Journal of Nuclear and Radiochemical Sciences, Vol. 1, No. 2, pp. 83−85, 2000

83

A Simple Derivation of the Formula of the Doppler Broadened 478 keV -Ray Lineshapefrom 7*Li and Its Analytical Application
Michael K. Kubo*,a and Yoichi SakaibaDepartment of Chemistry, School of Science, the University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
Department of Chemistry, Daido Institute of Technology, Hakusui-cho, Minami-ku, Nagoya 457-8532, Japanb

Received: September 19, 2000; In Final Form: December 11, 2000
A simple and intuitive derivation of the formula describing the Doppler broadened (± 7.6 keV wide) γ-ray shape at
478 keV of 7*Li was developed. In application to analysis of PGA spectra of multi-elemental targets, a computercode based on the formula efficiently deconvoluted the 472 keV γ-ray from 24Na superposed on the 7*Li line. The
code also provided peak areas and the degradation constants of 7*Li in the spectra of samples.

Introduction

Formulation

Neutron activation analysis (NAA) is a sensitive and versatile analytical technique especially in multielemental quantitative trace analysis of environmental, industrial, biologicalsamples etc. With the advent of the prompt γ-ray analysis
method (PGA) very short-lived nuclides have been coming
into general use in NAA. Improved sensitivity and the rangeof applicable elements have opened a wide window of utility.1
One of the elements which PGA has an ultimate sensitivity isboron by using the 10B (n, α)7*Li reaction (σ = 3838 barn forthermal neutrons2). Since the halflife of 7*Li (73 fs) is of acomparable order of the thermalization time of the recoiling
7*Li in condensed materials, the 478 keV γ-ray is emitted byfairly rapid 7*Li, leading to Doppler broadening of the γ-ray
whose detailed shape reflects the deceleration rate of the 7*Liin the medium. From a practical point of view the broadening
sometimes causes serious overlapping of the γ-ray from 7*Liwith the ones from 24Na (472 keV), 80Br (469 keV), 60Co (484keV) etc. Since 7*Li does not emit other γ-rays, it is essential
to deconvolute the complex spectrum to each component inquantitative analysis of boron in PGA. Deconvolution also
provides the information about the deceleration rate of 7*Li ina sample which we define a degradation constant D (Ref. 3) tobe the averaged stopping power of the environment of sub
micrometer vicinity of the boron atoms.
Conventional methods have been adopted to decompose theoverlapped peaks in a spectrum into components and to estimate the contribution of each nuclide,4 no fitting method had
been applied to simultaneous extraction of both the peak area
and the degradation constant from a γ-ray spectrum before theproposal of Sakai et al. 3 in PGA. While Sakai's method isuseful to obtaining degradation constants in materials, itscalculation time requirement is not adequate for multielemental analysis of many samples required in practical use.
Sometimes an apporximate lineshape formula has been used tofit the spectrum.5 In this study we reexamined the formulationand succeeded in reducing the multiplicity of numerical integration and computation time. Although similar formulae havebeen obtained by other authors based on different approaches6,7
and a pure mathematical procedure leads to the identicalformula starting from distributed δ functions within theDoppler shift range,8 in this NOTE we present a more simple
and intuitively understandable derivation of the formula
describing the Doppler boradened lineshape with emphasizing
physical meanings.

The Doppler energy shift is determined by the projectedvelocity (vz) of 7*Li along the measurement axis, the detecorsample line (Figure 1). A γ-ray spectrum reflects the velocity
distribution of 7*Li when it emits a γ quantum integrated over
its lifetime. Since the velocity of 7*Li relative to the detector
does not exceed the initial recoil velocity v0, 4.8 × 106 ms−1,
which is negligibly small compared with the speed of light,
everything is treated non-relativistically. The distribution ofthe projected velocity f(vz) is easily converted to a γ-ray energy
spectrum g(E) by variable transformation E = E0 (1 + vc ) = E0+ ∆E, where E0 is the unshifted γ-ray energy, c is the speed oflight, ∆E is the Dopper shift (for 7*Li ∆Emax = 7.6 keV).
We start with the special case when 7*Li experiences novelocity degradation. We observe γ-rays emitted from 7*Li
recoiling isotropically with a mean lifetime τ = 1/ λ . The
velocity distribution of 7*Li which emit a γ quantum at time t
is the product of the disintegration rate and the solid angle
fraction of 7*Li with vz,*Corresponding author. E-mail:kubo@chem.s.u-tokyo.ac.jp.
FAX: +81-3-5841-8324.

θ)f (vz) dvz = (λ N dt) 2 π v sinθ (−vd4π v 2= − 1 λ N dt sinθ dθ
2(1)
(2)

where θ is the angle between the motion of 7 *Li and themeasurement axis, N is the number of 7*Li at time t, v is thevelocity of 7*Li. Since v remains constant by assumption,vz = v cosθ = v0 cosθ(3)d (vz) = d (v0 cosθ) = −v0 sinθ dθ.

(4)

Combining eqs. 2 and 4, we obtainf (vz) dvz =λN
dt dv z.
2v0(5)

Figure 1. Detection of a γ-ray emitted from 7*Li in flight. The angle
between the velocity vector of 7*Li and the observed γ-ray is θ.

© 2000 The Japan Society of Nuclear and Radiochemical Sciences
Published on Web 12/28/2000

Kubo

84 J. Nucl. Radiochem. Sci., Vol. 1, No. 2, 2000
This means that f (v z ) is independent of v z , or the velocity
distribution is constant over the range (−v0, +v0), leading to arectangular shaped γ-ray energy spectrum,g (E) dE = g (E0 + ∆E) dE = c λ N dt dE.
E0 2v0(6)The overall spectrum can be obtained by summing up the timesliced spectrum obtained above, or integrating eq 6 withrespect to t. This process is stacking up the rectangles witharea dS = ε λ N dt, the number of emitted γ quanta times detection efficiency ε (hereafter we put ε = 1 for simplicity) (Figure2). Because N decreases with time as N = N0 exp (-λt) but thewidth of the rectangle which is equal to the Doppler shift range
keeps constant value of 2E0 v0 / c, the height of the rectangle
diminishes with time as
cλN0 exp (−λt) dtdS=(7)2E0 v0 /c
2E0 v0t=∞
cN0g (E) = ∫t = 0 dh =(|∆E| ≤ ∆Emax). (8)2E0 v0dh =All the γ quanta (N0) are distributed equally over the range(E0-∆Emax, E0 + ∆Emax).
Next, let us consider the γ-rays emitted from 7*Li producedin condensed materials where 7*Li experiences deceleration
within its lifetime. In such a case the shape of the γ-ray energy
spectrum is obtainable by substituting v0 in eq 7 with timedependent v. For 7 *Li the time dependence of v is welldescribed asv = v0 exp (−Dt),

E0v0
tE = 1 ln.
c|∆E|D()(11)

By integrating eq 10 one comes to the final form,g (E) =cN0
2E0v0

λ

λ−DD|(12)[ ( c|E−EEv ) ](|∆E| ≤ ∆E ).
01-0 0max

A real spectrum is obtained by convoluting eq 12 with an
instrumental response function, usually assumed to beGaussian, by one step numerical integration. It has reduced
computing time compared with our previous method3 which
required double numerical integration. From the exponent ofeq 12 we can derive a new characteristic of the spectrum which
eluded us before. The spectrum is upward concave and bellshaped when the degradation is slow, (λ−D) /D ≥ 1, or 2D ≤ λ,and downward concave when 7*Li decelerates rapidly, (λ−D)/D ≤ 1, or 2D ≥ λ (Figure 3). In our derivation this is clearlyshown through a consideration about the shape of the rectangles. As the width of a spectrum-building rectangle at time t is
E0v0 exp (−Dt) and its height is described as cλN0 exp (−λt) dt from
2E0v0 exp (−Dt)
ceq 7, the time dependence of the horizontal to vertical ratio of(−2Dt)
the rectangle is proportional to exp
exp (−λt) . Therefore the skewness of the lineshape is determined by the balance between 2Dand λ as mentioned above. As far as we know, no material hasever shown a downward concave spectrum for 7*Li case.

(9)

where D is the degradation constant characterizing the deceleration process of 7 *Li in the material. 3 The height of therectangle dh is larger in this case than the one in no deceleration case (Figure 2) compared at the same timing, because theDoppler shift range, the width of the rectangle, is smaller inthis case but the area of the rectangle which is proportional tothe γ-ray emission rate λ N dt is same in both the cases. Theoverall spectrum is given byg (E) = ∫t = 0 E dh =∫0 Et=t

the spectrum region where Doppler shift is larger than ∆E(Figure 2). Since |∆E| = E0v0 exp (−DtE) / c at t = tE, the limit tE
is written ast

cλ N0 exp(−λ t)dt.
2E0 v0 exp(−Dt)

Analytical Application
Experimental details were published previously.3 Gammaray spectra were obtained with the PGA system by using thethermal neutron beam guide at JRR-3M in Japan Atomic
Energy Research Institute.1 Typical thermal neutron flux was
2 × 107 cm−2 s−1, and cold neutron flux was 2 × 106 cm−2 s−1.
Measuring time varied from 10 minutes to 2 hours dependingon the boron content of the samples.
A usual procedure to fit a γ-ray spectrum obtained with a(10)Energy dependence is included in the upper limit of the integral tE as follows. Because of the deceleration 7*Li is so slow
at t = tE that the maximum Doppler shift can not exceed |E − E0|= |∆E|. Hence the emitted γ quanta after tE do not contribute toFigure 2. Doppler broadened γ-ray spectra from *Li: (a) without
deceleration; (b) with deceleration. The spectra can be obtained bystacking rectangles whose areas correspond to the sum of γ quantaemitted during dt distributed over the region (E0 (1 − v/c), E0 (1 + v/c)).
7

Figure 3. Lineshape dependence on degradation constant D for 7*Li
case. D's are 0.0, 1.0, 2.0, 3.0, 4.0, 5.0 and 6.0 ps−1 from top to bottom.
Lineshape changes from upward concave (bell-shaped) curve todownward concave one as deceleration becomes faster. Instrumental
response function is not convoluted.

A Simple Derivation of the Formulasemiconductor detector is to assume that a γ -ray line isobserved as a Gaussian function on a background, which is acombination of a linear function and an arctan type function.9
Since in our case eq 12 is interpreted as continuously
distributed Gaussian functions in the energy range (−∆Emax,+∆Emax), we can safely assume that the steep arctan type background jump is also distributed over the range and smeared out

J. Nucl. Radiochem. Sci., Vol. 1, No. 2, 2000 85
to become a linear gradual change. So we developed acomputer code to fit a spectrum as a superposition of the function of eq 12 from 7 *Li, several Gaussian type peaks fromnuclides other than 7*Li and a background composed of threelinear line segments. Decomposition examples are shown in
Figure 4. Sharp γ-rays from 24Na and broad lines from 7*Li areclearly separated in both cases. The code also yielded degradation constants, 2.54 ± 0.02 ps−1 for elemental boron and 1.52
± 0.03 ps −1 for boric acid, which are good agreement withthose for pure substances obtained by using our previous
analysis method.3 The calculation times were less than a few
minutes with a personal computer, which clearly proves theeffectiveness of the code based on the formulation of thepresent study.
Acknowledgements. We thank Drs. C. Yonezawa and H.
Matsue of Japan Atomic Energy Research Institute for PGA
measurements.
References

Figure 4. Decomposition of γ-ray spectra: (a) a mixture of elemental
boron and Na2CO3; (b) a mixture of boric acid and Na2CO3. Filled
circles are observed data, solid lines are fitted line, dotted Gaussian
curves are 472keV γ-ray from 24Na, dashed lines are Doppler broadened 478keV γ-ray from 7*Li (eq 12). For elemental boron and boric
acid D = 2.54 ± 0.02 ps−1 and D = 1.52 ± 0.03 ps−1 were obtained,respectively.

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