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Photon interaction studies in solutions of some alkali metalchloridesÐI
Gagandeep Kaura, Kulwant Singha,*, B.S. Larkb, H.S. Sahotac
aDepartment of Physics, Guru Nanak Dev University, Amritsar, IndiabDepartment of Chemistry, Guru Nanak Dev University, Amritsar, India
cDepartment of Physics, Punjabi University, Patiala, India
Received 27 July 1999; accepted 23 November 1999
Abstract
Attenuation coe�cients of aqueous solutions of alkali metal chlorides viz. LiCl, NaCl and KCl having di�erentconcentrations have been determined at 81, 356, 511, 662, 1173 and 1332 keV gamma photons using narrow beamtransmission method. The experimental results are analysed in terms of total cross sections, e�ective atomicnumbers and electron density values of these solutions. In addition, molar extinction coe�cients, interaction cross-
sections and interaction radii for the solid solutes in aqueous solutions have also been calculated at these energiesand concentrations. The mixture rule is applied to examine the compatibility of the results. 7 2000 Elsevier ScienceLtd. All rights reserved.
1. Introduction
Reliable values of attenuation coe�cients for
gamma rays are required in a variety of applications,
such as computerised tomography, dosimetry, radi-
ography etc. The study of these coe�cients is poten-
tially useful in the development of semi-empirical
formulations of high accuracy, for example along the
lines detailed by Jackson and Hawkes (1981). In order
to make use of the fact that scattering and absorption
of gamma radiations are related to the density and
e�ective atomic number of the material, a knowledge
of the mass attenuation coe�cient, m=r, is of prime im-
portance. Hubbell (1982) and Hubbell and Seltzer
(1995) have compiled the mass attenuation coe�cients
for a large number of compounds and mixtures ofdosimetric and biological importance.
Most of the previously determined coe�cients havebeen concerned with crystalline samples in the solidform. In their pioneering work, Teli et al. (1994a,
1994b, 1994c, 1996), and Teli and Chaudhary (1995a,1995b, 1996) have determined the gamma-ray attenu-ation coe�cients in dilute solutions of some salts. Den-
sities of the solutions, however, were estimated bytaking volumes of the solutions as the sum of thevolumes of solutes and solvents, which in the light ofprevalence of strong solute±solvent interactions is
untenable. The density of a solution should thus bemeasured experimentally.Gerward (1996) has elaborated the mixture rules for
linear and mass attenuation coe�cients in the generalcase as well as in the limit of extreme dilution and thushas developed the theory of X-ray and gamma-ray at-
tenuation in solutions.It was pointed out by Hine (1952) that the e�ective
Radiation Physics and Chemistry 58 (2000) 315±323
0969-806X/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved.
PII: S0969-806X(99 )00524-1
www.elsevier.com/locate/radphyschem
* Corresponding author. Fax.: +91-183-258819.
E-mail address: kulwant.singh@malexcite.com (K. Singh).
atomic numbers in materials composed of several el-ements can not be represented uniquely and the num-
ber for each partial process has to be determinedseparately. El-Kateb and Abdul Hamid (1991) pointedout the e�ective atomic numbers in case of hydro-
carbon materials tend to be constant as a function ofenergy when contributions from photoelectric e�ectand pair production are not signi®cant. Certain par-
ameters such as e�ective atomic number and electrondensity of a medium are essential for the determinationof energy absorption in a given medium. Seltzer (1993)
reported new calculations of the mass energy-transfercoe�cient and the mass energy-absorption coe�cientfor photon energies from 1 keV to 100 MeV based ona re-examination of the processes involved after the in-
itial photon interaction. These coe�cients are used torelate the photon energy ¯uence in an irradiated med-ium to quantities approximating the absorbed dose.
The present study covers the energy regions in whichthe in¯uence of all photon processes can be seen andthe investigation is expected to yield valuable infor-
mation.As a sequel to our previous study (Singh et al.,
1998, 1999; Kaur et al., 2000) of the absorption prop-
erties of some solutes in water at di�erent concen-trations, the attenuation coe�cients of LiCl, NaCl andKCl at six di�erent gamma ray energies in aqueousmedium as a function of concentration are reported in
this paper. Densities required for their estimationshave been measured experimentally. However, in thispaper we have also reported a quantity which is more
practically used when we want to determine the inter-action of photons for solutions in terms of molar con-centration and that is molar decadic extinction
coe�cient e which is dependent upon the photonenergy and nature of the dissolved substance. Theenergy absorbed from radiation may produce a chemi-cal change in the absorbing medium and the amount
of this may be used to measure dose. The values ofmolar extinction coe�cients for the solid solutes inaqueous solutions have also been reported for the ®rst
time for these compounds in aqueous medium.
2. Theory
2.1. Attenuation coe�cients:
A material of thickness x, placed in the path of a
beam of gamma radiation will e�ect the transmittedintensity according to `Lambert±Beer' law, whichstates that a beam of monoenergetic gamma rays is
attenuated according to the equation:
I � I0eÿmrrx �1�
where I0 and I are the incident and transmitted intensi-ties, respectively, m=r (cm2/g) is the mass attenuation
coe�cient; r (g/cm3) is the density of the solution andx (cm) is the thickness of the sample. If the absorbingmaterial under experimentation is a binary solution, its
mass absorption coe�cient is given by the mixture ruleas:
mr��mr
�S
wS ��mr
�W
�1ÿ wS � �2a�
mr��mr
�W
�"�m
r
�S
ÿ�mr
�W
#wS, �2b�
where �m=r�S and �m=r�W are the mass attenuation
coe�cients of the solute and water, respectively, andwS is the weight fraction of the solute. Eq. (2a) can berewritten in the following form:
mr� mS
rS
�M1
M1 �M2
�� mW
rW
�M2
M1 �M2
��3a�
where M1 and M2 are the masses of solute and waterand mS and mW are their linear attenuation coe�cients,respectively. If rS, rW (=1 g/cm3) are the densities and
V1��M1=rS� and V2��M2=rW� are the volumes of thesolute and water, respectively, a � V1=V2 and c��M1=V2 � arS� is the concentration of the solution,
then Eq. (3a) becomes:
mr� mSa� mW
arS � rW
�4�
Neglecting the change in volume of the solution andmultiplying Eq. (4) by
r � M1 �M2
V1 � V2� arS � rW
1� a�5�
we get
m � �mS ÿ mW �a
1� a� mW �6�
A plot of m versus a=�1� a� yields a straight line with
the intercept being mW and the slope `m' equal to thedi�erence �mS ÿ mW� of linear attenuation coe�cientsof solute and water. From the slope and mW, mS thus
can be calculated. Knowing the experimental value of`m' and density of the solute its corresponding mass at-tenuation coe�cient �mS=rS� can be determined.
2.2. E�ective atomic numbers
Values of mass attenuation coe�cients can be usedto determine the total molecular cross-section st, m, bythe following relation.
G. Kaur et al. / Radiation Physics and Chemistry 58 (2000) 315±323316
st, m ��mr
�At
NA
�7�
where At �P
i niAi is the molecular weight, NA is theAvogadro's number and ni is the total number ofatoms (with respect to mass number) in the molecule.
The total atomic cross-section st, a can be easily deter-mined from Eq. (7) as:
st, a � st, m
1Xi
ni
�8�
Also the total electronic cross-section st, el for the indi-vidual element is expressed by the following formula:
st, el � 1
NA
Xi
fiAi
Zi
�mr
�i
�9�
where fi is the number of atoms of element i relative to
the total number of atoms of all elements in the mix-ture. The total atomic and electronic cross-sections arerelated to the e�ective atomic number �Zeff� throughthe following relation:
Zeff � st, a
st, el
�10�
The attenuation cross section in barns/atom can also
be obtained by dividing the mass attenuation coe�-cient m=r (cm2/g) by the total number of atoms presentin one unit of mass (1 g) of the material. Thus, one
has
s � m=r
NA
Xi
wi
Ai
�11�
where wi is the weight fraction of element i, and Ai isthe atomic weight (g/mol) of element i. Another physi-
cal quantity the electronic density NE (electrons/g) isgiven by:
NE ��Zeff
Ai
�NA
Xi
ni �12�
Also the number NE of electrons per gram is deter-mined by the measure of mass attenuation coe�cientat a given energy by the following relation:
NE � m=res
KN �13�
where esKN
is the Klein±Nishina cross-section. Theabove Eq. (13) is valid for energies where the mass at-tenuation coe�cient is mainly due to Compton scatter-
ing, and thus described by the Klein±Nishina formula.Except for 81 keV energy, this condition is very wellful®lled for the energies used in the present work.
2.3. Molar extinction coe�cients
The absorbance A or optical density of a solution is
de®ned by the equation:
A � logI0I
�14�
where I0 is the intensity of the incident beam imping-ing on a cell containing the solution and I that of thetransmitted beam. Since the optical density is pro-
portional to the amount of material in a solution, itmay be expressed as:
A � exMc �15�
where x (cm) is the path length of the cell, Mc (moles/l) is the molar concentration (number of moles of thesolute dissolved per litre of the solution) of the absorb-
ing species in the solution and e is a constant of pro-portionality called the molar extinction coe�cient.Since A is dimensionless, e has dimensions of area/mol.It is normally convenient to express e in l molÿ1 cmÿ1,but alternative units are cm2/mol. This change of unitsemphasizes the point that e is a molar attenuationcoe�cient, analogous to the mass attenuation coe�-
cient m=r: The latter has dimensions of area/mass andis typically reported in cm2/g. If e is known, the con-centration may be determined from a measurement of
A using Eq. (15).The change in the radiation intensity dI due to inter-
actions occurring during the passage is given by:
ÿdI � sIN dx �16�
where N is the number of interaction centres per unitvolume and s is the interaction cross section having
the dimensions of area (m2) called the probability ofinteraction and may be visualised as the area, whichhas to be hit by the photons in order to cause inter-
action. Eq. (16) may be written in terms of molar con-centration by using N � NAMc, where NA is theAvogadro's constant.
ÿdI � sINAMc dx �17�
Integration leads to
I � I0eÿsNAMcx �18�
This expression is essentially identical to the so-called`Lambert±Beer law' which is used to describe radiation
attenuation in homogeneous solutions. For practicalpurposes, the following form is preferred:
I � I0 � 10ÿeMcx �19�
Comparing Eqs. (18) and (19), we have
G. Kaur et al. / Radiation Physics and Chemistry 58 (2000) 315±323 317
e � sNAlog10 e �20�or numerically
sm2� 3:82� 10ÿ25
emolÿ1 dm3 cmÿ1
�21�
The values of interaction cross sections can then beused to calculate the interaction radii of the solutes in
solution, using s � pr2: The mixture rule can be incor-porated for the molar extinction coe�cients de®ned bythe following relation:
logI0I� e1Mc1x� e2Mc2x �22�
where e1, e2 are the molar extinction coe�cients of the
solid solute and pure water and Mc1, Mc2 are theirmolar concentrations, respectively.
3. Experimental details
The details of the experimental arrangement aregiven in our previous paper (Singh et al., 1998). A 1.50� 1.50 NaI(Tl) crystal having energy resolution of 12%
at 662 keV was used for the measurement of attenu-ation coe�cients. The 137Cs, 133Ba, 22Na and 60Co
sources of strength 5 mci were obtained from Bhabha
Atomic Research Centre, Trombay, Bombay, India.The signal from the detector after suitable ampli®ca-tion was recorded in an EG & G ORTEC Plug-in-card
coupled to a PC/AT. The stability and reproducibilityof the arrangement was tested before and after eachset of runs in the usual manner. The experimental pro-cedure records the intensity of gamma radiation pas-
sing through each sample of known thickness. Theintensity of the transmitted radiation was measured bygating the channels at the FWHM position of the
photo peak to minimise the contributions of bothsmall angle and multiple scattering events to themeasured intensity. A su�cient number of counts were
collected under the photo peak to limit the statisticalerror to less than 1%. The samples contained in per-spex boxes of di�erent thicknesses were placed one by
one between the source and detector.The intensity of the gamma radiation, passing
through a standard material can be written as:
lnISTD
I0� ÿ
�mrrx�
STD
�23�
where `STD' stands for the standard material which inthe present case is an aluminium slab of known thick-
ness. Intensity of the radiation passing through perspex(the material used for the solution container) is givenas:
lnI
I0� ÿ
�mrrx�
P
�24�
where `P' stands for perspex material.Subtracting Eq. (23) from Eq. (24) for eliminating
the need for measuring radiation intensity passing
through air I0, we have
lnI
ISTD
� ÿ�mrrx�
P
��mrrx�
STD
�25�
Similarly, the intensity of the radiation passing
through the solution can be written as:
lnI
ISTD
� ÿ�mrrx�
SOL
�"�m
rrx�
STD
ÿ�mrrx�
P
#�26�
A plot of `ÿln�I=ISTD�' versus �rx�SOL gives a straightline with the slope being the mass attenuation coe�-cient m=r of the solution. The terms in the bracket givethe intercept. The following procedure was used to
determine the value of m=r for solutions of di�erentconcentrations.(i) The perspex containers of di�erent path lengths
were ®lled with solutions under study. (ii) Each samplewas placed one by one between the source and thedetector and the corresponding spectra were recorded.
(iii) The solution sample was replaced with the alu-minium standard to determine the ISTD:For the determination of molar extinction coe�-
Fig. 1. Plot of linear attenuation coe�cient of the solution
(LiCl) as a function of a=�1� a�:
G. Kaur et al. / Radiation Physics and Chemistry 58 (2000) 315±323318
cients of solutes, the intensity of radiation passingthrough the container (®xed size) containing only dis-
tilled water was taken as I0, whereas that passingthrough a solution as I.
4. Results and discussion
4.1. Attenuation coe�cients
For the determination of attenuation coe�cients,
several measurements were carried out for the givensolutions of lithium chloride, sodium chloride and pot-assium chloride by putting them in perspex containers.
The experimental values of linear attenuation coe�-cients were calculated by using Eq. (26) for aqueoussolutions of these compounds in the concentration
range of 0.5 to 0.25 g/cm3. A plot of the linear attenu-ation coe�cient m as a function of a=�1� a� is shownin Fig. 1. It is clear from this plot that the values oflinear attenuation coe�cients vary linearly with the
concentration of the solution and increase with theincrease in concentration of the solution. The sensi-tivity of m against a=�1� a� and concentration at
di�erent photon energies is correlated with the slopesof the curves which are positive. From the slope ofthis plot, the linear attenuation coe�cient of the corre-
sponding solute mS was obtained. The mS values for thevarious solid solutes so arrived are shown in Table 1.The mS values were also calculated by using the weight
fractions of the solid solutes and a very accurate com-puter programme and data base named XCOM pho-ton cross-sections on a personal computer was
prepared by Berger and Hubbell (1987). The corre-sponding standard deviations in the attenuation coe�-
cients of the solutes are also been listed in Table 1.From the precisely measured densities of these sol-
utions at 258C (Singh et al., 1998), mass attenuation
coe�cients of the solutions were then obtained and aresummarised in Table 2. For a given salt, the graphbetween m=r and wS did give perfectly linear plots (a
typical plot for LiCl has been shown as an illustrationin Fig. 2). Also from the slope �m=r�S ÿ �m=r�W, (Eq.(2b)) the mass attenuation coe�cient of the corre-
sponding solute was obtained. These �m=r�S values forthe three solid solutes so arrived have been comparedin Table 3 with those calculated from XCOM calcu-lations of the solutes in their solid form. The di�erence
lies between 0.2 and 0.5% which is much within theexperimental uncertainty of measurement of the �m=r�Svalues. Thus, this method gives us an alternative
method for the direct determination of attenuationcoe�cients of solutes without preparing them in purecrystalline form.
4.2. E�ective atomic number and electron densities
By taking the experimental data of �m=r� for Li, Naand K at di�erent energies from El-Kateb and AbdulHamid (1991) the e�ective atomic numbers of solutions
were determined with the help of piece-wise interp-olation computer programme. The theoretical valuesof e�ective atomic numbers were also obtained from
calculated cross-sections. The e�ective atomic numbersalmost tend to be a constant function of energy whichis in line with the view point of El-Kateb and Abdul
Table 1
Linear attenuation coe�cients mS (cmÿ1) of the solutes in solid form
Solute Energy (keV) XCOM value Experimental value Slope
LiCl 81 0.5037 0.5193 Slope = 0.5467
356 0.1985 0.2608
511 0.1700 0.1798 Intercept = 0.4096
662 0.1515 0.1623
1173 0.1150 0.1220 Standard Deviation = 0.0009
1332 0.1078 0.1139
NaCl 81 0.4994 0.5057 Slope = 0.5340
356 0.2104 0.2119
511 0.1804 0.1859 Intercept = 0.3722
662 0.1608 0.1657
1173 0.1221 0.1262 Standard Deviation = 0 0004
1332 0.1078 0.1183
KCl 81 0.5824 0.6032 Slope = 0.6030
356 0.1967 0.2013
511 0.1679 0.1766 Intercept = 0.5844
662 0.1494 0.1576
1173 0.1132 0.1198 Standard Deviation = 0.0024
1332 0.1062 0.1127
G. Kaur et al. / Radiation Physics and Chemistry 58 (2000) 315±323 319
Hamid (1991) in case of materials containing hydro-
gen, carbon and oxygen. The signi®cant variations inZeff were found with concentration of the solution
which is due to the variations in weight fractions ofNa and Cl in the solutions. The behaviour of all sol-
utions is identical with energy and concentration of the
solution. These values of e�ective atomic numberswere used to calculate the electron densities of sol-
utions at di�erent concentrations. The electron densityvalues determined from Eqs. (12) and (13) are shown
in Table 4. A very good linear relationship wasachieved between measured and theoretical electron
density values. These results clearly reveal the analyti-cal capability of our method and support the mixture
rule. The average values of molar extinction coe�-
cients obtained for the solutes in solution are shown in
Table 5. For a given solute and at a ®xed energy thesecoe�cients remain constant with change in concen-
tration of the solution as the molar extinction coe�-cient is a characteristic of the solute and depends on
the nature of the dissolved substance and not upon itsamount. However, the e values decrease with increas-
ing energy as e depends upon wavelength.
The values of interaction radii determined from thecalculated interaction cross-sections for solutes in sol-
ution are shown in Table 6. It is observed that the in-teraction radii of a particular alkali halide remain
constant with the increase in concentration and
decrease with energy and as we are taking the higheralkali halide the interaction radii values show an
Table 2
Mass attenuation coe�cients m=p (cm2/g) of solutions of alkali chlorides
Solution Density of the solution
(g/cm3Concentration
(g/cm3)
Mass attenuation coe�cient m=r (cm2/g)a
81
keV
356
keV
511
keV
662
keV
1173
keV
1332
keV
Lithium chloride
(LiCl)
1.0275698 0.05 a 0.1850 0.1100 0.09529 0.08514 0.06473 0.06069
b 0.1857 0.1104 0.09532 0.08510 0.06485 0.06075
1.0546711 0.10 a 0.1879 0.1092 0.09469 0.08453 0.06439 0.06027
b 0.1884 0.1097 0.09473 0.08460 0.06443 0.06036
1.0815910 0.15 a 0.1899 0.1087 0.09407 0.08404 0.06397 0.05999
b 0.1908 0.1091 0.09418 0.08411 0.06404 0.06009
1.1069374 0.20 a 0.1924 0.1080 0.09359 0.08357 0.06357 0.05960
b 0.1930 0.1086 0.09368 0.08365 0.06369 0.05967
1.1267312 0.25 a 0.1946 0.1075 0.09319 0.08336 0.06329 0.05931
b 0.1950 0.1080 0.09322 0.08342 0.06337 0.05937
Sodium chloride
(NaCl)
1.0273692 0.05 a 0.1847 0.1100 0.09529 0.08513 0.06481 0.06070
b 0.1851 0.1104 0.09538 0.08520 0.06489 0.06079
1.0543111 0.10 a 0.1863 0.1091 0.09475 0.08465 0.06443 0.06038
b 0.1872 0.1098 0.09483 0.08470 0.06450 0.06043
1.0794619 0.15 a 0.1899 0.1089 0.09428 0.08419 0.06409 0.06006
b 0.1893 0.1093 0.09434 0.08425 0.06415 0.06010
1.1065312 0.20 a 0.1899 0.1080 0.09377 0.08378 0.06378 0.05971
b 0.1908 0.1088 0.09388 0.08383 0.06383 0.05980
1.1263340 0.25 a 0.1917 0.1076 0.09339 0.08339 0.06349 0.05947
b 0.1924 0.1083 0.09346 0.08345 0.06354 0.05953
Potassium chloride
(KCl)
1.0259239 0.05 a 0.1873 0.1102 0.09537 0.08519 0.06484 0.06072
b 0.1881 0.1105 0.09543 0.08523 0.06491 0.06081
1.0525964 0.10 a 0.1923 0.1097 0.09483 0.08469 0.06447 0.06039
b 0.1929 0.1102 0.09493 0.08478 0.06455 0.06048
1.0779160 0.15 a 0.1969 0.1091 0.09439 0.08427 0.06413 0.06011
b 0.1972 0.1095 0.09448 0.08436 0.06422 0.06017
1.1037360 0.20 a 0.2066 0.1084 0.09397 0.08388 0.06378 0.05980
b 0.2012 0.1091 0.09406 0.08397 0.06392 0.05989
1.1210717 0.25 a 0.2041 0.1079 0.09359 0.08357 0.06359 0.05959
b 0.2049 0.1086 0.09367 0.08362 0.06364 0.05963
a Values in rows `a' and `b' stand for experimental and theoretical values, respectively.
G. Kaur et al. / Radiation Physics and Chemistry 58 (2000) 315±323320
Table 3
Mass attenuation coe�cients m=r (cm2/g) of the solutes in solid form
Solute Density of the salt (g/cm3) Energy (keV) Experimental value XCOM value
LiCl 81 0.2456 0.2436
356 0.0947 0.0959
2.068 511 0.0822 0.0822
662 0.0743 0.0733
1173 0.0558 0.0556
1332 0.0521 0.0522
NaCl 81 0.2286 0.2308
356 0.0954 0.0972
2.164 511 0.0835 0.0834
662 0.0743 0.0743
1173 0.0566 0.0564
1332 0.0531 0.0529
KCl 81 0.2928 0.2930
356 0.0960 0.0989
1.988 511 0.0843 0.0844
662 0.0751 0.0751
1173 0.0571 0.0569
1332 0.0537 0.0534
Fig. 2. Plot of mass attenuation coe�cient of the solution (LiCl) as a function of weight fraction of the solute.
G. Kaur et al. / Radiation Physics and Chemistry 58 (2000) 315±323 321
Table 4
Electron densities of solutions of alkali chlorides
Sample Concentration (g/cm3) Electron density NE � �1023� (electrons/g)a
81 keV 356 keV 511 keV 662 keV 1173 1332 keV
Lithium chloride (LiCl) 0.05 a 3.600 3.332 3.328 3.325 3.326 3.326
b 3.587 3.320 3.327 3.323 3.320 3.323
0.10 a 3.653 3.311 3.307 3.304 3.304 3.305
b 3.643 3.296 3.306 3.301 3.302 3.300
0.15 a 3.699 3.293 3.288 3.285 3.284 3.290
b 3.682 3.281 3.284 3.282 3.281 3.285
0.20 a 3.742 3.278 3.270 3.267 3.268 3.267
b 3.730 3.260 3.267 3.264 3.260 3.263
0.25 a 3.780 3.260 3.254 3.258 3.250 3.251
b 3.773 3.245 3.253 3.253 3.246 3.247
Sodium chloride (NaCl) 0.05 a 3.589 3.332 3.330 3.327 3.328 3.328
b 3.581 3.320 3.327 3.325 3.324 3.323
0.10 a 3.629 3.314 3.311 3.308 3.308 3.309
b 3.612 3.293 3.308 3.306 3.304 3.306
0.15 a 3.670 3.299 3.293 3.290 3.290 3.291
b 3.662 3.287 3.291 3.288 3.287 3.288
0.20 a 3.699 3.284 3.277 3.273 3.273 3.274
b 3.682 3.260 3.274 3.272 3.271 3.269
0.25 a 3.730 3.269 3.263 3.259 3.259 3.259
b 3.716 3.248 3.260 3.257 3.256 3.256
Potassium chloride (KCl) 0.05 a 3.647 3.335 3.331 3.329 3.329 3.329
b 3.631 3.326 3.329 3.327 3.325 3.324
0.10 a 3.740 3.326 3.314 3.311 3.310 3.311
b 3.728 3.311 3.311 3.307 3.306 3.306
0.15 a 3.823 3.305 3.298 3.295 3.293 3.294
b 3.817 3.293 3.295 3.291 3.289 3.291
0.20 a 3.901 3.293 3.284 3.279 3.278 3.279
b 3.889 3.272 3.280 3.276 3.275 3.274
0.25 a 3.972 3.278 3.270 3.266 3.264 3.265
b 3.957 3.257 3.267 3.264 3.261 3.263
a Values in rows `a' and `b' stand for experimental and theoretical values, respectively.
Table 5
Molar extinction coe�cients e1 (molÿ1 l cmÿ1) of solid solutes in aqueous solutions of some alkali chlorides
Solute Concentration (mol/l) Energy (keV)
81 356 511 662 1173 1332
Lithium chloride (LiCl) 1.157±5.328 0.00293 0.00080 0.00075 0.00067 0.00049 0.00046
Standard deviation 0.00068 0.00099 0.00010 0.00092 0.00063 0.00063
Sodium chloride (NaCl) 1.838±3.860 0.00368 0.00107 0.00103 0.00093 0.00068 0.00064
Standard deviation 0.00008 0.00007 0.00002 0.00002 0.00003 0.00004
Potasium chloride (KCl) 0.503±2.305 0.00861 0.00186 0.00171 0.00151 0.00113 0.00014
Standard deviation 0.00029 0.00014 0.00008 0.00004 0.00001 0.00005
G. Kaur et al. / Radiation Physics and Chemistry 58 (2000) 315±323322
increase. This behaviour may be due to the chemicalbinding of the solute with solvent. At high concen-trations, the average distance between the species re-
sponsible for absorption is diminished to the pointwhere the charge distribution of the neighbours isa�ected. This interaction, in turn, can alter the species
ability to absorb a given wavelength of radiationbecause the extent of interaction depends upon concen-tration, the occurrence of this phenomenon causes de-viations from the linear relationships between
absorbance and concentration. The close proximity ofions to the absorber alters the molar absorptivity ofthe latter by electrostatic interactions; the e�ect is les-
sened by dilution. Deviations from Lambert±Beer lawalso arise because e is dependent upon the refractiveindex of the solution which is a function of concen-
tration of the solution.
Acknowledgements
The authors are thankful to U.G.C. for providing®nancial assistance to carry out the research project.
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Table 6
Average values of interaction radii of solutes in aqueous solutions of alkali chlorides
Solute Concentration range (mol/l) Interaction radii (m)� 10ÿ15
81 keV 356 keV 511 keV 662 keV 1173 keV 1332 keV
Lithium chloride (LiCl) 1.157±5.328 18.88 9.849 9.517 9.009 7.736 7.507
Standard deviation 0.007 0.006 0.002 0.002 0.003 0.004
Sodium chloride (NaCl) 0.838±3.860 21.17 11.51 11.19 10.60 9.118 8.790
Standard deviation 0.005 0.007 0.002 0.001 0.005 0.005
Potasium chloride (KCl) 0.503±2.305 132.35 15.02 14.42 13.54 11.70 11.26
Standard deviation 0.001 0.009 0.007 0.001 0.004 0.005
G. Kaur et al. / Radiation Physics and Chemistry 58 (2000) 315±323 323
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