct oversights in the previous
Report and in some cases provide additional clarifi-
cation. It was felt, however, that a new Report would
establish a complete current reference for funda-
mental quantities and units for ionizing radiation.
This Report does not deal with quantities and units
specifically intended for use in radiation protection,
which were covered in ICRU Report 51 (1993a) and
which are currently under review by the ICRU.
The quantities and units for ionizing radiation
dealt with in this Report include those for radiome-
try, interaction coefficients, dosimetry, and radioac-
tivity. The ICRU appreciates the assistance rendered
by scientific bodies and individuals who commented
on aspects of ICRU Report 60, and a number of
these comments have informed the refinements
included in the present Report. The remainder of
this Report is structured in five subsequent major
Sections, each of the last four of which is followed by
tables summarizing for each quantity its symbol,
unit, and the relationship used in its definition.
Section 2 deals with terms and mathematical
conventions used throughout the Report.
Section 3, entitled Radiometry, presents quan-
tities required for the specification of radiation
fields. Two classes of quantities are used, referring
either to the number of particles or to the energy
transported by them. Accordingly, the definitions of
radiometric quantities are grouped into pairs. Both
scalar and vector quantities are defined.
Interaction coefficients and related quantities
are covered in Section 4. The fundamental inter-
action coefficient is the cross section. All other
coefficients defined in this Section can be
expressed in terms of cross section or differential
cross section. The current Report adds the defi-
nition of the ionization yield, Y, to provide a
logical connection between the radiation chemical
yield and the mean energy expended in a gas per
ion pair formed. Explicit language is now included
in the definition of W, the mean energy expended
in a gas per ion pair formed, to clarify what is
meant by “ion pair” (ICRU, 1979) in order to avoid
possible confusion.
Section 5 deals with dosimetric quantities that
describe the results of processes by which particle
energy is converted and finally deposited in
matter. Accordingly, the definitions of dosimetric
quantities are presented in two parts entitled
Conversion of Energy and Deposition of Energy,
respectively. The first part includes refinement of
the definition of g as the fraction of the kinetic
energy of liberated charged particles that is lost
in radiative processes in the material, and explicit
specification of dry air in the definition of
exposure. In the second part on deposition of
energy, the definitions of kerma and of exposure
are refined to explicitly include the kinetic energy
of all charged particles emitted in the decay of
excited atoms/molecules or nuclei, and consider-
ation of elapsed time is acknowledged in the spe-
cifications of energy imparted, particularly in
cases involving the production of radionuclides by
the incident particle(s). Energy deposit, i.e., the
energy deposited in a single interaction, is the
basis in terms of which all other quantities pre-
sented in the Section can be defined. These are
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the traditional stochastic quantities (which are
now so labeled as they are defined): energy
imparted, lineal energy, and specific energy, the
latter leading to the non-stochastic quantity
absorbed dose.
Quantities related to radioactivity are defined in
Section 6.
The current document strives for an incremental
improvement in scientific rigor and yet to remain
as consistent as possible with earlier ICRU Reports
in this series and in similar publications used in
other fields of physics. It is hoped that this Report
represents a modest step toward a universal scien-
tific language.
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2. General Considerations
This Section deals with terms and mathematical
conventions used throughout the Report.
2.1 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 quantities of the same kind are compared.
Every quantity is expressed as the product of a
numerical value and a unit. As a quantity remains
unchanged when the unit in which it is expressed
changes, its numerical value is modified accordingly.
Quantities can be multiplied or divided by one
another resulting in other quantities. Thus, all
quantities can be derived from a set of base quan-
tities. The resulting quantities are called derived
quantities.
A system of units is obtained in the same way by
first defining units for the base quantities, the base
units, and then forming derived units. A system is
said to be coherent if no numerical factors other
than the number 1 occur in the expressions of
derived units.
The ICRU recommends the use of the
International System of Units (SI) (BIPM, 2006). In
this system, the base units are meter, kilogram,
second, ampere, kelvin, mole, and candela, for the
base quantities length, mass, time, electric current,
thermodynamic temperature, amount of substance,
and luminous intensity, respectively.
Some derived SI units are given special names,
such as coulomb for ampere second. Other derived
units are given special names only when they are
used with certain derived quantities. Special
names pertaining to ionizing radiation currently in
use in this restricted category are becquerel (equal
to reciprocal second for activity of a radionuclide),
gray (equal to joule per kilogram for absorbed dose,
kerma, cema, and specific energy), and sievert
(equal to joule per kilogram for dose equivalent,
ambient dose equivalent, directional dose equival-
ent, and personal dose equivalent). Some examples
of SI units are given in Table 2.1.
There are also a few units outside of the inter-
national system that may be used with SI. For
some 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 the system but, because of long usage, are per-
mitted to be used with SI (see Table 2.2).
Decimal multiples and submultiples of SI units
can be formed using the SI prefixes (see Table 2.3).
2.2 Ionizing Radiation
Ionization produced by particles is the process by
which one or more electrons are liberated in col-
lisions of the particles with atoms or molecules.
This can be distinguished from excitation, which is
a transfer of electrons to higher energy levels in
atoms or molecules and generally requires less
energy.
When charged particles have slowed down suffi-
ciently, ionization becomes less likely or impossible,
and the particles increasingly dissipate their
remaining energy in other processes such as exci-
tation or elastic scattering. Thus, near the end of
their range, charged particles that were ionizing
can be considered to be non-ionizing.
The term ionizing radiation refers to charged
particles (e.g., electrons or protons) and uncharged
particles (e.g., photons or neutrons) that can
produce ionizations in a medium or can initiate
nuclear or elementary-particle transformations
that then result in ionization or the production of
ionizing radiation. In the condensed phase, the
difference between ionization and excitation can
become blurred. A pragmatic approach for dealing
with this ambiguity is to adopt a threshold for the
energy that can be transferred to the medium. This
implies cutoff energies below which charged par-
ticles may be assumed not to be ionizing (unless
they can initiate nuclear or elementary-particle
transformations). Below such energies, their
ranges are minute. Hence, the choice of the cutoff
energies does not materially affect the spatial
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distribution of energy deposition except at the
smallest distances that can be of concern in micro-
dosimetry. The choice of the threshold value
depends on the application; for example, a value of
10 eV might be appropriate for radiobiology.
2.3 Stochastic and Non-Stochastic
Quantities
Differences between results from repeated obser-
vations are common in physics. These can arise
from imperfect measurement systems, or from the
fact that many physical phenomena are subject to
inherent fluctuations. Quantum-mechanical issues
aside, one often needs to distinguish between a
non-stochastic quantity with its unique value and a
stochastic quantity, the values of which follow a
probability distribution. In many instances, this
distinction is not significant because the probability
distribution is very narrow. For example, the
measurement of an electric current commonly
involves so many electrons that fluctuations con-
tribute negligibly to inaccuracy in the
Table 2.1. SI units used in this Report
Category of units Quantity Name Symbol
SI base units length meter m
mass kilogram kg
time second s
amount of substance mole mol
SI derived units electric charge coulomb C
with special names energy joule J
(general use) solid angle steradian sr
power watt W
SI derived units activity becquerel Bq
with special names absorbed dose, kerma, gray Gy
(restricted use) cema, specific energy
Table 2.3. SI prefixesa
Factor Prefix Symbol Factor Prefix Symbol
1024
yotta Y 1021
deci d
1021
zetta Z 1022
centi c
1018
exa E 1023
milli m
1015
peta P 1026
micro m
1012
tera T 1029
nano n
109
giga G 10212
pico p
106
mega M 10215
femto f
103
kilo k 10218
atto a
102
hecto h 10221
zepto z
101
deca da 10224
yocto y
a
The prefix symbol attached to the unit symbol constitutes a new symbol, e.g., 1 fm2
¼ (10215
m)2
¼ 10230
m2
.
Table 2.2. Some units used with the SI
Category of units Quantity Name Symbol
Units widely used time minute min
hour h
day d
Units whose values in SI are
obtained experimentally
energy electron volta
eV
mass (unified) atomic mass unita
u
a
1 eV ¼ 1.602176487(40) 10219
J, and 1 u ¼ 1.660538782(83) 10227
kg. The digits in parentheses are the one-standard-deviation
uncertainty in the last digits of the given value (Mohr et al., 2008).
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measurement. However, as the limit of zero electric
current is approached, fluctuations can become
manifest. This case of course requires a more
careful measurement procedure, but perhaps more
importantly illustrates that the significance of sto-
chastic variations of a quantity can depend on the
magnitude of the quantity. Similar considerations
apply to ionizing radiation; fluctuations can play a
significant role, and in some cases need to be con-
sidered explicitly. On a practical level, this Report
adopts the convention that a quantity whose under-
lying distribution at microscopic levels is normally
of little interest and that is customarily expressed
in terms of a mean value will not be defined expli-
citly as stochastic.
Certain stochastic processes follow a Poisson dis-
tribution, a distribution uniquely determined by its
mean value. A typical example of such a process is
radioactive decay.1
However, more complex distri-
butions are involved in energy deposition. In this
Report, because of their relevance, four stochastic
quantities are defined explicitly, namely energy
deposit, 1i (see Section 5.2.1), energy imparted, 1
(see Section 5.2.2), lineal energy, y (see Section
5.2.3), and specific energy, z (see Section 5.2.4). For
example, the specific energy, z, is defined as the
quotient of the energy imparted, 1, and the mass,
m. Repeated measurements would provide an esti-
mate of the probability distribution of z and of its
first moment or mean,
z, the latter approaching the
absorbed dose, D (see Section 5.2.5), as the mass
becomes small. Knowledge of the distribution of z
is not required for the determination of the
absorbed dose, D. However, knowledge of the distri-
bution of z corresponding to a known D can be
important because in the irradiated mass element,
m, the effects of radiation can be more closely
related to z than to D, and the values of z can differ
greatly from D for small values of m (e.g., biological
cells).
2.4 Mathematical Conventions
To permit characterization of a radiation field
and its interactions with matter, many of the quan-
tities defined in this Report are considered as func-
tions of other quantities. For simplicity in
presentation, the arguments on which a quantity
depends often will not be stated explicitly. In some
instances, the distribution of a quantity with
respect to another quantity can be defined. The dis-
tribution function of a discrete quantity, such as
the particle number N (see Section 3.1.1), will be
treated as if it were continuous, as N is usually a
very large number. Distributions with respect to
energy are frequently required, as are other differ-
ential forms of defined quantities. This Report
follows the previous practice of often introducing
integral forms of quantities prior to their represen-
tation in differential forms. For example, fluence is
defined at the start of Section 3.1.3, followed by
that of the distribution of fluence with respect to
particle energy, given by (see Eq. 3.1.8a)
FE ¼ dF=dE; ð2:4:1Þ
where dF is the fluence of particles of energy
between E and E þ dE. Such distributions with
respect to energy are denoted in this Report by
adding the subscript E to the symbol of the distrib-
uted quantity. This results in a change of physical
dimensions; thus, the unit of F is m22
, whereas the
unit of FE is m22
J21
(see Tables 3.1 and 3.2).
Quantities related to interactions, such as the
mass attenuation coefficient, m/r, (see Section 4.2)
or the mass stopping power, S/r (see Section 4.4),
are functions of the particle type and energy, and
one might, if necessary, use a more explicit notation
such as m(E)/r or S(E)/r. For a radiation field with
an energy distribution, mean values such as
m=r
and
S=r, weighted according to the distribution of
the relevant quantity, are often useful. For
example,

m
r
¼
Ð
mðEÞ=r
½ FE dE
Ð
FE dE
¼
1
F
ð
mðEÞ=r
½ FE dE

ð2:4:2Þ
is the fluence-weighted mean value of m/r.
Stochastic quantities are associated with prob-
ability distributions. Two types of such distri-
butions are considered in this Report, namely the
distribution function (symbol F) and the probability
density (symbol f). For example, F(y) is the prob-
ability that the lineal energy is equal to or less
than y. The probability density f(y) is the derivative
of F(y), with f(y) being the probability that the
lineal energy is between y and y þ dy.
1
Radioactive decay is inherently governed by the binomial
distribution. However, for sufficiently large values of the
number of radioactive atoms that one usually deals with, the
Poisson distribution is an excellent approximation of the
binomial distribution.
General Considerations
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3. Radiometry
Radiation measurements and investigations of
radiation effects require various degrees of specifi-
cation of the radiation field at the point of interest.
Radiation fields consisting of various types of par-
ticles, such as photons, electrons, neutrons, or
protons, are characterized by radiometric quan-
tities that apply in free space and in matter.
Two classes of quantities are used in the charac-
terization of a radiation field, referring either to
the number of particles or to the energy trans-
ported by them. Accordingly, most of the definitions
of radiometric quantities given in this Report can
be grouped into pairs.
Both scalar and vector 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.
Equivalent definitions that are used in particular
applications are given in the text. Distributions of
some radiometric quantities with respect to energy
are given when they will be required later in the
Report. This extended set of quantities relevant to
radiometry is summarized in Tables 3. 1 and 3. 2.
3.1 Scalar Radiometric Quantities
Consideration of radiometric quantities begins
with the definition of the most general quantities
associated with the radiation field, namely the par-
ticle number, N, and the radiant energy, R (see
Section 3.1.1). The full description of the radiation
field, however, requires information on the type
and the energy of the particles as well as on their
distributions in space, direction, and time. This
most differential form is the basic field quantity
from a radiation-transport perspective, and defi-
nitions of quantities in terms of integrals of such a
quantity might seem to better represent mathemat-
ical rigor. However, the physical processes relevant
here do not preclude differentiation. Thus, in the
present Report, the specification of the radiation
field is achieved with increasing detail by defining
scalar radiometric quantities through successive
differentiations of N and R with respect to time,
area, volume, direction, or energy.2
Thus, these
quantities relate to a particular value of each vari-
able of differentiation. This procedure provides the
simplest definitions of quantities such as fluence
and energy fluence (see Section 3.1.3), often used
in the common situation in which radiation inter-
actions are independent of the direction and time
distribution of the incoming particles.
The scalar radiometric quantities defined in this
Report are used also for fields of optical and ultra-
violet radiations, sometimes under different names.
In this Report, the equivalence between the
various terminologies is often noted in connection
with the relevant definitions.
3.1.1 Particle Number, Radiant Energy
The particle number, N, is the number of par-
ticles that are emitted, transferred, or received.
Unit: 1
The radiant energy, R, is the energy (excluding
rest energy) of the particles that are emitted,
transferred or received.
Unit: J
For particles of energy E (excluding rest energy),
the radiant energy, R, is equal to the product NE.
The distributions, NE and RE, of the particle
number and the radiant energy with respect to
energy are given by
NE ¼ dN=dE; ð3:1:1aÞ
and
RE ¼ dR=dE; ð3:1:1bÞ
where dN is the number of particles with energy
between E and E þ dE, and dR is their radiant
energy. The two distributions are related by
RE ¼ ENE: ð3:1:2Þ
2
Mathematically, the differentials are understood to be of
expected or mean values of the quantities.
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The particle number density3
, n, is given by
n ¼ dN=dV; ð3:1:3aÞ
where dN is the number of particles in the volume dV.
Similarly, the radiant energy density, u, is given by
u ¼ dR=dV: ð3:1:3bÞ
The distributions, nE and uE, of the particle number
density and the radiant energy density with respect
to energy are given by
nE ¼ dn=dE; ð3:1:4aÞ
and
uE ¼ du=dE: ð3:1:4bÞ
The two distributions are related by
uE ¼ EnE: ð3:1:5Þ
3.1.2 Flux, Energy Flux
The flux, _
N, is the quotient of dN by dt, where
dN is the increment of the particle number in
the time interval dt, thus
_
N ¼
dN
dt
:
Unit: s2l
The energy flux, _
R, is the quotient of dR by dt,
where dR is the increment of radiant energy in
the time interval dt, thus
_
R ¼
dR
dt
:
Unit: W
These quantities frequently refer to a limited
spatial region, e.g., the flux of particles emerging
from a collimator. For source emission, the flux of par-
ticles emitted in all directions is generally considered.
For visible light and related electromagnetic radi-
ations, 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 in other texts
for the quantity termed fluence rate in the present
Report (see Section 3.1.4). This usage is discour-
aged because of the possible confusion with the
above definition of flux.
3.1.3 Fluence, Energy Fluence
The fluence, F, is the quotient of dN by da,
where dN is the number of particles incident on
a sphere of cross-sectional area da, thus
F ¼
dN
da
:
Unit: m22
The energy fluence, C, the quotient of dR by da,
where dR is the radiant energy incident on a
sphere of cross-sectional area da, thus
C ¼
dR
da
:
Unit: J m22
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
of each particle. The quantities fluence and energy
fluence are applicable in the common situation in
which radiation interactions are independent of the
direction of the incoming particles. In certain situ-
ations, quantities (defined below) involving the
differential solid angle, dV, in a specified direction
are required.
In dosimetric calculations, fluence is frequently
expressed in terms of the lengths of the particle trajec-
tories. It can be shown (Papiez and Battista, 1994;
and references therein) that the fluence, F, is given by
F ¼
dl
dV
; ð3:1:6Þ
where dl is the sum of the lengths of particle trajec-
tories in the volume dV.
For a radiation field that does not vary over the
time interval, t, and is composed of particles with
velocity v, the fluence, F, is given by
F ¼ nvt; ð3:1:7Þ
where n is the particle number density.
3
This quantity was previously termed volumic particle number
(ICRU, 1998). Recognizing that the terms volumic and massic
are not commonly used in English, this Report reverts to the
convention (IUPAC, 1997) that density indicates per volume and
specific indicates per mass (additionally, surface . . . density
indicates per area, linear . . . density indicates per length, and
rate indicates per time). However, the adjectives massic ¼ per
mass, volumic ¼ per volume, areic ¼ per area, and lineic ¼ per
length (ISO, 1993) are acceptable and recognized for their
convenience.
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The distributions, FE and CE, of the fluence and
energy fluence with respect to energy are given by
FE ¼
dF
dE
; ð3:1:8aÞ
and
CE ¼
dC
dE
; ð3:1:8bÞ
where dF is the fluence of particles of energy
between E and E þ dE, and dC is their energy
fluence. The relationship between the two distri-
butions is given by
CE ¼ EFE: ð3:1:9Þ
The energy fluence is related to the quantity
radiant exposure defined, for fields of visible light,
as the quotient of the radiant energy incident on a
surface element by the area of that element (CIE,
1987). When a parallel beam is incident at an angle
u with the normal direction to a given surface
element, the radiant exposure is equal to C cos u.
3.1.4 Fluence Rate, Energy-Fluence Rate
The fluence rate, Ḟ, is the quotient of dF by dt,
where dF is the increment of the fluence in the
time interval dt, thus
Ḟ ¼
dF
dt
:
Unit: m22
s2l
The energy-fluence rate, Ċ, is the quotient of dC
by dt, where dC is the increment of the energy
fluence in the time interval dt, thus
Ċ ¼
dC
dt
:
Unit: W m22
In radiation-transport literature, these quantities
have also been termed particle flux density and
energy flux density, respectively. Because the word
density has several connotations, the term fluence
rate is preferred.
For a radiation field composed of particles of vel-
ocity v, the fluence rate, Ḟ, is given by
Ḟ ¼ nv; ð3:1:10Þ
where n is the particle number density.
3.1.5 Particle Radiance, Energy Radiance
The particle radiance, ḞV, is the quotient of dḞ
by dV, where dḞ is the fluence rate of particles
propagating within a solid angle dV around a
specified direction, thus
ḞV ¼
dḞ
dV
:
Unit: m22
s21
sr21
The energy radiance, ĊV, is the quotient of dĊ
by dV, where dĊ is the energy fluence rate of
particles propagating within a solid angle dV
around a specified direction, thus
ĊV ¼
dĊ
dV
:
Unit: W m22
sr21
The specification of a direction requires two
variables. In a spherical coordinate system with
polar angle u and azimuthal angle w, dV is equal to
sin u du dw.
For visible light and related electromagnetic
radiations, the particle radiance and energy radi-
ance are termed photon radiance and radiance,
respectively (CIE, 1987).
The distributions of particle radiance and energy
radiance with respect to energy are given by
ḞV; E ¼
dḞV
dE
; ð3:1:11aÞ
and
ĊV; E ¼
dĊV
dE
; ð3:1:11bÞ
where dḞ V is the particle radiance for particles of
energy between E and E þ dE, and dĊ V is their
energy radiance. The two distributions are related
by
ĊV; E ¼ EḞV; E: ð3:1:12Þ
The quantity Ḟ V;E is sometimes termed angular
flux or phase flux in radiation-transport theory.
Apart from aspects that are of minor importance
in the present context (e.g., polarization), the field
of any radiation of a given particle type is comple-
tely specified by the distribution, ḞV; E, of the par-
ticle radiance with respect to particle energy, as
this defines number, energy, local density, and
arrival rate of particles propagating in a given
Radiometry
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direction. This quantity, as well as the distribution
of the energy radiance with respect to energy, can
be considered as basic in radiometry.
3.2 Vector Radiometric Quantities
As radiometric quantities are primarily con-
cerned with the flow of radiation, it is appropriate
to consider some of them as vector quantities.
Vector quantities are not needed in those cases for
which the corresponding scalar quantities are
appropriate, e.g., in deriving dosimetric quantities
that are independent of the particle direction. In
other instances, vector quantities are useful, and
they are important in theoretical considerations
related to radiation fields and dosimetric quan-
tities. There is, in general, no simple relationship
between the magnitudes of a scalar quantity and
the corresponding vector quantity. However, in the
case of a unidirectional field, they are of equal mag-
nitude. The use of boldface symbols distinguishes
the vector quantities introduced in this Section
from the corresponding scalar quantities.
The vector quantities, defined in this Section, are
obtained by successive integrations of the quan-
tities vector particle radiance and vector energy
radiance (see Section 3.2.1). Vector quantities are
used extensively in radiation-transport theory, but
often with a different terminology. The equival-
ences are indicated for the convenience of the
reader.
3.2.1 Vector Particle Radiance, Vector
Energy Radiance
The vector particle radiance, ḞV, is the product
of V by ḞV, where V is the unit vector in the
direction specified for the particle radiance ḞV,
thus
ḞV ¼ VḞV:
Unit: m22
s2l
sr21
The vector energy radiance, ĊV, is the product
of V by ĊV, where V is the unit vector in the
direction specified for the energy radiance ĊV ,
thus
ĊV ¼ VĊ V:
Unit: W m22
sr2l
The magnitudes jḞVj and jĊVj are equal to ḞV
and ĊV, respectively.
The distributions ḞV; E and ĊV; E of the vector
particle radiance and the vector energy radiance,
with respect to energy, are given by
ḞV; E ¼ VḞV; E; ð3:2:1aÞ
and
ĊV; E ¼ VĊV; E; ð3:2:1bÞ
where ḞV; E and ĊV; E are the distributions of the
particle radiance and the energy radiance, respect-
ively, with respect to energy.
In radiation-transport theory, ḞV; E is sometimes
called angular current density, phase-space current
density, or directional flux.
3.2.2 Vector Fluence Rate, Vector
Energy-Fluence Rate
The vector fluence rate, Ḟ, is the integral of ḞV
with respect to solid angle, where ḞV is the
vector particle radiance in the direction specified
by the unit vector V, thus
Ḟ ¼
ð
ḞV dV:
Unit: m22
s21
The vector energy-fluence rate, Ċ, is the integral
of ĊV with respect to solid angle, where ĊV is
the vector energy radiance in the direction speci-
fied by the unit vector V, thus
Ċ ¼
ð
ĊV dV:
Unit: W m22
The vectorial integration determines both direc-
tion and magnitude of the vector fluence rate and
of the vector energy-fluence rate. The scalar quan-
tities fluence rate and energy-fluence rate can be
obtained in a similar way according to
Ḟ ¼
ð
ḞV dV; ð3:2:2aÞ
and
Ċ ¼
ð
ĊV dV: ð3:2:2bÞ
It is important that these quantities not be con-
fused with the vector counterparts. In particular, it
needs to be recognized that the magnitude of vector
fluence rate and of vector energy-fluence rate
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change from zero in an isotropic field to Ḟ and Ċ
in a unidirectional field.
The vector fluence rate is sometimes referred to
as current density in radiation-transport theory.
3.2.3 Vector Fluence, Vector Energy Fluence
The vector fluence, F, is the integral of Ḟ with
respect to time, t, where Ḟ is the vector fluence
rate, thus
F ¼
ð
Ḟdt:
Unit: m22
The vector energy fluence, C, is the integral of
Ċ with respect to time, t, where Ċ is the vector
energy-fluence rate, thus
C ¼
ð
Ċdt:
Unit: J m22
The distributions FE and CE of the vector fluence
and vector energy fluence, with respect to energy
are given by
FE ¼
dF
dE
¼
ð
ḞEdt; ð3:2:3aÞ
and
CE ¼
dC
dE
¼
ð
ĊEdt; ð3:2:3bÞ
where dF is the vector fluence of particles of
energy between E and E þ dE, and dC is their
vector energy fluence.
The vector energy fluence, C, can be obtained
from the distribution ḞV;E according to
C ¼
ð
ð
ð
VEḞV; Edt dV dE: ð3:2:4Þ
It should be noted that the (vectorial) integration
of ḞV; E over time, energy, and solid angle results
in a point function in space, but this is not the
case when the integration is over area. It is
meaningful to integrate, for instance, the scalar
product C . da over a given area a to obtain the
net flow of radiant energy through this area.
Integration with respect to a particular surface
must take account of the (three-dimensional)
shape of the surface and its orientation, because
the number of particles in a given direction,
intercepted by a surface, depends on the angle of
incidence.
Table 3.1. Scalar radiometric quantities
Namea
Symbol Unit Definition Appearance in the Report
particle number N 1 – Section 3.1.1
radiant energy R J – Section 3.1.1
energy distribution of particle number NE J21
dN/dE Eq. 3.1.1a
energy distribution of radiant energy RE 1 dR/dE Eq. 3.1.1b
particle number density n m23
dN/dV Eq. 3.1.3a
radiant energy density u J m23
dR/dV Eq. 3.1.3b
energy distribution of particle number density nE m23
J21
dn/dE Eq. 3.1.4a
energy distribution of radiant energy density uE m23
du/dE Eq. 3.1.4b
flux _
N s21
dn/dt Section 3.1.2
energy flux _
R W dR/dt Section 3.1.2
energy distribution of flux _
NE s21
J21
dN/dE –
energy distribution of energy flux _
RE s21
dR/dE –
fluence F m22
dN/da Section 3.1.3
energy fluence C J m22
dR/da Section 3.1.3
energy distribution of fluence FE m22
J21
dF/dE Eq. 3.1.8a
energy distribution of energy fluence CE m22
dC/dE Eq. 3.1.8b
fluence rate Ḟ m22
s21
dF/dt Section 3.1.4
energy-fluence rate Ċ W m22
dC/dt Section 3.1.4
energy distribution of fluence rate ḞE m22
s21
J21
dḞ=dE –
energy distribution of energy-fluence rate ĊE m22
s21
dĊ=dE –
particle radiance ḞV m22
s21
sr21
dḞ=dV Section 3.1.5
energy radiance _
CV W m22
sr21
dĊ=dV Section 3.1.5
energy distribution of particle radiance ḞV; E m22
s21
sr21
J21
dḞV=dE Eq. 3.1.11a
energy distribution of energy radiance ĊV; E m22
s21
sr21
dĊV=dE Eq. 3.1.11b
a
The expression “distribution of a quantity with respect to energy” has been replaced in this table by the shorthand expression “energy
distribution of the quantity.”
Radiometry
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Table 3.2. Vector radiometric quantities
Namea
Symbol Unit Definition Appearance in the Report
vector particle radiance ḞV m22
s21
sr21
VḞV Section 3.2.1
vector energy radiance ĊV W m22
sr21
VĊV Section 3.2.1
energy distribution of vector particle radiance ḞV; E m22
s21
sr21
J21
V _
FV; E Eq. 3.2.1a
energy distribution of vector energy radiance ĊV; E m22
s21
sr21
VĊV; E Eq. 3.2.1b
vector fluence rate Ḟ m22
s21
Ð
ḞVdV Section 3.2.2
vector energy-fluence rate Ċ W m22
Ð
ĊVdV Section 3.2.2
energy distribution of vector fluence rate ḞE m22
s21
J21
Ð
ḞV; EdV –
energy distribution of vector energy-fluence rate ĊE m22
s21
Ð
ĊV; EdV –
vector fluence F m22
Ð
Ḟdt Section 3.2.3
vector energy fluence C J m22
Ð
Ċdt Section 3.2.3
energy distribution of vector fluence FE m22
J21
Ð
ḞEdt Eq. 3.2.3a
energy distribution of vector energy fluence CE m22
Ð
ĊEdt Eq. 3.2.3b
a
The expression “distribution of a quantity with respect to energy” has been replaced in this table by the shorthand expression “energy
distribution of the quantity.”
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4. 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 particle is altered
or the particle is absorbed. The interaction might
be followed by the emission of one or several sec-
ondary particles. The likelihood of such inter-
actions is characterized by interaction coefficients.
They refer to a specific interaction process, type
and energy of radiation, and target or material.
The fundamental interaction coefficient is the
cross section (see Section 4.1). All other interaction
coefficients defined in this Report can be expressed
in terms of cross sections or differential cross
sections.
Interaction coefficients and related quantities
discussed in this Section are listed in Table 4.1.
4.1 Cross Section
The cross section, s, of a target entity, for a
particular interaction produced by incident
charged or uncharged particles of a given type
and energy, is the quotient of N by F, where N is
the mean number of such interactions per target
entity subjected to the particle fluence F, thus
s ¼
N
F
:
Unit: m2
A special unit often used for the cross section is
the barn, b, defined by
1 b ¼ 1028
m2
¼ 100 fm2
:
A full description of an interaction process requires,
among other things, the knowledge of the distri-
butions of cross sections in terms of energy and
direction of all emergent particles resulting from
the interaction. Such distributions, sometimes
called differential cross sections, are obtained by
differentiations of s with respect to energy of emer-
gent particles and solid angle.
If incident particles of a given type and energy
can undergo different and independent types of
interaction in a target entity, the resulting cross
section, sometimes called the total cross section, s,
is expressed by the sum of the component cross sec-
tions, sj, hence
s ¼
X
J
sJ ¼
1
F
X
J
NJ; ð4:1:1Þ
where NJ is the mean number of interactions of
type J per target entity subjected to the particle
fluence F, and sJ is the component cross section
relating to an interaction of type J.
4.2 Mass Attenuation Coefficient
The mass attenuation coefficient, m/r, of a
material, for uncharged particles of a given type
and energy, is the quotient of dN/N by rdl,
where dN/N is the mean fraction of the particles
that experience interactions in traversing a dis-
tance dl in the material of density r, thus
m
r
¼
1
r dl
dN
N
:
Unit: m2
kg21
The quantity m is the linear attenuation coeffi-
cient. The probability that at normal incidence an
uncharged particle undergoes an interaction in a
material layer of thickness dl is mdl.
The reciprocal of m is called the mean free path of
an uncharged particle.
The linear attenuation coefficient, m, depends on
the density, r, of the material. This dependence is
largely removed by using the mass attenuation
coefficient, m/r.
The mass attenuation coefficient can be
expressed in terms of the total cross section, s. The
mass attenuation coefficient is the product of s and
NA/M, where NA is the Avogadro constant, and M is
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the molar mass of the target material, thus
m
r
¼
NA
M
s ¼
NA
M
X
J
sJ; ð4:2:1Þ
where sJ is the component cross section relating to
an interaction of type J.
Relationship 4.2.1 can be written as
m
r
¼
nt
r
s; ð4:2:2Þ
where nt is the number density of target 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,
m
r
¼
1
r
X
L
ðntÞL sL ¼
1
r
X
L
ðntÞL
X
J
sL;J; ð4:2:3Þ
where (nt)L is the number density of target entities
of type L, sL the total cross section for an entity L,
and sL,J the cross section of an interaction of type
J for a single target entity of type L. Relationship
4.2.3, which ignores the effects on the cross sec-
tions of the molecular, chemical, or crystalline
environment of an atom, is justified in most cases,
but can occasionally lead to errors, for example, in
the interaction of low-energy photons with mol-
ecules (Hubbell, 1969) and in the interaction of
slow neutrons with molecules, particularly those
containing hydrogen (see, e.g., Caswell et al., 1982;
Houk and Wilson, 1967; Rauch, 1994).
4.3 Mass Energy-Transfer Coefficient
The mass energy-transfer coefficient, mtr /r, of a
material, for uncharged particles of a given type
and energy, is the quotient of dRtr/R by rdl,
where dRtr is the mean energy that is trans-
ferred to kinetic energy of charged particles by
interactions of the uncharged particles of inci-
dent radiant energy R in traversing a distance
dl in the material of density r, thus
mtr
r
¼
1
r dl
dRtr
R
:
Unit: m2
kg21
The binding energies of the liberated charged
particles are not to be included in dRtr. However,
binding energies are usually assumed negligible
and thus included in calculations of the mass
energy-transfer coefficient for photons. In materials
consisting of elements of modest atomic number,
such an inconsistency with the definition can
become important for photons with energies below
1 keV. In addition, the decay of excited nuclear
states produced by interactions can contribute
charged particles; this process is usually not
included in evaluation of the mass energy-transfer
coefficient for photons, but rather is treated as a
separate source term in dosimetry calculations.
If incident uncharged particles of a given type
and energy can produce several types of indepen-
dent interactions in a target entity, the mass
energy-transfer coefficient can be expressed in
terms of the component cross sections, sJ, by the
relationship
mtr
r
¼
NA
M
X
J
fJsJ; ð4:3:1Þ
where fJ is the quotient of the mean energy trans-
ferred to kinetic energy of charged particles in an
interaction of type J by the kinetic energy of the
incident uncharged particle, NA is the Avogadro
constant, and M is the molar mass of the target
material.
The mass energy-transfer coefficient is related to
the mass attenuation coefficient, m/r, by the
equation
mtr
r
¼
m
r
f; ð4:3:2Þ
where
f ¼
P
J fJsJ
P
J sJ
:
The mass energy-transfer coefficient of a compound
material is usually treated as if the latter consisted
of independent atoms. Thus,
mtr
r
¼
1
r
X
L
ðntÞL
X
J
fL;JsL;J; ð4:3:3Þ
where (nt)L and sL, J have the same meaning as in
Eq. 4.2.3, and fL, J is the quotient of the mean
energy transferred to kinetic energy of charged par-
ticles in an interaction of type J with a target entity
of type L by the kinetic energy of the incident
uncharged particle. Relationship 4.3.3 implies the
same approximations as relationship 4.2.3.
A fraction g of the kinetic energy transferred to
charged particles is subsequently lost on average in
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radiative processes (bremsstrahlung, in-flight
annihilation, and fluorescence radiations) as the
charged particles slow to rest in the material, and
this fraction g is specific to the material. The
product of mtr/r for a material and (1 2 g) is called
the mass energy-absorption coefficient, men/r, of the
material for uncharged particles,
men
r
¼
mtr
r
ð1 gÞ: ð4:3:4Þ
The mass energy-absorption coefficient of a com-
pound material depends on the stopping power (see
Section 4.4) of the material. Thus, its evaluation
cannot, in principle, be reduced to a simple sum-
mation of the mass energy-absorption coefficient of
the atomic constituents (Seltzer, 1993). Such a
summation can provide an adequate approximation
when the value of g is sufficiently small.
4.4 Mass Stopping Power
The mass stopping power, S/r, of a material, for
charged particles of a given type and energy, is
the quotient of dE by rdl, where dE is the mean
energy lost by the charged particles in traversing
a distance dl in the material of density r, thus
S
r
¼
1
r
dE
dl
:
Unit: J m2
kg2l
The quantity E may be expressed in eV, and
hence S/r may be expressed in eV m2
kg2l
or some
convenient multiples or submultiples, such as MeV
cm2
g2l
.
The quantity S ¼ dE/dl denotes the linear stop-
ping power.
The mass stopping power can be expressed as a
sum of independent components by
S
r
¼
1
r
dE
dl

el
þ
1
r
dE
dl

rad
þ
1
r
dE
dl

nuc
; ð4:4:1Þ
where
1
r
dE
dl

el
¼
1
r
Sel is the mass electronic (or collision4
)
stopping power due to interactions
with atomic electrons resulting in
ionization or excitation,
1
r
dE
dl

rad
¼
1
r
Srad is the mass radiative stopping
power due to emission of brems-
strahlung in the electric fields of
atomic nuclei or atomic elec-
trons, and
1
r
dE
dl

nuc
¼
1
r
Snuc is the mass nuclear stopping
power5
due to elastic Coulomb
interactions in which recoil
energy is imparted to atoms.
In addition, one can consider energy losses due to
nonelastic nuclear interactions, but such processes
are not usually described by a stopping power.
The separate mass stopping-power components
can be expressed in terms of cross sections. For
example, the mass electronic stopping power for an
atom can be expressed as
1
r
Sel ¼
NA
M
Z
ð
1
ds
d1
d1; ð4:4:2Þ
where NA is the Avogadro constant, M the molar
mass of the atom, Z its atomic number, ds/d1 the
differential cross section (per atomic electron) for
interactions, and 1 is the energy loss.
Forming the quotient Sel/r greatly reduces, but
does not eliminate, the dependence on the density
of the material (see ICRU, 1984; 1993b, where the
density effect and the stopping powers for com-
pounds are discussed).
4.5 Linear Energy Transfer (LET)
The linear energy transfer or restricted linear
electronic stopping power, LD, of a material, for
charged particles of a given type and energy, is
the quotient of dED by dl, where dED is the
mean energy lost by the charged particles due to
electronic interactions in traversing a distance
dl, minus the mean sum of the kinetic energies
in excess of D of all the electrons released by the
charged particles, thus
LD ¼
dED
dl
:
Unit: J m21
The quantity ED may be expressed in eV, and
hence LD may be expressed in eV m2l
or some con-
venient multiples or submultiples, such as keV
mm2l
.
4
The older term was “collision stopping power.” Because all
interactions can be considered “collisions,” the more specific
term “electronic” is strongly preferred.
5
The established term “mass nuclear stopping power” can be
misleading because this quantity does not pertain to nuclear
interactions.
Interaction Coefficients and Related Quantities
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The linear energy transfer, LD, can also be
expressed by
LD ¼ Sel
dE ke;D
dl
; ð4:5:1Þ
where Sel is the linear electronic stopping power,
and dEke, D is the mean sum of the kinetic energies,
greater than D, of all the electrons released by the
charged particle in traversing a distance dl.
The definition expresses the following energy
balance: energy lost by the primary charged par-
ticle in interactions with electrons, along a distance
dl, minus energy carried away by energetic second-
ary electrons having initial kinetic energies greater
than D, equals energy considered as “locally trans-
ferred,” although the definition specifies an energy
cutoff, D, and not a distance cutoff.
Note that LD includes the binding energy of elec-
trons for all interactions. As a consequence, L0
refers to the energy lost that does not reappear as
kinetic energy of released electrons. Thus, the
threshold of the kinetic energy of the released elec-
trons is D as opposed to D minus the binding energy.
In order to simplify notation, D may be expressed
in eV. Then L100 is understood to be the linear
energy transfer for an energy cutoff of 100 eV. If no
energy cutoff is imposed, the unrestricted linear
energy transfer, L1, is equal to Sel, and may be
denoted simply as L.
4.6 Radiation Chemical Yield
The radiation chemical yield, G(x), of an entity,
x, is the quotient of n(x) by
1, where n(x) is the
mean amount of substance of that entity pro-
duced, destroyed, or changed in a system by the
mean energy imparted,
1, to the matter of that
system, thus
GðxÞ ¼
nðxÞ

1
:
Unit: mol J21
The mole is the amount of substance of a system
that contains as many elementary entities as there
are atoms in 0.012 kg of 12
C. The elementary enti-
ties must be specified and can be atoms, molecules,
ions, electrons, other particles, or specified groups
of such particles (BIPM, 2006).
A related quantity, called G value, has been
defined as the mean number of entities produced,
destroyed, or changed by an energy imparted of
100 eV. The unit in which the G value is expressed
is (100 eV)2l
. A G value of 1 (100 eV)21
corresponds
to a radiation chemical yield of approximately
0.1036 mmol J21
.
4.7 Ionization Yield in a Gas
The ionization yield in a gas, Y, is the quotient
of N by E, where N is the mean total liberated
charge of either sign, divided by the elementary
charge, when the initial kinetic energy E of a
charged particle of a given type is completely
dissipated in the gas, thus
Y ¼
N
E
:
Unit: J21
The quantity Y may also be expressed in eV21
.
The ionization yield in a gas is a particular case
of the radiation chemical yield. It follows from the
definition of Y that the charge produced by brems-
strahlung or other secondary radiation emitted by
the initial and secondary charged particles is
included in N. The charge of the initial charged
particle is not included in N, as this charge is not
liberated in the energy-dissipation process.
4.8 Mean Energy Expended in a Gas per Ion
Pair Formed
The mean energy expended in a gas per ion pair
formed, W, is the quotient of E by N, where N is
the mean total liberated charge of either sign,
divided by the elementary charge, when the
initial kinetic energy E of a charged particle
introduced into the gas is completely dissipated
in the gas, thus
W ¼
E
N
:
Unit: J
The quantity W may also be expressed in eV.
It follows from the definition of W that the ions
produced by bremsstrahlung or other secondary
radiation emitted by the initial and secondary
charged particles are included in N. The charge of
the initial charged particle is not included in N.
In certain cases, it could be necessary to focus
attention on the variation in the mean energy
expended per ion pair along the path of the
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particle; then the concept of a differential W is
required, as defined in ICRU Report 31 (ICRU,
1979). The differential value, w, of the mean
energy expended in a gas per ion pair formed is the
quotient of dE by dN; thus
w ¼
dE
dN
; ð4:8:1Þ
where dE is the mean energy lost by a charged par-
ticle of kinetic energy E in traversing a layer of gas
of infinitesmal thickness, and dN is the mean total
liberated charge of either sign divided by the
elementary charge when dE is completely dissi-
pated in the gas.
The relationship between W and w is given by
WðEÞ ¼
E
ÐE
I ½dE0=wðE0Þ
; ð4:8:2Þ
where I is the lowest ionization potential of the gas,
and E’ is the instantaneous kinetic energy of the
charged particle as it slows down.
In solid-state theory, a concept similar to W is the
average energy required for the formation of a
hole–electron pair.
Table 4.1. Interaction coefficients and related quantities
Name Symbol Unit Definition Appearance in
the Report
cross section s m2
N/F Section 4.1
mass attenuation coefficient m/r m2
kg21
dN/(N r dl) Section 4.2
linear attenuation coefficient m m21
dN/(N dl) Section 4.2
mean free path m21
m N dl/dN Section 4.2
mass energy-transfer coefficient mtr/r m2
kg21
dRtr/(R r dl) Section 4.3
mass energy-absorption coefficient men/r m2
kg21
(mtr/r)(1 2 g) Section 4.3
mass stopping power S/r J m2
kg21
dE/(r dl) Section 4.4
linear stopping power S J m21
dE/dl Section 4.4
linear energy transfer LD J m21
dED /dl Section 4.5
radiation chemical yield G(x) mole J21
n(x)/1̄ Section 4.6
ionization yield in a gas Y J21
N/E Section 4.7
mean energy expended in a gas per ion pair formed W J E/N Section 4.7
differential mean energy expended in a gas
per ion pair formed
w J dE/dN Eq. 4.8.1
Interaction Coefficients and Related Quantities
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5. Dosimetry
The effects of radiation on matter depend on the
radiation field, as specified by the radiometric
quantities defined in Sections 3.1 and 3.2, and on
the interactions between radiation and matter, as
characterized by the interaction quantities defined
in Sections 4.1 to 4.5. Dosimetric quantities, which
are selected to provide a physical measure to corre-
late with actual or potential effects, are products of
radiometric quantities and interaction coefficients.
In calculations, the values of these quantities and
coefficients must be known, while measurements
might not require this information.
Radiation interacts with matter in a series of pro-
cesses in which particle energy is converted and
finally deposited in matter. The dosimetric quantities
that describe these processes are presented below in
two Sections dealing with the conversion and with
the deposition of energy. The evaluation of the quan-
tities defined in this Section requires, in general, con-
sideration of elapsed time. Most applications and
measurements of ionizing radiation involve time
scales of the order of seconds or minutes, so at a prac-
tical level the decay of an excited atomic state
usually can be assumed to have occurred. This might
not be the case for nuclear de-excitations or spon-
taneous nuclear decays, with the obvious example of
a radionuclide produced in an interaction or pre-
existing in a volume of interest.
5.1 Conversion of Energy
The term conversion of energy refers to the trans-
fer of energy from ionizing particles to secondary
ionizing particles. The quantity kerma pertains to
the kinetic energy of the charged particles liberated
by uncharged particles; the energy expended to
overcome the binding energies, usually a relatively
small component, is, by definition, not included. In
addition to kerma, the quantity cema is defined that
pertains to the energy lost by charged particles (e.g.,
electrons, protons, alpha particles) in interactions
with atomic electrons; by definition, the binding
energies are included. Cema differs from kerma in
that cema involves the energy lost in electronic
interactions by the incoming charged particles,
while kerma involves the energy of outgoing
charged particles as a result of interactions by
incoming uncharged particles. Both quantities, in
conditions of charged-particle equilibrium, serve as
approximations to absorbed dose, kerma for
uncharged and cema for charged ionizing particles.
5.1.1 Kerma6
The kerma, K, for ionizing uncharged particles,
is the quotient of dEtr by dm, where dEtr is the
mean sum of the initial kinetic energies of all
the charged particles liberated in a mass dm of
a material by the uncharged particles incident
on dm, thus
K ¼
dEtr
dm
:
Unit: J kg21
The special name for the unit of kerma is gray
(Gy).
The quantity dEtr includes the kinetic energy of
the charged particles emitted in the decay of
excited atoms/molecules7
or in nuclear
de-excitation or disintegration.
For a fluence, F, of uncharged particles of energy
E, the kerma, K, in a specified material is given by
K ¼ FEmtr=r ¼Cmtr=r; ð5:1:1Þ
where mtr/r is the mass energy-transfer coefficient
of the material for these particles.
The kerma per fluence, K/F, is termed the kerma
coefficient for uncharged particles of energy E in a
specified material. The term kerma coefficient is
used in preference to the older term kerma factor,
as the word coefficient implies a physical dimension
whereas the word factor does not.
In dosimetric calculations, the kerma, K, is usually
expressed in terms of the distribution, FE, of the
6
Kinetic energy released per mass.
7
For example, Auger, Coster-Kronig, shake-off electrons.
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uncharged-particle fluence with respect to energy
(see Eq. 3.l.8a). The kerma, K, is then given by
K ¼
ð
FEE
mtr
r
dE ¼
ð
CE
mtr
r
dE; ð5:1:2Þ
where mtr/r is the mass energy-transfer coefficient of
the material for uncharged particles of energy E.
The expression of kerma in terms of fluence
makes it clear that one can refer to a value of
kerma or kerma rate for a specified material at a
point in free space, or inside a different material.
Thus, one can speak, for example, of the air kerma
at a point inside a water phantom.
Although kerma is a quantity that concerns the
initial transfer of energy to matter, it is sometimes
used as an approximation to absorbed dose. The
numerical value of the kerma approaches that of
the absorbed dose to the degree that charged-
particle equilibrium exists, that radiative losses are
negligible, and that the kinetic energy of the
uncharged particles is large compared with the
binding energy of the liberated charged particles.
Charged-particle equilibrium exists at a point if
the distribution of the charged-particle radiance
with respect to energy (see Eq. 3.1.11a) is constant
within distances equal to the maximum charged-
particle range.
A quantity related to the kerma, termed the col-
lision kerma, has long been in use as an approxi-
mation to absorbed dose (Attix, 1979a; 1979b)
when radiative losses are not negligible. The col-
lision kerma, Kcol, excludes the radiative losses by
the liberated charged particles, and – for a fluence,
F, of uncharged particles of energy E in a specified
material – is given by
Kcol ¼ FE
men
r
¼ FE
mtr
r
ð1 gÞ ¼ Kð1 gÞ; ð5:1:3Þ
where men/r is the mass energy-absorption coeffi-
cient of the material for uncharged particles of
energy E, and g is the fraction of the kinetic energy
of liberated charged particles that would be lost in
radiative processes in that material (see Section
4.3).
In dosimetric calculations, the collision kerma,
Kcol, can be expressed in terms of the distribution,
FE, of the uncharged-particle fluence with respect
to energy as
Kcol ¼
ð
FEE
men
r
dE ¼
ð
FEE
mtr
r
1 g
ð Þ dE
¼ K 1
g
ð Þ; ð5:1:4Þ
where
g is the mean value of g averaged over the
distribution of the kerma with respect to the elec-
tron energy.
The expression of collision kerma in terms of the
product of the kerma and a radiative-loss correction
factor evaluated for the same material as the
kerma suggests that one can refer to a value of col-
lision kerma or collision kerma rate for a specified
material at a point in free space, or inside a differ-
ent material.
5.1.2 Kerma Rate
The kerma rate, _
K, is the quotient of dK by dt,
where dK is the increment of kerma in the time
interval dt, thus
_
K ¼
dK
dt
:
Unit: J kg21
s21
If the special name gray is used, the unit of
kerma rate is gray per second (Gy s21
).
5.1.3 Exposure
The exposure, X, is the quotient of dq by dm,
where dq is the absolute value of the mean total
charge of the ions of one sign produced when all
the electrons and positrons liberated or created
by photons incident on a mass dm of dry air are
completely stopped in dry air, thus
X ¼
dq
dm
:
Unit: C kg21
The ionization produced by electrons emitted in
atomic/molecular relaxation processes is included
in dq. The ionization due to photons emitted by
radiative processes (i.e., bremsstrahlung and fluor-
escence photons) is not to be included in dq. Except
for this difference, significant at high energies, the
exposure, as defined above, is the ionization ana-
logue of the dry-air kerma. Exposure can be
expressed in terms of the distribution, FE, of the
fluence with respect to the photon energy, E, and
the mass energy-transfer coefficient, mtr/r, for dry
air and for that energy as follows:
X
e
W
ð
FEE
mtr
r
ð1 gÞdE

e
W
ð
FEE
men
r
dE; ð5:1:5Þ
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where e is the elementary charge, W is the mean
energy expended in dry air per ion pair formed, and
g is the fraction of the kinetic energy of the electrons
liberated by photons that is lost in radiative pro-
cesses in air. The approximation symbol in Eq. 5.1.5
reflects the fact that the exposure includes the
charge of electrons or ions liberated by the incident
photons whereas W pertains only to the charge pro-
duced during the slowing down of these electrons.8
For photon energies of the order of 1 MeV or
below, for which the value of g is small, Eq. 5.1.5
can be further approximated by X (e/W)Kair
(l 2
g) ¼ (e/W)Kcol,air, where Kair is the dry-air
kerma for primary photons and
g is the mean value
of g averaged over the distribution of the air kerma
with respect to the electron energy.
As in the case of collision kerma, it can be con-
venient to refer to a value of exposure or of exposure
rate in free 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.
5.1.4 Exposure Rate
The exposure rate, _
X, is the quotient of dX by dt,
where dX is the increment of exposure in the
time interval dt, thus
_
X ¼
dX
dt
:
Unit: C kg21
s21
5.1.5 Cema9
The cema, C, for ionizing charged particles, is
the quotient of dEel by dm, where dEel is the
mean energy lost in electronic interactions in a
mass dm of a material by the charged particles,
except secondary electrons, incident on dm, thus
C ¼
dEel
dm
:
Unit: J kg21
The special name of the unit of cema is gray
(Gy).
The energy lost by charged particles in electronic
interactions includes the energy expended to over-
come the binding energy and the initial kinetic
energy of the liberated electrons, referred to as sec-
ondary electrons. Thus, energy subsequently lost
by all secondary electrons is excluded from dEel.
The cema, C, can be expressed in terms of the dis-
tribution, FE, of the charged-particle fluence, with
respect to energy (see Eq. 3.1.8a). According to the
definition of cema, the distribution FE does not
include the contribution of secondary electrons to the
fluence, but the contributions of all other charged
particles, such as secondary protons, alpha particles,
tritons, and ions produced in nuclear interactions, are
included in the cema. The cema, C, is thus given by
C ¼
ð
FE
Sel
r
dE ¼
ð
FE
L1
r
dE; ð5:1:6Þ
where Sel/r is the mass electronic stopping power of a
specified material for charged particles of energy E,
and L1 is the corresponding unrestricted linear
energy transfer. In general, the cema is evaluated as
the sum of contributions by all species of charged par-
ticles, except liberated secondary electrons.
For charged particles of high energies, it might
be undesirable to disregard energy transport by
secondary electrons of all energies. A modified
concept, restricted cema, CD, (Kellerer et al., 1992)
is then defined as
CD ¼
ð
F0
E
LD
r
dE: ð5:1:7Þ
This differs from the integral in Eq. 5.1.6 in that
L1 is replaced by LD and that the distribution F
0
E
now includes secondary electrons liberated in dm
with kinetic energies greater than D. For D ¼ 1,
restricted cema is identical to cema.
The expression of cema and restricted cema in
terms of fluence makes it clear that one can refer
to their values for a specified material at a point in
free space, or inside a different material. Thus, one
can speak, for example, of tissue cema in air
(Kellerer et al., 1992).
The quantities cema and restricted cema can be
used as approximations to absorbed dose from
charged particles. Equality of absorbed dose and
cema is approached to the degree that secondary-
charged-particle equilibrium exists and that radia-
tive losses and those due to elastic nuclear inter-
actions are negligible. Secondary-charged-particle
equilibrium is achieved at a point if the fluence of
secondary charged particles is constant within dis-
tances equal to their maximum range. For restricted
8
This difference, although relatively small, tends to become
more significant as the photon energy decreases. Additionally, W
is not constant as perhaps implied in Eq. 5.1.5, but known to
increase at low energies (ICRU, 1979). At energies for which the
variation of W with energy becomes important, one should
consider the effect of this increase on the relationship between
exposure and air kerma.
9
Converted energy per mass.
Dosimetry
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cema, only partial secondary-charged-particle equi-
librium, up to kinetic energy D, is required.
5.1.6 Cema Rate
The cema rate, _
C, is the quotient of dC by dt,
where dC is increment of cema in the time inter-
val dt, thus
_
C ¼
dC
dt
:
Unit: J kg21
s21
cema rate is gray per second (Gy s21
).
5.2 Deposition of Energy
In this Section, certain stochastic quantities are
introduced. Energy deposit is the basis in terms of
which all other quantities presented here can be
defined.
5.2.1 Energy Deposit
The energy deposit, 1i, is the energy deposited in
a single interaction, i, thus
1i ¼ 1in 1out þ Q;
where 1in is the energy of the incident ionizing
particle (excluding rest energy), 1out is the sum of
the energies of all charged and uncharged ioniz-
ing particles leaving the interaction (excluding
rest energy), and Q is the change in the rest ener-
gies of the nucleus and of all elementary par-
ticles involved in the interaction (Q . 0: decrease
of rest energy; Q , 0: increase of rest energy).
Unit: J
The quantity 1i may also be expressed in eV.
Note that 1i is a stochastic quantity.
Interactions with atomic electrons resulting in
atomic excitation (and subsequent de-excitation)
do not involve a change in rest energies of the
nucleus or of elementary particles and thus have
Q ¼ 0. The restriction of 1in and 1out to energies of
ionizing particles can in principle lead to a slight
energy imbalance by ignoring the net energy
transport by non-ionizing particles; thus, energy of
non-ionizing particles, e.g., very-low-energy
photons, that can leave the interaction are
included in the energy deposit. The dimensions
over which that energy is re-absorbed can be
thought to define a spatial region associated with
a single interaction.
5.2.2 Energy Imparted
The energy imparted, 1 , to the matter in a given
volume is the sum of all energy deposits in the
volume, thus
1 ¼
X
i
1i;
where the summation is performed over all
energy deposits, 1i, in that volume.
Unit: J
The quantity 1 may also be expressed in eV. Note
that 1 is a stochastic quantity.
The energy deposits over which the summation
is performed can belong to one or more energy-
deposition events; for example, they might belong
to one or several independent particle trajectories.
The term energy-deposition event denotes the
imparting of energy to matter by correlated par-
ticles. Examples include a proton and its second-
ary electrons, an electron-positron pair, or the
primary and secondary particles in nuclear
reactions.
If the energy imparted to the matter in a given
volume is due to a single energy-deposition event,
it is equal to the sum of the energy deposits in the
volume associated with the energy-deposition
event. If the energy imparted to the matter in a
given volume is due to several energy-deposition
events, it is equal to the sum of the individual
energies imparted to the matter in the volume due
to each energy-deposition event.
The mean energy imparted,
1, to the matter in a
given volume equals the mean radiant energy, Rin,
of all charged and uncharged ionizing particles
that enter the volume minus the mean radiant
energy, Rout, of all charged and uncharged ionizing
particles that leave the volume, plus the mean
sum,
P
Q, of all changes of the rest energy of
nuclei and elementary particles that occur in the
volume (Q . 0: decrease of rest energy; Q , 0:
increase of rest energy); thus

1 ¼ Rin Rout þ
X
Q: ð5:2:1Þ
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5.2.3 Lineal Energy
The lineal energy, y, is the quotient of 1s by
l,
where 1s 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 that
volume, thus
y ¼
1s

l
:
Unit: J m21
The quantity y is a stochastic quantity. The
numerator 1s may be expressed in eV; hence y may
be expressed in multiples and submultiples of eV
and m, e.g., in keV mm2l
.
The mean chord length of a volume is the mean
length of randomly oriented chords (uniform isotropic
randomness) through that volume. For a convex body,
it can be shown that the mean chord length,
l, equals
4V/A, where V is the volume and A is the surface
area (Cauchy, 1850; Kellerer, 1980); thus, for a sphere
the mean chord length is 2/3 of the sphere diameter.
It is useful to consider the probability distribution
of y. The value of the distribution function, F(y), is the
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 of F(y); thus
fðyÞ ¼
dFðyÞ
dy
: ð5:2:2Þ
F(y) and f(y) are independent of absorbed dose and
absorbed-dose rate, but are dependent on the size
and shape of the volume.
5.2.4 Specific Energy
The specific energy (imparted), z, is the quotient
of 1 by m, where 1 is the energy imparted by
ionizing radiation to matter in a volume of mass
m, thus
z ¼
1
m
:
Unit: J kg21
The special name for the unit of speific energy is
gray (Gy).
The quantity z is a stochastic quantity. The
specific energy can be due to one or more energy-
deposition events. The distribution function, F(z),
is the probability that the specific energy is equal
to or less than z. The probability density, f(z), is the
derivative of F(z); thus
fðzÞ ¼
dFðzÞ
dz
: ð5:2:3Þ
F(z) and f(z) depend on absorbed dose in the mass,
m. The probability density f(z) includes a discrete
component (in terms of a Dirac delta function) at
z ¼ 0 for the probability of no energy deposition.
The distribution function of the specific energy
deposited in a single energy-deposition event, Fs(z),
is the conditional probability that a specific energy
less than or equal to z is deposited if one energy-
deposition event has occurred. The probability
density, fs (z), is the derivative of Fs(z); thus
fsðzÞ ¼
dFsðzÞ
dz
: ð5:2:4Þ
For convex volumes, y and the increment, z, of
specific energy due to a single energy-deposition
event are related by
y ¼
rA
4
z; ð5:2:5Þ
where A is the surface area of the volume, and r is
the density of matter in the volume.
5.2.5 Absorbed Dose
The absorbed dose, D, is the quotient of d
1 by
dm, where d
1 is the mean energy imparted by
ionizing radiation to matter of mass dm, thus
D ¼
d
1
dm
:
Unit: J kg21
The special name for the unit of absorbed dose
is gray (Gy).
In the limit of a small domain10
, the mean
specific energy
z is equal to the absorbed dose D.
10
The absorbed dose, D, is considered a point quantity, but it
should be recognized that the physical process does not allow dm
to approach zero in the mathematical sense.
Dosimetry
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5.2.6 Absorbed-Dose Rate
The absorbed-does rate, _
D, is the quotient of dD
by dt, where dD is the increment of absorbed
does in the time interval dt, thus
_
D ¼
dD
dt
:
Unit: J kg21
s21
absorbed-does rate is gray per second (Gy s21
).
Table 5.1. Dosimetric quantities: conversion of energy
Name Symbol Unit Definition Appearance in the Report
kerma K J kg21
Gy dEtr/dm Section 5.1.1
collision kerma Kcol J kg21
Gy K(1 2 g) Section 5.1.1
kerma coefficient – J m2
kg21
Gy m2
K/F Section 5.1.1
kerma rate K̇ J kg21
s21
Gy s21
dK/dt Section 5.1.2
exposure X C kg21
dq/dm Section 5.1.3
exposure rate Ẋ C kg21
s21
dX/dt Section 5.1.4
cema C J kg21
Gy dEel/dm Section 5.1.5
restricted cema CD J kg21
Gy – Section 5.1.5
cema rate Ċ J kg21
s21
Gy s21
dC/dt Section 5.1.6
Table 5.2. Dosimetric quantities: deposition of energy
energy deposit 1i J 1in 2 1out þ Q Section 5.2.1
energy imparted 1 J X
i
1i
Section 5.2.2
lineal energy y J m21
1s=
l Section 5.2.3
specific energy z J kg21
Gy 1/m Section 5.2.4
absorbed dose D J kg21
Gy d
1=dm Section 5.2.5
absorbed-dose rate Ḋ J kg21
s21
Gy s21
dD/dt Section 5.2.6
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6. Radioactivity
The term radioactivity refers to the phenomena
associated with spontaneous transformations that
involve changes in the nuclei of atoms or of the
energy states of the nuclei of atoms. The energy
released in such transformations is emitted as
nuclear particles (e.g., alpha particles, electrons,
and positrons) and/or photons.
Such transformations represent a stochastic
process. The whole atom is involved in this process
because nuclear transformations can also affect the
atomic shell structure and cause emission or
capture of electrons, the emission of photons, or
both.
A nuclide is a species of atoms having a specified
number of protons and neutrons in its nucleus.
Unstable nuclides, which transform to stable or
unstable progeny, are called radionuclides. The
transformation results in another nuclide or in a
transition to a lower energy state of the same
nuclide.
6.1 Decay Constant
The decay constant, l, of a radionuclide in a
particular energy state is the quotient of
2dN/N by dt, where dN/N is the mean
fractional change in the number of nuclei
in that energy state due to spontaneous
nuclear transformations in the time interval
dt, thus
l ¼
dN=N
dt
:
Unit: s2l
The quantity (ln 2)/l, commonly called the
half life, Tl/2, of a radionuclide, is the mean time
taken for the radionuclides in the particular
energy state to decrease to one half of their initial
number.
6.2 Activity
The activity, A, of an amount of a radionuclide
in a particular energy state at a given time is
the quotient of 2dN by dt, where dN is the
mean change in the number of nuclei in that
energy state due to spontaneous nuclear trans-
formations in the time interval dt, thus
A ¼
dN
dt
:
Unit: s2l
The special name for the unit of activity is bec-
querel (Bq).
The “particular energy state” is the ground
state of the radionuclide unless otherwise specified.
The activity, A, of an amount of a radionuclide in
a particular energy state is equal to the product of
the decay constant, l, for that state, and the
number N of nuclei in that state, thus,
A ¼ lN: ð6:2:1Þ
6.3 Air-kerma-Rate Constant
The air-kerma-rate constant, Gd, of a radio-
nuclide emitting photons is the quotient of l2 _
Kd
by A, where _
Kd is the air-kerma rate due to
photons of energy greater than d, at a distance l
in vacuo from a point source of this nuclide
having an activity A, thus
Gd ¼
l2 _
Kd
A
:
Unit: m2
J kg2l
If the special names gray (Gy) and becquerel
(Bq) are used, the unit of air-kerma-rate con-
stant is m2
Gy Bq21
s2l
.
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The photons referred to in the definition include
gamma rays, characteristic x rays, and internal
bremsstrahlung.
The air-kerma-rate constant, a characteristic
of a radionuclide, is defined in terms of an
ideal point source, and the term is not strictly
applicable to a source of finite extent. In a
source of finite size, attenuation and scattering
occur, and annihilation radiation and external
bremsstrahlung can be produced. In many
cases, these processes require significant
corrections.
Any medium intervening between the source and
the point of measurement will give rise to absorption
and scattering for which corrections are needed.
The selection of the value of d depends upon the
application. To simplify notation and ensure uniform-
ity, it is recommended that d be expressed in keV. For
example, G5 is understood to be the air-kerma-rate
constant for a photon-energy cutoff of 5 keV.
Table 6.1. Quantities related to radioactivity
decay constant l s21
2(dN/N)/dt Section 6.1
half life T1/2 s (ln 2)/l Section 6.1
activity A s21
Bq 2dN/dt Section 6.2
air-kerma-rate constant Gd m2
J kg21
m2
Gy Bq21
s21
l2 _
Kd=A Section 6.3
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References
Attix, F. H. (1979a). “The partition of kerma to account
for bremsstrahlung,” Health Phys. 36, 347–354.
Attix, F. H. (1979b). “Addendum to ‘The partition of kerma
to account for bremsstrahlung’,” Health Phys. 36, 536.
BIPM (2006). Bureau International des Poids et
Mesures, Le Systéme International d’Unités (SI), The
Initernational System of Units (SI), 8th ed. (Bureau
International des Poids et Mesures, Sèvres).
Caswell, R. S., Coyne, J. J., and Randolph, M. L. (1982).
“Kerma factors of elements and compounds for
neutron energies below 30 MeV,” Int. J. Appl. Radiat.
Isot. 33, 1227–1262.
Cauchy, A. (1850). “Mémoire sur la rectification des
courbes et la quadrature des surfaces courbes,” Mém.
Acad. Sci. XXII, 3.
CIE (1987). Commission Internationale de l’Eclairage,
Vocabulaire Electrotechnique International, CIE
Publication 50 (845) (Bureau Central de la
Commission Electrotechnique Internationale, Genève).
Houk, T. L., and Wilson, R. (1967). “Measurements of the
neutron–proton and neutron–carbon total cross sec-
tions at electron-volt energies,” Rev. Mod. Phys. 39,
546–547.
Hubbell, J. H. (1969). Photon Cross Sections, Attenuation
Coefficients, and Energy Absorption Coefficients from
10 keV to 100 GeV, NSRDS-NBS 29 (National Bureau
of Standards, Washington, DC).
ICRU (1962). International Commission on Radiation
Units and Measurements. Radiation Quantities and
Units, ICRU Report 10a (Published as National
Bureau of Standards Handbook 84, U.S. Government
Printing Office, Washington, DC).
Units, ICRU Report 11 (International Commission on
Radiation Units and Measurements, Washington, DC).
Radiation Units and Measurements, Washington, DC).
Units and Measurements. Average Energy Required to
Produce an Ion Pair, ICRU Report 31 (International
Commission on Radiation Units and Measurements,
Bethesda, MD).
Radiation Units and Measurements, Bethesda, MD).
Units and Measurements. Stopping Powers for
Electrons and Positrons, ICRU Report 37
(International Commission on Radiation Units and
Measurements, Bethesda, MD).
ICRU (1993a). International Commission on Radiation
Units and Measurements. Quantities and Units in
Radiation Protection Dosimetry, ICRU Report 51
ICRU (1993b). International Commission on Radiation
Units and Measurements. Stopping Powers and
Ranges for Protons and Alpha Particles, ICRU Report
49 (International Commission on Radiation Units and
Units and Measurements. Fundamental Quantities
and Units for Ionizing Radiation, ICRU Report 60
IS0 (1993). International Organization for Standardization.
IS0 Standards Handbook, Quantities and Units, 3rd ed.
(International Organization for Standardization, Geneva).
IUPAC (1997). International Union of Pure and Applied
Chemistry. Compendium of Chemical Terminology, 2nd
ed., McNaught, A. D., and Wilkinson, A., Eds.
(Blackwell Science, Oxford).
Kellerer, A. M. (1980). “Concepts of geometrical prob-
ability relevant to microdosimetry and dosimetry,”
p. 1049 in Proc. 7th Symp. Microdosimetry, Booz, J.,
Ebert, H. G., and Hartfiel, H. D., Eds. (Harwood
Academic Publishers, Chur, Switzerland).
Kellerer, A. M., Hahn, K., and Rossi, H. H. (1992).
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15–25.
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633–730.
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ticle fluence,” Phys. Med. Biol. 39, 1053–1062.
Rauch, H. (1994). “Hydrogen detection by neutron optical
methods,” in Neutron Scattering from Hydrogen in
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Scattering, 14–20 August 1994, Zuoz, Switzerland,
Furrer, A., Ed. (World Scientific, Singapore).
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transfer and mass energy-absorption coefficients,”
Rad. Res. 136, 147–170.
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Index
absorbed dose 27
absorbed-dose rate 28
activity 29
air-kerma-rate constant 29
cema 25
cema rate 26
collision kerma 24
cross section 17
decay constant 29
energy deposit 26
energy fluence 12
energy flux 12
energy imparted 26
energy radiance 13
energy-fluence rate 13
exposure 24
exposure rate 25
fluence 12
fluence rate 13
flux 12
half life 29
ionization yield in a gas 20
kerma 23
kerma coefficient 23
kerma rate 24
lineal energy 27
linear attenuation coefficient 17
linear energy transfer 19
linear stopping power 19
mass attenuation coefficient 17
mass electronic (or collision) stopping power 19
mass energy-absorption coefficient 19
mass energy-transfer coefficient 18
mass nuclear stopping power 19
mass radiative stopping power 19
mass stopping power 19
mean energy expended in a gas per ion
pair formed
20
mean free path 17
particle number 11
particle number density 12
particle radiance 13
radiant energy 11
radiant energy density 12
radiation chemical yield 20
restricted cema 25
specific energy 27
vector energy fluence 15
vector energy radiance 14
vector energy-fluence rate 14
vector fluence 15
vector fluence rate 14
vector particle radiance 14
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