Review Article RADIATION SHIELDING TECHNOLOGY

Review Article

RADIATION SHIELDING TECHNOLOGY

J. Kenneth Shultis and Richard E. Faw*

Abstract—An historical review of the development of shieldingtechniques for indirectly ionizing radiation is presented, alongwith a summary of techniques at various levels of sophistica-tion for shielding design and analysis.Health Phys. 88(4):297–322; 2005

Key words: reviews; shielding; historical profiles; HealthPhysics Society

INTRODUCTION

THIS IS a review of the technology of shielding against theeffects of indirectly ionizing x rays, gamma rays, andneutrons produced by sources of commercial, industrial,and military concern. Not addressed is shielding forradiation sources used exclusively for medical applica-tions and sources associated with high-energy charged-particle accelerators.

There are two parts to the review. The first treats theevolution of radiation-shielding technology from thebeginning of the 20th century, when radium emanationsand low energy x rays were the only concerns and designmethods were primitive, to the beginning of the 21stcentury, when radiation shielding needs are very diverseand when computational technology gives broad rein todesign methods. The second part of the review addressesthe foundations of radiation shielding analysis and de-sign, with emphasis on the characterization of radiationsources and radiation fields, the transport of radiationthrough matter, and the conversion of radiation intensi-ties into radiation doses meaningful in the assessment ofrisk.

An extensive bibliography is provided in an effort toaddress both documents of historical interest and docu-mentation of the state-of-the-art of photon and neutronshielding. For historical documentation, the authors areindebted to Goldstein’s 1967 tribute to Everitt Blizard, to

Schaeffer’s text and his own historical review (1973), toTaylor’s review of the work of the NCRP and the ICRP(1979), and to experiences related by Simpson (1995)and Rockwell (2004).

As compilers of this review, we have tried to beobjective in our selection of topics. Clearly, we could notacknowledge all the many contributors to the advance ofradiation transport theory and radiation shielding designand analysis, much less the entire body of publishedwork. An historical review, like beauty, is in the eye ofthe beholder. Our eyes were opened to this field in thelate 1950’s, and we can’t comment at first hand to earlierwork. Those whose work we recognize are mostly thosewhose work intersected in some way with our own. Tothose we may have slighted, we offer not only ourapologies but also our thanks. Radiation shielding designand analysis has become a mature discipline with foun-dations established over many decades by many persons,all of whom can take pride in their contributions.

HISTORY OF RADIATION SHIELDING

The early years

Shielding of x-ray generators. The hazards of xrays were recognized within months of Roentgen’s 1895discovery, but dose limitation by time, distance, andshielding was at the discretion of the individual practi-tioner until about 1913. Only then were there organizedprofessional efforts to establish guides for radiationprotection, and not until about 1925 were there instru-ments available to quantify radiation exposure. In hismonumental survey of organization for radiation protec-tion, Taylor (1979) begins with British and Germanefforts at establishing guidance for x-ray shielding. In1913, the German Radiological Society on X-Ray Pro-tection Measures issued recommendations that 2 mm oflead shielding was needed, regardless of generator volt-age, workload, or filtration. In Britain, the RoentgenSociety addressed radiation protection, stressing operatorprotection, the need for beam collimation, and the im-portance of scattered x rays. No explicit recommenda-tions on shielding requirements were issued.

* Department of Mechanical and Nuclear Engineering, KansasState University, Manhattan, KS 66506.

For correspondence or reprints contact: J. K. Shultis at the aboveaddress, or email at [email protected].

(Manuscript received 25 September 2004; accepted 9 December2004)

0017-9078/05/0Copyright © 2005 Health Physics Society

297

In 1921, the British X-Ray and Radium ProtectionCommittee issued broad guidelines, both physical andadministrative, on radiation protection in x-ray facilities.For diagnostic examinations, 2 mm of lead screening wasrecommended for the operator, as well as gloves witheffectively 0.5 mm of lead shielding. For superficialtherapy (up to 100 kV x rays), 2 mm of lead shieldingwas recommended. For deep therapy (in excess of 100kV x rays) 3 mm of lead shielding was recommended.Again, filtration and workload were not addressed.

Mutscheller (1925) introduced important conceptsin x-ray shielding. He expressed the erythema dose,† ED,quantitatively in terms of the beam current i (mA),exposure time t (min), and source-to-receiver distance r(m), namely,

ED � 0.00368it

r2 (1)

independent of x-ray energy. Years later, as observed byTaylor (1981), unit erythema dose was equated to anexposure X of about 600 R. Thus, in modern terms,

X � Ko

it

r2 � 2.2it

r2 . (2)

Mutscheller also published attenuation factors inlead as a function of lead thickness and x-ray averagewavelength. This is the standard approach for assessmentof x-ray exposure rates, with Ko taking on values of 0.24to 0.97 R m2 mA�1 min�1 for well-filtered x-rays withpeak energies ranging from 30 to 140 kV (Simpkin 1987,1989).

Evolutionary changes in shielding recommendationswere made during the decades preceding World War II.These included consideration of scattered x rays, refine-ments in shielding requirements in terms of x-ray tubevoltages, recommendations for use of goggles (0.25-mmlead equivalent) and aprons (0.5-mm lead equivalent) forfluoroscopy, and specifications for tube-enclosure shield-ing and structural shielding for control rooms.

In the United States, the Advisory Committee onX-Ray and Radium Protection was established in 1929,under the auspices of the National Bureau of Standards.In 1964, the Committee was congressionally chartered asthe National Council on Radiation Protection and Mea-surements (NCRP). NCRP Reports 1 and 3 were issuedin 1931 and 1936. The former recommended a tolerancedose of 0.2 R per day, the second 0.1 R per day foroccupational exposure.

For additional information on the history of x-rayshielding, the reader is referred to excellent reviews

published by this journal. Archer (1995) addresses diag-nostic x-ray installations; McGinley and Miner (1995)address the shielding of therapeutic x-ray installations.For more information on the organizational aspects ofx-ray shielding, the reader is referred to the aforemen-tioned compendium and article by Taylor (1979, 1981).

Shielding of radium sources. In 1927, the Interna-tional Committee on X-Ray and Radium Protectionissued the following recommendations for storage ofradium sources. Tubes and applicators should have atleast 5 cm of lead shielding per 100 mg of radium.Radium solutions required lead shielding ranging from15 cm for a 0.5 g source to 30 cm for a 2 g source. NCRPReport 2 (1934) specified a 3-m protective zone aroundstored Ra sources, and recommended exhaust fans orhoods for removal of radon and decay products escapingfrom unsealed sources. Shielding of stored sources wasrevised to 4 cm of lead for 100 mg to 6 cm for 300 mg.NCRP Report 4 (1938) addressed dosimetry for gammarays emitted from radium sources. It was not until 1941that there was established a tolerance dose for radium,expressed in terms of a maximum permissible bodyburden of 0.1 �Ci. This was done largely in consider-ation of the experiences of early “radium-dial” paintersand the need for standards on safe handling of radioactiveluminous compounds (NCRP 1941).

Manhattan Project and early postwar period

Wartime. During World War II, research on nuclearfission, construction of nuclear reactors, production ofenriched uranium, generation of plutonium and separat-ing it from fission products, and the design, construction,testing, and deployment of nuclear weapons—all wereaccomplished at breakneck speed in the ManhattanProject. The project was a scientific, engineering, man-agement, and construction project of extraordinary mag-nitude and success. The pace of the project and theintroduction of many new and dangerous materials andprocesses demanded much of industrial hygiene. Radia-tion sources new in type and magnitude demanded notonly protective measures such as shielding but alsoexamination of biological effects and establishment ofwork rules.

Jones (1985) describes health and safety programswithin the Manhattan Engineering District. In June 1943,a medical section was established at District headquartersunder the direction of Professor Stafford Warren. Medi-cal research programs were established to address issuessuch as metabolism and dosimetry for ingested or inhaledradionuclides. Industrial hygiene programs were estab-lished to deal with many new and hazardous substances

† An ED value of unity represents a combination of time, distance,and beam current just leading to a first-degree burn.

298 Health Physics April 2005, Volume 88, Number 4

such as plutonium and uranium hexafluoride. What wemight call operational health physics got its start. Min-ing, milling, processing and enriching uranium involvednew risks. Processing spent fuel from nuclear reactors,dealing with waste fission and activation products, andextracting actinides also involved new risks. Bombdevelopment required dealing with radiation risks as wellas risks of high explosives. These health and safetyefforts were very successful. The Manhattan EngineeringDistrict operated with occupational injury rates muchless than those for comparable private industry.

Uncertainties in radiation transport made shieldingdesign very conservative. Shielding for the 1943graphite-moderated reactor in Chicago, constructed ofconcrete and paraffinized wood and based on attenuationmeasurements made by Fermi and Zinn, was adequatefor gamma rays and over designed for neutrons (Schaef-fer 1973). After this reactor operated successfully, theX-10 reactor was built at what would become the OakRidge National Laboratory to provide data for the designof plutonium-production reactors. The X-10 reactor wasconstructed of graphite and was air cooled. Its shieldingwas 2.1-m-thick made of baryte and haydite concretes,baryte being of high density and haydite being high inhydrogen content. This shielding also was adequate forgamma rays and over designed for neutrons (Schaeffer1973; Jaeger 1975). Operation of the X-10 reactorrevealed problems with streaming of gamma rays andneutrons around access holes in the shield. The water-cooled graphite plutonium production reactors at Han-ford, Washington, used iron thermal shields and highdensity limonite and magnetite concrete biologicalshields. Concrete compositions and design methods arediscussed in the Engineering Compendium on RadiationShielding (Jaeger 1975) and in National Standards (ANS1997).

By the 1940’s, the importance of scattered gammarays was certainly known from measurements, and use ofthe term buildup factor to characterize the relativeimportance of scattered and unscattered gamma rays hadits origin during the days of the Manhattan Project(Goldstein 1959). Neutron diffusion theory and Fermiage theory were established, but shielding requirementsfor high-energy neutrons were not well understood.Wartime radiation shielding was an empirical, rule-of-thumb craft (Goldstein 1967).

Nuclear reactors for propulsion. The Atomic En-ergy Act of 1946 transferred control of nuclear mattersfrom the Army to the civilian Atomic Energy Commis-sion (AEC). That same year, working with the AEC,the U.S. Navy began development of a nuclear pow-ered submarine and the U.S. Air Force a nuclear

powered aircraft. Both of these enterprises demandedminimization of space and weight of the nuclear reactorpower source. Such could be accomplished only by mini-mizing design margins and that required knowledge ofmechanical, thermal, and nuclear properties of materialswith greater precision than known before. Goldstein (1967)and Schaeffer (1973) describe how the challenge was takenup in 1947 at Oak Ridge National Laboratory under thedirection of Everitt Blizard in the first organized researchprogram in nuclear reactor shielding. The X-10 graphitereactor had a 2-ft square aperture in its shielding from whicha neutron beam could be extracted, the intensity beingaugmented by placement of fuel slugs in front of theaperture. Attenuation of neutrons could then be measuredwithin layers of shielding materials placed against the beamaperture. Early measurements revealed the importance ofcapture gamma rays produced when neutrons were ab-sorbed. Improved experimental geometry was obtained byusing a converter plate instead of relying on fission neutronsfrom fuel slugs. The converter plate was a thin plate, ineffect a plane of well defined shape (disk or rectangle),containing enriched uranium. A broadly uniform beam ofthermal neutrons incident on the plate generated a well-defined source of fission neutrons. C.E. Clifford then set upa water tank adjacent to the fission source, with shieldingslabs and instrumentation within the tank. This Lid TankShielding Facility, LTSF, was the precursor of many so-called bulk shielding facilities incorporated into manywater-cooled research reactors. Early among these was theLTSF at Brookhaven National Laboratory, the Bulk ShieldReactor at Oak Ridge National Laboratory, and the Britishbulk-shielding reactor LIDO, completed in 1956. The Brit-ish research reactor BEPO, similar to the X-10 reactor,began operation in 1948.

Streaming of radiation through shield penetrationsand heating in concrete shields due to neutron andgamma-ray absorption were early shielding studies con-ducted in support of gas-cooled reactor design. This workis described in the text by Price et al. (1957). Additionalefforts were undertaken soon thereafter at universities aswell as government and industrial laboratories. Shieldingmaterial properties, neutron attenuation, the creation ofcapture and inelastic scattering gamma rays, reflectionand streaming of neutrons and gamma rays through ductsand passages, and radiation effects on materials weremajor research topics.

Although a nuclear-powered aircraft never tookflight, the wealth of information gained on the thermal,mechanical, and shielding properties of many specialmaterials is a valuable legacy. Development of a nuclear-powered aircraft required examination of the relativeimportance of shielding the reactor and the crew com-partment. It was also necessary to perform measurements

299Radiation shielding technology ● J. K. SHULTIS AND R. E. FAW

absent ground reflection or in such a way as to allowcorrection for such reflection. This was accomplished inseveral ways. According to Schaeffer (1973), a groundtest reactor was suspended by crane for tests of groundreflection. Then an aircraft shield test reactor was flownin the bomb-bay of a B-36 aircraft to allow measure-ments at altitude. The Oak Ridge tower shielding facilitywent into operation in 1954, and remained in operationfor almost 40 y. Designed for the aircraft nuclear pro-pulsion program, the facility allowed suspension of areactor hundreds of feet above grade and separate sus-pension of aircraft crew compartments. In its long life,the TSF also supported nuclear defense and space nu-clear applications.

The decade of the 1950’sThis era saw the passage in the U.S. of the Atomic

Energy Act of 1954, the Atoms for Peace program, andthe declassification of nuclear data. Advances in neutronshielding technology came together in the treatment ofneutron attenuation in hydrogenous media. Advances ingamma-ray transport methodology permitted calculationof buildup factors for treatment of scattered photons. Allthese technological advances enabled design work fornuclear reactors to proceed well before widespread use ofdigital computers. The NERVA program (nuclear energyfor rocket vehicle applications) began in 1955 under thesponsorship of NASA and the Atomic Energy Commis-sion. The first nuclear rocket (KIWI) was tested in 1959.The Atmospheric Nuclear Test Ban Treaty of 1963prevented the deployment of a nuclear rocket system andthe program was ended in 1972. Atmospheric testing ofnuclear weapons continued through the decade, with theU.S. testing a fusion device in November 1952 and theUSSR in August 1953. The first experimental breederreactor, EBR-I, was operational and generated electricityin December 1951, construction having begun in May1949. Construction of the first commercial nuclear powerplant, Shippingport, was begun in 1954 and powergeneration commenced in December 1957.

Advances in neutron shielding methods. Theseadvances resulted from measurements at the LTSF andother bulk-shielding facilities. One advancement was themeasurement of point kernels, or Green’s functions, forattenuation of fission neutrons in water. The other wasthe discovery that the effect of water-bound oxygen,indeed the effect of homogeneous or heterogeneousshielding materials in hydrogenous media, could bemodeled by exponential attenuation governed by effec-tive “removal” cross sections for the non-hydrogencomponents.

Because the hydrogen cross section increases withdecreasing fast-neutron energies, Albert and Welton(1950) recognized the first scatter with hydrogen, onaverage, results in a large energy loss for very energeticand very penetrating fission neutrons, which are thenvery quickly removed from the fast energy region by thenow much more probable scattering interactions withhydrogen. Thus, the first scatter with hydrogen effec-tively removes the neutron from the fast energy region.In an hydrogenous medium, fission-neutron attenuationhence could be treated as attenuation in hydrogen super-imposed with energy-independent absorption, i.e., re-moval from the fast energy region by the non-hydrogencomponents. Albert and Welton determined that the totaluncollided fast-neutron flux density should vary withdistance r from a point source of fission neutrons in wateras

��r��r�/ 2

4�r2 exp��r��exp��R,Or�. (3)

Here the constants � and � are based on empiricalapproximations for the energy spectrum of fission neu-trons and the energy-dependent scattering cross sectionfor hydrogen, and take on values 0.928 and 0.58, respec-tively, for fission neutrons produced by thermal fission of235U. The exponential-attenuation term, in which �R,O iscalled the macroscopic removal cross section for oxygen,accounts simply and entirely for the effect of oxygen onfast-neutron attenuation, and empirically has the value0.0308 cm�1 for water at density 1 g cm�3. The corre-sponding microscopic removal cross section is �R,O �0.921 b. This formula, and the Casper formula to follow,are easily modified to apply to hydrogenous media otherthan water, to account for different hydrogen atomicdensities, and to account for neutron removal by ele-ments or compounds other than oxygen. Many removalcross sections were measured by Chapman and Storrs(1955) using the LTSF at Oak Ridge National Labora-tory. Selected values are given in Table 1.

In 1960, Casper published the following point-kernel equation for fast-neutron absorbed dose in tissuedue to a point 235U fission source in water:

Table 1. Selected removal cross sectionsa

Material �r (b/atom)

Aluminum 1.31 � 0.05Beryllium 1.07 � 0.06Carbon 0.81 � 0.05Iron 1.98 � 0.08Lead 3.53 � 0.30Oxygen 0.99 � 0.10Uranium 3.6 � 0.4

a Source: Chapman and Storrs (1955).


GH2O�r� �5.39 � 10�11

4�r2 r0.349

exp��0.422r0.698 0.0308r�, (4)

where r has units of centimeters and G has units of Gy fora source strength of one fission neutron. Suppose thatthere is a point fission source surrounded by a fixedshield of thickness t composed of a material with removalcross section �r followed by water of thickness r. Then,provided that r is at least 50 cm, the absorbed dose, perfission neutron, is given by

D �r2

�r t�2GH2O�r�e��Rt . (5)

Advances in gamma-ray shielding methods. Asthe decade began, a major program at the NationalBureau of Standards was underway in research onelectron and photon transport (Spencer and Fano 1951).The photon transport effort is described by Fano et al.(1959). Much of the effort dealt with the momentsmethod of solving the transport equation describing thespatial, energy, and angular distributions of particlefluences arising from fixed sources. The method appliesto infinite homogeneous media, with well defined mo-noenergetic sources such as point or infinite-plane iso-tropic sources. The angular dependence of the particlefluence, �(r,E,�), is expressed as a Legendre polyno-mial expansion, each term of which is expressed in termsof spatial moments. The energy dependence of eachmoment is then determined numerically based on atten-uation and scattering properties of the medium. Then theprocess may be reversed, yielding the energy spectrum ofthe angular distribution of the fluence, or an integral overthe energy spectrum, weighted by a fluence-to-doseconversion factor R(E), yielding the angular distributionof the dose rate. From these results buildup factors maybe obtained. These represent ratios of the total dose D(r),scattered dose Ds(r) plus uncollided dose Do(r), to thatfrom uncollided particles only, namely,

B�r� �D�r�

Do�r�� 1

Ds�r�

Do�r�. (6)

For a monoenergetic source,

B�Eo, r� � 1 1

�o�r��0

Eo

dER�E�

R�Eo��s�r, E�. (7)

In this case, the nature of the dose or response is fullyaccounted for in the ratio R(E)/R(Eo).

In a collaborative effort with the Bureau of Stan-dards, Goldstein and Wilkins (1954) made use of the

SEAC digital computer at the Bureau in a thoroughevaluation of energy spectra and buildup factors formany materials and a broad range of photon energies.‡

The results of this collaborative effort, known widely asNYO-3075, served for many years as the prime source ofbuildup factor data for use in shielding design. The use ofbuildup factors in shielding design and analysis wasgreatly facilitated by interpolation methods devised byTaylor (1954), Berger (1956), and Capo (1959). For agiven material, for a point isotropic source in an infinitemedium, and in terms of the number � of mean free pathsat source energy, these interpolation formulas are, re-spectively,

B�Eo, �� Ae��1� �1 Ae��2� , (8)

B�Eo, �� 1 C�eD� , (9)

and

B�Eo, �� n�0

3

�n�n , (10)

in which parameters A, �1, �2, C, D, and �n depend on thematerial, the photon energy, and, in principle, the natureof the response.

Advances in Monte Carlo computational meth-ods. The Monte Carlo method of simulating radiationtransport computationally has its roots in the work ofJohn von Neumann and Stanislaw Ulam at Los Alamosin the 1940’s. Neutron transport calculations were per-formed in 1948 using the ENIAC digital computer,which had commenced operations in 1945. Major ad-vances were made in the 1950’s by Kahn (1950),Goertzel and Kalos (1958), and Cashwell and Everett(1959). Berger (1955, 1956) began a Monte Carlo effortat the National Bureau of Standards that thrived forseveral decades.

The decade of the 1960’sThe 1960’s saw the technology of nuclear reactor

shielding consolidated in several important publications.Blizard and Abbott (1962) edited and released a revisionof a portion of the 1955 Reactor Handbook as a separatevolume on radiation shielding, recognizing that reactorshielding had emerged from nuclear reactor physics intoa discipline of its own. In a similar vein, the first volumeof the Engineering Compendium on Radiation Shielding(Jaeger 1968) was published. These two volumes broughttogether contributions from scores of authors and had agreat influence on both practice and education in the field

‡ Eight energies from 0.5 to 10 MeV and the following materials:free electrons only, water, Al, Fe, Sn, W, Pb and U.


of radiation shielding. Radiation shielding in the contextof nuclear plant design was the theme of Hungerford’s1966 review of the design of the sodium-cooled Fermiplant outside Detroit.

This exciting decade also saw the beginning of theApollo program, continuation of the NERVA program,the deployments in space of SNAP-3, a radioisotopethermoelectric generator in 1961, and SNAP-10Anuclear-reactor power system in 1965. It also saw theCuban missile crisis in October 1962, and a majorincrease in the cold-war apprehension about possible useof nuclear weapons. The Apollo program demandedattention to solar-flare and cosmic radiation sources andthe shielding of space vehicles. Cold-war concerns de-manded attention to nuclear-weapon effects, particularlystructure shielding from nuclear-weapon fallout. Reflec-tion of gamma rays and neutrons and their transmissionthrough ducts and passages took on special importance instructure shielding. The rapid growth in access to digitalcomputers allowed introduction of many computer codesfor shielding design and fostered advances in solvingvarious approximations to the Boltzmann transport equa-tion for neutrons and gamma rays. Similar advances weremade in treating the slowing-down and transport ofcharged particles.

Space shielding. Data gathered over many yearsrevealed a very complicated radiation environment inspace. Two trapped-radiation belts had been found tosurround the earth, an inner proton belt and an outerelectron belt. Energy spectra and spatial distributions inthese belts are determined by the earth’s magnetic fieldand by the solar wind, a plasma of low-energy protonsand electrons. The radiations pose a risk to astronauts andto sensitive electronic equipment. Uniform intensities ofvery high-energy galactic cosmic rays demand charged-particle shielding for protection of astronauts in longduration missions. The greatest radiation risk faced byApollo astronauts was from solar flare protons and alphaparticles with energies as great as 100 MeV for theformer and 400 MeV for the latter. As is well known,flares occur in an 11-y cycle. They can be detected inadvance by sun-spot observation and by advanced receiptof electromagnetic radiation. The overall subject of spaceradiation shielding is treated by Haffner (1967). Dataneeded for shielding calculations were provided byHubble (1969), Barkas and Berger (1964), and Bergerand Seltzer (1964). Shielding methodology is describedby Leimdorfer et al. (1967) and Alsmiller (1967). Ad-vances in electron transport theory are described byZerby and Keller (1967) and Berger and Seltzer (1968).

Structure shielding. Structure shielding againstnuclear-weapon fallout required careful examination ofthe atmospheric transport of gamma rays of a wide rangeof energies and expression of angular distributions andrelated data in a manner easily adapted to analysis ofstructures. There was a need to assess, at points within astructure, the ratio of interior dose rates to that outsidethe building, called reduction factors. Calculations werecompleted at the National Bureau of Standards by LewisSpencer (1962) using the moments method of transportcalculations which had been used so successfully incalculation of buildup factors. The engineering method-ology was developed by Eisenhauer (1964) and rapidlydeployed by the Office of Civil Defense for shelteranalysis. The body of theoretical, analytical, and exper-imental support was documented by Spencer et al.(1980).

Of great importance to structure shielding, but alsoof interest in reactor and plant shielding, was the need toquantify neutron and gamma-ray reflection and stream-ing through ducts and voids in shields. Advances throughthe 1960’s and early 1970’s have been described thor-oughly by Selph (1973). The central concept is theparticle albedo, defined for a broad parallel beam ofmonoenergetic particles passing through a non-attenuating medium and striking a thick reflecting half-space at a fixed polar angle o measured from a normalto the reflecting surface. If the incident current or flow,measured per unit area in the surface, is Jo(Eo, o) inunits, say, cm�2 and the energy and angular distributionof the reflected current is Jr(E, , �) in units, say, cm�2

MeV�1 sr�1 measured at polar angle and a zimuthalshift �, the differential number albedo is defined as

��Eo, o; E, , �� Jr�E, , ��

Jo�Eo, o��

cos �r�E, , ��

cos o�o�Eo, o�.

(11)

For practical purposes, the albedo commonly used is thedose albedo, the ratio of the emergent flow per steradianin dose units to that of the incident radiation. With R(E)representing the fluence to dose conversion factor, thedose albedo is defined as

�D�Eo, o; , ��

� dER�E�Jr�E, , ��

R�Eo�Jo�Eo, o�.

(12)

Characterizing neutron albedos is more complicated. Forincident fast neutrons, four albedos may be required: onefor fast neutrons, one for intermediate energy neutrons,


one for thermal neutrons, and one for capture gammarays. Early gamma ray albedo measurements were madeby Haggmark et al. (1965) and calculations were madeby Berger and Doggett (1956), Berger and Raso (1960),and Raso (1963). Early neutron measurements and cal-culations were made by Wells (1964) and Maerker andMuckenthaler (1965, 1966). Use of albedos in shieldingdesign and analysis requires analytical approximations toalbedo data. Contributors for neutron albedos includeFrench and Wells (1963), Coleman et al. (1967), andSong et al. (1969). Contributors for gamma rays includeChilton and Huddleston (1963), Chilton et al. (1965), andChilton (1967).

Digital computer applications. Radiation transportcalculations are by nature very demanding of computerresources. The community of interest in radiation trans-port and shielding has been served magnificently formore than four decades by the Radiation Safety Infor-mation Computational Center (RSICC). Established in1962 as the Radiation Shielding Information Center(RSIC) at Oak Ridge National Laboratory, RSICC’smission is to provide in-depth coverage of the radiationtransport field to meet the needs of the internationalshielding community. RSICC collects, organizes, evalu-ates, and disseminates technical information, includingsoftware, involving shielding and protection from theradiation associated with fission and fusion reactors,outer space, accelerators, weapons, medical facilities,and nuclear waste management.

The 1960’s saw many new “mainframe” computercodes developed and disseminated. Among these codeswere gamma-ray “point-kernel” codes such as ISOSHLD(Engle et al. 1966) and QAD (Malenfant 1967), withversions of both still in use after almost four decades.The discrete-ordinates method of solving the Boltzmanntransport equation was devised in the 1950’s (Carlson1955) and put into practice in the 1960s in a series ofcomputer codes, DTF (Lathrop 1965), DOT (Mynatt etal. 1969), and ANISN (Engle 1967). The sphericalharmonics method of treating neutron spatial and energydistributions in shields was advanced by Shure in one-dimensional P3 calculations (1964). Progress in MonteCarlo methods advanced in pace with discrete ordinatesmethods, and the multi-group Monte Carlo code forneutron and gamma ray transport, MORSE, was intro-duced at the end of the decade (Straker et al. 1970).

The Monte Carlo method has a rich heritage at LosAlamos. Efforts began in 1947, inspired by John vonNeumann and Enrico Fermi. As related in the MCNPoperating manual issued by the X-5 Monte Carlo Team(2003), a general-purpose particle-transport code MCS

was written in 1963. This was followed by the MCNcode for three-dimensional calculations written in 1965.

The decade of the 1970’sThe Nuclear Non-Proliferation Treaty (NPT) of

1968 and the National Environmental Policy Act(NEPA) of 1969 had major impacts on the radiation-shielding field in the 1970’s and succeeding decades. TheNPT precluded nuclear fuel reprocessing and led to everincreasing needs for on-site storage of spent fuel atnuclear power plants. NEPA required exhaustive studiesof off-site radiation doses around nuclear power plantsand environmental impacts of plant operations. Early inthe 1970’s there were major disruptions in oil suppliescaused by the OPEC embargo. The response in theUnited States was an energy policy that forbade electric-ity production using oil or natural gas. The result wasplacement of many orders for nuclear power plantsdespite NPT and NEPA constraints. In the field ofradiation shielding special attention was given to plantdesign issues such as streaming of neutrons and gammarays through voids, passageways, and shield penetra-tions, to operational issues such as fission productinventories in fuels and gamma ray skyshine, particularlyassociated with 16N sources. The end of the decade wasmarked by the 28 March 1979 accident at the Three MileIsland, Unit 2 power plant.

Information essential for plant design, fuel manage-ment, and waste management is data tracking radio-nuclide activities in reactor fuel and process streams, andcorresponding strengths and energy spectra of sources,including fission products, activation products, and ac-tinides. To accomplish this, the ORIGEN codes weredeveloped at Oak Ridge National Laboratory (Bell 1973;Croff 1980) and the CINDER code was developed at LosAlamos National Laboratory (England et al. 1976).Assessment of radiation doses from airborne beta-particle emitters was one of the motivations for Berger’swork on cloud doses (1974). Although the ETRANMonte Carlo code for electron transport was available atthe National Bureau of Standards, work began in the mid1970’s at Sandia Laboratory on the TIGER code and atStanford Linear Accelerator Center on the EGS code,both for coupled photon and electron transport by theMonte Carlo method. The former evolved to the Inte-grated Tiger Series (ITS) of codes (Halbleib et al. 1992),the later to the EGS4 code (Nelson et al. 1985).

Design needs brought new attention to buildupfactors and to attenuation of broad beams of neutrons andgamma rays. Eisenhauer and Simmons (1975) andChilton et al. (1980) published definitive tables ofbuildup factors. Doggett and Bryan (1970) and Fournie


and Chilton (1980) both addressed attenuation of ob-liquely incident photons, and the former addressed re-flection as well. Roussin and Schmidt (1971) and Rous-sin et al. (1973) addressed transmission of neutrons andsecondary gamma rays through shielding barriers. Thisdecade also saw the publication of two important NCRPreports (1971, 1976) dealing with neutron shielding anddosimetry and with design of medical facilities that pro-tected against effects of gamma rays and high energy x rays.

Design and analysis needs also fostered continuingattention to computer codes for criticality and neutron-transport calculations. Lathrop and Brinkley (1970), Hill(1975), Mynatt et al. (1969), Mynatt and Rhoades (1973),and Rhoades et al. (1979) all introduced discrete ordinatestransport codes. Advances in Monte Carlo calculations werealso made. The MCN code was merged with the MCG codein 1973 to form the MCNG code for treating coupledneutron-photon transport. Another merger took place withthe MCP code in 1977, allowing detailed treatment ofphoton transport at energies as low as 1 keV. This new codewas known, then and now, as MCNP.

The 1980’s and 1990’sThese years saw the consolidation of resources for

design and analysis work. In the 1980’s, personal computersallowed methods such as point-kernel calculations to beprogrammed. In the 1990’s, personal computers took overfrom the main-frame computers in even the most demand-ing shielding design and analysis. Comprehensive sets offluence-to-dose conversion factors became available forwide spread use. Radionuclide decay data became availablein data bases easily used for characterizing sources.Gamma-ray buildup factors were computed with precisionand a superb method of data fitting was devised. All thesecarried point-kernel as well as more advanced shieldingmethodology to a new plateau.

Data bases. Long a staple in health physics andnuclear engineering, the Table of the Isotopes (Firestoneand Shirley 1996), while unsurpassed in documentingenergy levels, is difficult to use in characterizing decaydata. Kocher (1981) published data for shielding designand analysis that largely supplanted the data of Martinand Blichert-Toft (1970), Dillman and Von der Lage(1975), and Martin (1976). Then a new MIRD compen-dium (Weber et al. 1989) and ICRP-38 data base (ICRP1983) became the norms, with the latter especially usefulfor characterizing low-energy x ray and Auger electronemission. A wealth of nuclear structure and decay data is

available from the National Nuclear Data Center atBrookhaven National Laboratory.§

Advances in buildup factors. Refinements in thecomputation of buildup factors continued to be made overthe years. An adjoint moments method code had come intouse (Simmons 1973) and treatment of positron annihilationhad been incorporated (Morris et al. 1975). The ASFIT code(Subbaiah et al. 1982; Gopinath et al. 1987), introduced in1971, and the PALLAS code (Takeuchi et al. 1981, 1984;Tanaka and Takeuchi 1986), introduced in 1973, led ulti-mately to a comprehensive set of precise buildup factorsstandardized for use in design and analysis (ANS 1991).The discrete ordinates ASFIT code and the integral-transport PALLAS code account for not only Comptonscattering and photoelectric absorption but also positroncreation and annihilation, fluorescence, and bremsstrahlung.The evolution of buildup-factor measurements and calcula-tions from 1950 until 1993 was thoroughly documented byHarima (1993).

Working with buildup factors computed using thePALLAS code, Harima developed a data fit in thefollowing form, called the geometric progression for-mula (Harima 1983; Harima et al. 1986, 1991):

B�Eo, �r� � 1 �b 1��K�r 1�/�K 1�

(13)

for K 1, and

B�Eo, �r� � 1 �b 1��r

for K � 1, where

K��r� � c��r�a dtanh��r/� 2� tanh��2�

1 tanh��2�,

and where a, b, c, d, and � are parameters that depend onthe gamma-ray energy, the attenuating medium, and thenature of the response. This seems to be a very unusual,even bizarre, fitting formula. That may be so, but it isindeed an extraordinarily precise formula, as is illustratedin Fig. 1. Both the results of PALLAS calculations andthe coefficients for the geometric progression buildupfactors are tabulated in design standards (ANS 1991).

Cross sections and dose conversion factors. Twomore foundation stones need to be in place to support amature radiation shielding technology. One is a compre-hensive set of cross sections, or interaction coefficients,accounting for not only reactions but also dosimetry relatedcoefficients such as those for energy deposition. Another is

§ http://www.nndc.bnl.gov/index.jsp.


a set of fluence-to-dose conversion factors applicable to acomprehensive array of dosimetry conditions.

Authoritative cross section data are available in theENDF/B-6 (evaluated nuclear data file)** database con-taining evaluated cross sections, spectra, angular distri-butions, fission product yields, photo-atomic and thermalscattering law data, with emphasis on neutron inducedreactions. Data are analyzed and evaluated by experiencednuclear physicists and stored in the internationally adoptedformat (ENDF-6) maintained by CSEWG, the Cross Sec-tion Evaluation Working Group, a cooperative effort ofnational laboratories, industry, and universities in the UnitedStates. ENDF data are processed for use in modern designand analysis computer codes such as MCNP (X-5 2003),DANTSYS (Alcouffe et al. 1995), and TORT (Rhoades andSimpson 1997) and into general purpose packages such asVITAMIN B-6 and BUGLE-96 (White et al. 1995).

The National Institute of Science and Technology(NIST) has long been the repository for gamma-rayinteraction coefficients. A sampling of the published datasets may be found in the work of Hubble and Berger(1966), Hubble (1969, 1982), Hubble and Seltzer (1995),and Seltzer (1993). The Institute also sponsors theXCOM cross-section code which may be executed on theNIST internet site†† or downloaded for personal use. Twoother sets of photon cross sections (Storm and Israel1967; Plechaty et al. 1981) have found wide use over theyears because they included energy transfer and energy

absorption cross sections, essential for fluence-to-doseconversion factors.

Gamma-ray fluence-to-dose conversion factors forlocal values of exposure or kerma may be computed directlyfrom readily available energy transfer or energy absorptioncoefficients for air, tissue, etc. Neutron conversion factorsfor local values of tissue kerma were computed by Caswellet al. (1980). One very important set of conversion factorsfrom the 1970’s remains in use. That set is to be used for thedeep dose equivalent index‡‡ for neutrons incident on a30-cm diameter tissue equivalent phantom (NCRP 1971).That set of factors remains in use in the U.S. NuclearRegulatory Commission regulations for radiation protec-tion, Title 10, Part 20 of the Code of Federal Regulations.As we begin the second century of radiation protection,there are two classes of fluence-to-dose conversion factorsin use for neutrons and gamma rays. One very conservativeclass is to be used for operational purposes at doses wellbelow regulatory limits. This class is based on doses at fixeddepths in 30-cm diameter spherical phantoms irradiated invarious ways. The other class is to be used for doseassessment purposes, and not for personnel dosimetry. Thisclass is based on the anthropomorphic human phantom andweight factors for effective dose equivalent (ICRP 1987) oreffective dose (ICRP 1996).

Computer applications. The 1980’s and 1990’swere decades of revolution for the computational aspectsof radiation shield design and analysis. The advent ofinexpensive personal computers with rapidly increasingspeeds and memory freed the shielding analyst fromdependence on a few supercomputers at national labora-tories. Many shielding codes that could previously runonly on large mainframe computers were reworked to runon small personal computers thereby allowing anyshielding analysts to perform detailed calculations thatonly a privileged few were able to do previously.

At the same time many improvements were made tothe transport codes and their algorithms. MCNP has gonethrough a series of improvements adding new capabilitiesand improvements, such as new variance reduction meth-ods, tallies, and physics models. It was translated to theFORTRAN-77 standard in 1983, and to the FORTRAN-90standard in 2003. New releases appear every few years, andthe version as of this writing is MCNP-5. It has also spunoff a second version MCNPX with a capability of treating34 types of particles with energies up to 150 MeV.

General purpose discrete-ordinates codes were exten-sively improved with many novel acceleration schemesintroduced to improve their speeds. An excellent review ofmany such improvements is given by Adams and Larsen

** http://www.nndc.bnl.gov/exfor/endf00.htm.†† http://physics.nist.gov/PhysRefData/Xcom/Text/XCOM.html. ‡‡ Maximum dose equivalent at depth of 1 cm or greater.

Fig. 1. Comparison of buildup factors computed by the PALLAScode and fit by the geometric progression method.


(2002). Some of the discrete-ordinates codes introduced inthese two decades included ONEDANT (O’Dell et al.1982), TWODANT (Alcouffe et al. 1984), TORT (Rhoadesand Childs 1987), DANTSYS (Alcouffe et al. 1995), andPARTISN (Alcouffe et al. 2002).

The futureIn many respects, radiation shielding is a mature

technological discipline. It is supported by a comprehen-sive body of literature and a diverse selection of compu-tational resources. The fundamentals and the methodol-ogy are readily incorporated into health physics andnuclear engineering education and training programs. Itis important that we not allow this maturity to fostercomplacency. Advances continue to be made in compu-tational resources as well as information on cross sec-tions and material properties, especially radiation resis-tance. These advances as well as advances in ourknowledge of dosimetry and health effects of radiationrequire continuing attention and adoption into theradiation-shielding discipline. We see this taking place inthe efforts of bodies such as the NCRP, ICRP, ICRU, andthe BEIR Committees of the National Research Council.We also see it taking place in the maintenance ofindustrial and manufacturing standards, especially thosefostered by the Health Physics Society and the AmericanNuclear Society.

We close this introduction by reminding readers thatthere are some important gamma-ray shielding problemsthat we have not been able to treat using hand calcula-tions or using the design and analysis codes employingpoint-kernel methods. These include transmission ofgamma rays through ducts and passages in structures,reflection of gamma rays from shielding walls and otherstructures, and transmission of beams of gamma raysobliquely incident on shielding slabs. These problems,like comparable neutron-shielding problems, require full-scale treatment of scattered particles as is done in theMonte Carlo and discrete ordinates computer codes. Ascomputational resources grow, these more advancedtransport methods become available on the desks ofoffice, classroom, and home. We expect this trend tocontinue, and we expect continuing roles for both MonteCarlo and discrete ordinates methods. We take note ofefforts at developing hybrid techniques employing bothmethods and look forward to continued exploitation ofthese hybrids. We also take note of development ofgraphical user interfaces to assist users of codes such asthe MCNP code. We applaud these efforts and lookforward to continuing advances.

PRACTICE OF RADIATION SHIELDING

First it must be said that shielding design andshielding analysis are complementary activities. In de-sign, the source is identified and a target dose goal isspecified. The task is to determine the nature of theshielding required to achieve the goal. In analysis, thesource and shielding are identified and the task is todetermine the consequent dose. Whether one is engagedin a hand calculation or in a most elaborate Monte Carlosimulation, one is faced with the tasks of (1) character-izing the source, (2) characterizing the nature and atten-uating properties of the shielding materials, (3) evaluat-ing at a target location the radiation intensity and perhapsits angular and energy distributions, and (4) convertingthe intensity to a dose or response meaningful in terms ofradiation effects.

Basic concepts

Source characterization. Source geometry, energy,and angular distribution are required characteristics.Radionuclide sources, with isotropic emission andunique energies of gamma and x rays are relatively easyto characterize. Activity and source strength must becarefully distinguished, as not every decay results inemission of a particular gamma or x ray. Careful consid-eration must be given to a low-energy limit below whichsource particles may be ignored, else computation re-sources may be wasted. Similarly, when photons of manyenergies are emitted, as in the case of fission-productsources, one is compelled to use a group structure insource characterization, and much care is needed inestablishing efficient and appropriate group energy limitsand group average energies. When the source energiesare continuously distributed, as is the case with fissionneutrons and gamma rays, one option is to use a multigroup approach, as might be used in point-kernel calcu-lations. Another option, useful in Monte Carlo simula-tions, is to sample source energies from a mathematicalrepresentation of the energy spectrum.

A point source is very often an appropriate approx-imation of a physical source of small size. It is alsoappropriate to represent a line, plane, or volume source asa collection of point sources, as is done in the point-kernel method of shielding analysis. Radionuclide andfission sources are isotropic in angular distribution;however, there are cases for which it is efficient toidentify a surface and to characterize the surface as asecondary source surface. Such surface sources are veryoften non-isotropic in angular distribution. For example,consider the radiation emitted into the atmosphere from alarge body of water containing a distributed radiationsource. The interface may be treated approximately, but


very effectively, as a plane source emitting radiation notisotropically, but with an intensity varying with the angleof emission from the surface.

Attenuating properties. The total microscopiccross section for an element or nuclide, �(E), multipliedby the atomic density, is the linear interaction coefficient�(E), also called the macroscopic cross section, theprobability per unit (differential) path length that aparticle of energy E interacts with the medium in someway. Its reciprocal, called the mean free path, is theaverage distance traveled before interaction. Usually theratio �/�, called the mass interaction coefficient, istabulated because it is independent of density. Varioussubscripts may be used to designate particular types ofinteractions, e.g., �a(E) for absorption or �f(E) forfission. Likewise, additional independent variables maybe introduced, with, e.g., �s(E, E�)dE� representing thecross section for scattering from energy E to an energybetween E� and E� � dE�. Information resources forattenuating properties are described in this paper’s his-torical review, as are resources for radionuclide decaydata.

Intensity characterization. The intensity of a neu-tron or photon field is ordinarily described in terms ofparticles crossing the surface of a small spherical volumeV. The fluence � is defined, in the limit V 3 0, as theexpected or average sum of the path lengths in V traveledby entering particles divided by the volume V. Equiva-lently, � is, again in the limit V 3 0, the expectednumber of particles crossing the surface of V divided bythe cross sectional area of the volume. The time deriva-tive of the fluence is the fluence rate or flux density �.Note that the fluence, though having units of reciprocalarea, has no reference area or orientation. Note too thatthe fluence is a point function. The fluence, a function ofposition, may also be a distribution function for particleenergies and directions. For example, �(r, E, �)dEd isthe fluence at r of particles with energies in dE about Eand with directions in d about �. When a particularsurface is used as a reference, it is useful to defineradiation intensity in terms of the flow Jn(r) across thereference surface, as is done in defining the albedo in eqn(11).

Fluence-to-dose conversion factors. Whether theshield designer uses the simplest of the point-kernelmethods or the most comprehensive of the Monte Carloor discrete ordinates methods, fluence-to-dose conver-sion factors invariably have to be used. The radiationattenuation calculation deals with the particle fluence, thedirect measure of radiation intensity. To convert that

intensity into a measure of radiation damage or heating ofa material, to a field measurement such as exposure, or toa measure of health risk, conversion factors must beapplied. Even in that humble health physics thumb rule6CEN/r2 for gamma-ray exposure rate,§§ the numericalfactor incorporates the fluence-to-exposure conversionfactor.

The shielding analysis ordinarily yields the energyspectrum �(r, E) of the photon or neutron fluence at apoint identified by the vector r. Use of a Monte Carlocode normally yields the energy spectrum as a continu-ous function of energy, whence the dose or, moregenerally, response D(r) is given by the convolution ofthe fluence with the fluence-to-dose factor, here calledthe response function R(E), so that

D�r� � �E

dER�r, E��r, E�. (14)

Point-kernel, or other energy-multigroup methods yieldthe energy spectrum at discrete energies, or in energygroups, and the dose convolution is a summation ratherthan an integration.

While the fluence is most always computed as apoint function of position, the response of interest may bea dose at a point, or it may be a much more complicatedfunction such as the average radiation dose in a physicalvolume such as an anthropomorphic phantom. We ex-amine local and phantom-related doses separately. Also,we assume that charged-particle equilibrium*** has beenattained, so that the kerma and absorbed dose may beequated.

The local dose. Suppose the local dose of interest isthe kerma. Then the response function is given by

R�E� � ��i

Ni

��

j

�ji�E��ji�E�. (15)

in which � is the mass density, Ni is the atoms of speciesi per unit volume (proportional to �), �ji(E) is the crosssection for the jth interaction with species i, and �ji(E) isthe average energy transferred to secondary chargedparticles in the jth interaction with species i. A unitsconversion factor � is needed to convert from, say, units

§§ The R/h exposure rate at a distance r (ft) in air from C curiesof a gamma-ray source emitting, per decay, N gamma rays of energyE (MeV).

*** Under conditions of charged-particle equilibrium, the neutronor gamma ray energy transferred to initial kinetic energy of secondarycharged particles is equal to the energy imparted to the medium asmanifested in ionization, excitation, chemical change, and heat.Equilibrium is approached in a region of homogeneity in compositionand uniformity in neutron or photon intensity.


of MeV cm�2 g�1 to units of rad cm�2 or Gy cm�2. Forneutrons, a quality factor multiplier Q(E) is needed toconvert to units of dose equivalent (rem or Sv).

Here we need to take note of one very importantlocal-dose response function—the tissue-kerma responsefunction for neutrons. This has been computed by Ca-swell et al. (1980) for a four-element tissue approxima-tion. Fig. 2 shows the response function based onENDF/B-V cross sections.

In shielding calculations, one must be very careful toselect a response function appropriate to the method usedin calculating the fluence energy spectrum �(r,E). Sup-pose, for example that the photon fluence is computedusing an approximation that pair production and thephotoelectric effect are purely absorption, i.e., annihila-tion and fluorescence photons and bremsstrahlung areexcluded from �. This was a common approximation inthe 1940’s and 1950’s. Then, for the photoelectric andpair production reactions, it is necessary to set �ji(E) � E.Then, in the usual gamma-ray notation, R(E) (Gycm�2) � 1.602 � 10�10E[�a(E)/�], in which �a/� is themass absorption coefficient. On the other hand, if thephoton fluence includes Compton-scattered photons, an-nihilation photons, and fluorescence photons, then thechoice is the mass energy transfer coefficient �tr(E)/�. Ifthe photon fluence includes bremsstrahlung as well, thenthe choice is the mass energy transfer coefficient�en(E)/�. For the special case of gamma or x-ray expo-sure, the fluence-to-dose conversion factor is given byR(E) (R cm�2) � 1.835 � 10�8E[�en

air(E)/�].

In point-kernel calculations for gamma rays, it iscommon practice to first compute the fluence of uncol-lided photons, then apply a response function to obtainthe “uncollided” dose rate, and then a buildup factor toobtain the total dose rate. In these circumstances, it isnecessary to apply a response function appropriate to themethod used in calculating the buildup factor. If thebuildup of secondary photons includes bremsstrahlung aswell as annihilation and fluorescence photons, then theappropriate dose conversion factor is based on �en(E)/�.

The phantom-related dose. By phantom-relateddose, we mean a local radiation dose within a simplegeometric phantom or some sort of average dose withinan anthropomorphic phantom. The phantom dose, in fact,is a point function and serves as a standardized referencedose for instrument calibration and radiation protectionpurposes. Even though the radiation fluence, itself apoint function, may have strong spatial and angularvariation as well as energy variation, it is still possible toassociate with the radiation fluence a phantom-relateddose. The procedure is as follows. The fluence is treated,for example, as a very broad parallel beam of the sameintensity as the actual radiation field, incident in somefixed way on the phantom. This is the so-called expandedand aligned field. For a geometric phantom, the dose iscomputed at a fixed depth. For an anthropomorphicphantom, the dose is computed as an average of doses toparticular tissues and organs, weighted by the suscepti-bility of the tissues and organs to radiation carcinogen-esis or hereditary illness.

Geometric phantoms are used for operationalradiation-protection applications, i.e., at doses well be-low regulatory limits. Slabs, cylinders, and spheres haveall been used as geometric phantoms, but the 30-cm-diameter “ICRU Sphere” has become the standard.†††

This is a sphere of density 1 g cm�3 comprised of afour-element tissue approximation, 76.2% by weightoxygen, 11.1% carbon, 10.1% hydrogen, and 2.6% ni-trogen. Fluence-to-dose conversion factors are availablefor four irradiation geometries: (PAR), a single planeparallel beam; this is the default geometry suitable forinstrument calibration; (OPP), two opposed plane paral-lel beams; (ROT), a rotating plane parallel beam (i.e., aplane parallel beam with the sphere rotating about an axisnormal to the beam); and (ISO), an isotropic radiationfield. Fluence-to-dose conversion factors are tabulated in

††† An exception is the use of a 30-cm cylindrical phantom for theneutron fluence-to-dose conversion factors listed in the regulations ofthe U.S. Nuclear Regulatory Commission, as recommended by theNCRP (1971). Furthermore, the dose computed is not the 10-mmambient dose, but rather the deep dose equivalent index, the maximumdose at a depth of 10 mm or greater in the phantom.

Fig. 2. Kerma response functions for neutron interactions in theICRU four-element approximation for tissue, with mass fractions0.101 H, 0.111 C, 0.026 N, 0.762 O. Computed using NJOY-processed ENDF/B-V data.


reports of the ICRP (1987, 1996) and reproduced byShultis and Faw (2000).

Dose conversion factors are also available for threedepths of penetration into the geometric phantom: (1) A10-mm depth, the dose being called the ambient dose, asurrogate to the earlier whole body dose and the dosesuitable for instrument calibration; (2) a 3-mm depth,suitable for representing the dose to the lens of the eye;and (3) a 0.07-mm depth, suitable for representing thedose to the skin.

Anthropomorphic phantoms are mathematical de-scriptions of the organs and tissues of the human body,formulated in such a way as to permit calculation ornumerical simulation of the transport of radiationthroughout the body. In calculations leading to responsefunctions, monoenergetic radiation is incident on thephantom in fixed geometry. One geometry leading toconservative values of response functions is anteropos-terior (AP), irradiation from the front to the back with thebeam at right angles to the long axis of the body. Othergeometries are posteroanterior (PA), right and left lateral(RLAT and LLAT), rotational (ROT), and isotropic(ISO). The AP case, being most conservative, is thechoice in the absence of particular information on theirradiation circumstances.

Figs. 3 and 4 compare response functions for pho-tons and neutrons, respectively. At energies above about0.1 MeV, the various photon response functions are verynearly equal. This is a fortunate situation for radiationdosimetry and surveillance purposes. Personnel dosime-ters are usually calibrated to give responses proportional

to the ambient dose. Both the ambient dose and the tissuekerma closely approximate the effective dose equivalent.

The comparison of response functions for neutronsis not so straightforward. The tissue kerma always hasthe smallest response function, largely because no qualityfactor is applied to the kinetic energy of a secondarycharged particle. Furthermore, the ambient dose alwaysexceeds the effective dose equivalent. Thus, calibrationof personnel dosimeters in terms of ambient dose is aconservative practice.

Basic analysis methodsIn this section fundamental methods for estimating

neutron or photon doses are reviewed. Such indirectlyionizing radiation is characterized by straight-line trajec-tories punctuated by “point” interactions. The basicconcepts presented here apply equally to all particles ofsuch radiation.

It should be noted that throughout this paper, calcu-lated doses are the expected or average value of thestochastic measured doses, i.e., the mechanistically cal-culated dose represents the statistical average of a largenumber of dose measurements which exhibit randomfluctuations as a consequence of the stochastic nature ofthe source emission and interactions in the detector andsurrounding material.

Uncollided radiation doses. In many situations thedose at some point of interest is dominated by particlesstreaming directly from the source without interacting inthe surrounding medium. For example, if only air sepa-rates a gamma-ray or neutron source from a detector,

Fig. 3. Comparison of photon response functions. Data are fromICRP (1987).

Fig. 4. Various neutron response functions.


interactions in the intervening air or in nearby solidobjects, such as the ground or building walls, are oftennegligible, and the radiation field at the detector is duealmost entirely to uncollided radiation coming directlyfrom the source.

In an attenuating medium, the uncollided dose at adistance r from a point isotropic source emitting Sp

particles of energy E is

Do�r� �SpR

4�r2e�� (16)

where � is the total number of mean-free-path lengths ofmaterial a particle must traverse before reaching thedetector, namely, �0

r ds ��s�. Here R is the appropriateresponse function. The 1/(4�r2) term in eqn (16) is oftenreferred to as the geometric attenuation and the e�� termthe material attenuation. Eqn (16) can be extended easilyto a source emitting particles with different discreteenergies or a continuous spectrum of energies.

Point kernel for uncollided dose. Consider anisotropic point source placed at rs and an isotropic pointdetector (or target) placed at rt in an homogeneousmedium. The detector response depends not on rs and rt

separately, but only on the distance �rs � rt� between thesource and detector. For a unit strength source thedetector response is (cf. eqn 16):

Go�rs, rt, E� �R�E�

4��rs rt�2e��E��rs�rt� . (17)

Here Go(rs, rt, E) is the uncollided dose point kernel andequals the dose at rt per particle of energy E emittedisotropically at rs. This result holds for any geometry ormedium provided that the material through which a rayfrom rs to rt passes has a constant interaction coefficient �.

With this point kernel, the uncollided dose due to anarbitrarily distributed source can be found by first de-composing (conceptually) the source into a set of con-tiguous effective point sources and then summing (inte-grating) the dose produced by each point source.

Applications to selected geometries. The resultsfor the uncollided dose from a point source can be usedto derive expressions for the uncollided dose arising froma wide variety of distributed sources such as line sources,area sources, and volumetric sources (Rockwell 1956;Blizard and Abbott 1962; Jaeger 1968; Schaeffer 1973).An example to illustrate the method is as follows.

A straight-line source of length L emitting isotropi-cally Sl particles per unit length at energy E is depicted inFig. 5. A detector is positioned at point P, a distance hfrom the source along a perpendicular to one end of the

line. Suppose the only material separating the line sourceand the receptor is a slab of thickness t and totalattenuation coefficient �s.

Consider a segment of the line source betweendistance x and x � dx measured from the bottom of thesource. The source within this segment may be treated asan effective point isotropic source emitting Sl dx parti-cles, which produces an uncollided dose at P of dDo. Theray from the source in dx must pass through a slantdistance of the shield t sec � so that the dose at P fromparticles emitted in dx about x is

dDo�P� �1

4�

SlR dx

x2 h2 exp��t sec �, (18)

where R and � generally depend on the particle energy E.To obtain the total dose at P from all segments of the linesource, one then must sum, or rather integrate, dDo overall line segments. As can be seen from Fig. 5, x � h tan �and x2 � h2 � h2 sec2 �. It then follows that dx � h sec2 � d�,so that the total uncollided dose at P becomes

Do�P� �SlR

4�h�0

�o

d�e��t sec � �SlR

4�hF��o, �st�, (19)

where the Sievert integral or the secant integral F isdefined as F(�, b) ' �0

� dy e�b sec y.

Intermediate methods for photon shieldingIn this section we summarize several special tech-

niques for the design and analysis of shielding for

Fig. 5. Isotropic line source with a slab shield.


gamma and x rays with energies from about 1 keV toabout 20 MeV. These techniques are founded on veryprecise radiation transport calculations for a wide rangeof carefully prescribed situations. These techniques,which rely on buildup factors, attenuation factors, albe-dos or reflection factors, and line-beam response func-tions, then allow estimation of photon doses for manycommon shielding situations without the need of trans-port calculations.

Buildup factor concept. The total photon fluence�(r, E) at some point of interest r is the sum of twocomponents: the uncollided fluence �o(r, E) of photonsthat have streamed to r directly from the source withoutinteraction, and the scattered or secondary photon flu-ence �s(r, E) consisting of source photons scattered oneor more times, as well as secondary photons such as xrays and annihilation gamma rays.

The buildup factor B(r) is defined in eqn (6) and, formonoenergetic sources, in eqn (7). By far the largestbody of buildup-factor data is for point, isotropic, andmonoenergetic sources of photons in infinite homoge-neous media. Calculation of buildup factors for high-energy photons requires consideration of the paths trav-eled by positrons from their creation until theirannihilation. Such calculations have been performed byHirayama (1987) and by Faw and Shultis (1993) forphoton energies as great as 100 MeV. Because incoher-ent scattering was neglected in many buildup factorcalculations, coherent scattering should also be neglectedin calculating the uncollided dose, a significant consid-eration only for low energy photons at deep penetration.

Buildup factor geometry. Buildup factors gener-ally depend on the source and shield geometries. For thesame material thickness between source and detector,buildup factors are slightly different for point isotropicsources in (a) an infinite medium, (b) at the surface of abare sphere, and for a slab shield between source anddetector. However, the use of buildup factors for a pointisotropic source is almost always conservative, i.e., theestimated dose is greater than that for a finite shield(Shultis and Faw 2000). Adjustment factors for buildupfactors at the surface of a finite medium in terms of theinfinite medium buildup factors are illustrated in Fig. 6.

Buildup factors are also available for plane isotopic(PLI) and plane monodirectional (PLM) gamma-raysources in infinite media. Indeed, Fano et al. (1959),Goldstein (1959), and Spencer (1962), in their moments-method calculations, obtained buildup factors for planesources first and, from these, buildup factors for pointsources. Buildup factors at depth in a half-space shieldare also available for the PLM source, that is, normally

incident photons (Takeuchi et al. 1981; Takeuchi andTanaka 1984; Hirayama 1987). Again, the use of buildupfactors for a point isotropic source in an infinite mediumfor these geometries is conservative.

Buildup in stratified media. The use of thebuildup-factor concept for heterogeneous media is ofdubious merit for the most part. Nevertheless, implemen-tation of point-kernel codes for shielding design andanalysis demands some way of treating buildup when thepath from source point to dose point is through more thanone shielding material. Certain regularities do exist,however, which permit at least approximate use ofhomogeneous-medium buildup factors for stratifiedshields. Many approximate methods have been sug-gested, as described by Shultis and Faw (2000); however,they are of little use in point-kernel calculations and of noneed in transport methods of shielding analysis.

Point-kernel computer codes. There are manycodes in wide use that are based on the point-kerneltechnique. In these codes a distributed source is decom-posed into small but finite elements and the dose at somereceptor point from each element is computed using theuncollided dose kernel and a buildup factor based on theoptical thickness of material between the source elementand the receptor. The results for all the source elementsare then added together to obtain the total dose. Somethat have been widely used are MicroShield (Negin andWorku 1998), the QAD series [QAD (Malenfant 1967),QAD-CG (Cain 1977), QADMOD (Price and Blattner1979)], and G3 (Malenfant 1973).

Broad-beam attenuation. Often a point radionu-clide or x-ray source in air is located sufficiently far from

Fig. 6. Adjustment factors for the buildup factor Bx at the boundaryof a finite medium in terms of the infinite-medium buildup factorB� for the same depth of penetration. EGS4 calculations courtesyof Sherrill Shue, Nuclear Engineering Department, Kansas StateUniversity.


a wall or shielding slab that the radiation reaches the wallin nearly parallel rays. Further, the attenuation in the airis quite negligible in comparison to that provided by theshielding wall. Shielding design and analysis for suchbroad-beam illumination of a slab shield are addressedby NCRP Report 49 (1976), Archer (1995), and Simpkin(1989). The dose at the surface of the cold side of thewall can be computed as

D � DoAf . (20)

For a radionuclide source of activity C, the dose Do

without the wall can be expressed in terms of the sourceenergy spectrum, response functions, and distance r fromthe source to the cold side of the wall. Then

D � DoAf �C

r2�Af , (21)

where � is a specific gamma-ray constant, that is, thedose rate in vacuum at a unit distance from a source withunit activity, and Af is an attenuation factor whichdepends on the nature and thickness of the shieldingmaterial, the source energy characteristics, and the angleof incidence (with respect to the wall normal). Valuesfor � and Af are provided by NCRP (1976).

Oblique incidence. Attenuation factors for ob-liquely incident beams are presented in NCRP Report 49(1976). For such cases, special three-argument slant-incidence buildup factors should be used (Shultis andFaw 2000). For a shield wall of thickness t mean freepaths, slant incidence at angle with respect to thenormal to the wall, and source energy Eo, the attenuationfactor is in function form Af(Eo, t, ). However, acommon, but erroneous, practice has been to use atwo-argument attenuation factor based on an infinitemedium buildup factor for slant penetration distance t sec , in the form Af(Eo, t sec ). This practice can lead tosevere under prediction of radiation dose.

X-ray beam attenuation. For x-ray sources, theappropriate measure of source strength is the electron-beam current i, and the appropriate characterization ofphoton energies, in principle, involves the peak acceler-ating voltage (kVp), the wave form, and the degree offiltration (e.g., beam half-value thickness). If i is thebeam current (mA) and r is the source-detector distance(m), the dose behind a broadly illuminated shield wall is

D�P� �i

r2KoAf , (22)

in which Ko is the radiation output (factor), the dose ratein vacuum (or air) per unit beam current at unit distance

from the source in the absence of the shield. Empiricalformulas for computing Af are available for shield design(Archer et al. 1983, 1994; Simpkin 1995).

Intermediate methods for neutron shieldingShielding design for fast neutrons is far more com-

plex than shielding design for photons. Besides having toprotect against the neutrons emitted by some source,there are primary gamma rays produced by most neutronsources, and secondary photons produced by inelasticneutron scattering and from radiative capture. There mayalso be secondary neutrons produced from (n,2n) andfission reactions. In many instances, secondary photonsproduce greater radiological risks than do the primaryneutrons. Fast-neutron sources include spontaneous andinduced fission, fusion, (�, n) reactions, (�, n) reactions,and spallation reactions in accelerators, each producingneutrons with a different distribution of energies.

Unlike photon cross sections, neutron cross sectionsusually vary greatly with neutron energy and among thedifferent isotopes of the same element. Large cross-section data bases are needed. Also, because of the erraticvariation of the cross sections with energy, it is difficultto calculate uncollided doses needed in order to use thebuildup factor approach. Moreover, buildup factors arevery geometry dependent and sensitive to the energyspectrum of the neutron fluence and, consequently,point-kernel methods can be applied to neutron shieldingonly in very limited circumstances.

Early work led to kernels for fission sources inaqueous systems, as described by eqns (3) and (4), and tothe use of removal cross sections, eqn (5), to account forshielding barriers. Over the years, the methodology wasstretched to apply to non-aqueous hydrogenous media,then to non-hydrogenous media, then to sources otherthan fission. Elements of diffusion and age theory weremelded with the point kernels. Today, with the availabil-ity of massive computer resources, neutron shieldingdesign and analysis is largely done using transportmethods. Nevertheless, the earlier methodology offersinsight allowing more critical interpretation of results oftransport calculations.

Also, unlike ratios of different photon responsefunctions, those for neutrons vary, often strongly, withneutron energy (see Fig. 4). Hence neutrons doses cannotbe converted to different dose units by simply multiply-ing by an appropriate constant. The energy spectrum ofthe neutron fluence is needed to obtain doses in differentunits. Consequently, many old measurements or calcula-tions of point kernels, albedo functions, transmissionfactors, etc., made with obsolete dose units cannot beconverted to modern units because the energy spectrum


is unknown or lost. In this case there is no recourse butto repeat the measurements or calculations.

Capture-gamma photons. A significant, oftendominant, component of the total dose at the surface of ashield accrues from capture gamma photons produceddeep within the shield arising from neutron absorption.Of lesser significance are secondary photons produced inthe inelastic scattering of fast neutrons. Secondary neu-trons are also produced as a result of (�, n) reactions.Thus, in transport methods, gamma-ray and neutrontransport are almost always coupled.

Historically, capture gamma-ray analysis was ap-pended to neutron removal calculations. Most neutronsare absorbed only when they reach thermal energies, and,consequently, only the absorption of thermal neutronswas considered.‡‡‡ For this reason it is important tocalculate accurately the thermal neutron fluence �th(r) inthe shield. The volumetric source strength of capturephotons per unit energy about E is then given by

S��r, E� � �th�r��r� f�r, E�, (23)

where ��(r) is the absorption coefficient at r for thermalneutrons and f(r, E) is the number of photons produced inunit energy about E per thermal neutron absorption at r.

Once the capture-gamma ray source term S�(r,E) isknown throughout the shield, point-kernel techniquesusing exponential attenuation and buildup factors can beused to calculate the gamma-ray dose at the shieldsurface.

Neutron shielding with concrete. Concrete is prob-ably the most widely used shielding material because ofits relatively low cost and the ease with which it can becast into large and variously shaped shields. However,unlike that for photon attenuation in concrete, the con-crete composition, especially the water content, has astrong influence on its neutron attenuation properties.Other important factors that influence the effectivenessof concrete as a neutron shield include type of aggregate,the dose response function, and the angle of incidence ofthe neutrons.

Because concrete is so widely used as a shieldmaterial, its effectiveness for a monoenergetic, broad,parallel beam of incident neutrons has been extensivelystudied, both for normal and slant incidence, and manytabulated results for shields of various thickness areavailable (Chilton 1969, 1971; Roussin and Schmidt1971; Roussin et al. 1973; Wyckoff and Chilton 1973;

Wang and Faw 1995). These tabulated results are ex-tremely useful in the preliminary design of concreteshields.

Gamma-ray and neutron reflectionSo far in this discussion, we have dealt with shield-

ing situations in which, for radiation reaching a target,there is a component of uncollided radiation. Then, inprinciple, point-kernel approximations may be used andconcepts such as particle buildup may be applied. Inmany problems of shielding design and analysis, onlyscattered radiation may reach a target. Radiation dosedue to reflection from a surface is an example that arisesin treatment of streaming of radiation through multi-legged ducts and passageways. Treatment of radiationreflection from structure surfaces is also a necessaryadjunct to precise calibration of nuclear instrumentation.Skyshine, i.e., reflection in the atmosphere of radiationfrom fixed sources to distant points, is another exampleof this class of problem. All such reflection problems areimpossible to treat using elementary point-kernel meth-ods and very difficult and inefficient to treat usingtransport methods. For reflection from a surface ofradiation from a point source to a point receiver, thealbedo function has come to be very useful in design andanalysis. The same can be said for use of the line beamresponse function in treatment of skyshine.

Albedo methods. There are frequent instances forwhich the dose at some location from radiation reflectedfrom walls and floors may be comparable to the line-of-sight dose. The term reflection in this context does notimply a surface scattering. Rather, gamma rays or neu-trons penetrate the surface of a shielding or structuralmaterial, scatter within the material, and then emergefrom the material with reduced energy and at somelocation other than the point of entry.

In many such analyses, a simplified method, calledthe albedo method, may be used. The albedo method isbased on the following approximations: (1) the displace-ment between points of entry and emergence may beneglected; (2) the reflecting medium is effectively ahalf-space, a conservative approximation; (3) scatteringin air between a source and the reflecting surface andbetween the reflecting surface and the detector may beneglected.

Application of the albedo method. Radiation re-flection may be described in terms of the geometryshown in Fig. 7 and eqns (11) and (12). Suppose that apoint isotropic and monoenergetic source is locateddistance r1 from area dA along incident direction �o andthat a dose point is located distance r2 from area dA along

‡‡‡ Exceptional cases include the strong absorption of epithermalneutrons in fast reactor cores or in thick slabs of low-moderating,high-absorbing material.


emergent direction �. Suppose also the source has anangular distribution such that S( o) is the source intensityper steradian, evaluated at the direction from the sourceto the reflecting area dA. Then the dose dDr at thedetector from particles reflected from dA can be shown tobe (Shultis and Faw 2000)

dDr � Do�D�Eo, o; , ��dA cos o

r22 , (24)

in which Do is the dose at dA due to incident particles.Determination of the total reflected dose Dr requiresintegration over the area of the reflecting surface. Doingso must acknowledge that as the location on the surfacechanges, all the variables o, , �, r1, and r2 change aswell. Also, it is necessary to know �D(Eo, o; ,�) or,more usefully, to have some analytical approximation forthe dose albedo so that integration over all areas can beperformed efficiently.

Gamma-ray dose albedo approximationsA two-parameter approximation for the photon dose

albedo was first devised by Chilton and Huddleston(1963) and later extended by Chilton et al. (1965).Chilton (1967) later proposed a more accurate7-parameter albedo formula for concrete. Brockhoff(2003) published seven-parameter fit data for albedosfrom water, concrete, iron, and lead. Two examples ofthis dose albedo approximation are shown in Fig. 8.

Neutron dose albedo approximations. The dosealbedo concept is very useful for streaming problems thatinvolve “reflection” of neutrons or photons from somematerial interface. However, unlike photon albedos, theneutron albedos are seldom tabulated or approximatedfor monoenergetic incident neutrons because of the rapid

variation with energy of neutron cross sections. Rather,albedos for neutrons with a specific range of energies(energy group) are usually considered, thereby averagingover all the cross section resonances in the group. Alsounlike photon albedos, neutron albedos involve reflecteddose from both neutrons and secondary capture gammarays.

There are many studies of the neutron albedos inthe literature. Selph (1973) published a detailed re-view. Extensive compilations of neutron albedo dataare available, for example, SAIL (Simmons et al.1979) and BREESE-II (Cain and Emmett 1979). Ofmore utility are analytic approximations for the albedobased on measured or calculated albedos. Neutronsalbedos are often divided into three types: (1) fastneutron albedos (E � 0.2 MeV), (2) intermediateenergy albedos, and (3) thermal-neutron albedos.Selph (1973) reviews early approximations for neutronalbedos, among which is a 24-parameter approxima-tion developed by Maerker and Muckenthaler (1965).

Fig. 7. Angular and energy relationships in the albedo formulation.

Fig. 8. Ambient-dose-equivalent albedos for reflection of 1.25-MeV photons from concrete, computed using the seven-termChilton-Huddleston approximation.


Newly computed fast-neutron albedos, also in a 24-parameter approximation, were computed by Brock-hoff (2003).

For neutrons with energy less than about 100 keV,the various dose equivalent response functions are veryinsensitive to neutron energy. Consequently, the dosealbedo �D is very closely approximated by the numberalbedo �N. Thus, for reflected dose calculations involv-ing intermediate or thermal neutrons, the number albedois almost always used. Coleman et al. (1967) calculatedneutron albedos for intermediate-energy neutrons (200keV to 0.5 eV) incident monodirectionally on reinforcedconcrete slabs and developed a 9-parameter formula forthe albedo.

Thermal neutrons entering a shield undergo isotro-pic scattering that, on the average, does not change theirenergies. For one-speed particles incident in an azimuth-ally symmetric fashion on a half-space of material thatisotropically scatters particles, Chandrasekhar (1960)derived an exact expression for the differential albedo. Apurely empirical and particularly simple formula, basedon Monte Carlo data for thermal neutrons, has beenproposed by Wells (1964) for ordinary concrete, namely,

�N� o; , �� 0.21 cos �cos o��1/3 . (25)

Radiation streaming through ducts. Except in thesimplest cases, the analysis of radiation streaming re-quires advanced computational procedures. However,even within the framework of Monte Carlo transportcalculations, albedo methods are commonly used, andspecial data sets have been developed for such use(Simmons et al 1979; Cain and Emmett 1979; Gomesand Stevens 1991).

Elementary methods for gamma-ray streaming arelimited to straight cylindrical ducts, with incident radia-tion symmetric about the duct axis and uniform over theduct entrance. Transmitted radiation generally may besubdivided into three components: line-of-sight, lip-penetrated, and wall scattered. The first two may betreated using point-kernel methodology. The last requiresuse of albedo methods to account for scattering over theentire surface area of the duct walls. Selph (1973)reviews the methodology of duct transmission calcula-tions and LeDoux and Chilton (1959) devised a methodof treating two-legged rectangular ducts, which is impor-tant in analysis of structure shielding.

Neutron streaming through gaps and ducts in ashield is much more serious for neutrons than for gammaphotons. Neutron albedos, especially for thermal neu-trons, are generally much higher than those for photons,and multiple scattering within the duct is very important.Placing bends in a duct, which is very effective for

reducing gamma-ray penetration, is far less effective forneutrons. Fast neutrons entering a duct in a concreteshield become thermalized and thereafter are capable ofscattering many times, allowing the neutrons to streamthrough the duct, even those with several bends. Also,unlike gamma-ray streaming, the duct need not be a void (orgas filled) but can be any part of a heterogeneous shield thatis “transparent” to neutrons. For example, the steel walls ofa water pipe embedded in a concrete shield (such as thecooling pipes that penetrate the biological shield of anuclear reactor) act as an annular duct for fast neutrons.

There is much literature on experimental and calcu-lational studies of gamma-ray and neutron streamingthrough ducts. In many of these studies empirical formu-las, obtained by fits to the data, have been proposed.These formulas are often useful for estimating duct-transmitted doses under similar circumstances. As astarting point for finding such information, the interestedreader is referred to Rockwell (1956), Selph (1973), andNCRP (2003).

Gamma-ray and neutron skyshine. For manyintense localized sources of radiation, the shieldingagainst radiation that is directed skyward is usually farless than that for the radiation emitted laterally. How-ever, the radiation emitted vertically into the air under-goes scattering interactions and some radiation is re-flected back to the ground, often at distances far from theoriginal source. This atmospherically reflected radiation,referred to as skyshine, is of concern both to workers ata facility and to the general population outside thefacility site.

As alternatives to rigorous transport-theory treat-ment of the skyshine problem, several approximateprocedures have been developed for both gamma-photonand neutron skyshine sources (Shultis et al. 1991). Thissection summarizes one approximate method, which hasbeen found useful for bare or shielded skyshine sources.The integral line-beam skyshine method is based on theavailability of a line-beam response function R(E, �, x),which gives the dose (air kerma or ambient dose) at adistance x from a point source emitting a photon orneutron of energy E at an angle � from the source-to-detector axis into an infinite air medium. The air-groundinterface is neglected in this method. This responsefunction can be fit over a large range of x to the followingthree-parameter empirical formula, for a fixed value of Eand � (Lampley et al. 1988):

R�E, �, x� � ��/�o�2E�x��/�o�

b exp�a cx��/�o�,

(26)


in which � is the air density in the same units as thereference density �o � 0.0012 g cm�3. The constant �depends on the choice of units.

The parameters a, b, and c in eqn (26) depend on thephoton or neutron energy and the source emission angle.These parameters have been estimated and tabulated forfixed values of E and � by fitting eqn (26) to values ofthe line-beam response function, at different x distances,usually obtained by Monte Carlo calculations. Gamma-ray response functions have been published by Lampley(1979) and Brockhoff et al. (1996). Neutron and second-ary gamma-ray response functions have been publishedby Lampley (1979) and Gui et al. (1997). These data andtheir method of application are presented by Shultis andFaw (2000).

To obtain the skyshine dose D(d) at a distance dfrom a bare collimated source, the line-beam responsefunction, weighted by the energy and angular distributionof the source, is integrated over all source energies andemission directions. Thus, if the collimated source emitsS(E, �) photons, the skyshine dose is

D�d� � �0

�

dE� s

d S�E, ��R�E, �, d�, (27)

where the angular integration is over all emission direc-tions �s allowed by the source collimation. Here � is afunction of thee mission direction �. To obtain thisresult, it has been assumed that the presence of anair-ground interface can be neglected by replacing theground by an infinite air medium. The effect of theground interface on the skyshine radiation, except atpositions very near a broadly collimated source, has beenfound to be very small.

The presence of a shield over a skyshine source, forexample, a building roof, causes some of the sourceparticles penetrating the shield to be degraded in energyand angularly redirected before being transportedthrough the atmosphere. The effect of an overhead shieldon the skyshine dose far from the source can be accu-rately treated by a two-step hybrid method (Shultis et al.1991; Stedry 1994). First a transport calculation isperformed to determine the energy and angular distribu-tion of the radiation penetrating the shield, and then, withthis distribution as an effective point, bare skyshinesource, the integral line-beam method is used to evaluatethe skyshine dose.

The integral line-beam method for gamma-ray andneutron skyshine calculations has been applied to avariety of source configurations and found to givegenerally excellent agreement with benchmark calcula-tions and experimental results (Shultis et al. 1991). It has

been used as the basis of the microcomputer codeMicroSkyshine (Negin 1987) for gamma rays. A codepackage for both neutron and gamma-ray calculations isavailable from the Radiation Safety Computation Infor-mation Center.§§§

Transport theoryFor difficult shielding problems in which simplified

techniques such as point kernels with buildup correctionscannot be used, calculations based on transport theorymust often be used. There are two basic approaches fortransport calculations: deterministic transport calcula-tions in which the linear Boltzmann equation is solvednumerically, and Monte Carlo calculations in which asimulation is made of how particles migrate stochasti-cally through the problem geometry. Both approacheshave their advantages and weaknesses. Because of spacelimitations, we are unable to give a detailed review of thevast literature supporting both approaches. Below a briefexplanation of the basic ideas involved and some generalreferences are presented.

Deterministic transport theory. The neutron orphoton flux �(r, E, �) for particles with energy E anddirection � is rigorously given by the linear Boltzmannequation or, simply, the transport equation

� � ƒƒ��r, E, �� r, E��r, E, �� S�r, E, ��

�0

�

dE��4�

d � �s�r, E�, �� 3 E, ��r, E�, ��,

(28)

where S is the volumetric source strength of particles.This equation can be formally integrated to yield theintegral form of the transport equation, namely,

��r, E, �� r R�, E, �� f�R�

�0

R

dR� q�r R��, �� f�R��, (29)

where f�x� � exp��0

x

��r R��, E�dR�

and q is given by

§§§ Code package CCC-646: SKYSHINE-KSU: Code System toCalculate Neutron and Gamma-Ray Skyshine Doses Using the IntegralLine-Beam Method, and data library DLC-188: SKYDATA-KSU:Parameters for Approximate Neutron and Gamma-Ray SkyshineResponse Functions and Ground Correction Factors.


q�r, E, �� S�r, E, ��

�0

�

dE��4�

d � �s�r, E�, �� 3 E, ��r, E�, ��.

(30)

Unfortunately, neither of these formulations of thetransport equation can be solved analytically except foridealistic cases, e.g., infinite medium with monoener-getic particles. Numerical solutions must be used for allpractical shielding analyses. Many approximations of thetransport equation are used, such as diffusion theory, toallow easier calculations. The energy multigroup approx-imation is almost always used in which the groupaveraged cross sections depend on an assumed energyspectrum of the radiation. Even with an energy multi-group approximation, numerical solutions are still com-putationally formidable.

The most widely used deterministic transport ap-proach is the discrete-ordinates method. In this method aspatial and directional mesh is created for the problemgeometry, and the multigroup form of the transportequation is then integrated over each spatial and direc-tional cell. The solution of the approximating algebraicequations is then accomplished by introducing anotherapproximation that relates the cell-centered flux densitiesto those on the cell boundaries, and an iterative procedurebetween the source (scattered particles and true sourceparticles) and flux density calculation is then used tocalculate the fluxes at the mesh nodes. For details of thismethod the reader is referred to Carlson and Lathrop(1968), Duderstadt and Martin (1979), and Lewis andMiller (1984).

Discrete ordinates calculations can be computation-ally expensive because of the usually enormous numberof mesh nodes and the fact that the convergence of aniterative solution is often very slow. A subject of greatinterest in the last thirty years has been the developmentof numerous methods to accelerate convergence of theiterations. Without convergence acceleration schemes,discrete ordinate solutions would be computationallyimpractical for many shielding problems. An excellentdescription of the various acceleration schemes that havebeen used is provided by Adams and Larsen (2002).

Mature computer codes based on the discrete-ordinates method are widely available to treat one-, two-,and three-dimensional problems in the three basic geom-etries (rectangular, spherical, and cylindrical) with anarbitrary number of energy groups (Rhoades and Childs1987; Alcouffe et al. 2002).

Although discrete-ordinates methods are widelyused by shielding analysts, these methods do have their

limitations. Most restrictive is the requirement that theproblem geometry must be one of the three basic geom-etries (rectangular, spherical, or cylindrical) with bound-aries and material interfaces placed perpendicular to acoordinate axis. Problems with irregular boundaries andmaterial distributions are difficult to solve accuratelywith the discrete-ordinates method. Also, in multidimen-sional geometries, the discrete-ordinates method oftenproduces spurious oscillations in the flux densities (theray effect) as an inherent consequence of the angulardiscretization. Finally, the discretization of the spatialand angular variables introduces numerical truncationerrors, and it is necessary to use sufficiently fine angularand spatial meshes to obtain flux densities that areindependent of the mesh size. For multidimensionalsituations in which the flux density is very anisotropic indirection and in which the medium is many mean-free-path lengths in size, typical of many shielding problems,the computational effort to obtain an accurate discrete-ordinates solution can become very large. However,unlike Monte Carlo calculations, discrete-ordinatesmethods can treat very deep penetration problems, i.e.,the calculation of fluxes and doses at distances manymean-free-path lengths from a source.

Monte Carlo transport theory. In Monte Carlocalculations particle tracks are generated by simulatingthe random nature of the particle interactions with themedium. One does not even need to invoke the transportequation; all one needs are complete mathematical ex-pressions of the probability relationships that govern thetrack length of an individual particle between interactionpoints, the choice of an interaction type at each suchpoint, the choice of a new energy and a new directions ifthe interaction is of a scattering type, and the possibleproduction of additional particles. These are all stochas-tic variables, and in order to make selections of specificvalues for these variables, one needs a complete under-standing of the various processes a particle undergoes inits lifetime from the time it is given birth by the sourceuntil it is either absorbed or leaves the system underconsideration.

The experience a particle undergoes from the time itleaves its source until it is absorbed or leaves the systemis called its history. From such histories expected oraverage values about the radiation field can be estimated.For example, suppose the expected energy �E� absorbedin some small volume V in the problem geometry is beingsought. There is a probability f(E)dE that a particledeposits energy in dE about E. Then the expected energydeposited is simply �E� � � Ef(E)dE. Unfortunately, f(E)is not known a priori and must be obtained from atransport calculation. In a Monte Carlo analysis, f(E) is


constructed by scoring or tallying the energy deposited Ei

in V by the ith particle history. Then in the limit of a largenumber of histories N

�E� � �Ef�E�dE � E� �1

N�i�1

N

Ei . (31)

The process of using a computer to generate particlehistories can be performed in a way completely analo-gous to the actual physical process of particle transportthrough a medium. This direct simulation of the physicaltransport is called an analog Monte Carlo procedure.However, if the tally region is far from the sourceregions, most analog particle histories will make zerocontribution to the tally, and thus a huge number ofhistories must be generated to obtain a statisticallymeaningful result. To reduce the number of histories,nonanalog Monte Carlo procedures can be used wherebycertain biases are introduced in the generation of particlehistories to increase the chances that a particle reachesthe tally region. For example, source particles could beemitted preferentially towards the tally region instead ofwith the usual isotropic emission. Of course, whentallying such biased histories, corrections must be madeto undo the bias so that a correct score is obtained. Manybiasing schemes have been developed, and are generallycalled variance reduction methods since, by allowingmore histories to score, the statistical uncertainty orvariance in the average score is reduced.

The great advantage of the Monte Carlo approach,unlike discrete-ordinates, is that it can treat complexgeometries. However, Monte Carlo calculations can becomputationally extremely expensive, especially fordeep penetration problems. The stochastic contribution asingle history makes to a particular score requires that agreat many histories be simulated to achieve a goodestimate of the expected or average score. If a tallyregion is many mean-free-path lengths from the source,very few histories reach the tally region and contribute tothe score. Even with powerful variance reduction tech-niques, enormous numbers of histories often are requiredto obtain a meaningful score in deep-penetration prob-lems.

Those readers interested in more comprehensivetreatments of the Monte Carlo method will find richresources. A number of monographs address MonteCarlo applications in radiation transport. Those designedfor the specialists in nuclear reactor computations areGoertzel and Kalos (1958), Kalos (1968), Kalos et al.(1968), and Spanier and Gelbard (1969). More generaltreatments will be found in the books by Carter andCashwell (1975) and Lux and Koblinger (1991). Coupled

photon and electron transport are addressed in the com-pilation edited by Jenkins et al. (1988). A very great dealof practical information can be gleaned from the manualsfor Monte Carlo computer codes. Especially recom-mended are those for the EGS4 code (Nelson et al. 1985),the TIGER series of codes (Halbleib et al. 1992), and theMCNP code (X-5 2003).

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