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Internal-source method of measuring absolute pair-production cross sections
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Radiation Physics and Chemistry 75 (2006) 644–655 www.elsevier.com/locate/radphyschem
Internal-source method of measuring absolute pair-production cross sections F.T. Avignone III Department of Physics and Astronomy, University of South Carolina, Columbia, SC 29208, USA
Abstract The history, detailed description and results obtained using a unique but simple method of measuring absolute pairproduction cross sections is reviewed. These activities which occurred between 1973 and 1985 yielded results for photons on targets of Z ¼ 26; 29; 50; 82, and 92 for photons with energies of 1:1205; 1:3325; 1:8362, and 2:6145 MeV. They were measured for these energies and for 1:077 MeV for targets up to Z ¼ 82. The cross sections deviated significantly from theoretical calculations for energies near threshold. The discrepancy was later confirmed and an explanation suggested. r 2005 Elsevier Ltd. All rights reserved.
1. Introduction In a volume like this one, the following historical commentary is potentially repetitive; however, it is appropriate in the context of this article. The major breakthrough accomplished by P.A.M. Dirac’s relativistic formulation of quantum mechanics in 1928 (Dirac, 1928) was questioned for several years mainly because of the interpretation of the negative energy solutions. The now familiar particle–hole interpretation given by Dirac and Oppenheimer drew the following remark from Pauli in 1933 (Motz et al., 1969):‘‘– It follows that g-ray photons might spontaneously convert into an electron and anti-electron. Thus, we do not believe that the attempt can be taken seriously.’’ The attempt, meaning that to explain and interpret Dirac’s negative energy solutions. This quotation was taken from a longer one given in reference (Motz et al., 1969). Of course, a short time after this remark, the discovery of the positron by Anderson settled the issue.
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A relativistic theoretical treatment of pair production was later given by Bethe and Heitler in a famous article in 1934 (Bethe and Heitler, 1934). To emphasize how recent these early historical events were, I will admit that Pauli’s remark was made in the year that the author of this review article was born. In the early treatments of pair-production the cross section was given in Bethe and Heitler (1934) by s ¼ aZ2 r20 GðZ; E 0 Þ,
(1)
where a is the fine structure constant ð1=137Þ, r0 is the classical electron radius, e2 =mc2 , Z is the atomic number of the target atoms, and GðZ; E 0 Þ is a slowly varying function of Z and the incident photon energy E 0 . It is only approximately true that experiments carried out with the same photon energy, E 0 , will yield crosssections approximately varying by Z2 (Heitler, 1954). The modern attempts to describe the creation of eþ e pairs in the coulomb field of the nucleus of an atom are described in detail in the other articles in this volume. They involve complex computational methods of accounting for the atomic structure, and in particular the screening of the nuclear charge by the atomic
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ARTICLE IN PRESS F.T. Avignone III / Radiation Physics and Chemistry 75 (2006) 644–655
electrons. In this article, the aim is to review the history, methodology and results of the internal-source method of measuring absolute pair-production cross section measurements. This method was developed and applied by the author, his graduate students and colleagues, and described in 11 articles between 1973 and 1985 (Avignone III, 1973; Avignone III and Blankenship, 1974a,b,c; Girard et al., 1978, 1979; Avignone and Khalil, 1981; Avignone III and Khalil, 1982; Avignone III and Khalil, 1982; Avignone III et al., 1984, 1985).
2. The first experiments The invention of this technique was nothing more than an attempt to introduce the basic concept of pair production into an undergraduate nuclear physics laboratory. The author was stimulated by the excitement and enthusiasm of physics students when they were told that the creation of eþ e -pairs could be observed in an undergraduate laboratory. The first attempts were simply accomplished with a collimated source of 228 Th and a thin lead foil and two NaI(Tl) detectors. While successful as a demonstration, the extraction of a cross section was cumbersome and unconvincing. The production and hence counting rates were very low and much of the student excitement suffered. The initial concept arose in conversations with the author’s graduate research assistant S.M. Blankenship in which the notion of surrounding the g-ray source with the target material surfaced. The first experiments were written as pedagogical articles and published in the American Journal of Physics (Avignone III, 1973; Avignone III and Blankenship, 1974a,b,c). The geometry consisted of a small wire source of 228 Th wrapped in a sealed copper tube and placed in a cylinder of lead or other target material as shown in Fig. 1. The source flux, f, was measured with a 7:62 cm diameter 7:62 cm long sodium iodide, NaI(Tl), detector. The only g-ray in the decay cascade above the pair-
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production threshold is the 2:6145 MeV g-ray in 208 Pb, following the decay of the final b-emitting nuclide in the thorium chain, 208 Tl. The detection rate was then simply expressed as R ¼ foð1 emx Þ,
(2)
where x is the distance the g-ray traversed through the target material, is the probability that the annihilation photons will both escape the target, and o is the detection efficiency. The absorbtion coefficient, m, has the standard definition m ðsN 0 rÞ=A, where s is the pair-production cross section integrated over all lepton momenta, N 0 is Avagadro’s number, r is the density of the target material, and A is the atomic mass of the target. The ratio of the average flux through the target and the incident flux, f=f, was measured with and without the target. With this approximate method, the ratio was measured as 0:83. The absorption coefficient, m, to absorb g-rays from all processes was available in a number of published tables. The experiment was repeated for targets of Al, Fe, Sn, and Pb. Using this technique, the following relative cross sections, s, were measured with 2:6145 MeV g-rays: sðAlÞ¼ 0:03 0:03; sðFeÞ ¼ 0:27 0:03; sðSnÞ ¼ 1:08 0:11, and sðPbÞ ¼ 2:49 0:13. A plot demonstrating the approximate Z 2 dependence is shown in Fig. 2. An improved version of the student laboratory experiment was published (Avignone III and Blankenship, 1974) later using a (10–30) mCi source of 88 Y using the 1:836 MeV g-ray. It was at this point that we noticed that the 1972 calculations of Tseng and Pratt (Tseng and Pratt, 1972) resulted in cross sections significantly different in Z-dependence from Z2 . It was then that serious interest in this subject as a research topic began with a literature search beginning with the work of Dayton (1953) and Titus and Levy (1966).
3.0
TARGET AND SOURCE
σj (z)
2.0
Nal(TI)
Nal(TI)
1.0
LUCITE STAND
0 20
40
60
80
z Fig. 1. Experimental geometry showing the target cylinder with vertical source hole and stand.
Fig. 2. Sample data with s plotted vs. Z of the target. The solid curve is the parabolic function s ¼ const: Z2 .
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3. The internal source method of research quality In the early 1970s the emergence of more powerful computers stimulated interest in computational solutions of problems in many body physical systems in atomic and nuclear physics. These led to the early computations of Tseng and Pratt (1972) which showed a departure in the pair-production cross sections from early measurements. Improved computational powers, and the advent of common use of Monte-Carlo calculations inspired S.M. Blankenship and the author to consider elevating their method used in student experiments discussed above, to the level of a serious research tool. There were discrepancies between the earlier measurements and the then current computations (Tseng and Pratt, 1972). The usual experiments consisted of strong collimated sources and flat metal targets with two g-ray detectors (usually NaI(Tl)) facing each SPHERICAL TARGET
0.511 keV Nal(TI)
Ge(Li) 0.511 keV RADIO ACTIVE SOURCE BEAD
Fig. 3. Experimental geometry used in the first research level internal-source pair-production measurements.
other. This simple set-up has very difficult geometric corrections which in some cases were neglected. The first serious attempt in our laboratory is discussed in detail in Nuclear Instruments and Methods (Avignone III and Blankenship, 1974). The first improvement was in the symmetry of the experiment. A 20 mCi source of 88 Y was prepared in a plastic bead of approximately 2 mm in diameter. The targets used in these demonstrations were a tin sphere of 1:80 cm diameter and a lead sphere 1:55 cm in diameter. They were cut in hemispheres with hemispherical holes for the plastic bead. The hemispheres were placed in a plastic stand that clamped them together with the bead enclosed. The second improvement was the replacement of one of the NaI(Tl) detectors with a lithium-drifted Ge detector, Ge(Li). This geometry is shown in Fig. 3. The Ge(Li) detector was soon replaced with an intrinsic Ge detector with an efficiency of 30% of that of a 7:62 cm diameter 7:62 cm NaI(Tl) detector at 1:33 MeV. The electronic system consisted of a standard NIM bin and ORTEC modular coincidence system with a resolving time of 100 ns. The resolving time and coincidence efficiency were measured with a calibrated 2 mm bead of 22 Na inside of a 1:8 cm plastic sphere. The decay of 88 Y contains a weak positron branch which gives rise to annihilation radiations that interfere with the measurements of pair-production cross sections. This decay branch is weak enough so that accurate corrections could be made. Fig. 4 shows coincidence spectra taken with the 88 Y source in the Sn target, and also with no target. By moving the detectors to positions
Sn TARGET 180° DATA 90° DATA
511 keV
THOUSANDS OF COINCIDENCES
THOUSANDS OF COINCIDENCES
4
3
2
1
CHANNEL NUMBER
NO TARGET 3
2 511 keV 1
CHANNEL NUMBER
Fig. 4. Typical coincidence spectra recorded during the measurement of the pair production cross-section of 1:8362 MeV g-rays on a tin target, and without the tin target. The lower points of the left hand figure were taken at 90 .
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90% to one another, relative to a vertical axis through the target, it is clear from the figure that essentially all of the coincidence rate vanishes. The spherical symmetry of the target and source configuration results in the following simple relationship between the observed annihilation–radiation coincidence rate, C, and the absolute pair-production cross section, s:
volume of the sphere, fs is the flux of pair producing grays at the sphere’s surface, and of4 is the flux of the source g-rays averaged over the target sphere. Substituting the known relations for these qualities:
s ¼ CðNhIiÞ1 ,
where m is the total linear attenuation coefficient of the pair-producing g-rays in the sphere, re is the positron range in the sphere, and R is the radius of the sphere. The results of these calculations are f ¼ 0:02; 0:06; 0:18; 0:81 and 2:04%, positron escapes for initial g-ray energies of 1:12; 1:17; 1:33; 1:84 and 2:61 MeV, respectively, for the 1:8 cm diameter tin target used in these developmental experiments. In addition, these calculations were repeated for 1:84 MeV g-rays in the 1:8 cm lead sphere; the result was f ¼ 0:58%. The Monte-Carlo computational technique was also used to evaluate the probability of pair production following Compton scattering of the initial g-ray as it emerges from the source at the center. The scattered photon energy is reduced according to well known kinematics, but if still above 1:022 MeV, can produce a eþ e -pair. This correction was calculated for the same initial energies (1:12; 1:17; 1:33; 1:84; and 2:61 MeV) incident on targets of Al, Fe, Cu, Sn, and Pb spherical targets from the center. The probability for this process in all cases was found to be less than 102 times as probable as for pair production by the unscattered incident gamma ray. Finally, the contribution to the total coincidence rate, as well as losses due to small angle scattering of one or both annihilation photons out of the solid angle of the detectors, was computed using a third Monte-Carlo code, for all of the targets and incident g-ray energies given above. In this process, it is also possible for annihilation photons to be scattered into the solid angle subtended by the detectors; however these corrections are negligible. The most significant correction was found to be that due the scattering and absorption of the annihilation grays in the target material, and in no case investigated was that correction greater than 5%. Our computations revealed that target sizes of about 1:8 cm in diameter for Zp60, and 1.5 cm for Z460 are optimum for keeping the corrections small, while yielding high counting rates for a source bead of 88 Y. In addition, these target sizes do not produce counting geometries very different from those of point-sources, even when the detectors were within 6 cm.
(3)
where N is the number of target atoms per cubic volume, is the annihilation pair-detection efficiency and, hIi, is the average g-ray rate from the source which is obtained by integrating Aemr over the target volume. The quantity hIi is then simply given by hIi ¼ ðA=mÞfexpðmrs Þ expðmRÞg,
(4)
where A is the source activity in g-rays/s, m is the total gray absorption coefficient in cm1 and rs and R are the source-hole and target radii, respectively. The values of m used in these calculations were best fit values to the experimental data prepared by the National Bureau of Standards by White (1957), and Hubbell (1969). 3.1. Corrections to the data In addition to the first-order corrections above, it is necessary to evaluate the corrections necessary to account for the absorption and scattering of the annihilating radiations leaving the target as well as the scattering and subsequent pair-producing absorption of the incident g-ray. The correction for the scattering and absorption of the 511 keV annihilation photons was made in two steps. First, an accurate measurement of the absorption of coincident pairs of 511 keV photons from a 2 mm plastic bead of 22 Na in the center of the target sphere was made. Second, a Monte-Carlo calculation was made of the total probability of absorption and scattering of the 511 keV annihilation g-ray pairs which originate at points throughout the sphere. Each point was weighted by a factor expðmrÞ, where m is the total absorption coefficient, and r is the radial location of the emission. A direct comparison of the probability of escape of the pair thus calculated was made with that calculated with the source of 0:511 MeV photon pairs located at the center. In this way, only a very small calculated correction was made to the escape probability measured with the calibrated 22 Na source located at the center of the target. It is also necessary to correct for the finite probability that positrons produced near the outer surface of the target escape prior to annihilating. An approximate expression for the fraction of these escaping positions is f ’ V s fs =V hfi,
(5)
where V s is the volume of the sphere which lies within one-half of the positron range from the surface, V is the
f ’
mre , 2fexpðmRÞ 1g
(6)
3.2. The Monte-Carlo computations Although the Monte-Carlo codes written in our laboratory in 1973 and 1974 are now completely obsolete,
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source hole, thereby effectively reducing its distance d; out of the target. At this point d 1 þ d 2 is computed and after many trials, the average quantity hexp½mðd 1 þ d 2 Þi is computed and compared to the quantity exp½mðR rs Þ. The latter quantity was also carefully measured using the 22 Na source bead as described above. In this way, the correction is far less sensitive to the values of the attenuation coefficients used. A similar Monte-Carlo code was written to simulate the relative probability of pair production following Compton scattering of the initial g-ray. Fig. 6 depicts the geometry of the computation of the probability of this effect.
they were used much later for other applications and later compared to modern CERN codes. At the photon energies relevant here, the results were identical within statistical uncertainty. The calculation of the comparison of the escape probability of the annihilation photon pair created at the center, with that in which the pair is created at points distributed throughout the target can be understood by reference to Fig. 5. In the case depicted in the diagram on the left-hand side, the probability can be expressed simply as expð2mdÞ, where m is the total g-ray absorption coefficient and d is the distance the 511 keV photons must traverse to escape, or the effective radius. This probability was compared with that computed by the Monte-Carlo code based on the geometry shown in the diagram on the right-hand side of Fig. 5. In this case, an angle for one annihilation is randomly selected to emerge from the source ðg1 Þ. Next a random number, n1 , is selected on the integral of the probability exp½mðr rs Þ, where rs again is the source-hole radius. This is the survival probability of g1 to radius r. The radius at which the paircreation event occurs for this particular trial is obtained from Eq. (7). 1 c r ¼ rs þ ln , (7) m c n1
0.511
ec 1
where c ¼ f1 exp ½mðR rs Þg1 . This expression for c results from the normalization that requires the integral of exp½mðr rs Þ vary from 0 to 1, when r varies from rs to R. After solving for r, it is assumed that the creation and annihilation sites are the same. The code then selects random number, n2 , and chooses an angle y ¼ 12 ðn2 pÞ, for the direction of one annihilation photon relative to the direction of the incident photon. The code was also sensitive to the fact that one of the annihilation photons could possibly pass through the
0.511
Fig. 6. Geometrical depiction of the class of events in which a Compton scattered g-ray creates a eþ e -pair.
0.511 d1 0.511
d2 0.511
1
PAIR CREATION SITE
22 NO SOURCE 0.511
Fig. 5. Geometry of the Monte-Carlo calculation of the comparison of the absorption or scattering of annihilation photons from the center of the target to those created throughout the volume of the target.
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4. The first measurements to test the calculated cross sections Shortly after the publication of the method of absolute cross section measurements previously discussed in the last section, a significant effort was made to apply this technique to test several theoretical results. The measurements were made on the tin target described earlier. The measurements were made with the following incident photon energies: 1:1205; 1:3325; 1:8362 and 2:6145 MeV. One measurement with the source of 60 Co was made with the combined 1:1732 and 1:3325 MeV grays. In another, the cross section with incident 1:1732 MeV g-rays from 60 Co was measured in a 3detector configuration which tagged the 1:3325 MeV gray in a third detector. The radioactive sources were in the form of 2 mm diameter beads of: 46 Scð1:205 MeVÞ; 60 Coð1:1732; 1:3325 MeVÞ; 88 Yð1:8362 MeVÞ and 228 Thð2:6145 MeVÞ. The results are given in Table 1 below. The results are also plotted in Fig. 7 with the cross sections shown as ratios to the theory of Overbo et al. (1973), and Tseng and Pratt (1972).
Table 1 Experimental and theoretical cross sections E (MeV)
s (exp) (mb)
sðTPÞa (mb)
sðOMOÞb (mb)
sðBornÞc (mb)
1.120 (1.173 + 1.333) 1.333 1.836 2.615
5.470.5 4274 6573 419720 1090750
4.11 39.5 66.1 418 1063
3.54 38.2 64.6 402 1061
2.11 24.03 41.33 317.1 941.3
a
From the calculations of Tseng and Pratt (1972). From the calculations of Overbo et al. (1973). c From Henry and Kennett (1972).
649
These results gave us the first indication that our results were in excellent agreement with theory, except for a disagreement near threshold.
5. Extension of the measurements to Z ¼ 13; 26; 29; 50 and 82 It would be several years until we were convinced that there were no unforseen anomalies in the experimental technique which led to the departure from the theoretical predictions near threshold. The measurements were repeated for photon energies as 1:119 MeV from a source of 46 Sc; 2:6145 MeV from the decay of 228 Th, and the mixture of g-rays 1.1732 and 1:3325 MeV from the decay of 60 Co. The results were still in significant disagreement with both the screened and unscreened calculation as shown in Table 2. A more extensive comparison to the various theoretical treatments of the day is given in Table 3 relative to the Born approximation. It is clear that the internal source method experiments yielded cross sections in excellent agreement with the screened calculations of Tseng and Pratt at incident gray energies significantly above threshold. As the energy approaches threshold, the experimental cross sections were 50% above those of the unscreened calculations of Overbo et al. (1973), and 30% above those of Tseng and Pratt (1972). At this point it became highly desirable to make measurements closer to pair-production threshold and with a higher Z target to further explore this discrepancy.
6. Measurements of absolute pair-production cross sections on a depleted uranium (Z ¼ 92) target
b
(EXPT) / (THEORY)
1.5 1.4 1.3 1.2 1.1 1.0 0.9 1.00
2.00
3.00
PHOTON ENERGY (MeV) Fig. 7. Ratio of experimental to theoretical cross sections for pair production in a Pb target using the internal source method.
To extend the measurements to lower energies and higher atomic number, again a 2 mm diameter plastic source bead of 46 Sc was used. This nuclide decays by a low-energy b -branch to the 2:0099 MeV level of 46 Ti with a half-life of 84 days. This level reaches the ground state by a g–g cascade; the energies are 1:12051 and 0:88925 MeV. At the time of this experiment, the relevant g-ray energy was thought to be 1:119 MeV, a trivial difference from today’s value. The 228 Th bead source, as well as the 60 Co source was also used with their g-rays of 2:6145 MeV from the thorium, and 1:1732 and 1:3325 g-ray cascade from 60 Co. A 1 cm diameter sphere of depleted uranium was obtained in the form of two hemispheres, with the hemispherical holes for the bead sources as in the earlier experiments. The main issue at this time concerned the various methods of making the screening corrections. The work by Overbo yielded higher cross sections and the results of the measurements of spair ðg þ UÞ are compared to the
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Table 2 Experimental and theoretical pair-creation cross sections Eg
Z
1.119
13 29 50 82
0.23(0.03) 1.11(0.07) 4.66(0.21) 14.5(0.8)
1.1732 + 1.3325
13 26 29 50 82
2.05(0.11) 7.84(0.33) 10.5(0.5) 40.5(2.0) 161.0(6.0)
2.6145
13 26 29 50 82
65.2(2.9) 265.0(11.0) 328.0(14.0) 1046.0(36.0) 3357.0(107.0)
sexp (mb)
sPC O (mb)
sBorn (mb) 0.1427 0.7099 2.110 5.676
sSC O (mb)
sSC TP (mb)
0.164 1.11 4.32 13.3
0.164 1.11 4.11 11.2
0.1617 1.039 3.541 7.352
1.625 6.50 8.08 24.03 64.66 63.63 254.5 316.6 941.0 2532.0
1.69 7.73 9.95 38.2 137.9
1.72 7.82 10.10 39.9 157.2
64.22 263.8 331.0 1063.0 3309.0
64.21 263.7 330.8 1061.0 3306.0
1.72 7.81 10.0 39.5 151.2 64.2 265.0 332.0 1063.0 3340.0
SC SC sBorn : Born approximation, unscreened; sPC O : Overbo, point Coulimb; sO : Overbo, screened; sTP : Tseng and Pratt, screened.
Table 3 Experimental comparisons E inc g
s0 =sB
s1 =sB
s2 =sB
s3 =sB
s4 =sB
s5 =sB
s6 =sB
s7 =sB
s8 =sB
s9 =sB
1.1205
29 50 82
1.56 2.21 2.59
y 2.56 y
2.10 2.37 2.61
y y 1.59
y y y
y y y
y y 1.96
y y y
y y y
y y 2.36
1.1732 + 1.3325
13 26 29 50 82
1.26 1.21 1.27 1.69 2.49
y y y 1.75 y
y y y y y
y y 1.20 2.20
0.997 y 1.111 1.494 2.04
y y y y y
y y y y y
y y 1.18 1.38 2.02
y y y y 2.12
1.04 1.10 1.16 1.36 2.02
13 26 29 50 82
1.02 1.04 1.04 1.11 1.33
y y y 1.16 y
y y y y y
y y y y y
1.006 y 1.03 1.081 1.229
0.93
y y y y y
y y y y y
y y y y y
y y y y y
2.6145
y 1.04 y
sB , Born approximation; s0 , experimental results of USC group; s1 , (Avignone III and Khalil, 1982); s2 , (Jenkins, 1956); s3 , (Khalil and Avignone III, 1982); s4 , (Girard et al., 1978); s5 , (Girard et al., 1979); s6 , (Martin and Blichert-Toft, 1970); s7 , (Moltz et al., 1969); s8 , (Overbo, 1970); s9 , (Overbo, 1978); s9 , (Standil and Shkolnik, 1958).
calculation of Tseng and Pratt and to those of Overbo in Table 4.
7. Measurements with 1.1205 MeV photons on targets of Z ¼ 26; 29; 50; 82; and 92 The next series of measurements were made to more accurately measure the Z-dependence of the cross sections and to compare them to the calculations of
Table 4 Absolute pair production cross sections for photons incident on uranium E g ðMeVÞ
sTP ðmbÞ
sO ðmbÞ
sexp ðmbÞ
1.1205 (1.1732 + 1.3325) 2.6145
13.1 207.9 4395
17.0 217.3 4333
17.470.5 22279 42457137
sTP : calculated by Tseng and Pratt. s0 : calculated by Overbo (1970, 1978, 1979).
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Tseng and Pratt made subsequent to their work published in Tseng and Pratt (1972). These theoretical results appeared in 1980 and 1981 (Tseng and Pratt, 1980, 1981). To this point our results agreed well with the calculations of Tseng and Pratt (1972, 1980, 1981) for incident g-ray energies several hundred kilovolts above threshold, and for atomic numbers up to Z ¼ 50. Measurements by Coquette in 1979 and 1980 (Coquette, 1979) had shown that there may have been a deviation from theory for large atomic numbers and for g-ray energies near threshold. For this experiment, it was extremely important to have an accurate efficiency calibration by avoiding the use of an absolute activity measurement of the 46 Sc bead. Instead an accurate measurement was made of the relative intensities of the 1:1205 MeV g-ray from the 46 Sc source and the 1:275 MeV g-ray form our standard calibration bead of 22 Na. An expression for the absolute cross section was then easily derived in terms of the ratio N 22 =hN g i, the ratio of the number of counts under the 0:511 MeV line in the 22 Na calibration source to that of the 1:1205 MeV peak in the 46 Sc source in the same geometry, corrected for the energy dependence of efficiency of the detector. The average bracket indicates that a correction for the decay of the source between the calibration and the experiment has been made. The relative intensity ratio Ið0:511Þ=Ið1:275Þ in the decay of 22 Na is well known and is 1:812 0:002 (Standil and Shkolnik, 1958), which propagates an error of 0:14%. One of the Ge detectors, as well as the 4:62 cm 4:62 cm NaIðTlÞ spectrometer were relativeefficiency calibrated on an energy interval from 1:01 to 1:46 MeV using the well known intensities of the g-ray lines in the decay of 152;154 Eu, as well as the efficiency calculations of Yates for the NaI(Tl) detector (Yates, 1965). The data were least-squares fitted to polynomials and exponential functions over the 0:45 MeV range. The relative efficiencies, ð1:1205Þ and ð1:275Þ were then determined with an accuracy of less than 1% for the Ge detector, and the relative intensity of the 1:1205 MeV gray of the 46 Sc bead was determined relative to that of the 1:275 MeV g-ray from the 22 Na calibration source to an accuracy of 1%. They both have much smaller systematic errors than those involved in absolute activity measurements. The intensity ratio Ið1:1205Þ=Ið1:275Þ measured with the NaIðTlÞ spectrometer was within 1% of that measured with the Ge detector. The intensity of the annihilation photons from the 22 Na calibration source was then accurately known relative to that of the 1:1205 MeV g-ray from the 46 Sc bead source. With this technique, it was possible to write an explicit expression for the measured pair-production cross section spp as follows: spp ¼
C pp N 22 W m1 f1 exp½m1 ðrs r0 Þg1 , 2c22 hN g irA0 by
(8)
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Table 5 Absolute pair-production cross sections in millibarns at E g ¼ 1:1205 MeV Z
sðexpÞa (mb)
sðexpÞb (mb)
sðtheorÞc (mb)
sðexpÞ=sðtheorÞ
26 29 50 82 92
0.8770.02 1.1070.04 4.1670.15 12.770.5 15.670.5
1.1170.07 4.6670.21 14.570.80 17.470.5
1.11 4.11 11.2 13.1
0.99 1.01 1.13 1.19
a
Results using the methods Presented in this section. Results presented in Bethe and Heitler (1934) for Z ¼ 29; 50 and 82, and Davidson (1968) for Z ¼ 92. c Cross sections calculated by Tseng and Pratt (Sakharov, 1991; Schmidt and Huber, 1955). b
where C pp is the annihilation radiation coincidence rate with 46 Sc source in the target and c22 is that with the 22 Na calibration source with no target. The ratio N 22 =hN g i was determined in the relative intensity measurements discussed above. In Eq. (8), W is the atomic weight, r is the density, and m1 is the absorption coefficient of the target. The product by is the probability that both annihilation photons emerge from the target without interacting. This quantity was calculated by Monte-Carlo techniques as discussed earlier. The quantity hN g i was determined as follows: hN g i ¼
N 0 elt1 lt1 ðe elt2 Þ, lðt2 t1 Þ
(9)
where t1 and t2 are the beginning and ending times of the measurement, and N 0 was the quantity N g ðt ¼ 0Þ at the time of the measurement of N g =N 22 . The five resulting cross sections are listed in Table 5. There consistently existed a departure from the theoretical cross sections, and at this point it was decided to improve the experimental apparatus by adding a second intrinsic Ge detector and remeasure the cross sections. 8. Improved measurements in targets of Z ¼ 26; 29; 50; 82; and 92, with two Ge detectors At this point in the history of this series of measurements there was no clear understanding of the differences in our own measurements, though small, and the deviation from sophisticated numerical calculations of the cross sections near threshold (Tseng and Pratt, 1972, 1980, 1981; Overbo, 1970, 1978, 1979). The experiments described in Dayton (1953), and described in Section 4, concentrated on improving the techniques for calibrating the efficiencies for detecting the
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annihilation quanta, and for relating these to the absolute cross sections extracted from the data. There was one troublesome issue remaining. It was difficult to evaluate the systematic error associated with broad energy resolution of the NaI(Tl) detector. A second intrinsic Ge detector was obtained, with an absolute efficiency of 30% of that of a 7:62 cm 7:62 cm cylindrical NaI(Tl) detector at 1:3325 MeV. Since no deviation from the theoretical cross sections was observed for incident g-ray energies a few hundred keV above threshold, trial measurements were made using the source beads of 60 Co and 228 Th. All of the procedures used earlier for making the corrections to the data for absorption of annihilation radiations in the target, for pair creation following Compton scattering, and for the relative and absolute detection efficiencies were again used. These included the measurements described in Section 4, and MonteCarlo computations described in Section 3. All the results of these measurements were in excellent agreement with those of both Tseng and Pratt and of Overbo as shown in Table 6. These results were published in 1982 and the theoretical values were the latest computed with new screening corrections. They were transmitted to the authors privately by both authors and published later (Tseng and Pratt, 1981). At this point it was quite clear that for incident photon energies of 1:1732 MeV and above the screened nucleus calculations were experimentally confirmed. The next task was to extend these measurements to energies closer to threshold using the same experimental techniques that produced results in excellent agreement with theory at higher g-ray energies.
Table 6 Experimental and theoretical pair-production cross sections. so : Overbo screened calculation; sTP : Tseng and Pratt screened calculation E g ðMeVÞ
Z
1.1732 1.3325 +
26 29 50 82 92
2.6145
26 29 50 82 92
sexp ðmbÞ 7.78(0.30) 10.0(0.35) 39.0(1.7) 152.6(5) 209.4(8) 263(7) 327(10) 1051(22) 3331(75) 4345(96)
so ðmbÞa
sTP ðmbÞ
7.82 10.10 39.9 157.2 217.3
7.81 10.0 39.5 151.2 207.9
263.7 330.8 1061 3306 4332
265 332 1063 3340 4395
a All of the theoretical cross sections were calculated by the original authors and transmitted via private communication prior to publication.
9. Near-threshold measurements of absolute pairproduction cross sections of photons on lead In the work done previously, the experimental techniques of the internal source method were by 1984 well developed, while the corrections and systematic errors were clearly understood. A radio active source bead of 86 Rb was prepared by colleagues at the Los Alamos National Laboratory (Avignone III et al., 1984, 1985). Our measurements to this point were limited to photon energies down to 1:1205 MeV. However, 86 Rb, the only practical source with a single g-ray in the most interesting energy range, was not previously available to us with its 1:077 MeV g-ray, only 55 keV above threshold. This isotope was made by thermal-neutron capture on stable 85 Rb. But even a small component of highenergy neutrons can produce enough contamination with the positron emitter 84 Rb, via the reaction 85 Rbðn; 2nÞ 84 Rb, to preclude its use in the internalsource method. Any contamination with a positron emitter produces annihilation radiation, which if it comprises more than 20% of the total coincidence event rate, seriously degrades the accuracy of the measurement. The irradiation in a highly pure thermal neutron column, the separation chemistry and manufacturing of a 2 mm diameter plastic bead were accomplished at the Los Alamos National Laboratory, and represented the new breakthrough necessary to push our measurements closer to threshold. Attempts to use the uranium target failed due to the radioactivity remaining in the depleted uranium. Accordingly, the measurements were made in the lead target and are presented in Table 7 below, along with earlier measurements at higher energies. These results are also plotted in Fig. 7 where there is a clear departure from the screened calculations. These theoretical results were calculated especially for us, and sent to us by Professor Pratt. This was essentially a doubleblind comparison since the calculations were not sent until after the final analysis of the data.
Table 7 Experimental cross sections measured with the internal-source technique in Pb compared to the theoretical cross sections of Tseng and Pratt ðsTP Þ E g (MeV)
sexp (mb)
1.077 1.1205 1.1732 + 1.3325 2.6145
1.9270.11 12.7270.5
a
152.675.0 3331775
Calculated especially for our g-ray energies.
sTP (mb)a 1.34 11.2 151.2 3340
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At this point the discrepancy between theory and experiment appeared to be well established; however, it thus far defied explanation. Similar deviations from theory were deduced from the data published by Coquette et al., 1979). In Fig. 8, the data from Schmidt and Huber (1974) was divided by the calculations of Tseng and Pratt mentioned above, and plotted against photon energy. Other experimental papers which reported cross sections with near threshold discrepancies with theory were published during the period of our activities (1972–1985) and earlier by Yamazaki and Hollander (1965), Coquette (1979, 1977), and En‘yo et al. (1980). They all showed some excess over the various theoretical calculations. At the completion of our final experiments published in 1985 (Avignone III et al., 1985), in which the measurements were extended down to E g ¼ 1:077 MeV, we could not conceive of any further improvements that could be made that would make a significant difference. The USC group began to focus more on developing improved techniques to search for neutrinoless double beta decay and the issue remained unresolved until 1991 when Professor Eric Adelberger contacted the author
and discussed his interest in testing these results with an independent technique (De Braeckeleer et al., 1992). De Braeckeleer et al. (1992) investigated the pair production of g-ray on a germanium detector of a pair spectrometer. Two Ge detectors were used separately, a 40 cm3 coaxial detector, and a 10 cm3 planar detector. The Ge detector was surrounded by two 12:7 cm 15:2 cm NaIðTlÞ detectors to detect the annihilation photons. The double escape peaks were recorded in the Ge detector spectra. Monoenergetic g-rays were obtained from sources of 152 Eu ðE g ¼ 1:408; 1:299; 1:213; 1:112; and 1:086 MeVÞ; 60 Co ðE g ¼ 1:3325 and 1:1732 MeVÞ; 88 Y ðE g ¼ 1:836 MeVÞ; and 207 Bi ðE g ¼ 1:76971 MeVÞ. The results agreed with our results as well as those of En‘yo et al. (1980), Yamazaki and Hollander (1965) and Coquette (1977). They suggested an explanation of these discrepancies based on the fact that the kinetic energies of the eþ e pair were of order 30 keV, where the final state Coulomb interaction may be important. They estimated this effect using an expression derived by Sakharov (1991) for pairs produced with low relative velocity to each other. This yields the following multiplicative correction factor: F¼
1.7 z=32 1.6 1.5
(EXPT) / (THEORY)
1.4 1.3 1.2 1.1
653
2pa=v , 1 expð2pa=vÞ
(10)
where v is the relative velocity of the eþ e -pair, which was integrated over the electron-positron kinematics at each photon energy assuming the Bethe–Heitler angular and energy distribution (Davidson, 1968). This explanation gave reasonable agreement between experimental results and theoretical predictions corrected for this final state. De Braeckeleer et al. (1992) point out that this represents an approximate explanation; however, the perturbation of the lepton wave functions was neglected and welcomed a more realistic calculation. Nevertheless, the long standing discrepancy between experimental and theoretical pair production cross section was perhaps finally qualitatively understood and the present author had no more to contribute to the subject.
1.0
Acknowledgements
0.9 0.8 0.7 1.05 1.06
1.07 1.08
1.09 1.10
1.11 1.12
PHOTON ENERGY (MeV) Fig. 8. Ratio of experimental to theoretical cross sections for pair-production in a Ge target. The experimental data points were graphically measured from our enlargement of Fig. 1 of Schmidt and Huber (1974) and divided by the theoretical values of the screened cross sections of Tseng and Pratt (1981).
The author would like to recognize the experimental efforts of his excellent graduate students who contributed to the University of South Carolina experimental program: Samuel Blankenship, Craig Barker, Tom Girard, Terry Huntsberger, Ali Khalil and Harry Miley. He would also like to thank the University of Washington group for their efforts to explain the discrepancies between theory and experiment and for their private communications. The experimental effort at the University of South Carolina was supported by the National Science Foundation under Grants:
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PHY-7824885, PHY-8109038, and PHY-8306098. The Los Alamos effort was supported by the Office of Health and Environmental Research of the U.S. Department of Energy.
References Avignone III, F.T., 1973. Experiment to demonstrate the creation of positron–electron pairs by gamma rays. Am. J. Phys. 41, 71–74. Avignone III, F.T., Blankenship, S.M., 1974a. Recent improvements in the experiment to demonstrate the creation of positron–electron pairs. Am. J. Phys. 42, 698–699. Avignone III, F.T., Blankenship, S.M., 1974b. Internal source method for measuring absolute pair production crosssections. Nucl. Instrum. Methods 116, 515–519. Avignone III, F.T., Blankenship, S.M., 1974c. Low-energy pair production cross-section measurements for Z ¼ 50. Phys. Rev. A10, 793–796. Avignone III, F.T., Khalil, A.E., 1981. Cross section for the production of electron–positron pairs by 1.064- MeV photons on germanium. Phys. Rev. A24, 2920–2924. Avignone III, F.T., Khalil, A.E., 1982. Improved low-energy absolute pair production cross section measurements in targets of Z ¼ 26; 29; 50, and 92. Phys. Rev. A26, 825–830. Avignone III, F.T., Barker, W.C., Miley, H.S., O’Brian, H.A., Steinkruger, F.J., Wanek, P.M., 1984. Absolute pairproduction cross-section measurements in Z ¼ 82 with 1.077 MeV photons. Phys. Lett. 104A, 143–145. Avignone III, F.J., Barker, W.C., Miley, H.S., O’Brian Jr., H.A., Steinkruger, F.J., Wanek, P.M., 1985. Near threshold behaviour of pair production cross-sections in a lead target. Phys. Rev. A32, 2622–2627. Bethe, H.A., Heitler, W., 1934. On the stopping of fast particles and the creation of positive electrons. Proc. Roy. Soc. (London). A146, 83–112. Coquette, A., 1977. Methode de Mesure Absolue de la Section Efficace Totale de Creation de Paires Pres du Seuil. Nucl. Instrum. Methods 144, 571–577; Coquette, A., 1978. Methode de Mesure Absolue de la Section Efficace Totale de Creation de Paires Pres du Seuil. Journal de Physique 39, 1055–1058. Coquette, A., 1979. Methode de Mesure Absolue de Section Efficace Totale de Creation de Paires Pres du Seul, Au Moyen d’Une Source d’Energie Variable. Nucl. Instrum. Methods, 164, 337–342; Coquette, A., 1980. Methode de Mesure Absolue de Section Efficace Totale de Creation de Paires, Pres du Seul Au Moyen d’Une Source d’Energie Variable. Le Journal de Physique (Paris) 41, 97–99. Davidson, C.M., 1968. Gama-ray attenuation coefficients, in: Siegbahn, K. (Ed.), Alpha-, Beta-, and Gamma-Ray Spectroscopy, vol. 1, North Holland, Amsterdam, pp. 827–843. Dayton, I.E., 1953. Z-dependence of pair production cross sections at 1.33 and 2.62 MeV. Phys. Rev. 89, 544–551. De Braeckeleer, L., Adelberger, E.G., Garcia, A., 1992. Reexamination of an anomaly in near-threshold pair production. Phys. Rev. A46, R5324–R5326.
Dirac, P.A.M., 1928. The quantum theory of the electron. Proc. Roy. Soc. (London) A117, 610–624. En‘yo, H., Numao, T., Yamazaki, T., 1980. Cross Section of Pair Production Near Threshold. Phys. Rev. A21, 1439–1442. Girard, T.A., Avignone III, F.T., Blankenship, S.M., 1978. Low-energy pair-production measurements in targets of Z ¼ 13; 26; 29; 50, and 82. Phys. Rev. A17, 218–222. Girard, T.A., Avignone III, F.T., Hunstberger, T.L., 1979. Absolute pair-production cross-section measurements in Z ¼ 92 at energies near threshold. Phys. Lett. 71A, 33–34. Heitler, W., 1954. The Quantum Theory of Radiation, third ed. Oxford University Press, Oxford, p. 265. Henry, L.C., Kennett, T.J., 1972. Total pair production cross sections from 1,120 to 2.754 MeV. Can. J. Phys. 50, 2756–2759. Hubbell, J.H., 1969. Photon cross sections, attenuation coefficients, and energy absorption coefficients from 10 KeV to 100 GeV. Nat. Bur. Stand. Ref. Data Serv. 29. Jenkins, T.L., 1956. A Measurement of electron pair production cross sections in lead. Bull. Am. Phys. Soc. 1, 167. Khalil, A.E., Avignone III, F.T., 1982. Absolute pair-production cross-section measurements in targets of Z ¼ 26, 29, 50, 82, and 92 with 1.119-MeV photons. Phys. Rev. A26, 770–774. Martin, M.J., Blichert-Toft, P.H., 1970. Radioactive atoms auger-electrons, -,-,-, and X-ray data. Nucl. Data Tables 8, 1–198. Moltz, J.W., Olsen, H.A., Koch, H.W., 1969. Pair productions by photons. Rev. Mod. Phys. 41, 581–639. Overbo, I., 1970. Screening effects in pair production, Ark. Fys. Semin. I Trondheim. Overbo, I., 1978. Pair production by photons. Screening corrections for intermediate and high energies. Nuovo Cimento 47A, 43–58. Overbo, I., 1979. Screening corrections in electron pair production by intermediate and low energy photons. Phys. Scr. 19, 299–306. Overbo, I., Mork, K.J., Olsen, H.A., 1973. Pair Production by photons: exact calculations for an unscreened atomic field. Phys. Rev. A8, 668–685. Sakharov, A.D., 1991. Interaction of electron and positron in pair production. Sov. Phys. Usp. 34, 375–377. Schmidt, P., Huber, P., 1955. Absolute Paarbildungsquerschnitte von Blei fur Gammastrahlen von 60Co und 24Na und Paarerzeungung der RaC-GammaStrahlung in Blei. Helv. Phys. Acta. 28, 369–388. Schmidt, P., Huber, P., 1974. Z-Abahngigkeit des Paarbildungsquerschnittes fur 60Co-Gammastrahlen und Absorptionskoeffizienten fur Vernichtungsstrahlung. Helv. Phys. Acta. 27, 152–155. Standil, S., Shkolnik, V., 1958. Measurements of absolute pair production cross sections at low energies. Cana. J. Phys. 36, 1154–1160. Titus, F., Levy, A.J., 1966. Pair production in Al, Sn, Ta, and Au by 2.62 MeV Gamma Rays. Nucl. Phys. 80, 588–592.
ARTICLE IN PRESS F.T. Avignone III / Radiation Physics and Chemistry 75 (2006) 644–655 Tseng, H.K., Pratt, R.H., 1981. Comment on comparrison of theory and experiment for pair production cross sections near threshold. Phys. Rev. A24, 1127–1128. Tseng, H.K., Pratt, R.H., 1972. Screening effects in atomic pair production below 5 MeV. Phys. Rev. A6, 2049–2056. Tseng, H.K., Pratt, R.H., 1980. Screening effects in pair production near threshold. Phys. Rev. A21, 454–457.
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Yamazaki, T., Hollander, J., 1965. Measurement of pairproduction cross section near threshold. Phys. Rev. 140, 630–637. Yates, M.J.L., Finite solid angle corrections, in: Seigbahn, K. (Ed.), Alpha-,Beta-, and Gamma-Ray Spectroscopy, Interscience Publishing, New York, p. 1691–1703.
Radiation Physics and Chemistry 75 (2006) 644–655 www.elsevier.com/locate/radphyschem
Internal-source method of measuring absolute pair-production cross sections F.T. Avignone III Department of Physics and Astronomy, University of South Carolina, Columbia, SC 29208, USA
Abstract The history, detailed description and results obtained using a unique but simple method of measuring absolute pairproduction cross sections is reviewed. These activities which occurred between 1973 and 1985 yielded results for photons on targets of Z ¼ 26; 29; 50; 82, and 92 for photons with energies of 1:1205; 1:3325; 1:8362, and 2:6145 MeV. They were measured for these energies and for 1:077 MeV for targets up to Z ¼ 82. The cross sections deviated significantly from theoretical calculations for energies near threshold. The discrepancy was later confirmed and an explanation suggested. r 2005 Elsevier Ltd. All rights reserved.
1. Introduction In a volume like this one, the following historical commentary is potentially repetitive; however, it is appropriate in the context of this article. The major breakthrough accomplished by P.A.M. Dirac’s relativistic formulation of quantum mechanics in 1928 (Dirac, 1928) was questioned for several years mainly because of the interpretation of the negative energy solutions. The now familiar particle–hole interpretation given by Dirac and Oppenheimer drew the following remark from Pauli in 1933 (Motz et al., 1969):‘‘– It follows that g-ray photons might spontaneously convert into an electron and anti-electron. Thus, we do not believe that the attempt can be taken seriously.’’ The attempt, meaning that to explain and interpret Dirac’s negative energy solutions. This quotation was taken from a longer one given in reference (Motz et al., 1969). Of course, a short time after this remark, the discovery of the positron by Anderson settled the issue.
E-mail address: [email protected].
A relativistic theoretical treatment of pair production was later given by Bethe and Heitler in a famous article in 1934 (Bethe and Heitler, 1934). To emphasize how recent these early historical events were, I will admit that Pauli’s remark was made in the year that the author of this review article was born. In the early treatments of pair-production the cross section was given in Bethe and Heitler (1934) by s ¼ aZ2 r20 GðZ; E 0 Þ,
(1)
where a is the fine structure constant ð1=137Þ, r0 is the classical electron radius, e2 =mc2 , Z is the atomic number of the target atoms, and GðZ; E 0 Þ is a slowly varying function of Z and the incident photon energy E 0 . It is only approximately true that experiments carried out with the same photon energy, E 0 , will yield crosssections approximately varying by Z2 (Heitler, 1954). The modern attempts to describe the creation of eþ e pairs in the coulomb field of the nucleus of an atom are described in detail in the other articles in this volume. They involve complex computational methods of accounting for the atomic structure, and in particular the screening of the nuclear charge by the atomic
0969-806X/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.radphyschem.2005.09.002
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electrons. In this article, the aim is to review the history, methodology and results of the internal-source method of measuring absolute pair-production cross section measurements. This method was developed and applied by the author, his graduate students and colleagues, and described in 11 articles between 1973 and 1985 (Avignone III, 1973; Avignone III and Blankenship, 1974a,b,c; Girard et al., 1978, 1979; Avignone and Khalil, 1981; Avignone III and Khalil, 1982; Avignone III and Khalil, 1982; Avignone III et al., 1984, 1985).
2. The first experiments The invention of this technique was nothing more than an attempt to introduce the basic concept of pair production into an undergraduate nuclear physics laboratory. The author was stimulated by the excitement and enthusiasm of physics students when they were told that the creation of eþ e -pairs could be observed in an undergraduate laboratory. The first attempts were simply accomplished with a collimated source of 228 Th and a thin lead foil and two NaI(Tl) detectors. While successful as a demonstration, the extraction of a cross section was cumbersome and unconvincing. The production and hence counting rates were very low and much of the student excitement suffered. The initial concept arose in conversations with the author’s graduate research assistant S.M. Blankenship in which the notion of surrounding the g-ray source with the target material surfaced. The first experiments were written as pedagogical articles and published in the American Journal of Physics (Avignone III, 1973; Avignone III and Blankenship, 1974a,b,c). The geometry consisted of a small wire source of 228 Th wrapped in a sealed copper tube and placed in a cylinder of lead or other target material as shown in Fig. 1. The source flux, f, was measured with a 7:62 cm diameter 7:62 cm long sodium iodide, NaI(Tl), detector. The only g-ray in the decay cascade above the pair-
645
production threshold is the 2:6145 MeV g-ray in 208 Pb, following the decay of the final b-emitting nuclide in the thorium chain, 208 Tl. The detection rate was then simply expressed as R ¼ foð1 emx Þ,
(2)
where x is the distance the g-ray traversed through the target material, is the probability that the annihilation photons will both escape the target, and o is the detection efficiency. The absorbtion coefficient, m, has the standard definition m ðsN 0 rÞ=A, where s is the pair-production cross section integrated over all lepton momenta, N 0 is Avagadro’s number, r is the density of the target material, and A is the atomic mass of the target. The ratio of the average flux through the target and the incident flux, f=f, was measured with and without the target. With this approximate method, the ratio was measured as 0:83. The absorption coefficient, m, to absorb g-rays from all processes was available in a number of published tables. The experiment was repeated for targets of Al, Fe, Sn, and Pb. Using this technique, the following relative cross sections, s, were measured with 2:6145 MeV g-rays: sðAlÞ¼ 0:03 0:03; sðFeÞ ¼ 0:27 0:03; sðSnÞ ¼ 1:08 0:11, and sðPbÞ ¼ 2:49 0:13. A plot demonstrating the approximate Z 2 dependence is shown in Fig. 2. An improved version of the student laboratory experiment was published (Avignone III and Blankenship, 1974) later using a (10–30) mCi source of 88 Y using the 1:836 MeV g-ray. It was at this point that we noticed that the 1972 calculations of Tseng and Pratt (Tseng and Pratt, 1972) resulted in cross sections significantly different in Z-dependence from Z2 . It was then that serious interest in this subject as a research topic began with a literature search beginning with the work of Dayton (1953) and Titus and Levy (1966).
3.0
TARGET AND SOURCE
σj (z)
2.0
Nal(TI)
Nal(TI)
1.0
LUCITE STAND
0 20
40
60
80
z Fig. 1. Experimental geometry showing the target cylinder with vertical source hole and stand.
Fig. 2. Sample data with s plotted vs. Z of the target. The solid curve is the parabolic function s ¼ const: Z2 .
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3. The internal source method of research quality In the early 1970s the emergence of more powerful computers stimulated interest in computational solutions of problems in many body physical systems in atomic and nuclear physics. These led to the early computations of Tseng and Pratt (1972) which showed a departure in the pair-production cross sections from early measurements. Improved computational powers, and the advent of common use of Monte-Carlo calculations inspired S.M. Blankenship and the author to consider elevating their method used in student experiments discussed above, to the level of a serious research tool. There were discrepancies between the earlier measurements and the then current computations (Tseng and Pratt, 1972). The usual experiments consisted of strong collimated sources and flat metal targets with two g-ray detectors (usually NaI(Tl)) facing each SPHERICAL TARGET
0.511 keV Nal(TI)
Ge(Li) 0.511 keV RADIO ACTIVE SOURCE BEAD
Fig. 3. Experimental geometry used in the first research level internal-source pair-production measurements.
other. This simple set-up has very difficult geometric corrections which in some cases were neglected. The first serious attempt in our laboratory is discussed in detail in Nuclear Instruments and Methods (Avignone III and Blankenship, 1974). The first improvement was in the symmetry of the experiment. A 20 mCi source of 88 Y was prepared in a plastic bead of approximately 2 mm in diameter. The targets used in these demonstrations were a tin sphere of 1:80 cm diameter and a lead sphere 1:55 cm in diameter. They were cut in hemispheres with hemispherical holes for the plastic bead. The hemispheres were placed in a plastic stand that clamped them together with the bead enclosed. The second improvement was the replacement of one of the NaI(Tl) detectors with a lithium-drifted Ge detector, Ge(Li). This geometry is shown in Fig. 3. The Ge(Li) detector was soon replaced with an intrinsic Ge detector with an efficiency of 30% of that of a 7:62 cm diameter 7:62 cm NaI(Tl) detector at 1:33 MeV. The electronic system consisted of a standard NIM bin and ORTEC modular coincidence system with a resolving time of 100 ns. The resolving time and coincidence efficiency were measured with a calibrated 2 mm bead of 22 Na inside of a 1:8 cm plastic sphere. The decay of 88 Y contains a weak positron branch which gives rise to annihilation radiations that interfere with the measurements of pair-production cross sections. This decay branch is weak enough so that accurate corrections could be made. Fig. 4 shows coincidence spectra taken with the 88 Y source in the Sn target, and also with no target. By moving the detectors to positions
Sn TARGET 180° DATA 90° DATA
511 keV
THOUSANDS OF COINCIDENCES
THOUSANDS OF COINCIDENCES
4
3
2
1
CHANNEL NUMBER
NO TARGET 3
2 511 keV 1
CHANNEL NUMBER
Fig. 4. Typical coincidence spectra recorded during the measurement of the pair production cross-section of 1:8362 MeV g-rays on a tin target, and without the tin target. The lower points of the left hand figure were taken at 90 .
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90% to one another, relative to a vertical axis through the target, it is clear from the figure that essentially all of the coincidence rate vanishes. The spherical symmetry of the target and source configuration results in the following simple relationship between the observed annihilation–radiation coincidence rate, C, and the absolute pair-production cross section, s:
volume of the sphere, fs is the flux of pair producing grays at the sphere’s surface, and of4 is the flux of the source g-rays averaged over the target sphere. Substituting the known relations for these qualities:
s ¼ CðNhIiÞ1 ,
where m is the total linear attenuation coefficient of the pair-producing g-rays in the sphere, re is the positron range in the sphere, and R is the radius of the sphere. The results of these calculations are f ¼ 0:02; 0:06; 0:18; 0:81 and 2:04%, positron escapes for initial g-ray energies of 1:12; 1:17; 1:33; 1:84 and 2:61 MeV, respectively, for the 1:8 cm diameter tin target used in these developmental experiments. In addition, these calculations were repeated for 1:84 MeV g-rays in the 1:8 cm lead sphere; the result was f ¼ 0:58%. The Monte-Carlo computational technique was also used to evaluate the probability of pair production following Compton scattering of the initial g-ray as it emerges from the source at the center. The scattered photon energy is reduced according to well known kinematics, but if still above 1:022 MeV, can produce a eþ e -pair. This correction was calculated for the same initial energies (1:12; 1:17; 1:33; 1:84; and 2:61 MeV) incident on targets of Al, Fe, Cu, Sn, and Pb spherical targets from the center. The probability for this process in all cases was found to be less than 102 times as probable as for pair production by the unscattered incident gamma ray. Finally, the contribution to the total coincidence rate, as well as losses due to small angle scattering of one or both annihilation photons out of the solid angle of the detectors, was computed using a third Monte-Carlo code, for all of the targets and incident g-ray energies given above. In this process, it is also possible for annihilation photons to be scattered into the solid angle subtended by the detectors; however these corrections are negligible. The most significant correction was found to be that due the scattering and absorption of the annihilation grays in the target material, and in no case investigated was that correction greater than 5%. Our computations revealed that target sizes of about 1:8 cm in diameter for Zp60, and 1.5 cm for Z460 are optimum for keeping the corrections small, while yielding high counting rates for a source bead of 88 Y. In addition, these target sizes do not produce counting geometries very different from those of point-sources, even when the detectors were within 6 cm.
(3)
where N is the number of target atoms per cubic volume, is the annihilation pair-detection efficiency and, hIi, is the average g-ray rate from the source which is obtained by integrating Aemr over the target volume. The quantity hIi is then simply given by hIi ¼ ðA=mÞfexpðmrs Þ expðmRÞg,
(4)
where A is the source activity in g-rays/s, m is the total gray absorption coefficient in cm1 and rs and R are the source-hole and target radii, respectively. The values of m used in these calculations were best fit values to the experimental data prepared by the National Bureau of Standards by White (1957), and Hubbell (1969). 3.1. Corrections to the data In addition to the first-order corrections above, it is necessary to evaluate the corrections necessary to account for the absorption and scattering of the annihilating radiations leaving the target as well as the scattering and subsequent pair-producing absorption of the incident g-ray. The correction for the scattering and absorption of the 511 keV annihilation photons was made in two steps. First, an accurate measurement of the absorption of coincident pairs of 511 keV photons from a 2 mm plastic bead of 22 Na in the center of the target sphere was made. Second, a Monte-Carlo calculation was made of the total probability of absorption and scattering of the 511 keV annihilation g-ray pairs which originate at points throughout the sphere. Each point was weighted by a factor expðmrÞ, where m is the total absorption coefficient, and r is the radial location of the emission. A direct comparison of the probability of escape of the pair thus calculated was made with that calculated with the source of 0:511 MeV photon pairs located at the center. In this way, only a very small calculated correction was made to the escape probability measured with the calibrated 22 Na source located at the center of the target. It is also necessary to correct for the finite probability that positrons produced near the outer surface of the target escape prior to annihilating. An approximate expression for the fraction of these escaping positions is f ’ V s fs =V hfi,
(5)
where V s is the volume of the sphere which lies within one-half of the positron range from the surface, V is the
f ’
mre , 2fexpðmRÞ 1g
(6)
3.2. The Monte-Carlo computations Although the Monte-Carlo codes written in our laboratory in 1973 and 1974 are now completely obsolete,
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source hole, thereby effectively reducing its distance d; out of the target. At this point d 1 þ d 2 is computed and after many trials, the average quantity hexp½mðd 1 þ d 2 Þi is computed and compared to the quantity exp½mðR rs Þ. The latter quantity was also carefully measured using the 22 Na source bead as described above. In this way, the correction is far less sensitive to the values of the attenuation coefficients used. A similar Monte-Carlo code was written to simulate the relative probability of pair production following Compton scattering of the initial g-ray. Fig. 6 depicts the geometry of the computation of the probability of this effect.
they were used much later for other applications and later compared to modern CERN codes. At the photon energies relevant here, the results were identical within statistical uncertainty. The calculation of the comparison of the escape probability of the annihilation photon pair created at the center, with that in which the pair is created at points distributed throughout the target can be understood by reference to Fig. 5. In the case depicted in the diagram on the left-hand side, the probability can be expressed simply as expð2mdÞ, where m is the total g-ray absorption coefficient and d is the distance the 511 keV photons must traverse to escape, or the effective radius. This probability was compared with that computed by the Monte-Carlo code based on the geometry shown in the diagram on the right-hand side of Fig. 5. In this case, an angle for one annihilation is randomly selected to emerge from the source ðg1 Þ. Next a random number, n1 , is selected on the integral of the probability exp½mðr rs Þ, where rs again is the source-hole radius. This is the survival probability of g1 to radius r. The radius at which the paircreation event occurs for this particular trial is obtained from Eq. (7). 1 c r ¼ rs þ ln , (7) m c n1
0.511
ec 1
where c ¼ f1 exp ½mðR rs Þg1 . This expression for c results from the normalization that requires the integral of exp½mðr rs Þ vary from 0 to 1, when r varies from rs to R. After solving for r, it is assumed that the creation and annihilation sites are the same. The code then selects random number, n2 , and chooses an angle y ¼ 12 ðn2 pÞ, for the direction of one annihilation photon relative to the direction of the incident photon. The code was also sensitive to the fact that one of the annihilation photons could possibly pass through the
0.511
Fig. 6. Geometrical depiction of the class of events in which a Compton scattered g-ray creates a eþ e -pair.
0.511 d1 0.511
d2 0.511
1
PAIR CREATION SITE
22 NO SOURCE 0.511
Fig. 5. Geometry of the Monte-Carlo calculation of the comparison of the absorption or scattering of annihilation photons from the center of the target to those created throughout the volume of the target.
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4. The first measurements to test the calculated cross sections Shortly after the publication of the method of absolute cross section measurements previously discussed in the last section, a significant effort was made to apply this technique to test several theoretical results. The measurements were made on the tin target described earlier. The measurements were made with the following incident photon energies: 1:1205; 1:3325; 1:8362 and 2:6145 MeV. One measurement with the source of 60 Co was made with the combined 1:1732 and 1:3325 MeV grays. In another, the cross section with incident 1:1732 MeV g-rays from 60 Co was measured in a 3detector configuration which tagged the 1:3325 MeV gray in a third detector. The radioactive sources were in the form of 2 mm diameter beads of: 46 Scð1:205 MeVÞ; 60 Coð1:1732; 1:3325 MeVÞ; 88 Yð1:8362 MeVÞ and 228 Thð2:6145 MeVÞ. The results are given in Table 1 below. The results are also plotted in Fig. 7 with the cross sections shown as ratios to the theory of Overbo et al. (1973), and Tseng and Pratt (1972).
Table 1 Experimental and theoretical cross sections E (MeV)
s (exp) (mb)
sðTPÞa (mb)
sðOMOÞb (mb)
sðBornÞc (mb)
1.120 (1.173 + 1.333) 1.333 1.836 2.615
5.470.5 4274 6573 419720 1090750
4.11 39.5 66.1 418 1063
3.54 38.2 64.6 402 1061
2.11 24.03 41.33 317.1 941.3
a
From the calculations of Tseng and Pratt (1972). From the calculations of Overbo et al. (1973). c From Henry and Kennett (1972).
649
These results gave us the first indication that our results were in excellent agreement with theory, except for a disagreement near threshold.
5. Extension of the measurements to Z ¼ 13; 26; 29; 50 and 82 It would be several years until we were convinced that there were no unforseen anomalies in the experimental technique which led to the departure from the theoretical predictions near threshold. The measurements were repeated for photon energies as 1:119 MeV from a source of 46 Sc; 2:6145 MeV from the decay of 228 Th, and the mixture of g-rays 1.1732 and 1:3325 MeV from the decay of 60 Co. The results were still in significant disagreement with both the screened and unscreened calculation as shown in Table 2. A more extensive comparison to the various theoretical treatments of the day is given in Table 3 relative to the Born approximation. It is clear that the internal source method experiments yielded cross sections in excellent agreement with the screened calculations of Tseng and Pratt at incident gray energies significantly above threshold. As the energy approaches threshold, the experimental cross sections were 50% above those of the unscreened calculations of Overbo et al. (1973), and 30% above those of Tseng and Pratt (1972). At this point it became highly desirable to make measurements closer to pair-production threshold and with a higher Z target to further explore this discrepancy.
6. Measurements of absolute pair-production cross sections on a depleted uranium (Z ¼ 92) target
b
(EXPT) / (THEORY)
1.5 1.4 1.3 1.2 1.1 1.0 0.9 1.00
2.00
3.00
PHOTON ENERGY (MeV) Fig. 7. Ratio of experimental to theoretical cross sections for pair production in a Pb target using the internal source method.
To extend the measurements to lower energies and higher atomic number, again a 2 mm diameter plastic source bead of 46 Sc was used. This nuclide decays by a low-energy b -branch to the 2:0099 MeV level of 46 Ti with a half-life of 84 days. This level reaches the ground state by a g–g cascade; the energies are 1:12051 and 0:88925 MeV. At the time of this experiment, the relevant g-ray energy was thought to be 1:119 MeV, a trivial difference from today’s value. The 228 Th bead source, as well as the 60 Co source was also used with their g-rays of 2:6145 MeV from the thorium, and 1:1732 and 1:3325 g-ray cascade from 60 Co. A 1 cm diameter sphere of depleted uranium was obtained in the form of two hemispheres, with the hemispherical holes for the bead sources as in the earlier experiments. The main issue at this time concerned the various methods of making the screening corrections. The work by Overbo yielded higher cross sections and the results of the measurements of spair ðg þ UÞ are compared to the
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650
Table 2 Experimental and theoretical pair-creation cross sections Eg
Z
1.119
13 29 50 82
0.23(0.03) 1.11(0.07) 4.66(0.21) 14.5(0.8)
1.1732 + 1.3325
13 26 29 50 82
2.05(0.11) 7.84(0.33) 10.5(0.5) 40.5(2.0) 161.0(6.0)
2.6145
13 26 29 50 82
65.2(2.9) 265.0(11.0) 328.0(14.0) 1046.0(36.0) 3357.0(107.0)
sexp (mb)
sPC O (mb)
sBorn (mb) 0.1427 0.7099 2.110 5.676
sSC O (mb)
sSC TP (mb)
0.164 1.11 4.32 13.3
0.164 1.11 4.11 11.2
0.1617 1.039 3.541 7.352
1.625 6.50 8.08 24.03 64.66 63.63 254.5 316.6 941.0 2532.0
1.69 7.73 9.95 38.2 137.9
1.72 7.82 10.10 39.9 157.2
64.22 263.8 331.0 1063.0 3309.0
64.21 263.7 330.8 1061.0 3306.0
1.72 7.81 10.0 39.5 151.2 64.2 265.0 332.0 1063.0 3340.0
SC SC sBorn : Born approximation, unscreened; sPC O : Overbo, point Coulimb; sO : Overbo, screened; sTP : Tseng and Pratt, screened.
Table 3 Experimental comparisons E inc g
s0 =sB
s1 =sB
s2 =sB
s3 =sB
s4 =sB
s5 =sB
s6 =sB
s7 =sB
s8 =sB
s9 =sB
1.1205
29 50 82
1.56 2.21 2.59
y 2.56 y
2.10 2.37 2.61
y y 1.59
y y y
y y y
y y 1.96
y y y
y y y
y y 2.36
1.1732 + 1.3325
13 26 29 50 82
1.26 1.21 1.27 1.69 2.49
y y y 1.75 y
y y y y y
y y 1.20 2.20
0.997 y 1.111 1.494 2.04
y y y y y
y y y y y
y y 1.18 1.38 2.02
y y y y 2.12
1.04 1.10 1.16 1.36 2.02
13 26 29 50 82
1.02 1.04 1.04 1.11 1.33
y y y 1.16 y
y y y y y
y y y y y
1.006 y 1.03 1.081 1.229
0.93
y y y y y
y y y y y
y y y y y
y y y y y
2.6145
y 1.04 y
sB , Born approximation; s0 , experimental results of USC group; s1 , (Avignone III and Khalil, 1982); s2 , (Jenkins, 1956); s3 , (Khalil and Avignone III, 1982); s4 , (Girard et al., 1978); s5 , (Girard et al., 1979); s6 , (Martin and Blichert-Toft, 1970); s7 , (Moltz et al., 1969); s8 , (Overbo, 1970); s9 , (Overbo, 1978); s9 , (Standil and Shkolnik, 1958).
calculation of Tseng and Pratt and to those of Overbo in Table 4.
7. Measurements with 1.1205 MeV photons on targets of Z ¼ 26; 29; 50; 82; and 92 The next series of measurements were made to more accurately measure the Z-dependence of the cross sections and to compare them to the calculations of
Table 4 Absolute pair production cross sections for photons incident on uranium E g ðMeVÞ
sTP ðmbÞ
sO ðmbÞ
sexp ðmbÞ
1.1205 (1.1732 + 1.3325) 2.6145
13.1 207.9 4395
17.0 217.3 4333
17.470.5 22279 42457137
sTP : calculated by Tseng and Pratt. s0 : calculated by Overbo (1970, 1978, 1979).
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Tseng and Pratt made subsequent to their work published in Tseng and Pratt (1972). These theoretical results appeared in 1980 and 1981 (Tseng and Pratt, 1980, 1981). To this point our results agreed well with the calculations of Tseng and Pratt (1972, 1980, 1981) for incident g-ray energies several hundred kilovolts above threshold, and for atomic numbers up to Z ¼ 50. Measurements by Coquette in 1979 and 1980 (Coquette, 1979) had shown that there may have been a deviation from theory for large atomic numbers and for g-ray energies near threshold. For this experiment, it was extremely important to have an accurate efficiency calibration by avoiding the use of an absolute activity measurement of the 46 Sc bead. Instead an accurate measurement was made of the relative intensities of the 1:1205 MeV g-ray from the 46 Sc source and the 1:275 MeV g-ray form our standard calibration bead of 22 Na. An expression for the absolute cross section was then easily derived in terms of the ratio N 22 =hN g i, the ratio of the number of counts under the 0:511 MeV line in the 22 Na calibration source to that of the 1:1205 MeV peak in the 46 Sc source in the same geometry, corrected for the energy dependence of efficiency of the detector. The average bracket indicates that a correction for the decay of the source between the calibration and the experiment has been made. The relative intensity ratio Ið0:511Þ=Ið1:275Þ in the decay of 22 Na is well known and is 1:812 0:002 (Standil and Shkolnik, 1958), which propagates an error of 0:14%. One of the Ge detectors, as well as the 4:62 cm 4:62 cm NaIðTlÞ spectrometer were relativeefficiency calibrated on an energy interval from 1:01 to 1:46 MeV using the well known intensities of the g-ray lines in the decay of 152;154 Eu, as well as the efficiency calculations of Yates for the NaI(Tl) detector (Yates, 1965). The data were least-squares fitted to polynomials and exponential functions over the 0:45 MeV range. The relative efficiencies, ð1:1205Þ and ð1:275Þ were then determined with an accuracy of less than 1% for the Ge detector, and the relative intensity of the 1:1205 MeV gray of the 46 Sc bead was determined relative to that of the 1:275 MeV g-ray from the 22 Na calibration source to an accuracy of 1%. They both have much smaller systematic errors than those involved in absolute activity measurements. The intensity ratio Ið1:1205Þ=Ið1:275Þ measured with the NaIðTlÞ spectrometer was within 1% of that measured with the Ge detector. The intensity of the annihilation photons from the 22 Na calibration source was then accurately known relative to that of the 1:1205 MeV g-ray from the 46 Sc bead source. With this technique, it was possible to write an explicit expression for the measured pair-production cross section spp as follows: spp ¼
C pp N 22 W m1 f1 exp½m1 ðrs r0 Þg1 , 2c22 hN g irA0 by
(8)
651
Table 5 Absolute pair-production cross sections in millibarns at E g ¼ 1:1205 MeV Z
sðexpÞa (mb)
sðexpÞb (mb)
sðtheorÞc (mb)
sðexpÞ=sðtheorÞ
26 29 50 82 92
0.8770.02 1.1070.04 4.1670.15 12.770.5 15.670.5
1.1170.07 4.6670.21 14.570.80 17.470.5
1.11 4.11 11.2 13.1
0.99 1.01 1.13 1.19
a
Results using the methods Presented in this section. Results presented in Bethe and Heitler (1934) for Z ¼ 29; 50 and 82, and Davidson (1968) for Z ¼ 92. c Cross sections calculated by Tseng and Pratt (Sakharov, 1991; Schmidt and Huber, 1955). b
where C pp is the annihilation radiation coincidence rate with 46 Sc source in the target and c22 is that with the 22 Na calibration source with no target. The ratio N 22 =hN g i was determined in the relative intensity measurements discussed above. In Eq. (8), W is the atomic weight, r is the density, and m1 is the absorption coefficient of the target. The product by is the probability that both annihilation photons emerge from the target without interacting. This quantity was calculated by Monte-Carlo techniques as discussed earlier. The quantity hN g i was determined as follows: hN g i ¼
N 0 elt1 lt1 ðe elt2 Þ, lðt2 t1 Þ
(9)
where t1 and t2 are the beginning and ending times of the measurement, and N 0 was the quantity N g ðt ¼ 0Þ at the time of the measurement of N g =N 22 . The five resulting cross sections are listed in Table 5. There consistently existed a departure from the theoretical cross sections, and at this point it was decided to improve the experimental apparatus by adding a second intrinsic Ge detector and remeasure the cross sections. 8. Improved measurements in targets of Z ¼ 26; 29; 50; 82; and 92, with two Ge detectors At this point in the history of this series of measurements there was no clear understanding of the differences in our own measurements, though small, and the deviation from sophisticated numerical calculations of the cross sections near threshold (Tseng and Pratt, 1972, 1980, 1981; Overbo, 1970, 1978, 1979). The experiments described in Dayton (1953), and described in Section 4, concentrated on improving the techniques for calibrating the efficiencies for detecting the
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652
annihilation quanta, and for relating these to the absolute cross sections extracted from the data. There was one troublesome issue remaining. It was difficult to evaluate the systematic error associated with broad energy resolution of the NaI(Tl) detector. A second intrinsic Ge detector was obtained, with an absolute efficiency of 30% of that of a 7:62 cm 7:62 cm cylindrical NaI(Tl) detector at 1:3325 MeV. Since no deviation from the theoretical cross sections was observed for incident g-ray energies a few hundred keV above threshold, trial measurements were made using the source beads of 60 Co and 228 Th. All of the procedures used earlier for making the corrections to the data for absorption of annihilation radiations in the target, for pair creation following Compton scattering, and for the relative and absolute detection efficiencies were again used. These included the measurements described in Section 4, and MonteCarlo computations described in Section 3. All the results of these measurements were in excellent agreement with those of both Tseng and Pratt and of Overbo as shown in Table 6. These results were published in 1982 and the theoretical values were the latest computed with new screening corrections. They were transmitted to the authors privately by both authors and published later (Tseng and Pratt, 1981). At this point it was quite clear that for incident photon energies of 1:1732 MeV and above the screened nucleus calculations were experimentally confirmed. The next task was to extend these measurements to energies closer to threshold using the same experimental techniques that produced results in excellent agreement with theory at higher g-ray energies.
Table 6 Experimental and theoretical pair-production cross sections. so : Overbo screened calculation; sTP : Tseng and Pratt screened calculation E g ðMeVÞ
Z
1.1732 1.3325 +
26 29 50 82 92
2.6145
26 29 50 82 92
sexp ðmbÞ 7.78(0.30) 10.0(0.35) 39.0(1.7) 152.6(5) 209.4(8) 263(7) 327(10) 1051(22) 3331(75) 4345(96)
so ðmbÞa
sTP ðmbÞ
7.82 10.10 39.9 157.2 217.3
7.81 10.0 39.5 151.2 207.9
263.7 330.8 1061 3306 4332
265 332 1063 3340 4395
a All of the theoretical cross sections were calculated by the original authors and transmitted via private communication prior to publication.
9. Near-threshold measurements of absolute pairproduction cross sections of photons on lead In the work done previously, the experimental techniques of the internal source method were by 1984 well developed, while the corrections and systematic errors were clearly understood. A radio active source bead of 86 Rb was prepared by colleagues at the Los Alamos National Laboratory (Avignone III et al., 1984, 1985). Our measurements to this point were limited to photon energies down to 1:1205 MeV. However, 86 Rb, the only practical source with a single g-ray in the most interesting energy range, was not previously available to us with its 1:077 MeV g-ray, only 55 keV above threshold. This isotope was made by thermal-neutron capture on stable 85 Rb. But even a small component of highenergy neutrons can produce enough contamination with the positron emitter 84 Rb, via the reaction 85 Rbðn; 2nÞ 84 Rb, to preclude its use in the internalsource method. Any contamination with a positron emitter produces annihilation radiation, which if it comprises more than 20% of the total coincidence event rate, seriously degrades the accuracy of the measurement. The irradiation in a highly pure thermal neutron column, the separation chemistry and manufacturing of a 2 mm diameter plastic bead were accomplished at the Los Alamos National Laboratory, and represented the new breakthrough necessary to push our measurements closer to threshold. Attempts to use the uranium target failed due to the radioactivity remaining in the depleted uranium. Accordingly, the measurements were made in the lead target and are presented in Table 7 below, along with earlier measurements at higher energies. These results are also plotted in Fig. 7 where there is a clear departure from the screened calculations. These theoretical results were calculated especially for us, and sent to us by Professor Pratt. This was essentially a doubleblind comparison since the calculations were not sent until after the final analysis of the data.
Table 7 Experimental cross sections measured with the internal-source technique in Pb compared to the theoretical cross sections of Tseng and Pratt ðsTP Þ E g (MeV)
sexp (mb)
1.077 1.1205 1.1732 + 1.3325 2.6145
1.9270.11 12.7270.5
a
152.675.0 3331775
Calculated especially for our g-ray energies.
sTP (mb)a 1.34 11.2 151.2 3340
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At this point the discrepancy between theory and experiment appeared to be well established; however, it thus far defied explanation. Similar deviations from theory were deduced from the data published by Coquette et al., 1979). In Fig. 8, the data from Schmidt and Huber (1974) was divided by the calculations of Tseng and Pratt mentioned above, and plotted against photon energy. Other experimental papers which reported cross sections with near threshold discrepancies with theory were published during the period of our activities (1972–1985) and earlier by Yamazaki and Hollander (1965), Coquette (1979, 1977), and En‘yo et al. (1980). They all showed some excess over the various theoretical calculations. At the completion of our final experiments published in 1985 (Avignone III et al., 1985), in which the measurements were extended down to E g ¼ 1:077 MeV, we could not conceive of any further improvements that could be made that would make a significant difference. The USC group began to focus more on developing improved techniques to search for neutrinoless double beta decay and the issue remained unresolved until 1991 when Professor Eric Adelberger contacted the author
and discussed his interest in testing these results with an independent technique (De Braeckeleer et al., 1992). De Braeckeleer et al. (1992) investigated the pair production of g-ray on a germanium detector of a pair spectrometer. Two Ge detectors were used separately, a 40 cm3 coaxial detector, and a 10 cm3 planar detector. The Ge detector was surrounded by two 12:7 cm 15:2 cm NaIðTlÞ detectors to detect the annihilation photons. The double escape peaks were recorded in the Ge detector spectra. Monoenergetic g-rays were obtained from sources of 152 Eu ðE g ¼ 1:408; 1:299; 1:213; 1:112; and 1:086 MeVÞ; 60 Co ðE g ¼ 1:3325 and 1:1732 MeVÞ; 88 Y ðE g ¼ 1:836 MeVÞ; and 207 Bi ðE g ¼ 1:76971 MeVÞ. The results agreed with our results as well as those of En‘yo et al. (1980), Yamazaki and Hollander (1965) and Coquette (1977). They suggested an explanation of these discrepancies based on the fact that the kinetic energies of the eþ e pair were of order 30 keV, where the final state Coulomb interaction may be important. They estimated this effect using an expression derived by Sakharov (1991) for pairs produced with low relative velocity to each other. This yields the following multiplicative correction factor: F¼
1.7 z=32 1.6 1.5
(EXPT) / (THEORY)
1.4 1.3 1.2 1.1
653
2pa=v , 1 expð2pa=vÞ
(10)
where v is the relative velocity of the eþ e -pair, which was integrated over the electron-positron kinematics at each photon energy assuming the Bethe–Heitler angular and energy distribution (Davidson, 1968). This explanation gave reasonable agreement between experimental results and theoretical predictions corrected for this final state. De Braeckeleer et al. (1992) point out that this represents an approximate explanation; however, the perturbation of the lepton wave functions was neglected and welcomed a more realistic calculation. Nevertheless, the long standing discrepancy between experimental and theoretical pair production cross section was perhaps finally qualitatively understood and the present author had no more to contribute to the subject.
1.0
Acknowledgements
0.9 0.8 0.7 1.05 1.06
1.07 1.08
1.09 1.10
1.11 1.12
PHOTON ENERGY (MeV) Fig. 8. Ratio of experimental to theoretical cross sections for pair-production in a Ge target. The experimental data points were graphically measured from our enlargement of Fig. 1 of Schmidt and Huber (1974) and divided by the theoretical values of the screened cross sections of Tseng and Pratt (1981).
The author would like to recognize the experimental efforts of his excellent graduate students who contributed to the University of South Carolina experimental program: Samuel Blankenship, Craig Barker, Tom Girard, Terry Huntsberger, Ali Khalil and Harry Miley. He would also like to thank the University of Washington group for their efforts to explain the discrepancies between theory and experiment and for their private communications. The experimental effort at the University of South Carolina was supported by the National Science Foundation under Grants:
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PHY-7824885, PHY-8109038, and PHY-8306098. The Los Alamos effort was supported by the Office of Health and Environmental Research of the U.S. Department of Energy.
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