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Precision measurement of beta-ray end-point energies: 60Co, 137Cs and 204Tl
I
1.E.2 I
Nuclear Physics All2 (1968) 156--160; (~) North-Holland Publishing Co., Amsterdam N o t to be reproduced b y p h o t o p r i n t or microfilm without written permission f r o m the publisher
PRECISION MEASUREMENT OF BETA-RAY END-POINT ENERGIES: 6°Co, lSTCs AND 2°~T1 J. L. WOLFSON and A. J. COLLIER Department of Physics, University of Saskatchewan, Regina Campus, Regina, Saskatchewan Received 30 January 1968
Abstract: Using a high resolving power, iron-free spectrometer, precise measurements have been performed of the end-point energies of the beta-ray spectra from ~°Co, 18vCs(low-energy spectrum) and ~°4T1.The values obtained were 317.884-0.10, 511.634-0.84, and 763.47±0.22 keV, respectively. E I
RADIOACTIVITY ~°Co, l~7Cs, 2°4T1;measured E/~•
[
1. Introduction The present paper gives an account of attempts to measure precisely the beta-ray end-point energies of the 60Co spectrum (318 keV), the lower-energy spectrum from 137Cs (512 keV) and the 2 O4T1 spectrum (763 keV). Recently, Park and Christmas 1) have reported a measurement of the end-point energy of the 204T1 beta spectrum performed on the same beta-ray spectrometer z) as the present measurement though under different conditions.
2. Apparatus and method 2.1. APPARATUS All sources were prepared by sublimation in vacuum on to backings of 200 #g/cm 2 aluminium foil. A backing of this thickness is transparent to electrons of energies greater than about 30 keV and is not expected to affect the measured energy of the end point if only the higher-energy electrons me used in the measurement. In these studies, only electrons of energies greater than half the end-point energy were so used. Park and Christmas 1) have actually shown that for 2°4T1 there is indeed little distortion of the spectrum to quite low energies for a source as thick as 150/zg/cm 2, which would also indicate that for 2°4T1 scattering from a backing of approximately this thickness is not serious. Care was taken to investigate sources of scattering within the spectrometer which t The experimental work on which this paper is based was performed by one of us (J.L.W.) during 1964 at the National Research Council of Canada laboratories, Division of Applied Physics, Ottawa, Canada. 156
t-RAY END-POINT ENERGIES
157
could have given rise to spurious results and to demonstrate that their effects were negligible. A study of the response of the detector, a proportional counter, gave no evidence of energy dependence. No corrections were applied for instrumental resolution. Following the suggestion of Paul 3), the centre of gravity of the calibration line profile was employed as calibration point. For calibration, the K internal conversion line of the 661.595_+0.076 keV gamma-ray transition 4) in 137Ba was used for the 1a7Cs and z O4Tl spectra, and the K internal conversion line of the 1173.226_+0.040 keV gamma-ray transition 5) in 6°Ni for the 6°Co spectrum. 2.2. T R E A T M E N T
OF DATA
A program was written to analyse all data in the form of Fermi-Kurie plots, which were fitted to straight lines by the method of least-squares. Since the spectrometer was an iron-free instrument, the voltage drop across a standard resistor in series with the magnet windings was proportional to electron momentum. Along with the calibration point, these voltage measurements were included in the program as well as the observed counting rate N at each of the voltages. The relationship between magnetic rigidity Bp and electron kinetic energy was also included using values for the fundamental constants as given by Cohen and DuMond 6). The quantity then calculated was
N_
l
3FoLo(S/Lo)A as a linear function of beta-ray energy in keV. The quantity p is electron momentum in units of me, (N/p) is proportional to the number of electrons emitted per unit momentum interval. Fo is the Fermi function, Lo an electron radial wave function and S/L 0 a shape distribution factor defined by Biihring 7). Each experimental point was weighted inversely as the square of its error. In general, the methods of Hildebrand s) were used in the treatment of the experimental data. Values of FoL 0 were calculated from the tables of Bhalla and Rose 9), and fourthorder interpolation used when interpolation was required. The influence of screening was taken into account using the tables of Biihring 10), though screening effects, as it turns out, are quite small. For distributions of the allowed shape, (SILo) is constant 7), while for first-forbidden unique spectra (SILo) is expected to have the value (q2+22pZ), where q is neutrino energy in mo ca units, 22 a function defined by BiihringT), which we calculated from the tables of Bhalla and Rose 9). Again, fourth-order interpolation was used when interpolation was required, and the values of 22 were corrected for screening using the tables of Biihring 10). 3. Results
For 6 ° C o , the data above 160 keV were fitted to the allowed shape with results shown in table 1. For ~37Cs, the transition ~ ) in ~37Ba is expected to be first-order
158
J. L. WOLFSONAND A. J. COLLIER
forbidden with unique shape. The shape factor is expected to be (p2 _[_22q2), and the data above 265 keV were fitted using this shape factor; the results are shown in table 2. The contribution from the higher-energy distribution had first to be subtracted, and this contribution was determined by fitting the data above 650 keV with a shape factor (q2+kp2) and an end-point energy 12) of 1.176 MeV. The value chosen for k was 0.010, the mean of the value found by Hsue, Langer and Tang 12) and the value 0.015 found by Daniel and Schmitt 13). The uncertainty in determining the contribution by the higher-energy spectrum is the largest source of error in measuring the end-point energy of the lower-energy spectrum. TABLE 1
Results for n°Co Run
1 2 3
Slit width (cm) 0.087 0.132 0.175
Transmission ( 70 o f 4 ~ sr)
Line width (~)
0.33 0.10 0.40 0.12 0.70 0.25 Mean value is 317.884-0.10 keV.
End-point energy (keV) 318.17-4-0.20 317.73 4-0.16 317.83 4-0.t4
Source: 2 cm×0.08 cm; 9/~g/cm2; 150 #Ci.
TABLE 2 Results for 13~Cs Run
1 2
Slit width (cm) 0.087 0.175
Transmission ( % o f 4z~ st)
Line width (%)
0.33 0.10 0.70 0.25 Mean value is 511.63 4-0.67 keV.
End-point energy (keV) 511.83 4-0.95 511.43 4-0.95
Source: 2 cm x0.08 cm; 4#g/cm2; 20/~Ci.
For 2°*T1, the shape factor (S/Lo) is expected to be (q2 +22p2) as the transition 11) is from a 2 - level in Z°4T1 to a 0 + level in z°4Pb. However, it is known that the spectrum is not adequately described by such a distribution [see, for instance, Park and Christmas 1)]. The spectrum has an excess of electrons at the lower energies. Park and Christmas 1) were able to obtain an adequate fit with a shape factor (bq z +22p z) with b determined from experiment equal to 1.161 +0.005. Park and Christmas also offer a theoretical argument for assigning b a value of about 1.1 rather than unity. The results obtained in the present experiment confirm those of Park and Christmas in that an excess of lower energy electron was observed, and a good fit could not be obtained with b equal to unity. A program was instituted to search for the value of b giving the best fit in each of the two runs. The criterion chosen for best fit was a mini-
159
fl-RAY END-POINT ENERGIES
mum value for the sum of the squares of the residuals. Only data for electrons of energies above 400 keV were used. The results are given in table 3. TABLE 3
Results for ~°4T1 Run 1 2
Slit width (cm) 0.13 0.25
Transmission ( 70 of 4n sr)
Line width (70)
End-point energy (keV)
0.40 0.13 763.334-0.25 0.70 0.27 763.674-0.30 Mean value is 763.47:t:0.20 keV.
b 1.203 1.191
Source: 2 cm×0.12 cm; 35/zg/cm2; 2.4 #Ci. 4. Discussion The errors quoted for the values of the end points in tables 1-3 do not include errors associated with the calibration lines. For the 6°C0 and 137Cs beta-ray end points, inclusion of these uncertainties does not affect the errors quoted appreciably. For the z°gTI beta-ray end point, the error is increased slightly. For the 137Cs beta-ray endpoint energy, there is an additional source of error associated with the shape factor of the high-energy component. As mentioned in subsect. 3.2, the shape factor used was (q2 +kpZ) with k = 0.010. Had either the value of Hsue et al. 12) (k = 0.004) or that of Daniel and Schmitt 13) (k = 0.015) been used, the end-point energy would have been changed by about 0.5 keV. Values for the end points are 6°Co 317.88-t-0.10 keV, 137Cs 511.63+__0.84 keV, 2°4T1 763.47+0.22 keV. The value obtained for 6 °Co may be compared with values averaging 315 keV previously obtained 11). For the lower-energy spectrum from 137Cs, the data obtained were consistent with the assumption of the shape factor (SILo) = (q2+2zp2). The value obtained for the end point is 2-3 keV lower than other recent determinations. Hsue, Langer and Tang 12) obtained a value of 514___1 keV, and Daniel and Schmitt [ref. 13)] obtained the same value with an error of ___2keV. The value obtained for the end-point energy of the 2 ogT1 spectrum is 763.47_ 0.22 keV, which is in good agreement with the value of 763.24___0.31 keV found by Park and Christmas 1). The value of b found, viz. 1.20, is to be compared with the value of 1.161 __+0.005 found by the same authors. However, b is quite sensitive to the energies of the beta rays studied. If measurements are confined to higher-energy electrons only, q is quite small, and b may undergo relatively large variations without affecting the shape factor markedly. The shape factor so modified accounts for the abnormally intense lower-energy
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J. L. WOLFSON AND A. J. COLLIER
p o r t i o n of the spectrum. The theoretical basis for the modification is, however, n o t firmly established, a n d accordingly some u n c e r t a i n t y attaches to a n e n d - p o i n t value based o n a n application of this shape factor. Some a d d i t i o n a l confidence m a y be gained i n the d e t e r m i n a t i o n b y a n e x a m i n a t i o n of how the e n d - p o i n t value o b t a i n e d varies with electron energy for b = 1. I n table 4, the results of e n d - p o i n t determinations are presented for electrons o f energies larger t h a n 400 keV, 472 keV, 545 keV TABLZ 4 ~°~T1end-point energy as a function of beta-ray energy for b = 1 End-point energy (keV)
Beta-ray energies larger than (keV)
run 1
run 2
mean
400 472 545 619
760.32 761.94 762.94 765.13
761.27 762.13 763.36 762.87
760.80 762.04 763.03 764.00
a n d 619 keV. Both runs are included, a n d i n all cases b = 1. E x a m i n a t i o n of the results indicates that the e n d - p o i n t energy shows a g r a d u a l increase as the data are successively restricted to higher-energy electrons. The results are consistent with a n endp o i n t between 763 a n d 764 keV.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)
J. J. H. Park and P. Christmas, Can. J. Phys. 45 (1967) 2621 J. L. Wolfson, W. J. King and J. J. H. Park, Can. J. Phys. 41 (1963) 1489 H. Paul, Nucl. Instr. 31 (1964) 307 J. S. Geiger, R. L. Graham and F. Brown, Can. J. Phys. 40 (1962) 1258 G. Murray, R. L Graham and J. S. Geiger, Nucl. Phys. 63 (1965) 353 E. R. Cohen and J. W. M. DuMond, Revs. Mod. Phys. 37 (1965) 537 W. Biahring, Nucl. Phys. 40 (1963) 472 F. B. Hildebrand, Introduction to numerical analysis (McGraw-Hill, New York, 1956) chapt. 7 C. P. Bhalla and M. E. Rose, ORNL Report No. 3207 (Office of Technical Services, Department of Commerce, Washington, D.C., 1961) W. Btihring, Nucl. Phys. 61 (1965) 110 C. M. Lederer, J. M. Hollander and I. Pedman, Table of isotopes, sixth ed. (Wiley, New York, 1967) S. T. Hsue, L. M. Langer and S. M. Tang, Nucl. Phys. 86 (1966) 47 H. Daniel and H. Schmitt, Z. Phys. 168 (1962) 292
1.E.2 I
Nuclear Physics All2 (1968) 156--160; (~) North-Holland Publishing Co., Amsterdam N o t to be reproduced b y p h o t o p r i n t or microfilm without written permission f r o m the publisher
PRECISION MEASUREMENT OF BETA-RAY END-POINT ENERGIES: 6°Co, lSTCs AND 2°~T1 J. L. WOLFSON and A. J. COLLIER Department of Physics, University of Saskatchewan, Regina Campus, Regina, Saskatchewan Received 30 January 1968
Abstract: Using a high resolving power, iron-free spectrometer, precise measurements have been performed of the end-point energies of the beta-ray spectra from ~°Co, 18vCs(low-energy spectrum) and ~°4T1.The values obtained were 317.884-0.10, 511.634-0.84, and 763.47±0.22 keV, respectively. E I
RADIOACTIVITY ~°Co, l~7Cs, 2°4T1;measured E/~•
[
1. Introduction The present paper gives an account of attempts to measure precisely the beta-ray end-point energies of the 60Co spectrum (318 keV), the lower-energy spectrum from 137Cs (512 keV) and the 2 O4T1 spectrum (763 keV). Recently, Park and Christmas 1) have reported a measurement of the end-point energy of the 204T1 beta spectrum performed on the same beta-ray spectrometer z) as the present measurement though under different conditions.
2. Apparatus and method 2.1. APPARATUS All sources were prepared by sublimation in vacuum on to backings of 200 #g/cm 2 aluminium foil. A backing of this thickness is transparent to electrons of energies greater than about 30 keV and is not expected to affect the measured energy of the end point if only the higher-energy electrons me used in the measurement. In these studies, only electrons of energies greater than half the end-point energy were so used. Park and Christmas 1) have actually shown that for 2°4T1 there is indeed little distortion of the spectrum to quite low energies for a source as thick as 150/zg/cm 2, which would also indicate that for 2°4T1 scattering from a backing of approximately this thickness is not serious. Care was taken to investigate sources of scattering within the spectrometer which t The experimental work on which this paper is based was performed by one of us (J.L.W.) during 1964 at the National Research Council of Canada laboratories, Division of Applied Physics, Ottawa, Canada. 156
t-RAY END-POINT ENERGIES
157
could have given rise to spurious results and to demonstrate that their effects were negligible. A study of the response of the detector, a proportional counter, gave no evidence of energy dependence. No corrections were applied for instrumental resolution. Following the suggestion of Paul 3), the centre of gravity of the calibration line profile was employed as calibration point. For calibration, the K internal conversion line of the 661.595_+0.076 keV gamma-ray transition 4) in 137Ba was used for the 1a7Cs and z O4Tl spectra, and the K internal conversion line of the 1173.226_+0.040 keV gamma-ray transition 5) in 6°Ni for the 6°Co spectrum. 2.2. T R E A T M E N T
OF DATA
A program was written to analyse all data in the form of Fermi-Kurie plots, which were fitted to straight lines by the method of least-squares. Since the spectrometer was an iron-free instrument, the voltage drop across a standard resistor in series with the magnet windings was proportional to electron momentum. Along with the calibration point, these voltage measurements were included in the program as well as the observed counting rate N at each of the voltages. The relationship between magnetic rigidity Bp and electron kinetic energy was also included using values for the fundamental constants as given by Cohen and DuMond 6). The quantity then calculated was
N_
l
3FoLo(S/Lo)A as a linear function of beta-ray energy in keV. The quantity p is electron momentum in units of me, (N/p) is proportional to the number of electrons emitted per unit momentum interval. Fo is the Fermi function, Lo an electron radial wave function and S/L 0 a shape distribution factor defined by Biihring 7). Each experimental point was weighted inversely as the square of its error. In general, the methods of Hildebrand s) were used in the treatment of the experimental data. Values of FoL 0 were calculated from the tables of Bhalla and Rose 9), and fourthorder interpolation used when interpolation was required. The influence of screening was taken into account using the tables of Biihring 10), though screening effects, as it turns out, are quite small. For distributions of the allowed shape, (SILo) is constant 7), while for first-forbidden unique spectra (SILo) is expected to have the value (q2+22pZ), where q is neutrino energy in mo ca units, 22 a function defined by BiihringT), which we calculated from the tables of Bhalla and Rose 9). Again, fourth-order interpolation was used when interpolation was required, and the values of 22 were corrected for screening using the tables of Biihring 10). 3. Results
For 6 ° C o , the data above 160 keV were fitted to the allowed shape with results shown in table 1. For ~37Cs, the transition ~ ) in ~37Ba is expected to be first-order
158
J. L. WOLFSONAND A. J. COLLIER
forbidden with unique shape. The shape factor is expected to be (p2 _[_22q2), and the data above 265 keV were fitted using this shape factor; the results are shown in table 2. The contribution from the higher-energy distribution had first to be subtracted, and this contribution was determined by fitting the data above 650 keV with a shape factor (q2+kp2) and an end-point energy 12) of 1.176 MeV. The value chosen for k was 0.010, the mean of the value found by Hsue, Langer and Tang 12) and the value 0.015 found by Daniel and Schmitt 13). The uncertainty in determining the contribution by the higher-energy spectrum is the largest source of error in measuring the end-point energy of the lower-energy spectrum. TABLE 1
Results for n°Co Run
1 2 3
Slit width (cm) 0.087 0.132 0.175
Transmission ( 70 o f 4 ~ sr)
Line width (~)
0.33 0.10 0.40 0.12 0.70 0.25 Mean value is 317.884-0.10 keV.
End-point energy (keV) 318.17-4-0.20 317.73 4-0.16 317.83 4-0.t4
Source: 2 cm×0.08 cm; 9/~g/cm2; 150 #Ci.
TABLE 2 Results for 13~Cs Run
1 2
Slit width (cm) 0.087 0.175
Transmission ( % o f 4z~ st)
Line width (%)
0.33 0.10 0.70 0.25 Mean value is 511.63 4-0.67 keV.
End-point energy (keV) 511.83 4-0.95 511.43 4-0.95
Source: 2 cm x0.08 cm; 4#g/cm2; 20/~Ci.
For 2°*T1, the shape factor (S/Lo) is expected to be (q2 +22p2) as the transition 11) is from a 2 - level in Z°4T1 to a 0 + level in z°4Pb. However, it is known that the spectrum is not adequately described by such a distribution [see, for instance, Park and Christmas 1)]. The spectrum has an excess of electrons at the lower energies. Park and Christmas 1) were able to obtain an adequate fit with a shape factor (bq z +22p z) with b determined from experiment equal to 1.161 +0.005. Park and Christmas also offer a theoretical argument for assigning b a value of about 1.1 rather than unity. The results obtained in the present experiment confirm those of Park and Christmas in that an excess of lower energy electron was observed, and a good fit could not be obtained with b equal to unity. A program was instituted to search for the value of b giving the best fit in each of the two runs. The criterion chosen for best fit was a mini-
159
fl-RAY END-POINT ENERGIES
mum value for the sum of the squares of the residuals. Only data for electrons of energies above 400 keV were used. The results are given in table 3. TABLE 3
Results for ~°4T1 Run 1 2
Slit width (cm) 0.13 0.25
Transmission ( 70 of 4n sr)
Line width (70)
End-point energy (keV)
0.40 0.13 763.334-0.25 0.70 0.27 763.674-0.30 Mean value is 763.47:t:0.20 keV.
b 1.203 1.191
Source: 2 cm×0.12 cm; 35/zg/cm2; 2.4 #Ci. 4. Discussion The errors quoted for the values of the end points in tables 1-3 do not include errors associated with the calibration lines. For the 6°C0 and 137Cs beta-ray end points, inclusion of these uncertainties does not affect the errors quoted appreciably. For the z°gTI beta-ray end point, the error is increased slightly. For the 137Cs beta-ray endpoint energy, there is an additional source of error associated with the shape factor of the high-energy component. As mentioned in subsect. 3.2, the shape factor used was (q2 +kpZ) with k = 0.010. Had either the value of Hsue et al. 12) (k = 0.004) or that of Daniel and Schmitt 13) (k = 0.015) been used, the end-point energy would have been changed by about 0.5 keV. Values for the end points are 6°Co 317.88-t-0.10 keV, 137Cs 511.63+__0.84 keV, 2°4T1 763.47+0.22 keV. The value obtained for 6 °Co may be compared with values averaging 315 keV previously obtained 11). For the lower-energy spectrum from 137Cs, the data obtained were consistent with the assumption of the shape factor (SILo) = (q2+2zp2). The value obtained for the end point is 2-3 keV lower than other recent determinations. Hsue, Langer and Tang 12) obtained a value of 514___1 keV, and Daniel and Schmitt [ref. 13)] obtained the same value with an error of ___2keV. The value obtained for the end-point energy of the 2 ogT1 spectrum is 763.47_ 0.22 keV, which is in good agreement with the value of 763.24___0.31 keV found by Park and Christmas 1). The value of b found, viz. 1.20, is to be compared with the value of 1.161 __+0.005 found by the same authors. However, b is quite sensitive to the energies of the beta rays studied. If measurements are confined to higher-energy electrons only, q is quite small, and b may undergo relatively large variations without affecting the shape factor markedly. The shape factor so modified accounts for the abnormally intense lower-energy
160
J. L. WOLFSON AND A. J. COLLIER
p o r t i o n of the spectrum. The theoretical basis for the modification is, however, n o t firmly established, a n d accordingly some u n c e r t a i n t y attaches to a n e n d - p o i n t value based o n a n application of this shape factor. Some a d d i t i o n a l confidence m a y be gained i n the d e t e r m i n a t i o n b y a n e x a m i n a t i o n of how the e n d - p o i n t value o b t a i n e d varies with electron energy for b = 1. I n table 4, the results of e n d - p o i n t determinations are presented for electrons o f energies larger t h a n 400 keV, 472 keV, 545 keV TABLZ 4 ~°~T1end-point energy as a function of beta-ray energy for b = 1 End-point energy (keV)
Beta-ray energies larger than (keV)
run 1
run 2
mean
400 472 545 619
760.32 761.94 762.94 765.13
761.27 762.13 763.36 762.87
760.80 762.04 763.03 764.00
a n d 619 keV. Both runs are included, a n d i n all cases b = 1. E x a m i n a t i o n of the results indicates that the e n d - p o i n t energy shows a g r a d u a l increase as the data are successively restricted to higher-energy electrons. The results are consistent with a n endp o i n t between 763 a n d 764 keV.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)
J. J. H. Park and P. Christmas, Can. J. Phys. 45 (1967) 2621 J. L. Wolfson, W. J. King and J. J. H. Park, Can. J. Phys. 41 (1963) 1489 H. Paul, Nucl. Instr. 31 (1964) 307 J. S. Geiger, R. L. Graham and F. Brown, Can. J. Phys. 40 (1962) 1258 G. Murray, R. L Graham and J. S. Geiger, Nucl. Phys. 63 (1965) 353 E. R. Cohen and J. W. M. DuMond, Revs. Mod. Phys. 37 (1965) 537 W. Biahring, Nucl. Phys. 40 (1963) 472 F. B. Hildebrand, Introduction to numerical analysis (McGraw-Hill, New York, 1956) chapt. 7 C. P. Bhalla and M. E. Rose, ORNL Report No. 3207 (Office of Technical Services, Department of Commerce, Washington, D.C., 1961) W. Btihring, Nucl. Phys. 61 (1965) 110 C. M. Lederer, J. M. Hollander and I. Pedman, Table of isotopes, sixth ed. (Wiley, New York, 1967) S. T. Hsue, L. M. Langer and S. M. Tang, Nucl. Phys. 86 (1966) 47 H. Daniel and H. Schmitt, Z. Phys. 168 (1962) 292