The K/β^+ ratio in the decays of ^(19)Ne and ^(15)O have been measured as (9.6 ± 0.3) × 10^(-4) and (10.7 ± 0.6) × 10^(-4), respectively. A gas-flow proportional counter, operating in anticoincidence with the surrounding plastic scintillator, was used. Theoretical K/β^+ ratios for ^(19)Ne and ^(15)O were computed, using exchange-overlap corrections calculated by Vatai and, separately, exchange corrections extrapolated from the results of Bahcall for 14 ≤ Z ≤ 37. The experimental results were found to be in better agreement with Vatai's calculations

Drever, R. W. P.

Leiper, W.

English

Caltech Authors - Main

1132 D. ROBSON AND R. D. KOSHEL 6 
on in separate communications. Since the method-
ology presented here has been in part justified by 
a calculation, the extension of the formalism to 
more complicated problems is of importance. The 
*On leave from the Physics Department, Florida State 
University, Tallahassee, Florida 32306. 
tWork supported in part by the U. S. Atomic Energy 
Commission. 
1s. T. Butler, Proc. Roy. Soc. (London) A208, 559 
(1951); M. H. Macfarlane and J. B. French, Rev. Mod. 
Phys. 32, 567 (1960). 
2See;e.g., N. Austern, Direct Nuclear Reaction The-
ories (Wiley, New York, 1970). 
3R. J. Ascuitto and N. K. Glendenning, Phys. Rev. 181, 
1396 (1969). 
4P. K. Binda! and R. D. Koshel, Bull. Am. Phys. Soc. 
16, 623 (1971); P. K. Binda! and R. D. Koshel, to be pub-
lished. 
5See, e.g., J. R. Comfort, J. P. Schiffer, A. Richter, 
and M. M. stautberg, Phys. Rev. Letters 26, 1338 (1971). 
6See, e.g., R. Sherr, B. F. Bayman, E.Rost, M. E. 
Rickey, and C. G. Hood, Phys. Rev. 139, B1272 (1965). 
7N. Austern, R. M. Drisko, E. C. Halbert, and G. R. 
Satchler, Phys. Rev. 133, B3 (1964). 
8P. J. A. Buttle andhJ. B. Goldfarb, Proc. Phys. 
Soc. (London) 83, 701 (1964). 
applications presented in this paper are aimed at 
nuclear physics, but there seems to be no obvious 
reason why the theory should not prove equally 
useful in atomic or molecular collisions. 
9s. K. Penny and G. R. Satchler, Nucl. Phys. 53, 145 
(1964). 
10N. K. Glendenning and R. S. Mackintosh, Nucl. Phys. 
A168, 575 (1971). 
1IR. S. Mackintosh, Nucl. Phys. A170, 353 (1971). 
12A. M. Lane and D. Robson, Phys. Rev. 185, 1403 
(1969). 
13 A. M. Lane and D. Robson, Phys. Rev. 178, 1715 
(1969). 
14L. A. Charlton, private communication. 
15L. A. Charlton and D. Robson, Bull. Am. Phys. Soc. 
17' 508 (1972). 
"'11lT. Tamura, Rev. Mod. Phys. 37, 679 (1965). 
17c. M. Vincent and H. T. Fortune, Phys. Rev. C ~. 
782 (1970). 
18see, e.g., several references in Rev. Mod. Phys. 37, 
409-458 (1965). -
19L. D. Faddeev, Mathematical Problems of the Quan-
tum Theory of Scattering for a Three-Particle System 
(Israel Program for Scientific Translation, Jerusalem, 
1965). 
PHYSICAL REVIEW C VOLUME 6, NUMBER 4 OCTOBER 1972 
K -Electron-Capture-to-Positron-Emission Ratio in the Decays of 150 and 19Ne 
W. Leiper* and R. W. P. Drever 
Department of Natural Philosophy, University of Glasgow, Kelvin Laboratory, National Engineering Laboratory, 
East Kilbride, Lanarkshire, Scotland 
(Received 12 June 1972) 
TheK/{3+ ratio in the decays of 19Ne and 150 have been measured as (9.6±0.3)x 10-4 and 
(10. 7 ± 0 .6) x 10-4, respectively. A gas-flow proportional counter, operating in anticoincidence 
with the surrounding plastic scintillator, was used. Theoretical K f/3+ ratios for 19Ne and tso 
were computed, using exchange-overlap corrections calculated by Vatai and, separately, ex-
change corrections extrapolated from the results of Bahcall for 14 :sz :s 37. The experimental 
results were found to be in better agreement with Vatai's calculations. 
I. INTRODUCTION 
Electron exchange and overlap corrections, Bk, 
for K capture to positron emission ratios have 
been calculated by BahcalP for nuclides in the 
range 14 ""Z"" 37 and by Vatai2 for nuclides in the 
range 13 ""Z "" 37. The effect of the exchange -over-
lap correction is to reduce the theoretical value 
for K/{3+ ratios by a factor of (1 -Bk). The calcu-
lations of Bahcall and Vatai differ by about 7% in 
the value of Bk at Z = 14 and the discrepancy in-
creases on extrapolation into the region Z "" 14, as 
shown in Table I for Z = 8 and Z = 10. 
A recent measurement3 of the K/{3+ ratio in the 
decay of 30p is in agreement with Vatai's calcula-
tions. Ledingham et al. 3 also include a summary 
of other experimental measurements of K/{3+ 
ratios for Z "" 15, including the results given in 
this paper. We describe below measurements of 
K/{3+ ratios in the decays of 19Ne and 150. 
6 K -ELECTRON -CAPTURE -TO -POSITRON -EMISSION RATIO .•• 1133 
r----
1 
I 
I 
I 
I 
COPPER 
IRRADIATION 
VESSEL 
FIG. 1. Schematic outline of the gas-flow system. 
The transition 
is superallowed, with a half -life of 17.43 ± 0.06 
sec 4 and a positron end-point energy of 2.217 
± 0.005 MeV.5 The mean energy of the emitted 
radiations following K capture is about 700 eV.6 
The half-life of the 150 decay has been measured 
as 122.6± 1.0 sec. 7 The maximum energy of the 
emitted positrons is 1. 737 ± 0.004 MeV, 5 and the 
mean K peak energy is about 400 eV.6 
MAIN 
AMPLIFIER 
COUNTER 
TABLE I. Exchange and overlap correctionsBk, and 
comparison of experimental and theoretical values for 
K /{J+ ratios in the decays of 150 and 19Ne, 
Isotope Bk a Bk b K j[J+ (theor) K j[J+ (expt) 
tso 0.59 0.986 
19Ne 0.80 0.987 
9.2x 1o-4 
9.7x10-4 
(10. 7 ± o.6) x 1o-4 
(9.6 ± o.3) x 1o-4 
a Extrapolated from the calculations of Bahcall (Ref. 1). 
b From the extrapolation given by Vatai (Ref. 2). 
II. EXPERIMENTAL PROCEDURE 
1'Ne was produced by the 2~e(y, n) 19Ne reaction 
on neon gas which was allowed to flow through the 
x-ray beam of the Glasgow electron linear accel-
erator. The gas-flow system is shown in Fig. 1. 
The counter has been described in detail previous-
ly,8·9 and consists of a central cylindrical propor-
tional counter surrounded by a plastic scintillator. 
The scintillation and proportional counters were 
operated in anticoincidence, so as to reduce the 
intensity of low-energy signals from the propor-
tional counter owing to positrons emitted near the 
wall of the counter. The associated electronic ar-
rangement is outlined in Fig. 2 and is essentially 
a computerized version of the system described 
by Drever. 10 The positrons were detected as large 
amplitude pulses (Y analog-to-digital-converter 
overflow output) from the proportional counter. 
In addition to 19Ne, 150 and 11C were also pro-
duced by the accelerator x-ray beam. The 150 
was probably produced by the 2~e(y, na)150 reac-
tion. The source of the 11C is uncertain, but it 
MAIN 
AMPLIFIER 
SCALER 
ANTICOINCIDENCE 
INPUT 
OVERFLOW 
OUTPUT 
DEAD-TIME 
OUTPUT 
SIGNAL TO LINAC 
SIGNAL FROM LINAC 
CLOCK PULSES 
a: 
w 
1-
:::J 
a. 
~ 
0 
u 
00 
I 
a. 
0 
a. 
FIG. 2. Electronic arrangement, 
1134 W. LEIPER AND R. W. P. DREYER 6 
...J 
~ 
f:J1000 
1-
~ 
VI 
lz 200 
::;) 
0 
u 
0 10 
TIME (min) 
FIG. 3. Decay of activity trapped in the central count-
er following the measurement on 150. 
may have been produced from organic contamin-
ants in the neon gas, or by the 2~e(y, n2a)11C 
reaction. 
By varying the flow rate of the neon gas from the 
irradiation vessel to the counter to take advantage 
of the differences in half-lives of 19Ne and 150, it 
was possible to obtain in the counter activities con-
sisting mainly of either 19Ne or 150. K/{3+ ratio 
measurements on both isotopes were then possible. 
The amount of activity was estimated, for the 
150 runs, by closing off the counter and following 
the decay of the trapped activity (Fig. 3). Analysis 
of the decay curve indicated that about {7 .0± g:~)% 
of the activity in the counter was due to 11C. The 
percentage of 19Ne activity present during the 150 
measurements was estimated from the variation 
0 I 2 3 4 5 6 7 8 9 10 II 12 13 14 
SUBGROUP NUMBER 
70 sec 
FIG. 4. Variation of activity in the central counter 
when the accelerator x-ray beam is on for 20 sec in 
every 70 sec. Gas-flow conditions were maintained as 
during the !9NeKj,B+ measurements. The points on the 
curve are the experimental data. The smooth line was 
given by a computer simulation. 
2400'r-----~~~----------,-----------r-o 
...J 
::! 
a: 
LIJ 
1-
:!: 1600 
>-
<!) 
a: 
LIJ 
z 
LIJ 
a: 
~ BOO 
(/) 
1-
z 
::;) 
0 
u 
0o~--------~20~---------47o~--------~60~ 
ENERGY(ARBITRARY UNITS) 
FIG. 5. Pulse-height spectrum from the central count-
er, in anticoincidence with the scintillation counter, 
from measurements on 19Ne. 
in activity obtained by putting a vessel of known 
volume into the flow lines. The 19Ne contribution 
was calculated to be (3.0± 5:~)% of the total activity 
in the counter. 
Repeating this procedure for the 19Ne runs re-
sulted in large errors in the estimation of the 
amount of 19Ne due to its short half-life. A differ-
ent technique had to be adopted. The electron ac-
celerator was interfaced with the PDP -8 computer 
so that the accelerator could be operated under 
program control. The block of computer memory 
which was used for storing the proportional-count-
er spectrum and the total number of positrons 
emitted was extended into 14 subgroups of 64 
channels. These 14 subgroups were used for a 
cyclic sequential analysis of the activities pro-
duced in the counter when the accelerator was 
turned on by the PDP-8 computer for 20 sec in 
every 70 sec, with the flow rates in the gas-flow 
system set to the same value as in the 19Ne K/{3+ 
...J 
~ 
ffieoo 
I-
!: 
~600 
a: 
LIJ 
z 
LIJ 
a:400 
1&1 
D. 
(/) 
1- 200 
z 
::;) 
0 
u 0~--------~~--------~~--------~~J 0 ~ ~ ~ 
ENERGY(ARBITRARY UNITS) 
FIG. 6. Pulse-height spectrum from the central count-
er, in anticoincidence with the scintillation counter, 
from measurements on tso. 
6 K -ELECTRON -CAPTURE -TO-POSITRON -EMISSION RATIO ... 1135 
measurement. The number of positrons detected 
in the proportional counter as a function of sub-
group number is shown in Fig. 4. The fractions 
of the total activity in the proportional counter due 
to 150 and 11C were estimated from a computer 
simulation of the sequential analysis. The simu-
lated spectrum is given by the continuous line in 
Fig. 4. 
The weighted means of the results from 13 
cyclic sequential analyses were calculated as 
(4.9± 0.4)% of 150 and (2.0± 0.2)% of 11C. 
III. ANALYSIS OF EXPERIMENTAL RESULTS 
A. 1~e Analysis 
A preliminary estimate of the K/(3+ ratio in the 
decay of 150 was made based on the assumption 
that the peak obtained in the low-energy spectrum 
from the 150 measurements consisted entirely of 
150 K capture events. This estimate gave a result 
of 1.05 x 10-3 • The K/(3+ ratio in the decay of 11C 
was known. 9 Using the measured values for the 
amounts of 150 and 11C activity in the proportional 
counter during the 1~e K/(3+ measurement, the 
number of K capture events in the 1~e low-energy 
spectrum (Fig. 5) due to 11C and 150 were calcu-
lated. Poisson shapes of corresponding area at 
200 and 400 eV were subtracted from the 19Ne low-
energy spectrum. 
A computer program was written to subtract 
Poisson shapes corresponding to the 19Ne K capture 
peak from what remained of the low-energy spec-
trum after subtraction of the 150 and 11C K capture 
peaks, background spectrum, and a straight line 
extrapolated from a least-squares best fit to 
points on the spectrum above the main peak. The 
remaining spectrum was examined visually for 
any peaks or valleys. Remaining spectra which 
showed a significant lack of smoothness were re-
jected. 
The statistical errors in the computation of the 
final result arose from the estimation of the 
amounts of 11C and 150 and in the process of sepa-
rating out the 19Ne peak. These were added orthog-
*Present address: Dalhousie University, Halifax, 
Nova Scotia, Canada. 
1J. N. Bahcall, Phys. Rev. 129, 2683 (1963); 132, 362 
(1963). 
2E. Vatai, Nucl, Phys. A156, 541 (1970). 
3 ---K. W. D. Ledingham, J. Y. Gourlay, J. L. Campbell, 
M. L. Fitzpatrick, J. G. Lynch, and J. McDonald, Nucl. 
Phys.A170, 663 (1971). 
4L. G. Earwaker, J. G. Jenkin, and E. W. Titterton, 
Nature 195, 271 (1962). 
onally to give an experimental result of (9.6±0.3) 
x 10-4 for the K/(3+ ratio in the decay of 1~e. 
B. 150 Analysis 
The 150 low-energy spectrum is shown in Fig. 6. 
The analysis procedure was similar to that for 
19Ne. In this case, Poisson distributions corre-
sponding to the 19Ne and 11C contributions to the 
low-energy spectrum were subtracted off. The 
smoothness criterion was then applied to the re-
maining spectrum on further subtraction of a 
range of 150 K capture peaks. The final experi-
mental result was obtained as (10.7±0.6)x1Q-4 
for the K/(3+ ratio in the decay of 150, a correction 
of (1.2± 1)% having been made for positrons enter-
ing the proportional counter from the insensitive 
regions at each end of the counter. 
IV. DISCUSSION 
The theoretical values of the K/{3+ ratios in the 
decays of 150 and 19Ne were calculated using K 
electron wave functions derived from relativistic 
Hartree-Fock-Slater solutions of the Dirac equa-
tion.11'12 These wave functions were obtained for 
a nucleus of finite size and a diffuse surface and 
correspond, to an accuracy of better than 1%, 
with wave functions given by Suslov •13 The results 
of these calculations, without exchange or overlap 
corrections, are given in Table I. As these cor-
rections would reduce the theoretical result, it 
can be seen that the differences between theory 
and experiment would only be increased by the 
application of an exchange-overlap correction, and 
that the experimental results are in better agree-
ment with the calculations of Vatai than the results 
extrapolated from the work of Bahcall. 
The authors wish to express their gratitude to 
Professor G. R. Bishop and Professor P. I. Dee 
for their interest in this work. 
One of us (W.L.) is indebted to the University of 
Glasgow and the Science Research Council for 
financial support. 
5J. H. E. Mattauch, W. Thiele, and A. H. Wapstra, 
Nucl. Phys. 67, 32 (1965). 
6J. A. Bearden and A. F. Burr, Rev. Mod, Phys. 39, 
125 (1967). -
7J. W. Nelson, E. B. Carter, G. E. Mitchell, and R. H. 
Davis, Phys. Rev. 12 :l, 1723 (1963). 
8A. Moljk, R. W.P~Drever, and S.c. Curran, in 
Proceedings of the International Conference on Radio-
isotopes in Scientific Research, Paris, 1957 (Pergamon, 
London, 1958), Vol, II, p. 596. 
1136 W. LEIPER AND R. W. P. DREYER 6 
9J. L. Campbell, W. Leiper, K. W. D. Ledingham, and 
R. W. P. Drever, Nucl. Phys. A96, 279 (1967). 
toR. W. P. Drever, Ph.D. thesis, University of Glasgow, 
1958 (unpublished). 
11M. Martin, private communication. 
12c. W. Nestor, T. C. Tucker, T. A. Carlson, L. D. 
Roberts, F. B. Malek, and C. Froese, ORNL Report 
No. ORNL-4027, 1966 (unpublished). 
l3yu, P. Suslov, in Proceedings of the International 
Conference on Electron Capture and Higher-Order Pro-
cesses in Nuclear Decay, Debrecen, Hungary, 15-18 
July, 1968 (Eotvos L6rand Phys'ical Society, Budapest, 
Hungary, 1968), p. 51. 
PHYSICAL REVIEW C VOLUME 6, NUMBER 4 0 CTOBER 19 72 
Application of the Burnett-Kroll Soft-Proton Theorem 
to Nucleon-Nucleon Bremsstrahlung* 
Harold W. FearingH 
Los Alamos Scientific Laboratory, University of California, Los Alamos, New Mexico 87544 
(Received 5 June 1972) 
We give an explicit analytic formula for the leading terms in the square of the matrix ele-
ment for nucleon-nucleon bremsstrahlung in the relativistic, model-independent, soft-photon 
theory of Nyman. Such a formula should simplify direct and detailed comparisons of this soft-
photon theory with experiment and with the predictions of other models. These comparisons 
have been difficult because Nyman in his original calculation of NN- NN-y was forced by the 
extremely complicated matrix element to use entirely numerical procedures. To obtain an 
analytic result we have used the Burnett-Kroll theorem, which makes it possible to bypass 
many of the complications of a straightforward algebraic calculation. We also emphasize the 
general utility of the Burnett-Kroll theorem as a calculational tool for obtaining in an almost 
trivial fashion the leading terms in the squares of very complicated radiative matrix elements. 
I. INTRODUCTION 
A number of years ago it was suggested1 that an 
investigation of the nucleon-nucleon bremsstrah-
lung process N + N- N + N + y would give informa-
tion about the off-mass-shell components of the 
nucleon-nucleon scattering amplitude. More re-
cently, however, it has become clear, as are-
sult of various low-energy theorems2 - 6 that the 
leading terms in an expansion of the NNy ampli-
tude in powers of the photon momentum k depend 
only on the on-mass-shell parts of the NN ampli-
tude. Thus one must work harder than was origi-
nally supposed to get information about off-mass-
shell terms. However, experimental developments 
since the original suggestion have made it quite 
feasible to examine in detail regions where high-
er-order terms might be important, so nucleon-
nucleon bremsstrahlung remains a very interest-
ing process. 
There have been a large number of calculations 
of nucleon-nucleon bremsstrahlung, including, for 
example, those of Refs. 7-11. Typical ones in-
clude a number of calculations based on primarily 
nonrelativistic potential models, 8 the relativistic 
one-boson exchange model of Baier, K'uhnelt, and 
Urban9 and the completely relativistic calculation 
of Nyman10 which uses soft-photon techniques. 
This latter calculation is of particular interest be-
cause it is both relativistic and model-independent. 
By "model-independent" we mean that the radiative 
amplitude depends only on the on-mass-shell NN 
amplitude through the leading two orders, O(k-1) 
and 0(~), in the photon momentum k. This re-
sult2-6•12 follows directly from gauge invariance 
and certain smoothness and analyticity assump-
tions.12 Within the definition of model-indepen-
dent there exists the freedom to choose different 
continuations of the NN amplitude to the slightly 
unphysical point corresponding to the radiative 
process or to choose different sets of kinematic 
variables to describe the NN amplitude. One can 
show however, given the above assumptions, that 
this freedom alters the result only in order k. 3• 12 - 14 
Thus within the region of strict applicability of 
Nyman's calculation, i.e., when the O(k) terms 
are truly negligible, all other calculations must 
agree with it (provided of course that they repro-
duce the same NN scattering amplitude and satisfy 
the necessary smoothness conditions12). Thus the 
soft-photon theory provides a benchmark against 
which to check more elaborate models. 
From a practical point of view however, the 
region of strict applicability of the soft-photon 
theory is significantly smaller than the region 
which has been explored experimentally. The ex-


K-Electron-Capture-to-Positron-Emission Ratio in the Decays of ^(15)O and ^(19)Ne

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K-Electron-Capture-to-Positron-Emission Ratio in the Decays of ^(15)O and ^(19)Ne

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