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Interaction of photons with some solutions
Radiation Physics and Chemistry 61 (2001) 537–540
Interaction of photons with some solutions Kulwant Singh*, Gagandeep Kaur, G.K. Sandhu, B.S. Lark Department of Physics, Nuclear Spectroscopy Laboratory, Guru Nanak Dev University, Amritsar-143005, India
Abstract The linear attenuation coefficients in aqueous solutions of some chlorides and sulphates, viz. MgCl2 6H2O, CaCl2, SrCl2 6H2O, BaCl2 2H2O, Na2SO4, K2SO4 and MgSO4 7H2O were determined at 81, 356, 511, 662, 1173 and 1332 keV by the g-ray transmission method in a good geometry setup. From the precision measured densities of these solutions, mass attenuation coefficients were then obtained which varied systematically with the corresponding changes in the concentrations (g/cm3) of these solutions. A comparison between experimental and theoretical values of attenuation coefficients has shown that the study has potential application for the determination of attenuation coefficients of solid solutes from their solutions without obtaining them in pure crystalline form. r 2001 Elsevier Science Ltd. All rights reserved. Keywords: Attenuation coefficients; Aqueous solution; Solvent; Salts
1. Introduction The study of attenuation coefficients is potentially useful in the development of semi-empirical formulations of high accuracy, possibly along the lines detailed by Jackson and Hawkes (1981). Hubbell (1982) and Seltzer (1993) have compiled mass attenuation coefficients for a large number of compounds and mixtures of dosimetric and biological importance. An updated version of the attenuation coefficients for elements having atomic numbers from 1 to 92 and 48 additional substances of dosimetric interest has recently been compiled by Hubbell and Seltzer (1995). Most of the previous studies for the determination of these coefficients have been concerned with crystalline samples in the solid form. In their pioneer work, Teli et al. (1994) have determined the g-ray attenuation coefficients in dilute solutions of magnesium chloride. Gerward (1996) has determined linear and mass attenuation coefficients in the general case as well as in the limit of extreme dilution and in this way developed the theory of X-ray and g-ray attenuation in solutions. *Corresponding author. Tel.: +91-183-258840; fax: +91183-258819. E-mail address: k [email protected] (K. Singh).
As a sequel to our previous study (Singh et al., 1998; Gagandeep et al., 2000) on the absorption properties of some solutes in water at different concentrations, the attenuation coefficients of MgCl2 6H2O, CaCl2, SrCl2 6H2O, BaCl2 2H2O, Na2SO4, K2SO4 and MgSO4 7H2O at six different g-ray energies in an aqueous medium as a function of concentration are reported in this paper. Densities which have been experimentally determined, are required for the estimation of these mass attenuation coefficients.
2. Theory According to the Beer–Lambert’s law, a narrow beam linear attenuation coefficient m (cm@1), is given by the following relation: I ¼ I0 e@mx
ð1Þ
where I0 and I are the incident and transmitted photon intensities, respectively, and x the thickness of the material. A coefficient more accurately characterizing a given solution is the density-independent mass attenuation coefficient m=r (cm2/g). I ¼ I0 e@ðm=rÞrx :
0969-806X/01/$ - see front matter r 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 9 - 8 0 6 X ( 0 1 ) 0 0 3 2 5 - 5
ð2Þ
538
K. Singh et al. / Radiation Physics and Chemistry 61 (2001) 537–540
Table 1 Mass attenuation coefficients of aqueous solutions of some compoundsa Solution
Magnesium chloride MgCl2 6H2O
Calcium chloride CaCl2
Strontium chloride SrCl2 6H2O
Barium chloride BaCl2 2H2O
Sodium sulphate Na2SO4
Density of the solution (g/cm3)
Mass attenuation coefficient m=r (cm2/g)
Conc. (g/cm3)
1.021200
0.05
1.042107
0.10
1.064453
0.15
1.083726
0.20
1.100204
0.25
1.035680
0.05
1.072139
0.10
1.108080
0.15
1.141173
0.20
1.184648
0.25
1.050231
0.05
1.077728
0.10
1.096837
0.15
1.124722
0.20
1.190348
0.25
1.031267
0.05
1.067021
0.10
1.105524
0.15
1.137737
0.20
1.172951
0.25
1.0410180
0.05
1.0818457
0.10
1.1264854
0.15
81 keV
356 keV
511 keV
662 keV
1173 keV
1332 keV
a
0.184
0.110
0.095
0.085
0.065
0.061
b a b a b a b a b
0.184 0.185 0.186 0.186 0.187 0.187 0.188 0.188 0.189
0.111 0.110 0.111 0.110 0.110 0.110 0.110 0.109 0.110
0.096 0.095 0.095 0.095 0.095 0.095 0.095 0.094 0.095
0.086 0.085 0.085 0.085 0.085 0.084 0.085 0.084 0.085
0.065 0.064 0.065 0.064 0.065 0.064 0.065 0.064 0.065
0.061 0.060 0.061 0.060 0.061 0.060 0.061 0.059 0.060
a
0.187
0.110
0.095
0.085
0.065
0.061
b a b a b a b a b
0.188 0.192 0.193 0.197 0.198 0.201 0.202 0.205 0.206
0.111 0.110 0.110 0.109 0.110 0.109 0.109 0.108 0.109
0.095 0.095 0.095 0.094 0.095 0.094 0.094 0.094 0.094
0.085 0.085 0.085 0.084 0.084 0.084 0.084 0.084 0.084
0.065 0.064 0.065 0.064 0.064 0.064 0.064 0.064 0.064
0.061 0.060 0.061 0.060 0.060 0.060 0.060 0.060 0.060
a
0.203
0.110
0.095
0.085
0.065
0.061
b a b a b a b a b
0.204 0.222 0.222 0.239 0.240 0.255 0.255 0.269 0.270
0.111 0.110 0.111 0.110 0.111 0.110 0.110 0.110 0.110
0.096 0.095 0.095 0.095 0.095 0.095 0.095 0.094 0.095
0.085 0.085 0.085 0.085 0.085 0.084 0.084 0.084 0.084
0.065 0.064 0.065 0.064 0.064 0.064 0.064 0.063 0.064
0.061 0.060 0.061 0.060 0.060 0.060 0.060 0.059 0.060
a
0.290
0.115
0.099
0.088
0.067
0.062
b a b a b a b a b
0.291 0.397 0.397 0.502 0.502 0.602 0.603 0.702 0.703
0.115 0.120 0.120 0.125 0.125 0.129 0.130 0.134 0.1341
0.099 0.102 0.102 0.105 0.106 0.108 0.109 0.111 0.112
0.088 0.090 0.091 0.093 0.094 0.096 0.096 0.098 0.099
0.067 0.069 0.069 0.071 0.071 0.072 0.072 0.074 0.074
0.063 0.064 0.064 0.066 0.066 0.068 0.068 0.070 0.070
a
0.182
0.111
0.095
0.085
0.065
0.061
b a b a
0.183 0.182 0.184 0.181
0.111 0.109 0.110 0.109
0.096 0.095 0.095 0.094
0.085 0.085 0.085 0.084
0.065 0.064 0.065 0.064
0.061 0.061 0.061 0.060
539
K. Singh et al. / Radiation Physics and Chemistry 61 (2001) 537–540 Table 1 (continued) Solution
Potassium sulphate K2SO4
Magnesium sulphate MgSO4 7H2O
a
Density of the solution (g/cm3)
Mass attenuation coefficient m=r (cm2/g)
Conc. (g/cm3)
81 keV
356 keV
511 keV
662 keV
1173 keV
1332 keV
b
0.184
0.110
0.095
0.085
0.064
0.060
1.0365001
0.05
a
0.185
0.110
0.095
0.085
0.065
0.061
1.0736203
0.10
b a b
0.186 0.187 0.189
0.111 0.110 0.110
0.096 0.095 0.095
0.085 0.085 0.085
0.065 0.065 0.065
0.061 0.060 0.061
1.0212212
0.05
a
0.184
0.110
0.096
0.085
0.065
0.061
1.0415675
0.10
1.0623991
0.15
b a b a b
0.184 0.184 0.185 0.185 0.186
0.111 0.111 0.111 0.110 0.111
0.096 0.095 0.096 0.095 0.096
0.086 0.085 0.086 0.085 0.086
0.065 0.065 0.065 0.065 0.065
0.061 0.061 0.061 0.061 0.061
Note: a stands for experimental values, and b stands for theoretical
For a binary mixture, the mass attenuation coefficient of the solution is given by the mixture rule: m m m m wS þ @ ð3Þ ¼ r r W r S r W where ðm=rÞs and ðm=rÞw are the mass attenuation coefficients of the solute and water, respectively, wS the weight fraction of the solute and r the density of the solution (g/cm3). A plot of m=r versus wS gives a straight line with the intercept ðm=rÞW and slope ½ðm=rÞs @ðm=rÞW . The value of the slope may then be used to calculate the mass attenuation coefficient of the solid solute.
3. Experimental details The experimental setup was similar to the one used in our earlier paper (Singh et al., 1998). Radioactive sources 137Cs, 60Co, 133Ba and 22Na of strength 5 mCi each, were obtained from the Bhabha Atomic Research Centre, Trombay, Bombay, India. A 1.5 5 in. NaI(Tl) crystal having an energy resolution of 12% at 662 keV was used for the measurement of attenuation coefficients. The signal from the detector after a suitable amplification was recorded by means of an EGaG ORTEC 4 K MCA plug-in-card coupled to a PC/AT. The transmitted intensity was measured by gating the channels at the full-width at half-maximum position of the photopeak to minimise the contributions of both small angle and multiple scattering events to the measured intensity.
Fig. 1. Plot of mass attenuation coefficient versus weight fraction of the solute MgO2 6H2O.
4. Conclusions The mass attenuation coefficients of some aqueous solutions in the concentration range of 0.05 to 0.25 g/ cm3, are summarised in Table 1. These values have been compared with the values obtained from the use of the XCOM programme developed by Berger and Hubbell (1987). The values systematically increase with an
540
K. Singh et al. / Radiation Physics and Chemistry 61 (2001) 537–540
Table 2 Derived values of mass attenuation coefficients of the solutes of alkaline earth chlorides Solute
m=r (cm2/g) values of solid solutes
Density of the salt (g/cm3)
81 keV
356 keV
511 keV
662 keV
1173 keV
1332 keV
Magnesium chloride MgCl2 6H2O
1.569
a
0.214
0.105
0.090
0.080
0.057
0.053
Calcium chloride CaCl2
2.150
b a
0.213 0.299
0.105 0.099
0.090 0.085
0.081 0.076
0.061 0.057
0.058 0.054
Strontium chloride SrCl2 6H2O
1.980
b a
0.2980 0.616
0.099 0.105
0.085 0.089
0.076 0.079
0.057 0.057
0.054 0.053
Barium chloride BaCl2 2H2O
3.097
b a
0.616 2.259
0.106 0.128
0.089 0.091
0.078 0.077
0.059 0.056
0.055 0.053
b
2.226
0.127
0.093
0.078
0.055
0.052
increase in the concentration of the solute and agree well with those obtained from tabulated values. For a given solution, the slope of m=r as a function of weight fraction of the solute gives practically perfect linear plots (a typical plot for MgCl2 6H2O, has been given as an illustration in Fig. 1). From the slope ½ðm=rÞs @ðm=rÞW , the mass attenuation coefficient of the corresponding solute in aqueous solution was obtained. The ðm=rÞS values for the salts under study have been reported in Table 2 and are compared with those obtained from XCOM calculations. The agreement is excellent and the difference is within the experimental uncertainty. It is expected that the data presented in this paper will be useful in view of their importance in medical and biological applications. References Berger, M.J., Hubbell, J.H., 1987. XCOM: photon cross sections on a personal computer. National Bureau of Standards (now National Institute of Standards and Technology, NIST) Internal Report NBSIR 87-3597.
Gagandeep, K., Singh, K., Lark, B.S., Sahota, H.S., 2000. Attenuation measurements in solutions of some carbohydrates. Nucl. Sci. Eng. 134, 208–217. Gerward, L., 1996. On the attenuation of X-rays and g-rays in dilute solutions. Radiat. Phys. Chem. 48, 697–699. Hubbell, J.H., 1982. Photon mass attenuation and energy absorption coefficients from 1 keV to 20 MeV. Int. J. Appl. Radiat. Isot. 33, 1269–1290. Hubbell, J.H., Seltzer, S.M., 1995. Tables of X-ray mass attenuation coefficients and mass energy-absorption coefficients 1 keV–20 MeV for elements Z=1 to 92 and 48 an additional substances of dosimetric interest. NISTIR, 5632. Jackson, D.F., Hawkes, D.J., 1981. X-ray attenuation coefficients of elements and mixtures. Phy. Rep. 70, 169–233. Seltzer, S.M., 1993. Calculation of photon mass energy transfer and mass energy absorption coefficients. Rad. Res. 136, 147–170. Singh, Kulwant, Kaur, Gagandeep, Kumar, V., Dhami, A.K., Lark, B.S., 1998. Measurement of attenuation coefficients of some dilute solutions at 662 keV. Radiat. Phys. Chem. 53, 123–126. Teli, M.T., Chaudhary, L.M., Malode, S.S., 1994. Study of absorption of 123 keV gamma radiation by dilute solutions of magnesium chloride. Nucl. Instru. Meth. 346, 220–224.
Interaction of photons with some solutions Kulwant Singh*, Gagandeep Kaur, G.K. Sandhu, B.S. Lark Department of Physics, Nuclear Spectroscopy Laboratory, Guru Nanak Dev University, Amritsar-143005, India
Abstract The linear attenuation coefficients in aqueous solutions of some chlorides and sulphates, viz. MgCl2 6H2O, CaCl2, SrCl2 6H2O, BaCl2 2H2O, Na2SO4, K2SO4 and MgSO4 7H2O were determined at 81, 356, 511, 662, 1173 and 1332 keV by the g-ray transmission method in a good geometry setup. From the precision measured densities of these solutions, mass attenuation coefficients were then obtained which varied systematically with the corresponding changes in the concentrations (g/cm3) of these solutions. A comparison between experimental and theoretical values of attenuation coefficients has shown that the study has potential application for the determination of attenuation coefficients of solid solutes from their solutions without obtaining them in pure crystalline form. r 2001 Elsevier Science Ltd. All rights reserved. Keywords: Attenuation coefficients; Aqueous solution; Solvent; Salts
1. Introduction The study of attenuation coefficients is potentially useful in the development of semi-empirical formulations of high accuracy, possibly along the lines detailed by Jackson and Hawkes (1981). Hubbell (1982) and Seltzer (1993) have compiled mass attenuation coefficients for a large number of compounds and mixtures of dosimetric and biological importance. An updated version of the attenuation coefficients for elements having atomic numbers from 1 to 92 and 48 additional substances of dosimetric interest has recently been compiled by Hubbell and Seltzer (1995). Most of the previous studies for the determination of these coefficients have been concerned with crystalline samples in the solid form. In their pioneer work, Teli et al. (1994) have determined the g-ray attenuation coefficients in dilute solutions of magnesium chloride. Gerward (1996) has determined linear and mass attenuation coefficients in the general case as well as in the limit of extreme dilution and in this way developed the theory of X-ray and g-ray attenuation in solutions. *Corresponding author. Tel.: +91-183-258840; fax: +91183-258819. E-mail address: k [email protected] (K. Singh).
As a sequel to our previous study (Singh et al., 1998; Gagandeep et al., 2000) on the absorption properties of some solutes in water at different concentrations, the attenuation coefficients of MgCl2 6H2O, CaCl2, SrCl2 6H2O, BaCl2 2H2O, Na2SO4, K2SO4 and MgSO4 7H2O at six different g-ray energies in an aqueous medium as a function of concentration are reported in this paper. Densities which have been experimentally determined, are required for the estimation of these mass attenuation coefficients.
2. Theory According to the Beer–Lambert’s law, a narrow beam linear attenuation coefficient m (cm@1), is given by the following relation: I ¼ I0 e@mx
ð1Þ
where I0 and I are the incident and transmitted photon intensities, respectively, and x the thickness of the material. A coefficient more accurately characterizing a given solution is the density-independent mass attenuation coefficient m=r (cm2/g). I ¼ I0 e@ðm=rÞrx :
0969-806X/01/$ - see front matter r 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 9 - 8 0 6 X ( 0 1 ) 0 0 3 2 5 - 5
ð2Þ
538
K. Singh et al. / Radiation Physics and Chemistry 61 (2001) 537–540
Table 1 Mass attenuation coefficients of aqueous solutions of some compoundsa Solution
Magnesium chloride MgCl2 6H2O
Calcium chloride CaCl2
Strontium chloride SrCl2 6H2O
Barium chloride BaCl2 2H2O
Sodium sulphate Na2SO4
Density of the solution (g/cm3)
Mass attenuation coefficient m=r (cm2/g)
Conc. (g/cm3)
1.021200
0.05
1.042107
0.10
1.064453
0.15
1.083726
0.20
1.100204
0.25
1.035680
0.05
1.072139
0.10
1.108080
0.15
1.141173
0.20
1.184648
0.25
1.050231
0.05
1.077728
0.10
1.096837
0.15
1.124722
0.20
1.190348
0.25
1.031267
0.05
1.067021
0.10
1.105524
0.15
1.137737
0.20
1.172951
0.25
1.0410180
0.05
1.0818457
0.10
1.1264854
0.15
81 keV
356 keV
511 keV
662 keV
1173 keV
1332 keV
a
0.184
0.110
0.095
0.085
0.065
0.061
b a b a b a b a b
0.184 0.185 0.186 0.186 0.187 0.187 0.188 0.188 0.189
0.111 0.110 0.111 0.110 0.110 0.110 0.110 0.109 0.110
0.096 0.095 0.095 0.095 0.095 0.095 0.095 0.094 0.095
0.086 0.085 0.085 0.085 0.085 0.084 0.085 0.084 0.085
0.065 0.064 0.065 0.064 0.065 0.064 0.065 0.064 0.065
0.061 0.060 0.061 0.060 0.061 0.060 0.061 0.059 0.060
a
0.187
0.110
0.095
0.085
0.065
0.061
b a b a b a b a b
0.188 0.192 0.193 0.197 0.198 0.201 0.202 0.205 0.206
0.111 0.110 0.110 0.109 0.110 0.109 0.109 0.108 0.109
0.095 0.095 0.095 0.094 0.095 0.094 0.094 0.094 0.094
0.085 0.085 0.085 0.084 0.084 0.084 0.084 0.084 0.084
0.065 0.064 0.065 0.064 0.064 0.064 0.064 0.064 0.064
0.061 0.060 0.061 0.060 0.060 0.060 0.060 0.060 0.060
a
0.203
0.110
0.095
0.085
0.065
0.061
b a b a b a b a b
0.204 0.222 0.222 0.239 0.240 0.255 0.255 0.269 0.270
0.111 0.110 0.111 0.110 0.111 0.110 0.110 0.110 0.110
0.096 0.095 0.095 0.095 0.095 0.095 0.095 0.094 0.095
0.085 0.085 0.085 0.085 0.085 0.084 0.084 0.084 0.084
0.065 0.064 0.065 0.064 0.064 0.064 0.064 0.063 0.064
0.061 0.060 0.061 0.060 0.060 0.060 0.060 0.059 0.060
a
0.290
0.115
0.099
0.088
0.067
0.062
b a b a b a b a b
0.291 0.397 0.397 0.502 0.502 0.602 0.603 0.702 0.703
0.115 0.120 0.120 0.125 0.125 0.129 0.130 0.134 0.1341
0.099 0.102 0.102 0.105 0.106 0.108 0.109 0.111 0.112
0.088 0.090 0.091 0.093 0.094 0.096 0.096 0.098 0.099
0.067 0.069 0.069 0.071 0.071 0.072 0.072 0.074 0.074
0.063 0.064 0.064 0.066 0.066 0.068 0.068 0.070 0.070
a
0.182
0.111
0.095
0.085
0.065
0.061
b a b a
0.183 0.182 0.184 0.181
0.111 0.109 0.110 0.109
0.096 0.095 0.095 0.094
0.085 0.085 0.085 0.084
0.065 0.064 0.065 0.064
0.061 0.061 0.061 0.060
539
K. Singh et al. / Radiation Physics and Chemistry 61 (2001) 537–540 Table 1 (continued) Solution
Potassium sulphate K2SO4
Magnesium sulphate MgSO4 7H2O
a
Density of the solution (g/cm3)
Mass attenuation coefficient m=r (cm2/g)
Conc. (g/cm3)
81 keV
356 keV
511 keV
662 keV
1173 keV
1332 keV
b
0.184
0.110
0.095
0.085
0.064
0.060
1.0365001
0.05
a
0.185
0.110
0.095
0.085
0.065
0.061
1.0736203
0.10
b a b
0.186 0.187 0.189
0.111 0.110 0.110
0.096 0.095 0.095
0.085 0.085 0.085
0.065 0.065 0.065
0.061 0.060 0.061
1.0212212
0.05
a
0.184
0.110
0.096
0.085
0.065
0.061
1.0415675
0.10
1.0623991
0.15
b a b a b
0.184 0.184 0.185 0.185 0.186
0.111 0.111 0.111 0.110 0.111
0.096 0.095 0.096 0.095 0.096
0.086 0.085 0.086 0.085 0.086
0.065 0.065 0.065 0.065 0.065
0.061 0.061 0.061 0.061 0.061
Note: a stands for experimental values, and b stands for theoretical
For a binary mixture, the mass attenuation coefficient of the solution is given by the mixture rule: m m m m wS þ @ ð3Þ ¼ r r W r S r W where ðm=rÞs and ðm=rÞw are the mass attenuation coefficients of the solute and water, respectively, wS the weight fraction of the solute and r the density of the solution (g/cm3). A plot of m=r versus wS gives a straight line with the intercept ðm=rÞW and slope ½ðm=rÞs @ðm=rÞW . The value of the slope may then be used to calculate the mass attenuation coefficient of the solid solute.
3. Experimental details The experimental setup was similar to the one used in our earlier paper (Singh et al., 1998). Radioactive sources 137Cs, 60Co, 133Ba and 22Na of strength 5 mCi each, were obtained from the Bhabha Atomic Research Centre, Trombay, Bombay, India. A 1.5 5 in. NaI(Tl) crystal having an energy resolution of 12% at 662 keV was used for the measurement of attenuation coefficients. The signal from the detector after a suitable amplification was recorded by means of an EGaG ORTEC 4 K MCA plug-in-card coupled to a PC/AT. The transmitted intensity was measured by gating the channels at the full-width at half-maximum position of the photopeak to minimise the contributions of both small angle and multiple scattering events to the measured intensity.
Fig. 1. Plot of mass attenuation coefficient versus weight fraction of the solute MgO2 6H2O.
4. Conclusions The mass attenuation coefficients of some aqueous solutions in the concentration range of 0.05 to 0.25 g/ cm3, are summarised in Table 1. These values have been compared with the values obtained from the use of the XCOM programme developed by Berger and Hubbell (1987). The values systematically increase with an
540
K. Singh et al. / Radiation Physics and Chemistry 61 (2001) 537–540
Table 2 Derived values of mass attenuation coefficients of the solutes of alkaline earth chlorides Solute
m=r (cm2/g) values of solid solutes
Density of the salt (g/cm3)
81 keV
356 keV
511 keV
662 keV
1173 keV
1332 keV
Magnesium chloride MgCl2 6H2O
1.569
a
0.214
0.105
0.090
0.080
0.057
0.053
Calcium chloride CaCl2
2.150
b a
0.213 0.299
0.105 0.099
0.090 0.085
0.081 0.076
0.061 0.057
0.058 0.054
Strontium chloride SrCl2 6H2O
1.980
b a
0.2980 0.616
0.099 0.105
0.085 0.089
0.076 0.079
0.057 0.057
0.054 0.053
Barium chloride BaCl2 2H2O
3.097
b a
0.616 2.259
0.106 0.128
0.089 0.091
0.078 0.077
0.059 0.056
0.055 0.053
b
2.226
0.127
0.093
0.078
0.055
0.052
increase in the concentration of the solute and agree well with those obtained from tabulated values. For a given solution, the slope of m=r as a function of weight fraction of the solute gives practically perfect linear plots (a typical plot for MgCl2 6H2O, has been given as an illustration in Fig. 1). From the slope ½ðm=rÞs @ðm=rÞW , the mass attenuation coefficient of the corresponding solute in aqueous solution was obtained. The ðm=rÞS values for the salts under study have been reported in Table 2 and are compared with those obtained from XCOM calculations. The agreement is excellent and the difference is within the experimental uncertainty. It is expected that the data presented in this paper will be useful in view of their importance in medical and biological applications. References Berger, M.J., Hubbell, J.H., 1987. XCOM: photon cross sections on a personal computer. National Bureau of Standards (now National Institute of Standards and Technology, NIST) Internal Report NBSIR 87-3597.
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