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 MDDC - 880 
 
 UNITED STATES ATOMIC ENERGY COMMISSION 
 
 THE NUMBER OF NEUTRONS EMITTED BY A RA-BE SOURCE (SOURCE I) 
 
 by 
 
 H. L. Anderson 
 E. Fermi 
 J. H. Roberts 
 M. D. Whitaker 
 
 Date of Manuscript: March 21, 1942 
 Date Declassified: January 28, 1947 
 
 Issuance of this document does not constitute 
 
 authority for declassification of classified 
 
 copies of the same or similar content and title 
 
 and by the same authors. 
 
 Technical Information Division, Oak Ridge Operations 
 AEC, Oak Ridge, Tenn., 12-20-48 --850-A4648 
 
 Printed in U.S.A. 
 PRICE 5 CENTS 
 
THE NUMBER OF NEUTRONS EMITTED BY A RA-BE SOURCE (SOURCE I) 
 
 By H. L. Anderson, E. Fermi, J. H. Roberts, and M. D. Whitaker 
 
 In order to simplify the design of experiments directed toward the production of a chain re- 
 action involving uranium, it is desirable to know the actual number of neutrons emitted by the 
 primary sources used. A series of experiments have been planned whose aim is to determine 
 this number. The measurements described are a part of this series. Here, the number of neutrons 
 per square cm emerging from the top of a carbon parallelepiped is measured and calculated, and 
 from a knowledge of the neutron distribution in the carbon, with a given placement of source, this 
 measurement is reduced to the neutron emission from the source. 
 
 A radium -beryllium neutron source containing about 1.16 grams of radium was placed 28 
 inches from the bottom of a carbon parallelepiped five feet on a side and of variable height. The 
 -ource was placed on the vertical axis of the pile, and the number of neutrons emerging from the 
 top surface at the center was determined for two different heights. These neutrons were detected 
 by a BFj proportional counter of 3.62 cm inside diameter, which was filled with BFj gas to a 
 pressure of 12.1 cm of mercury at 0°C. A 10 cm section of the counter was exposed to the neutron 
 flux, the rest being shielded by a close fitting cadmium wrapper, in which a semicylindrical vin- 
 dow 10 cm long had been cut. The glass walls of the counter tube and the metal cathode (a nickel 
 cylinder .010 inches thick) were found, experimentally, to have an absorption factor of 1.13. This 
 absorption factor was measured before the assembly of the counter by observing the decrease in 
 the counting rate of a smaller BF3 counter placed near a graphite surface, when the glass tube 
 and nickel cylinder were slipped over it. 
 
 BFj COUNTER 
 
 3.62 CM = D 
 
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 [1 
 
2 ] MDDC - 880 
 
 A boron carbide shield, containing a circular hole 13 cm in diameter, was used to define the 
 part of the upper carbon surface from which the counter could receive C neutrons. The counter 
 tube was mounted horizontally on a vertical cadmium sheet cylinder, in such a manner, that the 
 semicylindrical opening in the cadmium wrapper around the counter was symmetrically placed, 
 with respect to the circular opening in the boron -carbide shield. The axis of the counter was 
 26 cm above the carbon surface. Additional cadmium shields were arranged so that it was im- 
 possible for the counter tube to receive C neutrons, other than those originating in the carbon 
 surface defined by the boron carbide shield. The number of disintegrations produced in the tube 
 was counted with and without a cadmium shield over the 13 cm opening. The difference between 
 the number of counts per minute observed with and without cadmium was taken with the carbon 
 surface 40 inches above the source, and again with the surface 44 inches above the source as a 
 check. These numbers were found to be 90/min and 63/min. These numbers must be reduced to 
 the number of neutrons per second, Jq, emerging from unit area of the carbon surface at its center 
 by suitable geometrical considerations, and by making use of the known angular distribution of 
 the neutrons coming from the carbon surface. After Jq is known, the number of neutrons emitted 
 by the source can be calculated from a knowledge of the neutron distribution in the carbon pile. 
 
 An approximate calculation of the number of neutrons producing disintegrations in the counter 
 tube in terms of J^ will be done, and the accuracy will be improved by determining correction 
 factors to take care of the approximations made. 
 
 The angular distribution of the neutrons coming from the surface varies with the angle 9, 
 which any neutron makes with the normal to the surface as cos + V^cos^ 9. The fraction of the 
 neutrons which come off at any angle 9 is then 
 
 cos 9 + ^ cos^ 9 
 
 IT (1 + _2_) 
 
 where the denominator is the definite integral of the numerator between the limits 0° and 90°. For 
 angles close to 90° this becomes 
 
 1 + v^ 
 
 IT (1 + _2_) 
 
 and the number of disintegrations taking place per second in the counter is approximately 
 T^^ Jo<y (kT) 1 + v^ , ,„„„ Ro' Joq(kT) 
 
 where uR^ is the area of the emitting carbon surface, D is the distance between surface and the 
 counter, and ctji^t) is the cross section of the counter for absorption of a neutron of energy kT. 
 "■(kT) will bs calculated later. 
 
 H. W. Ibser has calculated this numerical factor accurately for the actual geometry used and 
 finds it equal to .9264 of the value. His calculation is on file with the secret library copy of this 
 report. 
 
 We have assumed in the approximation that the neutron current is the same for all parts of 
 the carbon surface from which neutrons were counted in the above experiment. Since the neutron 
 flux falls off as cos 7rx/a cos iry/a, it follows that the average neutron current would not be J^, 
 which is the value at the center, but .9948 J^ for the surface of 13 cm diameter. 
 
I 
 
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 The approximation used does not take into account the fact that the window, through which 
 neutrons pass through the cadmium shield around the counter tube, is in the form of a semicylinder. 
 This means that the cadmium boundaries which define the length of the counter tube are at a lower 
 level than the level at which the width is defined, by an amount that has a maximum value equal to 
 the radius of the tube. This results in the tube having an effective length that is 1.07 times as great 
 as the length of the semicylindrical opening in the cadmium surrounding the tube. 
 
 Even before we calculate the cross section of the counter, which has been designated as <''(ijrT,\, 
 we will calculate the amount by which we must correct this cross section, in order to have it 
 apply to the Maxwellian neutron distribution which we actually have, instead of applying only to 
 the neutrons of energy kT. Since the cross section of the boron used varies as l/V, in order to 
 get a factor relating the cross section for neutrons of energy kT to the average cross section for 
 the neutrons having a Maxwellian distribution about the energy kT, we must get the ratio of the 
 velocity of a neutron of energy kT to the average velocity of the neutron having the Maxwellian 
 distribution. Or, if C = V2kT/m, then the cross section will vary as the average of C/V. 
 
 oo 
 
 /-^ V' e -VVC=' dV 
 
 i-TT-) ^-"-^ = -4 — = .8862 
 
 V 
 
 /v3 ,-vyc 
 
 2/^.2 
 
 dV 
 
 Now, if we apply all these correction factors to the first numerical constant 1.2679, we find 
 that this becomes 1.1218 and the number of neutron captures per second in the counter becomes 
 
 1.108 . ^'^°/ = .06925 V 
 
 where a(kT) has been changed to o' by the inclusion of the last correction factor. 
 
 If we neglect the attenuation of the neutron current by absorption in the BFj of the counter, 
 which we can do in this case with an error of less than 1%, then we can say that the total cross 
 section of the counter is the number of boron atoms in the tube times the cross section of each, 
 or, the number of moles of the gas times the cross section per mole, which is 473 cm^ for boron 
 and neutrons of energy kT. Then 
 
 /,_™,\ number of moles x 473 
 (T covmter (kT) = — — 
 
 L tXo 
 
 where 1.13 is the absorption factor of the wall material of the counter tube. 
 
 Volume of counter tube =—, — x 10 = -^ (3.62)^ x 10 = 102.92 cc 
 
 4 4 
 
 12.1 
 Pressure = _„ atmospheres at 0°C 
 7d 
 
 102 92 12 1 
 Number of moles = ' x = 7.311 x 10"'* moles 
 
 ^ ,,^, 7.311x10-^x473 ^-^ , 
 
 cr counter (kT) = —— = .306 cm^ 
 
4 ] MDDC - 880 
 
 With the top of the carbon 40 inches above the source, the number of disintegrations per second 
 due to C neutrons was observed to be 1.5. Therefore 
 
 1.5 = .06925 Jo X .306 
 
 and 
 
 T _ 1 5 1 70 78 "^"trons 
 
 "''' ' '•" •. 06925 X. 306 " '""'" cm^' sec 
 
 and for the case where the top of the carbon pile is 44 inches above the source 
 
 T - ^ i 49 54 "^"trons 
 
 ° 60 .06925 X .306 cm^ sec 
 
 In the report A-21, Formula 15 gives the density n of thermal neutrons as a function of z. 
 For a (Ra-Be) source, the neutrons may be represented as a super position of three groups 
 having ranges for slowing down to thermal energy and percentages as follows: 
 
 Range in graphite 
 
 27.1 cm 
 
 39.8 
 
 58.9 
 
 For large z and a 5 ft column the formula reduces to 
 
 XNV _. 008565 €-2/27.61 
 Q 
 
 Near the top of the column an additional term must be added to represent the effect of the nearby 
 boundary. We have then 
 
 Per cent 
 
 15.0 
 69.3 
 15.7 
 
 3iNV 
 
 -^^^^^ - .008565 
 
 Q 
 
 € 
 
 -E/27.61_^.Z/27.61 ^ -2/27.61 ^^^ ^ ^/^^ 
 
 where Zq is the ordinate of the top. Since the diffusion coefficient is xV/3 we have 
 
 z„ 
 
 = 1.9636 X 10"* g 
 
 27.61 
 
 -^ = 4.95 X 10"^ at Zo = 40 inches 
 W 
 
 -^- = 3.432 X 10"^ at Zo = 44 inches 
 
 •••Q= .c./Vn'-a -t4.29xl0°gg^^^Ig^^ 
 4.954 x 10 " sec 
 
MDDC - 880 [5 
 
 from the data taken at Zo = 40 inches and 
 
 49.54 ^ 14 43 ^ iqs neutrons 
 
 ^ 3.432 X 10-» ^^•^'^ "" ^" sec 
 
 from the data taken at Zq == 44 inches 
 
 .•. avg Q = 14.36 x 10° for source No. I 
 
 For the three sources I, U, and ni, we have the results given In the following tabulation in which 
 we combine the value of Q and the comparisons given in the report C -10. 
 
 Source No. Q Grams radium Q/gram 
 
 I 
 
 n 
 m 
 
 1.44 X 
 
 10' 
 
 1.16 
 
 1.24 
 
 X 10' 
 
 1.28 X 
 
 10-' 
 
 1.0308 
 
 1.24 
 
 X 10' 
 
 1.05 X 
 
 10'' 
 
 .84 
 
 1.25 
 
 xlO' 
 
 END OF DOCUMENT 
 
UNIVERSITY OF FLORIDA 
 
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