Simple Absolute Measurement Technique for Beta Radioactivity Application to Naturally Radioactive Rubidium W. F. LIBBY’ Geophysical Iaborafory, Carnegie lnsfifution o f Washingfon, Washington,
b Back-scattered radiation is somewhat softer than the original beta radiation and depends in both intensity and softness upon the atomic number of the back-scatterer. This effect has been included during this research. An important new effect was found in that the rough surface of a crystalline powder requires a larger correction for geometry than a smooth surface. This was shown b y direct calculation and experimentation. The geometrical effect of the surface roughness of a powdered solid is most marked for soft beta rays for which the surface looks much rougher than for hard beta rays. Empirically for powders as ordinarily prepared, a half thickness of about 7 mg. per sq. cm. seems to be a good dividing line, and beta radiations of smaller half-thickness require a geometry factor some 4oa/, larger than do those of larger half thickness. The geometry factor for hard betas i s the same one calculated for a smooth surface. The technique of measuring the absolute radioactivity of solids and liquids b y placing them in a cylindrical position around an ordinary Geiger counter gives results which agree with the true absolute assays, within 5%. The technique was applied to the measurement of the half life of naturally radioactive rubidium. This should have application in the development of new uses of isotopes, particularly in introducing isotopes into the chemistry classroom.
B
radiations resulting from a single transition between two nuclear energy states are absorbed exponentially even though the transition may be highly forbidden as in the case of potassium-40. This occurs under conditions of cylindrical geometry in which the sample lies on the surface of a cylinder whose axis is identical with that of the Geiger counter used to measure the radiation, as shown ETA
1 On leave of absence from the University of Chicago.
1566
ANALYTICAL CHEMISTRY
D. C.
by Suttle and Libby (SO). The cylindrical geometry is necessary to the control of the very large effects of beta ray scattering. Thus the ordinary end window type of counter with its flat sample does not give exponential absorption unless special orifice windows to control the scattering are used ( 1 1 ) . These popular counters can be used for absolute counting (4, 8, 11, l a ) . However, the controls and corrections necessary for this type of counter and disposition of sample are so rigorous and exacting that absolute counting by this technique is difficult in the average laboratory. The fact that absorption curves which are nearly exponential in nature can be obtained under certain conditions has long been known (6, 10, 18, $5, a?‘). As has been shown (18, 30) if the absorption of the radiation is exponential the total self-absorption in an ordinary solid or liquid sample, which has finite thickness and therefore can be readily made and handled, can be calculated easily and the relation between the absolute disintegration rate and the observed count rate obtained. No specific account \vas taken of the effects of self-scattering. It was assumed (SO) that the effects of the selfscattered radiation would be encompassed in the geometrical constant, G, used in the formula. This point is examined here. METHOD
Let the absolute specific radioactivity be LT disintegrations per minute per mg. of sample Absorption coefficient of the radiation in the material of the sample be l / X a , sq. em. per mg. Absorption coefficient in the counter wall material be l / & sq. cm. per mg. Wall thickness of the counter be I (mg. per sq. em.), including the air between the surface of the sample and the counter wall Geometry factor, the ratio of 4~ to the average solid angle subtended by the inner surface of the cylindrical counter wall a t the sample surface be G Sample thickness (less than saturation) be x mg. per sq. cm. Atomic number of the sample on a weight average basis (25) be 2
Back-scattering coefficient for close geometry be 7 ilrea of sample be A (sq. cm.). For a, layer of sample at depth y (mg. per sq. cm.) below the top and of thickness dy, the count rate will be: A d~ =
‘(1 + v ) e ( - u / A e - z/w G
dy
(1)
or integrating over the sample thickncss
Seliger ($4) has shown that the backscattered radiation is of lower energy and softer in penetrating power relative than the original radiation, the softening depending on the angle of scattering as well as on the atomic number of the back-scattering material. Muller (25) has studied the variation of back-scattering with the atomic number of the material causing the back-scattering under a particular set of geometrical conditions with applications to analytical chemistry in mind. Glendenin and Solomon (9) have experimented on the matter in considerable detail. It is generally agreed-cf. Steinberg (%)-that the back-scattered radiation is softer than the original. For m3terials of 2 below 15, the factor by which the absorption coefficient of the back-scattered radiation measured under 2~ conditions is increased is about 2. For larger values of 2, it decreases essentially linearly to a factor of about 1.2 a t atomic number 90. Therefore, for ordinary materials in which 2 is less than 15, a new equation can be written for the relation between the count rate and the absolute specific activity. ~(cpm= ) A * LG TX (1 + e - l / x w I x 2 e-l/Aw(l
-e-rh)
(1”)
I n this equation, the first term in parenthesis takes account of the fact that the counter wall and the air betwen the sample and the counter wall i d 1 absorb the back-scattered radiation more than they do the original. It also takes account of the magnitude of the back-scattered radiation.
The corresponding formula for larger values of Zis obtained easily by replacing the coefficient 2 in this term by the appropriately smaller softening factor (26) of the back-scattered radiation and including in the exponential term in the parenthesis the value of this new coefficicnt, less 1. For samples which are thick with respect t o Xo, the last parenthetical factor (the saturation term) disappears and for material of Z less than 15, the softening of back-scattering can be combined Kith the geometry factor into a new factor, G', to obtain the formula of Suttle and Libby (SO).
and Solomon (Q),and Steinberg (26). It is clear from the theory that the scattering effect for materials of low Z should be particularly simple, so it is not surprising that for these materials the very simple Equation 2 is nearly as accurate as the more detailed Equations 1' and 1". An empirical equation for q which has been used in this research and which fits quite well both the experimental and theoretical relations between q and Z for close geometry conditions approaching 2a in the solid angle subtended by the counter is q =
Bothe (2j and Danziger (6) in theoretical studies of beta absorption deduced the main features of the whole phenomenology and their curve for 7 versus Z agrees well with the experiniental data of Seliger (24), Glendenin
Table I.
X(mg./sq. cm.)
Spectrum, M. E. V. 0,0189 0.060
0,0755 0.155 0.167 0,270 0,255 0.296 0.762 0.716
A1 AI A1 Mylar plastic" AP 31yiar4 A1 -410 A1 -41 Ala
cua
9na Pb"
-11
A1 cua
Snn Ala
2,275
0
1 C
55 E3/2
Reciprocal of Absorption Half Thickness, Coeff. Mg./Sq. Cm. ?rIg./Sq. &In. 0 050 0 0720 0.35 0 506 0 63 0 91 1 9 2.74 2 2a 3 16" 2.35 3 35 2.7a 3 9s 4 85 7 0 4 9a 7.14 6 09 8 8 22 32 320 46" 260 370 21a 305 1% 26s 67 96 84 122 60" 72a 50a 1304 189a
150
ATOMIC WEIGHT OF ABSORBER
Figure 1.
Half thickness vs. atomic weight of absorber
C1'
155 k.e.v. 167 k.e.v. 0 C13@716 k.e.v.
@
T12a4 770 k.e.v. 1.708 m.e.v.
X P3*
Equation 5 has been used in this research and Figure 1 shows the adequate degree to which it fits the experimental data for A, given in Table I. The curre of Figure 1 is given by
(4) (4')
1
1 IDO
50
=
Absorption Data
Absorbing Material He
1.36 1.708
(3)
,4735)
liiz(rng./sq. c m . ) = 38E3'*
Max. Energy
of Beta
0.65 (1 -
The absorption coefficient, 1/X, has been found (6, 20, 65) to depend on the maximum energy of the beta spectrum, E , in the following way:
where Zliz is the half-thickness in aluminum, and X similarly refers to the reciprocal of the absorption coefficient in aluminum. Lerch (16) has shown that A, the reciprocal of the absorption coefficient, depends on the average atomic weight, -11,of the absorbing medium. The relation is
200
The fact that the back-scattered radiation is softened relative to the original radiation means that the absorption curve cannot be entirely exponential for the thinnest layers of absorber because the absorption relation must be as given in Equation 1". Figure 2 shows actual absorption curves of various soft beta emitters taken in the screen wall counter (19, 21) which effectively has no wall and which allows the very softest radiation to be measured, I n this case there is a soft component which is quickly absorbed out, leaving the long normal exponential absorption curve, which is the only curve observed with Geiger counters of usual wall thickness. Therefore, the exponential nature of the absorption is not strictly observed for soft beta emitters. Exponential curves will be found for hard beta emitters, for in this case the back-scattered radiation is lost in the large percentage of hard radiation which is present from thick solids. It is clear, however, that for hard radiations and thin sources the absorption term should not be strictly exponential. It is recalled (SO) that the absorption curves for the naturally radioactive element rubidium showed its nonexponential character in a way which was difficult to understand under the cylindrical geometry conditions which normally give exponential absorptions. This effect is the result of the large Zfor rubidium compounds in increasing the back-scattering coefficient 7, and the fact that the radiation from rubidium is soft (Table I). For counters with walls as thick as 2 mg. per sq. cm., the absorption curves normally are exponential as shown in Figure 3, in agreement with Figure 2 for the corresponding values of absorber plus counter wall thickness. The value of G, the geometry factor in the Equation l", is the ratio of 4a to the average solid angle subtended VOL. 29, NO. 11, NOVEMBER 1957
1567
ICOSO,)
\ ICCCO,)
0.1
I
I 5 MYLAR ABSORBER THICKNESS
0
IO (mpicm?
Figure 2. Absorption curves for bare soft betas
Sources in screen-wall counter a t the sample surface by the inner wall of the counter. If the counter is long relative to the sample, the radius of the inner m-all is p, and the radius of the sample surface relative to the counter wire is r, then it can be shon-n that
Results obtained for the specific radioactivity were normally and uniformly about 30% lorn for all isotopes with A values below about 10. The effect was the purely geometrical one of the roughness of the surface of a crystalline powder. For soft beta rays for which the range in the solid was less than the thickness of the crystals, the only radiation escaping was from the surfaces of the crystals. I n the case of harder betas, hovever, the entire crystal emitted and the surface effect roughness disappeared. The question of the effect of roughness of the sample is an important one. Equation 6 is applicable only to the case of a smooth sample constituting the wall of a cylinder of inner radius, r, or the portion of the wall of such a cylinder. Consider a normal crystalline powder consisting of cubes 50 microns on edge. If the density were 2 grams per cc., then the cube edge would correspond to an z value in Equations 1’ and I”of 10 mg. per sq. cm. Therefore, in the case of hard beta emitters with X values larger than 10 mg. per sq. em., the surface of the solid powder of randomly oriented cubes would appear to be smooth and the powder would have a G value close to that for a smooth surface as given by Equation 6. For 1568
ANALYTICAL CHEMISTRY
soft beta emitters, on the other hand, only the surfaces of the crystals can emit and the surface must appear rough. The fact that roughness causes a reduction in the total outward flux of radiation relative to that from a true smooth cylindrical surface of the same material a t the same specific radioactivity may not be obvious, but detailed calculation for various likely powders such as hexagonally packed spheres s h o w this to be a general result. The magnitude of the effect agrees with the results on the various beta radiation standards obtained from the K’ational Bureau of Standards and the Oak Ridge National Laboratory (through the kindness of W, B. Mann and S. A. Reynolds, respectively). Using Equations l”, l’, and 2, G was calculated from the observed count rate, R, and the known specific radioactivity, u. The results are given in Table 11. The counter used (29) consisted of a thin metallized plastic cylindrical wall inflated by the counting gas gently flowing through a t preqsure slightly in excess of atmospheric. The counter had a wall thickness of 1.8%mg. per sq. cni., and the sample was placed around the counter on the inner surface of a plastic cylinder on n-hich a sheet of rubber 1.5 mm. thick m-ith a square or circular hole punched in it of accurately knonn area was placed. The distance hetneen the counter vall and the surface of the sample as 0.27 mg. per sq. em. of air. By the use of thp rubber sheet, the sample area was accurately fixed. The sample powder was placed in the recess of the rubber shpet and smoothed with a spatula. Under these conditions the counter wall radius, p . was 1.5 cni. and the inner sample surface radius n-as 1.8 cin. The larger G factor for soft radiation as shown in Table I1 was further established experimentally by making a finely divided sodium carbonate containing carbon-14. Thi4 n-as done by powdering Na2C*01 HzO and then dehydrating it a t ion- w e n temperatures so sintering did not occur. T-ndpr these conditions the yalue of G was the smaller one for hard betas rather than the larger one found for the same salt hefore dehydration, as shown in Table 11. I n all other cases for soft beta radiations, the G was higher and the average for all soft betas with A values less than 10 was 3.9 2’s. 2.72 for the hard betas. Table I1 gives the final G TTalues as determined according to the three equations: l’, l”,or 2. It is necessary to decide which value of G applies to a particular solid sample and beta radioactivity being measured. For betas with X values well above 10 mg. per sq. em., the roughness necessary for the large value of G will be easily visible and easily destroyed by
grinding in a mortar and pestle. Therefore, for this class of radioactivity successive measurements after grinding will bring the count rate to a constant high value independent of the degree of fineness of the solid which is characteristic of the smooth surface and the lower G value. For the softer betas, however, it is necessary to groTr- the crystals larger and larger by sintering or other device, and thus to reach a constant count rate independent of crystalline size. Any doubt can be settled by a cursory examination n-it11 a microscope, the relative magnitude of the crystal size and X being borne in mind. It appears that the soft beta geometry factors for various powders are essentially the same. This can be seen in Table 11, though there is some evidence of scatter which could be due to the size or shape of the particular crystals. The procedure for converting the beta standards, a-hich were solutions of very high specific activity, to solid form for measurement was to add a solution of an appropriate salt to a known volume of the standard, mix, evaporate or precipitate chemically, and grind the resultant solid. Sometimes i t m-aq difficult to obtain
solids which were chemically identical with the radioactive molecules and attempts to use substitutes with which the radioactive species was likely to form mixed crystals were made. I n the case of the harder beta radiations, the requirement t h a t mixed crystals be formed seemed t o be less necessary. For example, radioactive phosphate containing P32(XA1= 122 mg./sq. cm.) was measured on sodium sulfate powder. It seems unlikely that a n y substrate not chemically identical can be used in the case of soft betas.
Table II.
Isotope K@
AgCl BaC12 TlCOzH
Ti204 Av.. . for .. > 10
~~
RESULTS
APPLICATION TO NATURALLY RADIOACTIVE RUBIDIUM
Aldrich, Wetherill, Tilton, and Davis (1) compared the ratio of radiogenic strontium-87 to rubidium-87 found in several rubidium minerals differing in rubidium content in rocks of known age as determined by the uranium lead method. They measured the half life of natural radioactive rubidium-87 to be 50 f 2 billion years. Strassman and Kalling (28) found 63 billion by a similar method. Direct counting experiments have given: 59.5 from 457 Geiger-AIuller counter ( I S ) , 41 from Screen-n all Geiger-I1Iiiller counter ( l b ) , 64.1 from Screen-wall proportional counter (S), 63.7 from i n Geiger-Muller counter (RS), 59.3 from RbI(T1) scintillation spectrometer ( l 7 ) , 43.0 from 457 Geiger-Muller counter ( 7 ) , and 49 (14). Because of the importance of this determination to geochronology, the technique described above was applied. The results are given in Table
NaCl
~ 1 3 6
Ca45
I n Table 111, the results of the application of the method to a series of beta radiation standards furnished b y the National Bureau of Standards and the Oak Ridge National Laboratory are shown. From these data it appears that the method is good to about 5%. Equation 2, which is the simplest, does nearly as n-ell as Equations 1’ and I ” except for the large atomic numbers.
G, Fully Corrected for Back-Scattering XI Mg./- and Softening, Sq. Cm. Equation 1”
Substrate ?JazS04 (NH,)H,PO, KzSOa
P32
535 C’4
Experimental Geometry Factors, G
GI Partly Corrected] Equation 1’
G, S o BackScattering Correction, Equation 2
129 141 98 45 34 34.4 27.1
2.65 f 0 . 0 2 2.87 f 0.03 2.73 f 0.06 2.85 f 0.03 2.74 f 0.03 2 . 6 1 1 . 0.03 3.09 & 0 . 0 4
2.73 f 0.02 2 . 8 5 f 0.03 2.83 f 0.06 2.97 =t 0 . 0 3 2.86 f 0.03 2 . 7 1 zk 0.03 3 , l l f 0.05
2.33 f 0.02 2.02 f 0 . 0 3 2.28f 0.05 2 . 4 3 =t 0 . 0 3 1 . 9 9 f 0.03 1 . 8 i z t 0.03 2 . 0 1 f 0.05
7.5 7.9 7.0 7.4 3.5 3.0 2.9 3.1
2.72 4.12 5 0 . 0 4 4 , 1 8 5 0.04 4 , 2 8 5 0.04 3 . 7 3 f 0 OS 3.46f0.34.4 k O . 2 3.4 f O . 1 4.15f0.16
2.82 4 . 4 1 1 . 0.05 4.47 5 0.04 4.60 zt 0.04 4.011.0.03 3.78zk0.35.0 x k O . 3 3,76& 0.1 4.56f0.18
2.20 3 . 7 2 f 0.04 3.76 5 0.04 3 . 7 1 1 . 0.04 3 . 2 8 1.0.03 3.16f0.3 3.5 1.0.3 3.16xkO.l 3.9550.1
3.0
2 55 xk 0.08
2.78 1 . 0 . 0 9
2.44 xk 0.09
3.9
4 2
3.5
,.. ?
CaC03 CaS04.2H20 CaO CaSOl KazSdc BaSO, CaCOa NazCO3.H2O Very fine Na2C0a
Av. for
