THE UNIVERSITY OF MIECHIGAN INDUSITRY PROGRAM OF THE COLLEGE OF ENGINEERING DOSE RATES PRODUCED FROM GAMM RAY SOURCES Jo G. LEWIS IP-158 Aprl, 1956

PREFACE This work was undertaken to provide a concise statement of certain methods and approximations useful in the estimation of physical dose rates in air in regions adjacent to sources of gamma radiation. The intent is primarily to provide working methods which yield approximate answers rapidly, To this end certain simplifying assumptions have been made and graphical methods of calculation have been included.

DOSE RATES PRODUCED FROM GAMMA RAY SOURCES Io ABSTRACT In this paper are presented some assumptions, methods, equations, and nomograms which have been employed to estimate physical dose rates in air due to sources of gamma radiation of sielected shapes. A method for estimating dose rates in air near hollow cylind.rical sources is reviewed briefly, and it is shown how a general equation degenerates to simpler forms for selected special shapes of the sources and for locations of the point at which dose is calculatedo Nomograms are presented for the simplified estimam!- of dose rates due to hollow cylindrical sources and poin.rt sourceso IIo IIN T R b' t (`-,, I' I T The emphasis in this work is to summarilze some methods for calculating problems which permit estimating gamma dose rates in configulrations and under conditions frequently encountered in irradiation work. One purpose ai; to show the relationships in special cases which have been treated previously in calculating gamma dose rates. Another purpose is to present nomograms -which permit rapid estimation of gamma dose rates in airo Rapid. estimatng procedures are of particular value to the engineer who does not have a wide background in nuclear studies, but who wishes to assess quickly the potent1ialities and requirements for. radiation effects and experimental Qr.stal lti ons requirie ng the use of gamma radiation. To this end., it is suggested that the dose rates in air calculated from the charts presented here may be comrected for abssorption of such things as pressure vessel walls, source con'tainers, etc., by rough approximations taking account of exponential decrease in dose rates with thickness of thin layers of absorbing materialso The use of sources of radiation has in=c =eased rap Idl.y duriLg the past few years because of the increased use of nuclear reactorso Reactors have made radiation avallable in a number of formso Among these are beams of neutrons 1,

emitted directly from reaetors, ani gamma anr.- ceta radciation emitted during fission or resulting from ielayes. emission from the radioacti.ile fission prcucts. In addition nonacti-Ae materials may te placea in nuclear reactors and, made radioactive by the absorption of neutrons, C"oralit-6; is procduced in this manner, for chemical, radiographic, radiotherapeutic, and other biological work. Many other isotopes such as rarioi-cline, radiophosphorus, and radiogold, may be produces by neutron a sorption for medical annrds ciher tracer work. Radiocesium anc rsa(icstrontium may be, sepa-rat'ed. from nuclear fission products and employed in a vari.ety of't;iological and c~hemical irradiations. The nuclear radiation emittes bty moss nucliLes of interest to irradiation work consists chiefly of elect lors (ibeta particles) and electromagnetic emissions (gamma radiation),'Jose rates Cue Lo beta rai. Cation are not treated here. Dose rates due to gamma radlsiation are considered in this paper. A knowledge of physiceal tse rates in air near sources of garmla radiation is of importance in handling an-' using many kirnd-s of radioac-Live Ua-l terials. Gamma radiation must be eant wiTh in such a way that personnel are not subjected to excessive exposures to the raLiationo I-n afddition, any applications of gamma radiation require some quantitattve krnowlege of the amounts of radiation received by a system of interest,, to permit correlation of observed effects with the radiationo Personnel protection requires shielding and problems encountered involve required thickness, shapes, anew. weighns of the shielcing materials. These in turn bring in consideration of abscrpticnn, scatteri.ng, dispersion, and other interactions of radiation andr matter, eg-, as discussed by Mcteff (7), and Goldstein and Wilkins (1). Appi(cations of gamma raCiation to irradiation of systems also involves these efiects, So However, the present d.iscussion neglects such effects arnc conceni-ratnes upon she problem of rapid esJimation of dose rates in air. Moderate treickesses of air urner ordinary conditions attenuate primary gamma ra*.'iation) so slightly that the effect of air can be ignored in approximate calcuLa icns. III, DOSE RAT:ES IN AIR I'f':...0 R..C.ES 2.F SEI..IE. SHAH-ES In this discussion various equations are revsewed., relating the gamma dose rate to the characteristics of tnhe source of radiation. ITne unit of gamma dose rate used is the "roentgen equivalent physical", often abbre-iated as "rep' as discussed by Parker (8). The rep will be considered to correspond to the absorption of 93 ergs of energy from gamma radiation per gram of absorber (i rep corresponds to about 6,4 x 10-12 tu/ilb of absorber). The precise Cdefinition of units of dose rate is eomplicatel by (:.cnsil!erations such as atomic nuir.ber of the absorber and energy spectrum of the gamma rad.iation, but these will not be considered further in this paper.o -he rep is only one of mar -- units which have been suggested for the reporting of dose rates. See Siri (10).

A brief review of units and conrversLorn 9 >etors:1. prese:nted here for convenient reference: 1 electron volt (ev) 1 59 x 1,-1'2 ergs 1 ev 1o59 x 10-C19 watt secon.~.ds 1 million electron volts (may), o59 x 106 ergs 1 meay = iL51 x 10-6 I-tu 1 mrev 4J43 x l0-20 kwh 1 kilowat t hour - 3410 Ltu I erg =iQ07 joui.es 1 watt second I- 1 joaLe I gram calorie - 418t joulees 1 3tu 252 gra calori es 1 curie =3 7 x iI0L d io n;grat c / S OL It may be noted that curies, a corrmoniy used measure of the adioactivity of a material, i;sanal.ogous o Current iL ordi —;. y'et+rical preactice, in that it gives a measure of the nn:ten= of nu1, r em-1.cn- pc- uni time, without regard to the energy of each emis:s o alny gamma rays a.-.re emitted from decaying nuclei with char{,c-. teriistic energies, analogous to *the spectral lines emitted -by luminou bodia-es The h i. cenergy of3 eom-is - sion of the gamra rays is often reported iln mi:~ll- ns -1 oo:''_ it is( o volts (me-v). Consequently, the emission energy is a98ialogous5 t~ voSlLDgRe n e ctricln prac2tice, and the product of curies and mnev giLves La thru~2eo powerL oututput of a source of radiation. A ra ioa,3%,ie maet al ofte;! e.mi-s'_otlt h }eta and gamma rays of different enrergies an d sometime- d:isrl:iKed among more thEa one decay scheme. Consauueoinly, caution mu.st be emrployed''L, n.rtawing an.logles such as these given atoveo Act ivities \en-aon' eS:in pn.rs''e may vy from microcuries (one micr.ocuAre equals iR06 cure) T- o megacurles (one> megasurie equals 106 curie). As an illustra-tive exampi., i ruie were emi12 t.in+,g only one gamma of 1 mev per disinrtegration at, a Eate of one rmilv 543 slion c ie, the power output of this source woul.d be: - 1 0 6 B.... I0 d- sineg..._ 36 sec_ 1 mev x 4.43 x 0 2 kw x i0 curie- x 307 x Lk xis.6e - e mayvd hrcv {~un

One million curies is a large source of radiation, and its power output of 5.9 kilowatts is not large compared with conventional sources of power. The relatively'small power available from what are now regarded as large sources of radiation illustrates the need of utilizing either very large sources of radiation or using radiation for applications not suited to other sources of energy where power is the primary consideration. It should be noted that the total power output of a radioactive source, 5.9 kilowatts in the example cited above, will not all be intercepted by a given specimen or absorber to be irradiated. This observation again leads to the discussion of dose rates, as follows. The radiation power is distributed about a radiation source in complicated ways, which depend largely upon the shape of the source and the location of the absorber. If the activity of the source is uniformly distributed within the source and of uniform composition, then the estimation of radiation power absorbed at a point in the absorber (equivalent to estimation of the dose rate at the point in the absorber) becomes chiefly a problem in relating geometrical variables of source and absorber. The following discussion is to relate the geometrical and radiation variables concerned. Lewis, et al. (4) have discussed equations for estimating dose rates near hollow cylindrical sources of gamma radiation. It is proposed in this section to discuss various forms into which the general equation may degenerate for certain restricted geometrical situations. Consider Fig. (1), a hollow cylindrical source having negligible wall thickness, The dose rate I at point P may be expressed by Equation (1). I_ MC [F (tan l Z1, k) - F (tan-1 Z1-L, k)] (1) L(R + r) ( r-RfI fr-RI for Z1 - O, R - O, r > O, R r, -ET/2 < tan-1 (Z1-L)/lr-RI < it/2, 0 tan-1 Z1 < i/2; k = 2 4Rr Ir-RI R + r It should be noted here that the expressions F (tan-1 Z1, k) (la) ir -RI and F (tan Z - L k) (lb Ir- RI

are geometrical functions of the bottom and top, respectively, of the hollow cylindrical radiation source as "seen" by the absorbing point at P(R,G0, Z1), and hence may be considered as analogous to the geometrical factors encountered in radiant heat transfer, cf. McAdams (6). Expressions (la) and (lb) are 7l-lip-t ic nlt;e. of 1ra -e first. lind-, tables of which appear in a number of sources, e g., Jalinke and Einde (2). It is interesting to note that the elliptic integrals of the first kind shown in Equations (la) and (lb) degenerate to circular angles expressed in radians when k - 0, corresponding in Equation (1) to the cases where R = 0 or R -eoo, i.e., lim [F (tan-' Z, k)] = tan-1 Z1 o (1c) Jr F,' Ir - RI k -0 It is of further interest to note that the ellipse to which these functions refer may have a major axis of R + r, parallel to the base of the source, arnd a minor axis of R - r, with center located at (R + r <, - ) The following are special cases which may be developed from Equation (1) under the conditions noted: A. Let R = O, ioe,, let point P be on the axis of the source. Therefore, k = 0, and consequently the elliptic integrals of the first kind degenerate to circular angles. I MC (tan-Z,/r- tan 1 Z1 L) (2) rL r B. Let r = O, i.e., let the cylindrical source degenerate to a line. I = (tan-' Z- tan1 Z1 -L) () LR R R C. Let Z1 = L/2, i.e,, let point P be always on the mid-plane of the cylindrical source. I = 2MC [F (an L (4) L (R + r) i.. Do Let L - oo i.e., let the source become very long compared with r and R. I 2MC [K (k)]. (5) L (R + r) E. Let R - oo, i.e., let R become very great compared with Z1, r, and L. Then lim k = 0, and the elliptic integrals again degenerate to circular anges, ain'A", above. For R> r, L, Z1, then tan-i Zi/lIr-RI approaches Z1/ r-RI and tan-1 Z1-L/l r-RI approaches Z!-L/lr=RI and the following relation results'

MC 1r2 - R21 which can be further reduced to: MC I =R2 (6) That is, the source behaves as a point source when the receiving point is remote in a direction transverse to the axis of the source. F. Let Z1 - oo, i.e., let Z1 become very great compared with R, r, and L. For convenience, put Equation (1) in the form: I =L(MC [F (tan-' R + r, k) - F(tan' R + r, k]. (I) L (R + r) Z1 -L ZAn analysis similar to that of "E" results in the following: I __ (7) which is also of the form for a point source. IVo SOME PROPERTIES OF RADIOACTIVE MATERIALS Various methods are available for reporting the ionizing power of radioisotopes. In the discussion of Section III, the quantity "M" was introduced, related to the methods of Marinelli, et al. (5). The quantity M is the "rolentgen equivalent physical" per hour produced in an absorber located one cm from a point source of radiation of one curie activity. Moteff (7) has reported plots of the gamma radiation in mev per square centimeter per second required to produce a dose rate of one roentgen per hour as a function of the energy of the radiation. A similar relation appears in Figure 2, showing the M value plotted as a function of mev of the radiation, assuming 93 ergs absorbed per gram per rep. Figure 2 was plotted on the assumption of one gamma ray emitted per disintegration. If more or fewer are emitted in the average disintegration, the M value to be used with this paper should be corrected by simple addition of the contributions of each ray emitted. 6

Vo GRAPHICAL ESTIMATION OF DOSE RATES FROM HOLLOW CYLINDRICAL SOURCES Figure 3 is a nomogram which provides a graphical means of solving Equation (1), above. Equation (1) relates activity in curies, C, specific ionizing power of the radioactive material in the source, M, and the various dimensions of the system comprising source and absorbing point. The hollow cylindrical source configuration was chosen for solution since it is a shape frequently encountered in practice and also lends itself well to the approximation of other shapes, For instance, if the radius, r of the source becomes small compared to other dimensions, the cylinder approaches a rod or line source. If the height, L, becomes small compared to other dimensions, the cylinder approaches a ring source. If all dimensions of the source become small compared with the distance between the absorbing point and the source, the cylinder appears to approach a point source. Provision is made in Figure 3 for direct solution with dimensions in either inches or centimeters. Scale factors are also introduced for taking account of ranges of variables greater than those scaled off directly on the chart. "Preferred ranges of variables" are indicated. If data are entered in these ranges, solutions for I, the dose rate, should not fall off the chart. An illustrative example is solved below. See Figure 1 for meanings of dimensional symbols, Dashed index lines indicate the mode of solution in Figure 3. Take a source with the following characteristics: M = 153,500 rep at 1 cm (cobalt-60), hr x curie C = 3000 curies, L = 10 inches, R = 0.3 inches, r = 4,,7 inches, Z1 = 15.5 inches. (Absorbing point 0,3 inch from axis 15.5 inches above bottom of source.) Compute R/r, here 0.0638; enter chart at upper right plot. Read up to curve and then to left, intersecting the index line at a = 28~ - 21'. (Note, a = sin-1 k - sin -1 0.4749.) This operation identifies the proper a line to use in the next step below~ Next compute Zl/lR-rl,3.5227; and Z1-L/JR-rj, (1.25). Enter both these values on the right side of the bottom 7

center ploto Project to the left to the tan-I ecurv'e, and. then project up to the plot above, intersecting the X = 28" - 29' line found in the pre-vious step. Project left from the -- 28~ - 2Yt line to the ordinate of the F(O,k) vs. 0 plot. Read F{Ok) for both Z/?R —r and Z —L/|R-ri, where, is tan-1 Zi/IR-r' or tan- Z:-Z/jR-r o Subtract F(tan-r Z-L</!R-r|, k) from F(tan-Z Zl/jR-rj, k) giving A F(i:~ k3 - F (s2, k) = 0k 3600 - 00 9207 0.4393 This value of Z, 0A43935, is now transferred to the left scale of the ordinate line of the F(jk) vso 0 piot. I:he determination of., (the term in brackets of Equation (1) ) just described is -the most ted!ious part of the solution. Next, the coefficient of the brackets in Equation (1) is combined with A to fin6 the dose rate, as follows: in the lower left- corner of'the chart, locate L, 10 inches, and project through the (R + r) scale att 5o0 inches, to find L (R + r), 50 inches, on the L (R + r) scale. Now move to the H./10s scale, lower ieft of chart, and locate M/103 = 13o5 for cobalt-60. Projec -th"hrough the C scale at, 3000 curies to find MC=40,500,000 in this example, Now conneet i with L (R + r). Locate the point of intersection of the conunecting line with the arbitrary irndex line. Connect this point on t~he arbitrary crmex in~e with -the;-alue of M(,C previously found, and project to the 1- scale, where the.cLse rate will'be fouren 3o(0o055 in this example), subject to correction cf the dLecimal point~ Mul-tiply the value (0.055) readc from the I scale by lQ[Q'>t-+j', (In -this example, 10[3+0+3-(0+') ] = 10 J, the cumula'ti-e correc-ion fr.actor resul-ting from cecimal point changes requireC. to get all variables on scale. The resulting c{ose rate is 0,055 x iV0. 55 000 rep,'hr in air at the location chosen. Some precaut.icns o observe are the following~ When R = 0, th thk = 0 Corsceqe.ie'- lyy-, -- sin-l, k 0,- and one does not need to use the upper right-hand. plo-; o k -vs R/r. One simply uses the a 0 line in the plot of F6,~k2 vs. rs In determining L: F({, k) - FP>, k), when C -- 0 - I., fthat is, the absorbing point lies between the top anr bottom altitudes of the source, then Z! - L is negati-ve This makes the seconl- term in bracke-ts in Equation (1) negative. 1The subtraction of the negative seconrd term then results in the addition of its absolute value to the first term in brackets. The ZA fourd by subtraction must:'be re-ent-ered7 on the left side of the ordinate line of the F({, k) vs. plo'to

A record must be kept of the exponents, N, P, Q, S, used to put all values on scale and used in the final correction factor. Answers may be read as described above if either inches or centimeters are used for dimensions, but the right scale of M/ O1C, must be used for inches and the left scale for centimeters. Any other units may be used for dimensions of length L, R, r, Z1 but the values of I calculated from the Figure 3 must then be corrected for the (length)2 factor in the denominator of m4:, e.go, if lengths used are in feet, and the M/103 scale for inches is used in reading Figure 3, then values of I read fromi Figure 3 must be divided by (12)-. = 144. Figure 3 may give inaccurate results if the absorber is remote compared with source dimensions, since it will be observed that the various values of a all give nearly the same (F (g, O) for the small values of 0 which result in this case. If values of dose rate on the midplane of the cylinder are desired, and R is sufficiently great that the included angle subtended by the top and bottom of the source is equal to or less than 45~, then one may treat the source as a point with an error no greater than about 11%. See below for a graphical solution of the point source caseo VI. GRAPHICAL ESTIMATION OF DOSE RA.ES FROKM P'N1TWIT SOURCE Figure 4 is a nomogram for the estimation of dose rates from a point source of radiation~ The symbols have the meanings given above, except that r, ZI, and L have no application, ar.d R is the distance from source to absorber. The solution of a case is illustrated by dashed index lines. M = 13,500 (cobalt-6o), C = 3,000 curies, R 24 inches. The resulting value of dose rate read from Figure 4 is 10,900 rep/hr. Numerical calculation yields a value of: MC R2 - (13,.500) rep at 1 cm in2 3000 curies hr curie 6.45 cm2 (24)2 ino 2 = 10,899 rep/hr in air, 9

VII. ACKNOWLEDGEMENTS The material contained in this paper was developed in part under the auspices of the Industry Program, College of Engineering, the University of Michigan. The author wishes to thank the Engineering Research Institute, University of Michigan for permission to present this paper. The author is grateful to H. A. Ohlgren, Professor, and J. J. Martin, Associate Professor, of the Department of Chemical and Metallurgical Engineering, the University of Michigan, for advice and encouragement in the preparation of this paper. Mr. Richard Dilley, Mr. Gilbert Marcus, Mr. Howard Silver, and Mr. Wayne Thiessen have assisted the author in calculations, drafting, and criticism of the nomograms presented. VIII. DEFINITIONS OF SYMBOLS I = dose rate, rep per hour, A = area of source, r = radius of source, R = radial distance of point at which I is taken from axis of source, G = central angle from R to r, Z = distance parallel to axis of source from base of source to element dA, Z. = Z-coordinate of point at which I is taken, k = k = 2 vRr/(R + r), a = sin-1 k, F(O,k) = elliptic integral of first kind of modulus k and amplitude O, 10

K(k) = complete elliptic integral of first kind of modulus k, C = total curies, M = (roentgen equivalent physical at 1 cm)/(hr. x curie point source), L = length of source. IX. BIBLIOGRAPHY, Goldstein, H., and Wilkins, J. E., NYO-3075 (1954). 2. Jahnke, E., and Emde, F., Tables of Functions, 4th ed., Dover, New York (1945). 3. Kaplan, W., Advanced Calculus, Addison-Wesley, Cambridge (1952). 4. Lewis, J. G., Nehemias, J. V., Harmer, D. E. and Martin, J, J., Nucleonics 12, No. 1, 40 (1954). 5. Marinelli, L. D., Quimby, E. H., and Hine, G. J., Am. J. Roentgenol. Radium Therapy 59, 260 (1948). 6. McAdams, W. H., Heat Transmission, 3rd Edition, McGraw-Hill, New York (1954). 7. Moteff, J., APEX-176 (1954). 8. Parker, H. M., Advances in Biological and Medical Physics, Vol. 1, Academic Press, New York (1948). 9. Sievert, R. M. Acta Radiologica I, 89 (1921). 10. Siri, W. E., Isotopic Tracers and Nuclear Radiations, McGraw-Hill, New York (1949). 11

P( ~ - 0-dA X I, X FIG. I-GEOMETRY OF CYLINDRICAL SOURCE HAVING NEGLIGIBLE WALL THICKNESS 12

FIG. 2- DOSE FACTORS FOR GAMMA RADIATION AS FUNCTIONS OF ENERGY 107 -06 Mev. per rep cm3 x sec. hr. / value, =.,_ / ~~rep I cm. hr. x curie pt. source 0.0I 01I11 10 104 0.01 0.1 1.0 10

/ C Z i~~~~~~~~~0~_ Lo.. 20 T2o.:2-2_ _ - N.. HHH0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ j~~~NE I I 1. ~~~~~~~~~~~~~~~~-7 0.7 ~e --'O. ~ - I4: / FF /~ ~ -LO I I02 ~~~~~~~/I ~~/ I_ 0.1 0.1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1::-~z;04. IIH A N.0 FROM THE LOIS OF A HOLLOW CYLINSER OF 510~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~EA HLLW CLIDRCA.TRNSARN COBALT HO H TISRI 0ICE 0.4a'0' H~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~OURE OF HAAN ARMOLT IHL i- H ONO -LO/ I —' / 2 (,0 ACTIRITHNG OFF00 CARERTOS. 0NNDATDDSHE-DARRW FROLTHEAISN ROTA HONR NRoLLOWEN CYLINDER ONFHN SAC -H THREN PREAFELEFT TOA RANGES.L TLARE VALUE AT ABS SUBTACTVAVEA 09, PLHACE.7IFFERNCHES ATI N ARA SCAL O~ N ~' NOW READ WEIHOT OH BOUNCE AT CONNECTINV WITH BOO. 00+4.7 AT 0 C URIES. Vo ~'U UTOO1 108TAO 0 I 1 4 H I N, iLTIP iOR I Y Is+3-("+ 4/ 003 12~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~- O I TH-NO OR LOCTIN 1.A NOW+ LOCATE CY-RLONINCHES AS POFNT SCALCE AT5 MAIMUM ORRON S,, OMGLUrlFRETMTION DS AE 100 4.0' - 6'.; N IALOCATING. NOW CURONNECT E WITH 5 10so2: ~ ~ ~ 1 0R NEAR OHOLLOW.CYLIDRICALR -RA. N RANDRDARIHNL BRA SRERAT EFT T O.O 10) AND OO RE/R HIIRONRHCUHRVE ITHERNI OP TIR IO 5~ ODARTONODR e WHC OND NAR IN DENED 23H NASHE 3. 10 -.4..~~~~~~~~~~~~~~~~~~~~~~~~~~~~NIH.RRILD SOURRCES OF GAMMA RDATIN' / 4 EXAMPLE' 0'~ + ~ Ou FIND DOSE RATE, REP/ HR IN AIR, AT A POINT EQUATION OF THE NOMOGRAM: NO 15.R INCHES ABOVE THE BASE AND 0.3 INCHES zSUN OOPANSDCT B 5. -r 3 ~ O FROM THE AXIS OF A HOLLOW CYLINDER OF I EUNEO PRTOSIDtTD R I~D ARW -_ // COBALT 60. THE CYLINDER IS 10 INC.ES 4C....O-GO..THR.E SEDE SF''E CALE RAN H............... t HIGH AND 4.7 INCHES IN RADIUS, WITH AN:0 TO CURIES TO...TI I'I,.LY DISTIBUTED ByOVENSU 0O CIT ACTIVITY OF 3000 CURIES. 0 IH O, LOCATING ~ WHERE WE FIND 0.055. SINCE Oe3, S=O, N=, PeO~ ~= Ion'' I'-"~-T — or Ion-' iE — L = ATITUDE CYLINDER OICAL SOURCEAT 4' SOLUTION IS ALLOWABLE READ R- AT A; READ UP TO K ON CURVE, THEN LEFT TO A ON INDEX LINE, 4 LOCATINR AT ~ 8,RADI ORC CX - N a28'- 23' CURVE. NEXT READ?L AND -Rlr A B:I ERROR 13 ALLOWABLE READ LEFT TO ~) AND (D ON CURVE THEN UP TO ~ AND~ ON -. 2 28 23' CURVE; THEN READ LEFT TO @ AND 0. TAKE VALUE AT 0 AND SUBTRACT VALUE AT ~.I.i ~ 0 PLACE DIFFERENCE AT ~ ON A SCALE. IDS AERP RI I C a ~~~~~~~~~~~~~~~~NOW READ HEIGHT OF SOURCE AT ~, CONNECTING WITH R4-r ~ 0.3+4.7 AT (, I I~DOERA I I/R IYI &5O6 6.U1.. RE A IT CENTIIEE THUS LOCATING O. NOW LOCATE CO - 60 ON INCHES SIDE OF f SA AT g(,rain HR r CURIE POINT i0UNCE ~; CONNECT WITH CURBLES AT 0, LOCATING ~. NOW CONNECT g WITH gCCRE OA CIIY SVE O,UIORL ITIUE VRSUC 2 103 LOCATING 0P WHERE WE FIND 0.055. SINCE 0-3, S-0, N-0. P.O. Z wLT_ T,'.-' gk=-,ZT L ALTITUDE OF CYLINDRICAL SOURCE DOE AT, -O.W CP- 5,00RE/ R F (0K);ICY LT ELLIPTIC INTEGRAL OF FIRST KIND F,~ HEIGHT OF POINIT (AT WHICH DOSE RAT I ERD a = F(%,KI) F(O. K) Rr RADIAL DISTANCE (IF POINT FROM AXIS SUC k J~- 2, RADIUS OF SOURCE R-r

INDEX LINE L C MC R H 103 IOY rc;Tc ~ I~0+Y-2Z 01A 100,000 T00,000 9,000 8- 8000 -0.8 8,0007 09e — 50000 $ 7,000 4000 6,000,9 1~~~~0,0,000 20 000 0 4,00 800 9,0 3,~~~~~~~~~~~~~~~~~O 000 04 6000 600 t 3 400~~~~~~~~~~~~~~~~~0 051 0 200 2,0004000 0.7Of 420000 1000 0 6 000,000 00 FIGURE 4 0 6 4-00 50002'+32 3. 400 40 0~ ~ ~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0.6" _T ORESO AMARD4TO 69~/ ~ O b 100 300 3 0 02000 2000 D ~~~-7 Soo 90~~~~~~~~~40 I:8 LZ- So 6 ~6 400 460 10 14 720 440 2NOSE RAME ORESIMTION OF DORT E N 1 50 REP40 w.E 600 ZOI4 POINT SOURCES OF GAMMA RADRPI S CO N UE T IDA`6 120 40 0A II Rz 90 0.0(SY-Z 7 20 140 2 T.DOSE RATE,~~~~~~~P IN AI40 128- oISAC RMSORET ON AT WHICH IS MN EAUE 6 00 06 0o 10 P70 7r f10 ~ ioo C-OURIES ACTIVITY OF SOURE0 oI 1007 o~s _60 13 92o~~~~~~~~~~ t tM 0.2~~~~~~~~~~~~~~~~~~1 EXAMPLE: 0 0.2 5 FINS DOSE RATE, REP PER HR. IN AIR AT 34 INCHES FROM POINT 40 SOURCE OF 3000 CURMED OF COBALT-D 6. 3+ -F200 170 + SOLUTION 30 0.4 B DRAW STRAINHT LINE FROM ~, AT Mo13,500 FOR COBALT -60, 0.00 THROUSH ~, AT 3000 CURIES TO LOCATE ~. CONNECT ~ WITH)~ AT 24", 40 160COD 2 4. 006 0~ ~ ~~~~~~1 LOCATING ~, WHERE -- IS READ AS 10.9. SINCE T'O, 50 tOO 0.04 02 2140 IN THIS EXAMPLE I 10,900 HR IN AIR. 60 400.0 1~~ 31 E 500 02 80 -~~~~~~~~~~~~~~~~~ Os~~~10E UI 0 7A600 1 O R30 100 07 0 01.04