Appl. Radiat. I.sor. Vol. 38, No. 12, pp. 1027-1031, 1987

hr. J. Rodiar. Appl. Inslrum. Par! A Printed in Great Britam. All rights reserved

0883-28X9/87 $3.00 + 0.00 Copyright c’ 1987 Pergamon Journals Ltd

A Method For Calculating Mass-Attenuation

Coefficients of Beta Particles

BHUPENDER SINGH and R. K. BATRA

Physics Department, Punjab Agricultural University, Ludhiana-141 004, India

(Received 19 November 1986; in revisedform 24 April 1987)

A new method to calculate mass-attenuation coefficients of /I and /I + particles of energies from 0.25 to 5.0 Met’ for elemental absorbers is described. This method is based upon the interactions of fly and p + particles with matter and the use of the varying transmission limit for different absorbers, The root-mean-square (r.m.s.) deviation of theory from 61 electron and 30 positron measurements in all elements is 6.5 and 9.7% respectively. Attenuation coefficients in rare earths calculated by this method are in excellent agreement with earlier measurements.

Introduction

Mass-attenuation coefficients of p _ and /I + particles in various media are useful for dosimetry studies and many nuclear physics experiments. These beta par- ticle coefficients have been determined at different energies for several absorbers, by measuring their transmissions through foils (Katz and Penfold, 1952; Cosslett and Thomas. 1964). The semi-log plot of the measured relative transmission was reported to be linear down to a few percent by many workers (Katz and Penfold, 1952; Cosslett and Thomas, 1964; Daddi and D’Angelo, 1963; Takhar, 1967). The observed transmission curves for j3 particles are of characteristic exponential shape. The shape of a transmission curve depends somwhat on the geo- metrical conditions. For instance, a plot of the meas- ured relative intensities of p and p’ particles at different energies vs thickness traversed in Al, Sn and Pb are shown in Fig. 1. Curve (b) in Fig. 1 depicts the transmission of p - and p + in Al and Pb around 1.8-MeV, due to Takhar (1967). Curve 1 of Fig. 1 shows the transmission of 0.324-MeV positrons in Al as reported by Patrick and Rupaal (1971). Spanel et al. (1973) studied the penetration of 0.544MeV posi- trons through thin metallic foils under the vacuum (Z 10 -3 torr). Such positron transmission curves in AI and Sn are displayed in curves 2 and 3 of Fig. 1. These studies reveal that the positron transmission is linear down to 0.1% and the nature of absorption of /? particles is approximately exponential. Recently, Nathu Ram et a/. (1982) have studied the trans- mission of /? particles of end point energies from 0.167 to 3.60 MeV in absorbers whose atomic num-

ber varied from 4 to 82. The experiments on positron attenuation are relatively scarce at present. Further, these workers developed the empirical relations suit- able to the experimental geometries used.

It is the usual practice to evaluate the mass- attenuation coefficient from the linear portion of measured transmission vs thickness curve. This ex- ponential behaviour of the attenuation of fi -- and B ’ particles, over a limited penetration range, is of the same form as that of photons. Following the nomen- clature due to Hubbell (1982), the mass-attenuation coefficient for photons is a measure of the average number of interactions between incident photons and matter that occur in a given mass per unit area thickness of material encountered. In the approxi- mate sense, we are also using the validity of the definition of photon mass-attenuation for p _ and /i + particles.

So far, there have been no attempts to understand the measured mass-attenuation coefficients for elec- trons and positrons. Recently, Batra and Sehgal (1981) have developed a theory to calculate the practical range of electrons and postrons based on the combined effects of their elastic and inelastic interaction with matter. When penetration param- eters such as range and the mass-attenuation coefficient of these particles in absorbers are meas- ured experimentally from the transmission curves, which depend upon the interplay of the elastic and inelastic interactions, it is thought that the practical range may be related to the mass-attenuation coefficient or vice versa. This paper contains a new method to calculate the mass-attenuation coefficient making use of the recent theory (Batra and Sehgal,

1027

1028 BHUPENDER SIN~H and R. K. BATRA

Thickness (mg/cm’)

Fig. 1. Relative measured transmission of positrons and electrons in Al, Sn and Pb at 0.324, 0.544. I .77 and 1.8%MeV. For the set of curves (b), the scale along the x--axis is different.

1981) for the evaluation of the practical range, and the exponential law of pm and fi + absorption. The proposed method is applicable for electrons and positrons in the energy region 0.2555.0 MeV for all elemental absorbers. The calculated mass-attenuation coefficients are compared with the experimental data for Al, Cu, Ag, Pb, Y. Nd, Ho and Yb.

Method

The ,!I particles (continuous-energy electrons and positrons). on passage through matter, lose energy as well as undergoing multiple Coulomb scattering. Therefore, the theory for determining the practical

range, R,, and the mass-attenuation coefficient, p, should account for both the processes. Batra and Sehgal (1981) have considered these processes while calculating the practical range of electrons and posi- trons in matter. Their penetration theory establishes

that the practical ranges Rl and R; are sufficient to understand the extrapolated ranges obtained from the observed number-transmitted vs thickness curves. Here, we use the penetration theory of electrons and positrons coupled with the knowledge of the ex- ponential law of absorption of /I particles to estimate the mass-attenuation coefficients of these particles. The method of calculation for p of electron and positron is based upon the following assumptions:

(1) The exponential law of absorption of jI particles is assumed to be valid down to a very low trans- mission. The lower limit of relative transmission of electrons and positrons varies with the atomic num- ber Z of the absorber and is of the order of 1% or less.

(2) The penetration theory (Batra and Sehgal, 1981) of monoenergetic electrons and positrons of energy T is also valid for a particles of end-point energy T,,,,, = T lying between 0.25 and 5.0 MeV. Batra and Sehgal(198 1) and Thontadarya (1985) give the experimental evidence in support of this assump- tion. The effect of energy straggling during pen- etration may be ignored.

The exponential law of the attenuation of /I _ and 8’ particles of energy Tin an absorber is written as

I(t) = I, exp(--r) (1)

where r is the thickness in gem-* penetrated to reduce the intensity from I, with zero absorber to I(t) and p denotes the mass-attenuation coefficient in cm2 gg’. Setting the value of Z(t)/& for an absorber such that the thickness t corresponds to the practical range R, of the j? particle, one writes

p + Rd = -In t/l,, (2)

The upper and the lower superscripts refer to posi- tron and electron respectively. Equation (2) implies that p is inversely proportional to R,, the constant of the inverse proportionality being In I/t,, which de- pends upon Z. Following the first assumption, the limit of relative transmission I/I, for electron and positron in an absorber is selected corresponding to the lowest value down to which the observed I/l,, vs thickness is linear under the ideal experimental conditions (Spanel et al., 1973; Thontadarya and Umakantha, 1971). As the experimental conditions used in many experiments are not identical, a large variety of datum on number-transmitted vs thickness is used to find an average value of lowest I/I,, for

Mass attenuation coefficients of beta particles 1029

different absorbers. To compute Jo- and p+, the values of I/Z0 used are 0.07, 0.15, 0.30 and 1.0% for

Al, Cu, Ag and Pb respectively. According to the penetration theory (Batra and

Sehgal, 1981) the practical range of positron and electron is given by

R;(T) = R ‘(T) - R ‘(T:) (3)

where R(T) is the continuous slowing-down- approximation (CSDA) range and is obtained by integrating the reciprocal of the expression for the mean total energy loss. T: and T; represent the characteristic energies corresponding to the random motion of positron and electron set in during the penetration. Using the multiple-scattering theory, the stopping-power theory and the idea of transport mean free path, it has been shown (Batra and Sehgal, 1981) that T: and T; can be obtained for the given T and 2.

Having estimated T: and T,- (Batra and Sehgal, 1981) the practical range R; and R; can be ob- tained for an energy and absorber using equation (3) and the procedure for calculating the CSDA range

Fig. 2. Calculated mass-attenuation coefficient for electrons in Al and Ag and the experimental values vs energy.

* The large uncertainties in experimental data are due to the different geometrical arrangements adopted by various workers (Daddi and D’Angelo, 1963; Takhar, 1967; Spanel et al., 1973; Nathu Ram et al., 1982; Thontadarya and Umakantha, 1971) the vacuum conditions @panel et al., 1973), the shape of the B-ray spectra of the sources (Cook, 1969). etc.

Expt

Theory

Fig. 3. Calculated mass-attenuation coefficient for electrons in Cu and Pb and the experimental values vs energy.

R + (T) and R (T) given by Batra and Sehgal (1973). Substituting R; (T) and R;(T) and the selected value of I/I0 for an element, one can compute p - and p + using equation (2) for different energies and absorbers.

Results and Discussion

Figures 2 and 3 show the present calculated values of electron mass-attenuation coefficient vs energy in Al, Cu, Ag and Pb. Available experimental data at different energies are also shown in these figures. The relative r.m.s. deviation of theory from the experi- ment is given by

6 r.m.r. = $ .$, [p(z, Tl)n,/p (G T,)exp - 111)” (4)

where n is the number of experimental data, ~((2, T,&, is the ith calculated value from equation (2) and ~(z, TJexp is the ith experimental value. The values of 6 r.m.s for electrons calculated using equation (4) and n for Al, Cu, Ag and Pb are given in columns 2 and 3 ofTable 1. The 6,,, for electrons is 8.6% when 73 experimental points in all absorbers studied are con- sidered (see the bottom row under electrons in Table 1). This is rather large and is mainly because of large variations* in the observations at some energies.

Among the different empirical relations beteen p and T for one element or the other, Thontadarya and Umakantha’s (1971) empirical relation for electron transmission measurements using “good geometry”

1030 BHUPENDER SINGH and R. K. BATRA

Table I. Experimental points n of mass-attenuation coefficient and

the root-mean-square deviation (S,, j ) of theory from experiment for electrons and positrons in different absorbers

Electrons Positrons

6 rms 6 Element n (%) n b”‘ ( /o)

Al 30 (25) 10.5 (7.5) 6 14.3 CU 10 (7) 8.9 (6.5) 5 9.2

Ag lO(9) 6.6 (5.6) 4 10.4 Pb lO(10) 6.8 (6.8)

8.6 (6.5) 3: 9.2

All elements* 73 (61) 9.7

*The overall data also includes absorbers other than those listed.

may be taken as most accurate. If we fix a criterion that only those measurements which do not deviate from the empirical relation (Thontadarya and Uma- kantha, 1971; Thontadarya, 1984) by more than 10% are admitted to estimate r.m.s. deviation for various absorbers, the 6,,,, becomes considerably smaller. Values within brackets of columns 2 and 3 of Table 1 contain the number of experimental points, n, admit- ted according to the above criterion and the corre-

sponding L,, s. for electrons in Al, Cu, Ag and Pb.

Thus L, deviation of electrons lies between 5.6 and 7.5% for various absorbers. The overall a,,, for 61 observations of electrons in different elements is found to be 6.5%.

The L b values for positrons are given in column 5 of Table 1 for Al, Cu, Ag and Pb. The overall 6,,,, for 30 positron measurements is found to be 9.7%. This is rather large because the positron data are sparse and the uncertainties in positron experiments are greater.

Table 2 contains the calculated values of mass- attenuation coefficients for positrons of energies from 0.25 and 5.0 MeV and the experimental values at particular energies in Al, Cu, Ag and Pb. The uncer- tainties in measurements of p + are usually between 5 and lo%, and even greater in some cases.

Figure 4 shows the dependence of calculated mass-

.Y’ 0 544 MeV b/’

0 1, ’ I I I

20 40 60 80

Atomic number

Fig. 4. Mass-attenuation coefficient p vs atomic number Z for positrons and electrons at different energies.

attenuation coefficient p and p + upon atomic num- ber Z at those energies where measurements are available. The measured values of p at 0.324, 0.544, 1.77 and 3.6 MeV and that of p + at 0.324, 0.544 and 1.88 MeV quoted by different authors (Takhar, 1967; Spanel et al., 1973; Nathu Ram et al., 1981) are also displayed in Fig. 4. It is apparent from Fig. 4 that at a particular energy, the mass-attenuation coefficient increases slowly with Z for electrons as well as for positrons. However, the attenuation coefficient is greater for electrons than for positrons for a given case. Also the mass-attenuation coefficient decreases with an increase of energy for both the particles in an absorber.

We have also calculated mass-attenuation coefficients for 1.77-MeV electrons and 1.88 MeV

Table 2. Calculated mass-attenuation coefficients (p) in cm2 g ’ for positrons in Al, Cu, Ag and Pb according to equation (2) vs energy and some experimental values

Element Aluminium Copper Silver Lead

T (MeV) Theory Expt. Theory Expt. Theory Exut. Theory Exot.

0.1 544.2 627.8 730.8 755. I 0.2 164.8 209.8 229.6 230.3 0.3 86.6 107.5 119.8 117.5 0.324 76.9 92.0’ 94.4 108.2’ 100.8 114.0’ 105.0 119.0’ 0.4 56.7 68.2 65.8 78.6 0.5 41.6 48.4 55.5 53.6 0.544 36.9 41.42 43.5 43.44 49.4 44.4’ 46.5 47.6’ 0.6 32.2 3X.6 43.4 39.7 0.7 26.2 30.9 33.4 32. I 1.0 16.7 18.3 20.5 20.7 1.5 9.6 10.8 11.5 11.4

1.88 7.3 5.6j3 8.2 7.153 8.6 9.39’ 8.4 9.53’ 2.0 6.8 7.5 8.0 7.7 3.0 4.3 4.6 - 4.8 4.6 4.0 3.2 3.3 3.4 3.2 5.0 2.5 2.6 2.6 2.5

References: ‘Nathu Ram et al., 1981; 2Thontadarya and Umakantha, 1971; ‘Takhar, 1967; 4Spanel et al., 1973.

Mass attenuation coefficients of beta particles 1031

Table 3. Comparison of our calculated values of mass-attenuation coefficient (cm*g~‘) of 1.77-MeV electrons and 1.88-MeV positrons with experimental values in rare earths

Electrons Positrons

Takhar’s Nathu Ram’s Present Takhar’s Present Element Exut. Exut. Theorv Expt. Theory

Yttrium 10.91 f 0.23’ 10.22 IO.51 Neodymium 13.60 + 0.23 II.38 II.44 Holmium 13.44 + 0.28 11.65 10.80 Ytterbium I I .98 + 0.24 II.91 II.75

*Uncertainties quoted are the standard deviations.

7.55 i 0.42* 8.3 8.16 _t 0.34 8.57 7.25 f 0.32 8.69 8.12+0.31 8.52

positrons in rare earths such as yttrium, neodymium, holmium and ytterbium using the present theory. Typical values of I(Z)/& for these elements are obtained by interpolation of the values used for Al, Cu, Ag and Pb. Table 3 gives a comparison of calculated mass-attenuation coefficients for rare earths with experimental data. The agreement be- tween experiment (Takhar, 1968) and theory is excel- lent. Our calculated values of p- in rare earths also agree well with the corresponding values estimated by Nathu Ram ef al. (1982) using their empirical formula.

We conclude that equation (2) provides a simple way to compute the attenuation coefficients of fi particles using the penetration theory of electrons and positrons. The dependences of p - and p + on

the energy and the nature of the absorber can be explained.

Acknowledgement-The authors express their thanks to Professor G. S. Chaddha, Head of the Physics Department, for his interest in this problem.

References Batra R. K. and Sehgal M. L. (1973) Approximate stopping

power law for electrons and positrons. Nucl. Instrum. Meth0d.y 109, 565.

Batra R. K. and Sehgal M. L. (1981) Range of electrons and positrons in matter. Phys. Rev. B23, 4448.

Cook C. S. (1969) Relative transmission of positive and negative electrons through aluminum. Phys. Rev. 178, 895.

Cosslett V. E. and Thomas R. N. (1964) Transmission of electrons in metallic foils. Br. J. Appl. Phys. 15, 883.

Daddi L. and D’Angelo V. (1963) On the absorption of the /? particles in aluminium. Int. J. Appl. Radiat. Isot. 14, 341.

Hubbell J. H. (1982) Photon mass-attenuation and energy absorption coefficients from 1 keV to 20 MeV. ht. J. Appl. Radiat. Isot. 32, 1269.

Katz L. and Penfold A. S. (1952) Range energy relations for electrons and the determinaton of beta ray end point energies by absorption. Reu. Mod. Phys. 24, 28.

Nathu Ram, Sundara Rao I. S. and Mehta M. K. (1981) Relative transmission of 0.324- and 0.544-MeV positrons and electrons in Be, Al, Cu, Ag and Pb. Phys. Rev. A 23, 1202.

Nathu Ram, Sundara Rao I. S. and Mehta M. K. (1982) Mass absorption coefficients and range of beta particles in Be, Al, Cu, Ag and Pb. Pramana 18, 121.

Patrick J. R. and Rupaal A. S. (1971) Transmission of low energy positrons and electrons through thin metallic foils. Phys. Lett. 35A, 235.

Spanel L. E., Rupaal A. S. and Patrick J. R. (1973) Penetration of low energy positrons through thin metallic foils. Phys. Rea. B 8, 4072.

Takhar P. S. (1967) Direct comparison of the penetration of solids and liquids by positrons and electrons. Phys. Rev. 157, 257.

Takhar P. S. (1968) Direct comparison of positron-electron ranges in rare earth metals and other solids. Phys. Lett. A 28, 423.

Thontadarya S. R. (1984) Effect of geometry on mass- attenuation coefficient of p particles. Inf. J. Appl. Radiat. Isot. 35, 981.

Thontadarya S. R. (1985) Determination of range of beta particles in aluminum from their mass-attenuation coefficients. ht. J. Appl. Radiat. Isot. 36, 251.

Thontadarya S. R. and Umakantha N. (1971) Comparison of mass absorption coefficients of positive and negative beta particles in aluminium and tin. Phys. Rev. B 4, 1632.