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Particle-size effects in non-destructive material assay by gamma-ray absorptiometry I. Theoretical model
Nuclear Instruments and Methods 212 (1983) 445-461 North-Holland Publishing Company
445
PARTICLE-SIZE EFFECTS IN NON-DESTRUCTIVE M A T E R I A L A S S A Y BY G A M M A - R A Y A B S O R P T I O M E T R Y I. T h e o r e t i c a l m o d e l
H.L. T H E , W. M I C H A E L I S a n d H.-U. F A N G E R Institut fi~r Physik, GKSS Forschungszentrum Geesthacht, Germany
Received 29 October 1982 To Professor Erich Bagge on the occasion of his 70th birthday
Gamma-ray absorptiometry has become a widely used method in non-destructive material assay. The procedure implicitly presumes a homogeneous, fine-grained distribution of the solid components of the material and is well described by a 'sandwich' model. However, due to the nonlinearity of the attenuation law, systematic errors arise if finite grain sizes are involved. The present paper deals with the influence of particle size and gives a thorough theoretical treatment of these effects, which allows the extension of the method to mixtures containing one or even several coarse-grained constituents. The derivation of suitable generalized transmission equations requires the construction of a particle-size dependent attenuation law. This is achieved by means of a 'cell' model which subdivides the irradiated volume into smallest possible 'cells' with representative parts by volume of the material components. Correction functions are deduced that can be used to take into account particle-size effects by properly modifying one of the relevant physical quantities in the conventional transmission equations. The properties of these functions are discussed with respect to their applicability in practice.
1. I n t r o d u c t i o n
G a m m a - r a y a b s o r p t i o m e t r y has p r o v e d to be a very powerful tool in n o n - d e s t r u c t i v e analysis of o p a q u e m a t e r i a l s [1]. P e r h a p s the m o s t i m p o r t a n t a p p l i c a t i o n s are the gauging a n d c o n t r o l of the h y d r a u l i c t r a n s p o r t of solids in pipes. In particular, t w o - c h a n n e l g a m m a - r a y transmission analysis m a y successfully be used for in-line q u a s i - c o n t i n u o u s and n o n - d e s t r u c t i v e m e a s u r e m e n t s of the individual p a r t s b y v o l u m e in t h r e e - c o m p o n e n t flows, b y p r o p e r l y choosing the two g a m m a - r a y energies a n d b y solving the p e r t i n e n t t r a n s m i s s i o n equations [2-4]. The o u t s t a n d i n g efficiency of the m e t h o d has been d e m o n s t r a t e d , for example, d u r i n g the first m a n g a n e s e n o d u l e s p i l o t - m i n i n g test in the Pacific [5], in the course of the p r e - p i l o t m i n i n g test for the haulage of h y d r o t h e r m a l slimes in the R e d Sea [6] a n d in c o n n e c t i o n with the h y d r a u l i c t r a n s p o r t of coal [7,8]. O n the basis of the d a t a o b t a i n e d , different m i n i n g facilities could be assessed with respect to efficiency, safety a n d economy, different o p e r a t i o n a l m o d e s were c o m p a r e d , and the most significant t r a n s p o r t p a r a m e t e r s could be optimized. So far, all a p p l i c a t i o n s have i m p l i c i t l y p r e s u m e d a homogeneous, fine-grained d i s t r i b u t i o n of the solid m a t e r i a l c o m p o n e n t s . The m e t h o d can therefore be well d e s c r i b e d b y m e a n s of a ' s a n d w i c h ' m o d e l in which the m i x t u r e u n d e r study is r e p r e s e n t e d b y successively a r r a n g e d slabs of the c o m p o n e n t materials, with the thicknesses of these slabs being chosen a c c o r d i n g to the i n d i v i d u a l parts b y volume. However, due to the n o n l i n e a r i t y of the g a m m a - r a y a t t e n u a t i o n law, systematic errors m a y arise if finite particle sizes are involved. The present p a p e r deals with the influence of particle size a n d gives a t r e a t m e n t of these effects which allows the extension of g a m m a - r a y t r a n s m i s s i o n analysis to mixtures c o n t a i n i n g c o a r s e - g r a i n e d c o m p o nents. In p a r t I the u n d e r l y i n g theoretical m o d e l is described. C o r r e c t i o n functions are d e d u c e d which can be used to take into account particle-size effects b y p r o p e r l y m o d i f y i n g p a r t i c u l a r quantities occurring in 0167-5087/83/0000-0000/$03.00
© 1983 N o r t h - H o l l a n d
446
H.L. The et al. / Particle size effects 1
the conventional transmission equations. The theoretical results were experimentally examined by means of a hydraulic loop which allows transmission measurements under a wide range of well-defined conditions. These experiments will be described in part II of this paper. As will be seen, quite satisfactory agreement was found between theory and experiment. Several attempts for tackling the problem of gamma-ray absorption by non-homogeneous materials have been reported in the literature. Berry [9] proposed a formula to explain the so-called "grain-size effect" in X- and gamma-ray fluorescence analysis. Starting from this formula, other authors [10,11] derived an expression for the effective absorption coefficient for a non-homogeneous material defined as a mixture of two kinds of spherically shaped grains of the same size. In comparison with the literature, the present paper gives a substantially more general treatment, in particular with respect to the number of components involved, the grain size, the occurrence of size distributions and the mutual obstruction of the particles.
2. Non-destructive measurement of material compositions by gamma-ray transmission analysis In practice, three-component mixtures represent the most important case of application. The principle of material analysis by gamma-ray absorptiometry will, therefore, be discussed in the following on the basis of a mixture consisting of three components. The stationary case of a material with a composition invariable with time may be regarded as a special case of the conveyor-flow analysis. In hydraulic transport systems the mixture, in general, consists of a carrier medium like water (w), a solid charge of primary interest (p), and a third solid or liquid dead component (q). Typical examples are the mixtures sea water manganese nodules-sediment [5], sea water-hydrothermal slimes-NaC1 brine [6], or water coal rocks [7,8]. Neglecting grain-size effects, the transmitted intensity I of radiation after passage through the attenuating medium is described by the equation I i = I0, e x p [ - / { t)p~pi +
l)q~qi + l)w~w,,)] ,
(1)
for any discrete energy E~. Here, t~j denotes the parts by volume of the components.,/= p , q , w ; / , , stands for the linear gamma-ray attenuation coefficients, and l is the transmission length. It is convenient to refer the intensity Ii to the corresponding quantity for uncharged water
lwi = Ioi exp[
-
II~wi ] .
Introducing t, = li/lw, one arrives at In t, l = l)pAjtLpi Jr- 12qA~[.Lqi ,
(2)
where the condition = l,
(3)
y and the abbreviation A#i~ = >j, - #w~ have been employed. If gamma-rays of two energies are used (i = 1,2), the associated eqs. (2) together with the sum rule (3) allow the determination of the parts by volume of the three components involved from the transmitted
intensities [2 4]: Vp = (IN)
' [A#q, in
A/.Zq2 In t,],
(4)
vq = ( I N ) - ] [A#p2 In t, - A/zp, in t=],
(5)
t 2
--
with N
= A [ . L p l A ~ q 2 -- A / - L p 2 A ~ q I .
H.L. The et al. / Particle size effects 1
447
The two energies have to be properly chosen, in order that N most strongly differs from zero. The radioisotopes 241Am(El = 60 keV) and 1 3 7 C s ( E 2 = 662 keV) turn out to be best suited both in this regard, and in respect of practicability. Possible systematic errors and the influence of counting statistics have been thoroughly discussed elsewhere [2-4,6]. Extension of the method to multi-component mixtures with n components and the usage of (n - 1) gamma-ray energies is straightforward.
3. The influence of particle size The outline of gamma-ray absorptiometry as given in the preceding section implicitly presumes a homogeneous, fine-grained spatial distribution of all components in the material to be analysed. The mixture may be equally well represented by a 'sandwich' set-up perpendicular to the gamma-ray beam, with slabs of thickness AXj = t~jl. Due to the nonlinearity of the gamma-ray attenuation law, particle-size dependent systematic errors may occur if coarse-grained materials are involved. In this case not only the concentration of the coarse-grained component, but also the number and correlated size of the particles,
V- radiation
F- detector
y-radiation
F-detector
Fig. 1. Schematic representation of a particle-size effect: Identical particles may cause different attenuation if irradiated in different spatial dispositions.
448
H.L. The et al. / Particle size effects 1
their shape, and their spatial distribution determine the gamma-ray attenuation. The latter effect, for example, can be understood with the aid of fig. 1 which shows schematically spherical particles of finite, but uniform size in two different spatial dispositions. Obviously, in the first case the attenuation approaches (1 - ~r/4)t, when the particles are assumed to tend towards opaqueness, while for the second configuration with mutual obstruction one obtains in the limit case ( 1 - ~r/16)t. Here, t denotes the attenuation factor for the medium surrounding the particles. To overcome this problem, the construction of a particle-size dependent attenuation law is required. This is achieved by means of a 'cell' model (sect. 4) which leads to a generalization of eq. (2) in such a way that grain-size effects caused by spherical particles of uniform size, including mutual obstruction, are taken into account in an approximate treatment [12]. From this new attenuation formula, appropriate correction functions can be deduced which allow the application of gamma-ray absorptiometry to materials containing coarse-grained constituents (sect. 5). Finally, the theory is extended to cover the occurrence of grain-size distributions (sect. 6).
4. 'Cell' model
4.1. Definitions Consider a multiple-component mixture with one component consisting of spherical particles of uniform size with radius r. The remaining fine-grained components may be treated together, for the present, and will be referred to in the following sections as the 'residual mixture'. As a first step, the total irradiated flow volume is subdivided into adjacent 'flow channels' with the particles on their axes, as shown in fig. 2. Let the channel axes be oriented parallel to the flow direction and the cross-sections chosen to be of square shape. Suppose that a particle just fits into such a channel, i.e., the lateral length in a plane perpendicular to the axis equals 2r. If it is now further assumed that the particles are distributed uniformly over the flow volume, insofar as for each of them the mean distance to both neighbours within the 'flow channel' is constant, then the flow may be subdivided into so-called 'cells', each of which contains just one particle (fig. 3). The cells defined in this way are characterized by the same parts by volume as the total flow. The model, which will be developed in the following subsections on the basis of this picture of idealized cells, thus presumes a certain order in the spatial distribution of the coarse-grained component. There is, however, no restriction on its universality, in that some order may be brought about, when particles are thought to be translated in parallel with the gamma-ray beam which leaves the attenuation of the gamma radiation unchanged.
detecto~/~ _._
---
spherical particle of radius r k~' ~ /
----t ;T . . . . . . . . . . . . . .
-~ . . . . . . . .
.
.
.
.
[d=2r
.
>k y - ray source
Fig. 2. Construction of 'flow channels'.
,, ce[L" Fig. 3. Conception underlying the construction of 'cells' (l ~- f i n
subsection 4.2).
H.L. The et al. / Particle size effects 1
449
4. 2. Hit probability and part by volume
The probability Pp that a photon strikes the spherical particle when penetrating a cell of lateral length/(fig. 3) is given by Pp = r r r / 2 L
On the other hand, according to subsection 4.1, the part by volume vp of the coarse-grained component may be deduced from the particle and cell volumes as Vp = ~rr/3[.
Hence, it follows that Pp = 3Vp.
(6)
4.3. Mean attenuation within the particle cross-sectional area in passage through a cell
If the particle volume is resolved into r i n g cylindrical volume elements, as shown in fig. 4, the attenuation factor for gamma radiation impinging at radius x on a cell of thickness 2r is given by
tx=
' / I 0 = exp[ -
{ ~ p ( 2 ~ - - X2 ) q- ~rm(2r -- 2~/~-r2- X2 )} ].
(7)
Here, ~p and /*rm denote the linear attenuation coefficients of the particle material and the residual mixture, respectively. Eq. (7) takes into account that the particle is always surrounded by the remainder components. The normalized probability that photons will strike the ring cylindrical elements of thickness Ax is
W(tx) = 2 x a x / r 2 .
(8)
Consequently, the attenuation averaged over the particle cross-section in passage through the cell is obtained from
= Y'. W( t )tx,
(9)
or, in integral form, using eqs. (7) and (8) tp/cell = ~ -2 e /*rm2rfor xe -("p "~m)2~/r2-x2dx. r Partial integration yields tp/ce,l = e - " " e r F ( r ) ,
Fig. 4. Ring cylindricalvolume elements of a sphere in a cell of thickness 2r.
(10)
450
H.L. The et al. / Particle size effects I
with
F(r)= 2{ eCr(Cr-1)+l ),
(ll)
c=
(12)
and
Since the first factor in eq. (10) describes the attenuation in passage through the cell in absence of coarse-grained particles, one may write /p/cell = t r m / c e l l F ( r ) .
(13)
4.4. Mutual obstruction of particles The attenuation of the gamma-rays is determined by the momentary spatial distribution of the particles in the flow volume. Due to the finite measurement time in transmission analysis, the measurement result is given by the time-averaged attenuation, i.e. all possible particle constellations have to be taken into account. If the total transmission length is l, then it follows from subsect. 4.1 that n = l/2r cells are penetrated by the gamma-ray beam. Considering also subsects. 4.1 and 4.3, the probability that a photon strikes a particle at radius x in passage through a cell may be written as
Px = (2xAx/r2)Pp = W(tx)Pp" Correspondingly, the hit probability for the residual mixture is Prm= 1 -Pp.
If now the probability is considered that a narrow beam, in passage through the total flow, strikes j particles, each at a different radius x i, and gets through (n - j ) cells with residual mixture, one obtains for the single event J J egmJ F I P~.~ = P/(1 -- e p ) " - J 1-I W(t~.,). i~l
i=1
According to the rules of combinatorial analysis, this event occurs in a (~)-fold manner where n!
(~) j!(n--j)!" Hence, the total probability is J
Wj = (~)Pr~(l - Pp)" JI-I w'(tx.i) dxi = wf d x I d x 2 . . , dxj,
(14)
W'( tx.i) = 2xi/r 2.
(15)
i=1
with
The corresponding attenuation factor tj is given by J tj=(trm/cell)n-JHtx,i i=l
[cf. eq. (7)].
H.L. The et al. / Particlesize effects I
451
4.5. Particle-size dependent attenuation law The mean total attenuation factor ttot is then obtained, with proper normalization, by summing up all weighted contributions
j W,.t, = ( " ) P ~ { ( 1 -ep)trm/¢e,,} "-J 1--I w'(tx,ilt~.idxi=tjdx, ' dx2.., dx,
(16)
i=1
over the n u m b e r j of struck particles and by simultaneously integrating over all possible radii xi: ~-0"" fx r/ tjl d x I d x 2 . . , dxj ~ fx~lI-- 0fx r2 --/=0 / t o t - j=0 j = 0 J x , "- 0 'x 2' --0f -
(17)
x,=0Wjdxldx2""dxj
Since the sums and all integrations are independent of each other, this formula can be simplified considerably. The denominator equals 1, because
ff
Zx d x = 1 r2
and, thus, with eqs. (14) and (15)
a=o ,-o 2-o"" --
--
/=0
Wjdx, dx2...dxj=
j=O
(,:)g(1-g)" ' = ( / ' p + l - g ) =l.
For the numerator of formula (17) one obtains by means of eqs. (16) and (9), remembering that a uniform size of the particles has been assumed: J
k (;)PpJ((lj=0
Pp)trm/cel,)
" J i~=l=f~i',_0W'(t~,)t~,dxs= ~ " '
_
=( Ppt-p/cen + ( 1 - Pp)trm/cell}
,-
(;)(Pptp/cell)J((1
Pp)trm/ce,l) n-j
j=0
t/
.
Hence, with n = l/2r and eqs. (6), (10) and (13) the mean total attenuation factor is finally given by t-to, = e-~rm'[~vp{F(r) - 1} + 1] ,/2,,
(18)
where F ( r ) follows from eqs. (11) and (12). Eq. (18) describes the gamma-ray attenuation by a multi-component material in the presence of a coarse-grained constituent with particles of uniform size with radius r under the assumptions made in subsect. 4.1. The formula represents a generalization of the equation t=e
~'rm/e-%(up
P'r m )/,
(19)
which, by analogy to eq. (2), is valid for a material fine-grained as a whole (r = 0) or, in other words, eq. (19) is a special case of formula (18) for the limit of vanishing particle radius. This is easily proved by equating the relations (18) and (19) and labelling the parts by volume of the coarse-grained component by the superscripts r and 0, respectively. One obtains 2 e ~°cr- 1
Vp- 3 F ( r ) - I
452
H.L. The et al. / Particle size effects I
By applying De l'Hospital's rule twice it is found that lim r = r ~ 0'Op
0 L~p,
q.e.d.
5. Corrections for particle-size effects 5.1. General outline
Owing to its complexity, eq. (18) is hardly accessible to a direct evaluation in practical applications of gamma-ray absorptiometry. Therefore, a simplified procedure for solving the transmission equations with respect to the parts by volume has to be found. This can be accomplished by transforming eq. (18), rewritten in exponential form,
=
111"
optionally into one of the following conventional relations that result from the 'sandwich' model [cf. eq. (19)]: t* = e ~,~/e-%~,l,
(21a)
t = e-~'rmte-~pJ~')*t,
(21b)
t = e-~rm/e-'~;~t,
(21c)
t = e-P'~m/e-%a~*l,
(21d)
t = e "'mte vd~3--Urm)l,
(21e)
or
with Z~# =/Xp - #rm' This means that the correction for grain-size effects is performed by modifying one of the physical quantities t, (vpA/~), Vp, A~ or /%, then labelled by an asterisk, with the aid of correction functions, which are obtained by comparison of eq. (20) with one of the eqs. (21a)-(21e), respectively. The optimum choice of these possible procedures depends upon the particular properties of the resulting correction functions. These properties, therefore, require a brief discussion. 5.2. Materials containing one coarse-grained constituent
As before, let the fine-grained constituents of a multi-component material be treated together as a homogeneous 'residual' mixture. By means of eqs. (20) and (21a) one obtains the correction function g(r)=
_
=
trot
Fig. 5 illustrates the dependence of this function on the particle radius for some two-component materials with gamma-ray energies of 60 keV and 662 keV and the part by volume Vp in the range 1-6% as parameters. A transmission length of 20 cm was assumed in these calculations. The relevant linear attenuation coefficients are shown in the insets. The graphs indicate a strong dependence of g ( r ) on Vp which makes this procedure only of limited usefulness for practical applications, since, in general, % is the quantity to be determined. From eqs. (20) and (21b)-(21d), respectively, identical correction functions are obtained, since the
H.L. The et a L / Particle size effects I
453
60*,k --
t'/,
- - ~ _ ~
.;:~ o. 9
~.j.
~>~ 0.9
~
'.L'
,, 1=20cm
i
o.8
I I
L
manganese nodutes in sea =wafer 0.6
60keV O.7z,8 662 keV 0122 t
coal
t=20cm
0195 0085L
1.o
0~92
662 keY: 0101
1.5
2.0
particle radius
- -
2.5
0,5
3.0
0085
t
~j [cm -1 ]
0.5
1.o
~j i[ c m - 1 ]
0.5
in wafer
60keY: 0.228
i
2.0 particle
662 k,V
radius
2.5
3.o
r[cm]
W,
it'/.
.;:~ O.9
~..~ 0.9 ,:,.,
--..~--------~ ~
2
i
1.5
r[cm]
~
~.,.
0.8
0.8
1:20cm
sfeafite spheres in wafer 60keV
055g
0192
60key
0664
0192
662key
O 182
0085
662keY:
0188
0085
~J Icm -1 ]
I
0.5
0.5
h0
J
I
1.5
~parficie
i=2o era
dead rocks in water
2.0
radius
2.5
3.0
o.s
r[cm]
# [cf 1 ] 1.o
1.5 ,
2.0 particle
2.5 radius
3.O
r[cm]
F i g . 5 ( a - d ) Correction function g(r) vs. particle radius r with the part by volume ranging from 1 to 6% for some two-component mixtures. Gamma-ray energies 60 and 6 6 2 k e V . Transmission length 20 cm. The linear attenuation coefficients used for the calculations are specified in the insets.
q u a n t i t i e s l)p
f(r)
-
and Aft appear as a product in the latter equations:
(VpAff)* f)pA]2
Vp _ Aft* t)p
Atx
ln[3vp{F(r)
-
1) + 1]
(23)
vpCr
In this formula the transmission length l is eliminated. However, the most characteristic feature o f f ( r ) is that this function exhibits only a very slight dependence on Vp, as can be seen from fig. 6. The same materials, gamma-ray energies, and range of volume concentration as in fig. 5 were taken as a basis. Thus, f(r) fulfills the most essential criterion for practical applications. The function can be calculated in advance for the material to be analysed. The measurement is performed as usual by solving the energy-dependent eqs. (19); however, in doing so, the product (vpA/~) or one of its factors is modified with the aid of eq. (23), which is also a function of the gamma-ray energies used. Because of the slight influence of Vp on f(r), a mean correction function f ( r ) may be employed in practice which is valid to a good approximation over the range of interest for %. In general, the errors arising from this procedure are negligible. A comprehensive discussion of this subject is included in ref. 12. Besides its dependence on the particle radius, f(r) depends also on the difference of the linear attenuation coefficients Aft =/% -/%m" Corresponding curves are shown in fig. 7. It should be pointed out that the absolute values of these coefficients do not enter.
H.L. The et al. / Particle size effects 1
454 i,o
662 keY--
~*,v
0.8
~ O.B
0.6
0.6 r'/,
o.&
0.2
I
I
COO[
manganese nodules in s e a - w a f e r 60keY: 0.748 0.196 662 keV: 0.122
in wafer
60k#4: 0.228 0.192
0.0854
662 keV: 0101
[cm-11 0.0
I
l
0,5
01085
p [cm -1 ]
1.0
1.5 --"
I
2.0 2.5 particle radius r [ c r n ]
I
1.0
0.5
1.5
2.0 porfic[e
1.0 ~
2.5
3.0
radiusr [ c r n l
1.0 ~
~
'
~
~
O.B
~
~
0.8
~
6
O.S 6%
0 '¢
0.4
dead rocks in wafer
sfeofite spheres in wafer 0.2
60keY 0.559
0.192
662keY: 0 179
0085
60keV
. . . .
[cr~- 1 ] o,o
1
[ 1.0
0.5
0 6 6 4 0192
662keV: 0.188 0085
1.5
2.0 2.5 particle radius r[crn]
o.o
3,0
i
~u [cm_ 11
0.5
1.0
I
1.5 ,
2.o 2.5 particle radius r [ c m )
3.
Fig. 6. ( a - d ) Correction f u n c t i o n / ( r ) vs. particle radius r with the part by v o l u m e ranging from 1 to 6% for some t w o - c o m p o n e n t mixtures. G a m m a - r a y energies 60 and 662 keV. The linear a t t e n u a t i o n coefficients used for the calculations are specified in the insets. 1.0
r 0.05 0.10
0.8
0.15 -0.20
v
:0.25
0.6
._.,..
"*---.,_.
0.4
Vp=5°lo
,0.30 '035 ~OJ,O :0,45 '0.50 ,0.55 ,060
AjJ [cm -1 ]
0.2
0.0 O.
1.0
2.
L.
=
2.5 pQrticie r a d i u s
3,0 r[crn]
Fig. 7. D e p e n d e n c e of the function f ( r ) on the particle radius r and the difference A/~ of the linear a t t e n u a t i o n coefficients/% and/L .... a s s u m i n g a p a r t by volume of 5%.
H.L. The et al. / Particle size effects 1
455
From eqs. (20) and (21e) the correction function /% -- btp ]'£rm
2r/)p
may be derived. Grain-size effects are considered by a procedure analogous to that described before. Again, the transmission length is eliminated and there is practically no dependence of h(r) on % (fig. 8). On the other hand, this function depends not only on A/~, but also in an explicit manner on /~rm which may prove to be a disadvantage. Therefore, the best choice between f(r) and h(r) is determined by the nature of the problem to be solved. Fig. 9 displays h(r) vs. the particle radius with A~t as a parameter, assuming /~rm = 0.192 cm ~ corresponding to water and Vp = 5%.
5.3. Transmission analysis of a three-component mixture The equations given in subsect. 5.2 apply to all multi-component systems containing one coarse-grained material provided that the remaining components can be treated as a unified homogeneous mixture. Formulae for special problems are easily derivable, although in most cases, additional assumptions have to be introduced. As an example, let the most important case of a three-component flow be discussed in some detail. Here, along with the percentage of the coarse-grained material, the individual parts by volume of the homogeneous components are of interest. Following the nomenclature of sects. 2 and 4, one obtains ~rm
l~/)p(~tq--~w)+~w ,
and thus
C= -2
I
(t~p-/~w)-
As a consequence, the functions F(r) and f(r) become dependent on the parts by volume in a rather complicated manner which requires cumbersome numerical solution procedures. For practical applications, therefore, a simplified treatment must be found in which the errors introduced remain negligible or, at least, tolerable. There are several possible approaches for solving this problem. The first procedure involves assigning mean values ~p(r) of the true linear attenuation coefficient/Xp(r, Vp, /)q), for the gamma-ray energies used, and, hence, mean values of the correction function h(r) over the expected ranges of variation of vp and Vq. The calculation of the data matrix is performed in advance for a proper set of particle radii. Extensive tables given in ref. 12 indicate that the resulting errors remain acceptable provided that the ranges of variation are not too large, which, for instance, is quite well valid for the transport of solids through pipes. A second rather efficient procedure may be derived by writing [cf. eq. (21d)] / ) p ( ~ p -- ~ r m ) *
q" ~ r m = /)p ~ p
=/)p
1- % - .w)*
- .w) + .w,
(25)
and correcting (t% -/~w) instead of (/~p -/~rm)" The pertinent correction function reads
f ( r ) = ln[~/)p{g'(r) - l} + 1] /)pC / r
with F ' ( r ) = 2 e C ' r ( C ' r - 1) + 1
and
C' = - 2 ( b t p - / x w ) .
(C'r) 2 Physically, this means that the problem is treated approximately within the framework of a mixed cell-sandwich model as illustrated in fig. 10. This can easily be shown by means of eq. (20). When applied
H.L. The et a L / Particle size effects 1
456
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radius
2.5
3.0
r[cm]
Fig. 8. (a-d) Correction function h(r) vs. particle radius r with the part by volume ranging from 1 to 6% for some two-component mixtures. Gamma-ray energies 60 and 662 keV. The linear attenuation coefficients used for the calculations are specified in the insets. 1.0
0.05 , 0.10
0.15 0.20 ..
0.8
0.25
0.30 0.35 O.t,O 0.45 050 0.55 0.60
ii
.~ o.~
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~,~u [cm-~]
0.2
0.0
1.0
1.5 -
2.0
parficle
2.5
radius
3.
r[cm
Fig. 9. Dependence of the function h(r) on the particle radius r and the difference A/~ of the linear attenuation coefficients/Xp and #rm assuming a part by volume of 5% and an absolute value of 0.192 c a - i for/~r,, corresponding to water.
H.L. The et al. / Particle size effects I
(£)
457
I. .: Ii..'
:
"
Fig. 10. Approximation of the pure cell model by a mixed cell-sandwich model.
to a mixture with the components p, w and a slab with component q, eq. (20) yields with
/)p q- 13w ~
1
/.tw(Vp + Vw) +f(r)Vp(~tp - btw)(Vp + Vw) + ~qL~q = ~tw(1 -- l)q) +f(r)vp(IXp -- ktw)(Vp + Vw) + #qVq
=f(r)vp(IXp - btw) + Vq(/Xq --/Zw) + ktw. The interpretation follows directly from comparison with relation (25). Again, detailed tables on the size of the resulting errors may be found in ref. 12. Concerning the dependence o f f ( r ) on Vp, we refer to subsect. 5.2. It is also shown in ref. 12 that employment of the maximum occurring value Vp max for calculatingf(r) may give the best results which further simplifies the evaluation. The parts by volume obtained from the preceding approximation method may also be used for recalculating the attenuation coefficients in an iteration procedure. In general, two steps already yield sufficient accuracy, even for larger particle sizes [12]. For small radii the product Cr becomes << 1. Then, the function F(r) can be expanded into a series: r(r)=
1 + ~ C r + (C-~2+ . . . .
It follows that f(r)=ln[1
+vpCr(l +~Cr)] l)pCr
and, hence, by expanding also the logarithmic expression
f ( r ) = (1 + 3 C r ) - ½vpCr(1 + 3Cr) 2 + . . . . The dependence on Vp occurs only in the terms of higher order. For sufficiently small values of r the second term may be neglected. The application of this approximation method in the case of fine-grained materials is, in particular, recommended if the evaluation is performed by means of microprocessors with a small word length, since rounding errors may possibly introduce inaccuracies when using the exact, but rather complex formulae.
5.4. Materials containing more than one coarse-grained constituent A stringent treatment of particle-size effects in materials with several coarse-grained components requires a theory based on complicated multinomial distributions. As a consequence, no closed analytical expressions can be derived. A more pragmatic approach, however, may be taken by using the mixed cell-sandwich model already utilized above with success. Consider first a three-component mixture with coarse-grained constituents p, q and a homogeneous residual constituent rm which in itself may be composed of several materials. Let the particles of components p and q be of different, but uniform size. Neglecting mutual obstruction of particles of
H.L. The et al. / Particle size effects I
458
different species and assuming %, Vq << 1, the transmission equations analogous to eqs. (20) and (2 ld) read
1 ln[~vp(Fp(rp)-1)+ _ ! 1 ln[(rp, rq)=P, rm_ 2-~p
1 ] - ~ r q1
ln[{vq(Fq(rq)_l)+l ]
1 and - 7 In t =/*r,n + VpA#~ + Vqa#q, with A g i = ~ , - - ~ r m (i = p, q) and corresponding substitution into the functions F,. Correction for particle size effects may thus be achieved by proper modification of the differences A/~i using separate functions f, (r,). Generalization to materials with n coarse-grained components yields 1 l n t - ( r 1 r2
,r,,)=/Zrm
~{
-
1
ln[~vi(Fi(r, )
1)+1]}
(26)
i=1
'
and - 7 In t = ~rm +
L V,A~a*, i=1
respectively. 6. Distributions of grain-sizes Since in many applications of the method the particle radii are not uniform, but follow more or less a distribution function, extension of the model to size distributions would be very attractive. One promising approach may be formulated as follows. By analogy to subsect. 5.2, consider a material with a coarse-grained component p and a homogeneous residual mixture. Let the particle radii obey a distribution function n(r). Then the associated normalized distribution function k(r) for the part by volume may be written in the sum representation as
1,(5) = n ( r, ) ri 3 i=1
and the differential part by volume is given by
vp(5) = k(5),
p0,
where %o is the total percentage of the constituent p. If each size interval is now thought of as a separate component, the problem becomes one of a multi-component mixture, which has already been discussed in subsect. 5.4. Since all particles belong to the same species, it follows that 1
t hii[k(r)l=.rm-- Z
{ 1
ln[3vpok(ri){F(r,) -
1)+ 11}
i=1
The rather cumbersome evaluation of this equation may be avoided if a representative particle radius can be found which leads to the same result as the stringent consideration of the size distribution. The proper choice depends on the distribution function presumed. A normal distribution of the grain radii hardly coincides with reality. Moreover, care must be taken that the condition r >/0 is fulfilled and, correspondingly, that the mean radius is not identical with the center of the distribution function. A more realistic description is provided by the Poisson distribution P(r), the asymmetric course of which is typical for many practical cases of application. Using Stirling's formula, P(r) may be written in the form P(F)--
l
e
-
~r
) .
H.L. The et aL / Particle size effects I
459
Fig. 1 la displays this function with the mean radius ? as a parameter. For comparison, in fig. 1 lb the associated distribution k(r) for the part by volume is shown. Here, rma~ denotes the grain radius at which the maximum in the distribution occurs. In search of a representative radius value on the basis of P(r) for the purpose given above, it is worthwhile comparing the correction function f(r), as obtained for the radius values ~ and rma ~, respectively, with that resulting from a stringent calculation on the basis of the complete distribution
0.20-
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o
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3. bo
.
+. bo
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5. oo
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Fig. 11. (a) Poisson distribution n(r) = P ( r ) of the particle radius r with the mean value ~ varying from 0.28 to 1.14 cm. (b) Associated differential distribution k ( r ) of the part by volume.
H.L. The et aL / Particle size effects 1
460
,,o
>~
0.20 I
o.o~.~o
O. 6B
• parficte
J
O. 91
radius
]. I¢
O.Ot) O.O0
O. ~2
F [Era]
i . 26
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I . 58
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2. I 0
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Fig. 12. (a-b) Comparison of the correction function f ( r ) as obtained by using the grain radii ~ and rmax, respectively, with the function f [ k (r)] resulting from a stringent consideration of the distribution function k (r).
function k ( r ) . As is shown in fig. 12 for the case of steatite particles in water, there are severe discrepancies between f ( ~ ) and f [ k (r)], while the function f ( r ma, ) coincides quite well with f [ k ( r )] for both gamma-ray energies assumed. Thus, rma, turns out to be sufficiently suited as a representative value for simplified correction procedures. In the presence of several coarse-grained components with individual grain-radius distributions n~(r), n2(r ) .... and parts by volume distributions k l ( r ), k 2 ( r ) .... the problem becomes very complex. Quite formally, the algorithms given above can be generalized without difficulty. If ~ denotes the representative particle radius of component i, the grain-size dependent transmission equation reads [cf. eq. (26)] 1 ln t [ k l ( r ) , k 2 ( r ) . . . . . k n ( r ) ] : ~ t r m - i =~1{ 1 l n [ ~ v , 0 { F , ( ~ , ) _ l ) + l ] } However, it should be pointed out that this general formula is based on several approximations. Its application, therefore, requires a detailed error analysis in each case.
7. Conclusions
In the present paper a 'cell' model has been developed which allows the construction of a particle-size dependent attenuation law for gamma radiation passing through materials containing one or more coarse-grained constituents. From this law, correction functions can be derived that permit the application and evaluation of the 'conventional' transmission equations in gamma-ray absorptiometry, by properly modifying one of the physical quantities involved. Both particles of uniform size, and those foll6wing a Poisson size distribution can be considered. Although certain approximation procedures had to be introduced in the course of the calculations, proper handling of the algorithms described ,ensures sufficiently accurate results. Experimental confirmation will be given in part II of the paper. The theoretical model presented considerably extends the possible range of application of gamma-ray absorptiometry in non-destructive material assay, since particle-size effects can now be taken into account in a quite straightforward manner. Of course, the use of the established correction procedures requires information on the grain radius or its distribution. These data if unknown may be obtained either by taking a sample from the material under study or, preferably, in a non-destructive way by gamma-ray transmission analysis as well. Pertinent suggestions have already been made by the authors elsewhere [3,12]. For example, particle-size information
H.L. The et al. / Particle size effects I
461
may be derived from the depths or widths of the counting rate 'dips' from a transmission gate which occur during passage of particles through the gate. If higher parts by volume of the coarse-grained constituent and, accordingly, increased mutual obstruction effects preclude such a direct electronic evaluation, the resulting complicated structures have to be resolved by means of a computer program similar to those employed for spectrum analysis in nuclear spectroscopy. Other more involved instruments may derive benefit from a spatial reduction of the measuring volume, for instance, by using two crossed gates or a conical multislit collimator.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
Proc. Int. Conf. on Industrial application of radioisotopes and radiation technology, Grenoble, 1981 (IAEA, Vienna 1982). H.-U. Fanger, R. Pepelnik and W. Michaelis, Nuclear techniques and mineral resources (IAEA, Vienna, 1977) p. 539. R. Pepelnik, H.-U. Fanger, W. Michaelis and E. B6ssow, 3R International 17 (1978) 289. H.-U. Fanger, W. Michaelis, R. Pepelnik and H.L. The, Hydrotransport 5, Proc. Int. Conf. on the Hydraulic transport of solids in pipes (1978), Hannover, Paper G4. H.-U. Fanger, R. Pepelnik and W. Michaelis, Annual Report (1978) GKSS Research Centre Geesthacht, p. 37. H.-U. Fanger and R. Pepelnik, Meerestechnik 10 (1979) 189. R. Pepelnik, E. B6ssow and H.-U. Fanger, GKSS 78/E/32. Bergbau-Forschung GmbH, Essen, Untersuchungsbericht 2/82. P.F. Berry, TND-70-6-6, Boston College, Massachusetts (1970). K. Umiastowski, Rapport INT 86/1, IFiTJ Krakbw (1975). K. Umiastowski, M. Buniak, J. Gyurcsak and P. Maloszewski, Nucl. Instr. and Meth. 141 (1977) 347. H.L. The, Thesis, University of Hamburg (1982).
445
PARTICLE-SIZE EFFECTS IN NON-DESTRUCTIVE M A T E R I A L A S S A Y BY G A M M A - R A Y A B S O R P T I O M E T R Y I. T h e o r e t i c a l m o d e l
H.L. T H E , W. M I C H A E L I S a n d H.-U. F A N G E R Institut fi~r Physik, GKSS Forschungszentrum Geesthacht, Germany
Received 29 October 1982 To Professor Erich Bagge on the occasion of his 70th birthday
Gamma-ray absorptiometry has become a widely used method in non-destructive material assay. The procedure implicitly presumes a homogeneous, fine-grained distribution of the solid components of the material and is well described by a 'sandwich' model. However, due to the nonlinearity of the attenuation law, systematic errors arise if finite grain sizes are involved. The present paper deals with the influence of particle size and gives a thorough theoretical treatment of these effects, which allows the extension of the method to mixtures containing one or even several coarse-grained constituents. The derivation of suitable generalized transmission equations requires the construction of a particle-size dependent attenuation law. This is achieved by means of a 'cell' model which subdivides the irradiated volume into smallest possible 'cells' with representative parts by volume of the material components. Correction functions are deduced that can be used to take into account particle-size effects by properly modifying one of the relevant physical quantities in the conventional transmission equations. The properties of these functions are discussed with respect to their applicability in practice.
1. I n t r o d u c t i o n
G a m m a - r a y a b s o r p t i o m e t r y has p r o v e d to be a very powerful tool in n o n - d e s t r u c t i v e analysis of o p a q u e m a t e r i a l s [1]. P e r h a p s the m o s t i m p o r t a n t a p p l i c a t i o n s are the gauging a n d c o n t r o l of the h y d r a u l i c t r a n s p o r t of solids in pipes. In particular, t w o - c h a n n e l g a m m a - r a y transmission analysis m a y successfully be used for in-line q u a s i - c o n t i n u o u s and n o n - d e s t r u c t i v e m e a s u r e m e n t s of the individual p a r t s b y v o l u m e in t h r e e - c o m p o n e n t flows, b y p r o p e r l y choosing the two g a m m a - r a y energies a n d b y solving the p e r t i n e n t t r a n s m i s s i o n equations [2-4]. The o u t s t a n d i n g efficiency of the m e t h o d has been d e m o n s t r a t e d , for example, d u r i n g the first m a n g a n e s e n o d u l e s p i l o t - m i n i n g test in the Pacific [5], in the course of the p r e - p i l o t m i n i n g test for the haulage of h y d r o t h e r m a l slimes in the R e d Sea [6] a n d in c o n n e c t i o n with the h y d r a u l i c t r a n s p o r t of coal [7,8]. O n the basis of the d a t a o b t a i n e d , different m i n i n g facilities could be assessed with respect to efficiency, safety a n d economy, different o p e r a t i o n a l m o d e s were c o m p a r e d , and the most significant t r a n s p o r t p a r a m e t e r s could be optimized. So far, all a p p l i c a t i o n s have i m p l i c i t l y p r e s u m e d a homogeneous, fine-grained d i s t r i b u t i o n of the solid m a t e r i a l c o m p o n e n t s . The m e t h o d can therefore be well d e s c r i b e d b y m e a n s of a ' s a n d w i c h ' m o d e l in which the m i x t u r e u n d e r study is r e p r e s e n t e d b y successively a r r a n g e d slabs of the c o m p o n e n t materials, with the thicknesses of these slabs being chosen a c c o r d i n g to the i n d i v i d u a l parts b y volume. However, due to the n o n l i n e a r i t y of the g a m m a - r a y a t t e n u a t i o n law, systematic errors m a y arise if finite particle sizes are involved. The present p a p e r deals with the influence of particle size a n d gives a t r e a t m e n t of these effects which allows the extension of g a m m a - r a y t r a n s m i s s i o n analysis to mixtures c o n t a i n i n g c o a r s e - g r a i n e d c o m p o nents. In p a r t I the u n d e r l y i n g theoretical m o d e l is described. C o r r e c t i o n functions are d e d u c e d which can be used to take into account particle-size effects b y p r o p e r l y m o d i f y i n g p a r t i c u l a r quantities occurring in 0167-5087/83/0000-0000/$03.00
© 1983 N o r t h - H o l l a n d
446
H.L. The et al. / Particle size effects 1
the conventional transmission equations. The theoretical results were experimentally examined by means of a hydraulic loop which allows transmission measurements under a wide range of well-defined conditions. These experiments will be described in part II of this paper. As will be seen, quite satisfactory agreement was found between theory and experiment. Several attempts for tackling the problem of gamma-ray absorption by non-homogeneous materials have been reported in the literature. Berry [9] proposed a formula to explain the so-called "grain-size effect" in X- and gamma-ray fluorescence analysis. Starting from this formula, other authors [10,11] derived an expression for the effective absorption coefficient for a non-homogeneous material defined as a mixture of two kinds of spherically shaped grains of the same size. In comparison with the literature, the present paper gives a substantially more general treatment, in particular with respect to the number of components involved, the grain size, the occurrence of size distributions and the mutual obstruction of the particles.
2. Non-destructive measurement of material compositions by gamma-ray transmission analysis In practice, three-component mixtures represent the most important case of application. The principle of material analysis by gamma-ray absorptiometry will, therefore, be discussed in the following on the basis of a mixture consisting of three components. The stationary case of a material with a composition invariable with time may be regarded as a special case of the conveyor-flow analysis. In hydraulic transport systems the mixture, in general, consists of a carrier medium like water (w), a solid charge of primary interest (p), and a third solid or liquid dead component (q). Typical examples are the mixtures sea water manganese nodules-sediment [5], sea water-hydrothermal slimes-NaC1 brine [6], or water coal rocks [7,8]. Neglecting grain-size effects, the transmitted intensity I of radiation after passage through the attenuating medium is described by the equation I i = I0, e x p [ - / { t)p~pi +
l)q~qi + l)w~w,,)] ,
(1)
for any discrete energy E~. Here, t~j denotes the parts by volume of the components.,/= p , q , w ; / , , stands for the linear gamma-ray attenuation coefficients, and l is the transmission length. It is convenient to refer the intensity Ii to the corresponding quantity for uncharged water
lwi = Ioi exp[
-
II~wi ] .
Introducing t, = li/lw, one arrives at In t, l = l)pAjtLpi Jr- 12qA~[.Lqi ,
(2)
where the condition = l,
(3)
y and the abbreviation A#i~ = >j, - #w~ have been employed. If gamma-rays of two energies are used (i = 1,2), the associated eqs. (2) together with the sum rule (3) allow the determination of the parts by volume of the three components involved from the transmitted
intensities [2 4]: Vp = (IN)
' [A#q, in
A/.Zq2 In t,],
(4)
vq = ( I N ) - ] [A#p2 In t, - A/zp, in t=],
(5)
t 2
--
with N
= A [ . L p l A ~ q 2 -- A / - L p 2 A ~ q I .
H.L. The et al. / Particle size effects 1
447
The two energies have to be properly chosen, in order that N most strongly differs from zero. The radioisotopes 241Am(El = 60 keV) and 1 3 7 C s ( E 2 = 662 keV) turn out to be best suited both in this regard, and in respect of practicability. Possible systematic errors and the influence of counting statistics have been thoroughly discussed elsewhere [2-4,6]. Extension of the method to multi-component mixtures with n components and the usage of (n - 1) gamma-ray energies is straightforward.
3. The influence of particle size The outline of gamma-ray absorptiometry as given in the preceding section implicitly presumes a homogeneous, fine-grained spatial distribution of all components in the material to be analysed. The mixture may be equally well represented by a 'sandwich' set-up perpendicular to the gamma-ray beam, with slabs of thickness AXj = t~jl. Due to the nonlinearity of the gamma-ray attenuation law, particle-size dependent systematic errors may occur if coarse-grained materials are involved. In this case not only the concentration of the coarse-grained component, but also the number and correlated size of the particles,
V- radiation
F- detector
y-radiation
F-detector
Fig. 1. Schematic representation of a particle-size effect: Identical particles may cause different attenuation if irradiated in different spatial dispositions.
448
H.L. The et al. / Particle size effects 1
their shape, and their spatial distribution determine the gamma-ray attenuation. The latter effect, for example, can be understood with the aid of fig. 1 which shows schematically spherical particles of finite, but uniform size in two different spatial dispositions. Obviously, in the first case the attenuation approaches (1 - ~r/4)t, when the particles are assumed to tend towards opaqueness, while for the second configuration with mutual obstruction one obtains in the limit case ( 1 - ~r/16)t. Here, t denotes the attenuation factor for the medium surrounding the particles. To overcome this problem, the construction of a particle-size dependent attenuation law is required. This is achieved by means of a 'cell' model (sect. 4) which leads to a generalization of eq. (2) in such a way that grain-size effects caused by spherical particles of uniform size, including mutual obstruction, are taken into account in an approximate treatment [12]. From this new attenuation formula, appropriate correction functions can be deduced which allow the application of gamma-ray absorptiometry to materials containing coarse-grained constituents (sect. 5). Finally, the theory is extended to cover the occurrence of grain-size distributions (sect. 6).
4. 'Cell' model
4.1. Definitions Consider a multiple-component mixture with one component consisting of spherical particles of uniform size with radius r. The remaining fine-grained components may be treated together, for the present, and will be referred to in the following sections as the 'residual mixture'. As a first step, the total irradiated flow volume is subdivided into adjacent 'flow channels' with the particles on their axes, as shown in fig. 2. Let the channel axes be oriented parallel to the flow direction and the cross-sections chosen to be of square shape. Suppose that a particle just fits into such a channel, i.e., the lateral length in a plane perpendicular to the axis equals 2r. If it is now further assumed that the particles are distributed uniformly over the flow volume, insofar as for each of them the mean distance to both neighbours within the 'flow channel' is constant, then the flow may be subdivided into so-called 'cells', each of which contains just one particle (fig. 3). The cells defined in this way are characterized by the same parts by volume as the total flow. The model, which will be developed in the following subsections on the basis of this picture of idealized cells, thus presumes a certain order in the spatial distribution of the coarse-grained component. There is, however, no restriction on its universality, in that some order may be brought about, when particles are thought to be translated in parallel with the gamma-ray beam which leaves the attenuation of the gamma radiation unchanged.
detecto~/~ _._
---
spherical particle of radius r k~' ~ /
----t ;T . . . . . . . . . . . . . .
-~ . . . . . . . .
.
.
.
.
[d=2r
.
>k y - ray source
Fig. 2. Construction of 'flow channels'.
,, ce[L" Fig. 3. Conception underlying the construction of 'cells' (l ~- f i n
subsection 4.2).
H.L. The et al. / Particle size effects 1
449
4. 2. Hit probability and part by volume
The probability Pp that a photon strikes the spherical particle when penetrating a cell of lateral length/(fig. 3) is given by Pp = r r r / 2 L
On the other hand, according to subsection 4.1, the part by volume vp of the coarse-grained component may be deduced from the particle and cell volumes as Vp = ~rr/3[.
Hence, it follows that Pp = 3Vp.
(6)
4.3. Mean attenuation within the particle cross-sectional area in passage through a cell
If the particle volume is resolved into r i n g cylindrical volume elements, as shown in fig. 4, the attenuation factor for gamma radiation impinging at radius x on a cell of thickness 2r is given by
tx=
' / I 0 = exp[ -
{ ~ p ( 2 ~ - - X2 ) q- ~rm(2r -- 2~/~-r2- X2 )} ].
(7)
Here, ~p and /*rm denote the linear attenuation coefficients of the particle material and the residual mixture, respectively. Eq. (7) takes into account that the particle is always surrounded by the remainder components. The normalized probability that photons will strike the ring cylindrical elements of thickness Ax is
W(tx) = 2 x a x / r 2 .
(8)
Consequently, the attenuation averaged over the particle cross-section in passage through the cell is obtained from
= Y'. W( t )tx,
(9)
or, in integral form, using eqs. (7) and (8) tp/cell = ~ -2 e /*rm2rfor xe -("p "~m)2~/r2-x2dx. r Partial integration yields tp/ce,l = e - " " e r F ( r ) ,
Fig. 4. Ring cylindricalvolume elements of a sphere in a cell of thickness 2r.
(10)
450
H.L. The et al. / Particle size effects I
with
F(r)= 2{ eCr(Cr-1)+l ),
(ll)
c=
(12)
and
Since the first factor in eq. (10) describes the attenuation in passage through the cell in absence of coarse-grained particles, one may write /p/cell = t r m / c e l l F ( r ) .
(13)
4.4. Mutual obstruction of particles The attenuation of the gamma-rays is determined by the momentary spatial distribution of the particles in the flow volume. Due to the finite measurement time in transmission analysis, the measurement result is given by the time-averaged attenuation, i.e. all possible particle constellations have to be taken into account. If the total transmission length is l, then it follows from subsect. 4.1 that n = l/2r cells are penetrated by the gamma-ray beam. Considering also subsects. 4.1 and 4.3, the probability that a photon strikes a particle at radius x in passage through a cell may be written as
Px = (2xAx/r2)Pp = W(tx)Pp" Correspondingly, the hit probability for the residual mixture is Prm= 1 -Pp.
If now the probability is considered that a narrow beam, in passage through the total flow, strikes j particles, each at a different radius x i, and gets through (n - j ) cells with residual mixture, one obtains for the single event J J egmJ F I P~.~ = P/(1 -- e p ) " - J 1-I W(t~.,). i~l
i=1
According to the rules of combinatorial analysis, this event occurs in a (~)-fold manner where n!
(~) j!(n--j)!" Hence, the total probability is J
Wj = (~)Pr~(l - Pp)" JI-I w'(tx.i) dxi = wf d x I d x 2 . . , dxj,
(14)
W'( tx.i) = 2xi/r 2.
(15)
i=1
with
The corresponding attenuation factor tj is given by J tj=(trm/cell)n-JHtx,i i=l
[cf. eq. (7)].
H.L. The et al. / Particlesize effects I
451
4.5. Particle-size dependent attenuation law The mean total attenuation factor ttot is then obtained, with proper normalization, by summing up all weighted contributions
j W,.t, = ( " ) P ~ { ( 1 -ep)trm/¢e,,} "-J 1--I w'(tx,ilt~.idxi=tjdx, ' dx2.., dx,
(16)
i=1
over the n u m b e r j of struck particles and by simultaneously integrating over all possible radii xi: ~-0"" fx r/ tjl d x I d x 2 . . , dxj ~ fx~lI-- 0fx r2 --/=0 / t o t - j=0 j = 0 J x , "- 0 'x 2' --0f -
(17)
x,=0Wjdxldx2""dxj
Since the sums and all integrations are independent of each other, this formula can be simplified considerably. The denominator equals 1, because
ff
Zx d x = 1 r2
and, thus, with eqs. (14) and (15)
a=o ,-o 2-o"" --
--
/=0
Wjdx, dx2...dxj=
j=O
(,:)g(1-g)" ' = ( / ' p + l - g ) =l.
For the numerator of formula (17) one obtains by means of eqs. (16) and (9), remembering that a uniform size of the particles has been assumed: J
k (;)PpJ((lj=0
Pp)trm/cel,)
" J i~=l=f~i',_0W'(t~,)t~,dxs= ~ " '
_
=( Ppt-p/cen + ( 1 - Pp)trm/cell}
,-
(;)(Pptp/cell)J((1
Pp)trm/ce,l) n-j
j=0
t/
.
Hence, with n = l/2r and eqs. (6), (10) and (13) the mean total attenuation factor is finally given by t-to, = e-~rm'[~vp{F(r) - 1} + 1] ,/2,,
(18)
where F ( r ) follows from eqs. (11) and (12). Eq. (18) describes the gamma-ray attenuation by a multi-component material in the presence of a coarse-grained constituent with particles of uniform size with radius r under the assumptions made in subsect. 4.1. The formula represents a generalization of the equation t=e
~'rm/e-%(up
P'r m )/,
(19)
which, by analogy to eq. (2), is valid for a material fine-grained as a whole (r = 0) or, in other words, eq. (19) is a special case of formula (18) for the limit of vanishing particle radius. This is easily proved by equating the relations (18) and (19) and labelling the parts by volume of the coarse-grained component by the superscripts r and 0, respectively. One obtains 2 e ~°cr- 1
Vp- 3 F ( r ) - I
452
H.L. The et al. / Particle size effects I
By applying De l'Hospital's rule twice it is found that lim r = r ~ 0'Op
0 L~p,
q.e.d.
5. Corrections for particle-size effects 5.1. General outline
Owing to its complexity, eq. (18) is hardly accessible to a direct evaluation in practical applications of gamma-ray absorptiometry. Therefore, a simplified procedure for solving the transmission equations with respect to the parts by volume has to be found. This can be accomplished by transforming eq. (18), rewritten in exponential form,
=
111"
optionally into one of the following conventional relations that result from the 'sandwich' model [cf. eq. (19)]: t* = e ~,~/e-%~,l,
(21a)
t = e-~'rmte-~pJ~')*t,
(21b)
t = e-~rm/e-'~;~t,
(21c)
t = e-P'~m/e-%a~*l,
(21d)
t = e "'mte vd~3--Urm)l,
(21e)
or
with Z~# =/Xp - #rm' This means that the correction for grain-size effects is performed by modifying one of the physical quantities t, (vpA/~), Vp, A~ or /%, then labelled by an asterisk, with the aid of correction functions, which are obtained by comparison of eq. (20) with one of the eqs. (21a)-(21e), respectively. The optimum choice of these possible procedures depends upon the particular properties of the resulting correction functions. These properties, therefore, require a brief discussion. 5.2. Materials containing one coarse-grained constituent
As before, let the fine-grained constituents of a multi-component material be treated together as a homogeneous 'residual' mixture. By means of eqs. (20) and (21a) one obtains the correction function g(r)=
_
=
trot
Fig. 5 illustrates the dependence of this function on the particle radius for some two-component materials with gamma-ray energies of 60 keV and 662 keV and the part by volume Vp in the range 1-6% as parameters. A transmission length of 20 cm was assumed in these calculations. The relevant linear attenuation coefficients are shown in the insets. The graphs indicate a strong dependence of g ( r ) on Vp which makes this procedure only of limited usefulness for practical applications, since, in general, % is the quantity to be determined. From eqs. (20) and (21b)-(21d), respectively, identical correction functions are obtained, since the
H.L. The et a L / Particle size effects I
453
60*,k --
t'/,
- - ~ _ ~
.;:~ o. 9
~.j.
~>~ 0.9
~
'.L'
,, 1=20cm
i
o.8
I I
L
manganese nodutes in sea =wafer 0.6
60keV O.7z,8 662 keV 0122 t
coal
t=20cm
0195 0085L
1.o
0~92
662 keY: 0101
1.5
2.0
particle radius
- -
2.5
0,5
3.0
0085
t
~j [cm -1 ]
0.5
1.o
~j i[ c m - 1 ]
0.5
in wafer
60keY: 0.228
i
2.0 particle
662 k,V
radius
2.5
3.o
r[cm]
W,
it'/.
.;:~ O.9
~..~ 0.9 ,:,.,
--..~--------~ ~
2
i
1.5
r[cm]
~
~.,.
0.8
0.8
1:20cm
sfeafite spheres in wafer 60keV
055g
0192
60key
0664
0192
662key
O 182
0085
662keY:
0188
0085
~J Icm -1 ]
I
0.5
0.5
h0
J
I
1.5
~parficie
i=2o era
dead rocks in water
2.0
radius
2.5
3.0
o.s
r[cm]
# [cf 1 ] 1.o
1.5 ,
2.0 particle
2.5 radius
3.O
r[cm]
F i g . 5 ( a - d ) Correction function g(r) vs. particle radius r with the part by volume ranging from 1 to 6% for some two-component mixtures. Gamma-ray energies 60 and 6 6 2 k e V . Transmission length 20 cm. The linear attenuation coefficients used for the calculations are specified in the insets.
q u a n t i t i e s l)p
f(r)
-
and Aft appear as a product in the latter equations:
(VpAff)* f)pA]2
Vp _ Aft* t)p
Atx
ln[3vp{F(r)
-
1) + 1]
(23)
vpCr
In this formula the transmission length l is eliminated. However, the most characteristic feature o f f ( r ) is that this function exhibits only a very slight dependence on Vp, as can be seen from fig. 6. The same materials, gamma-ray energies, and range of volume concentration as in fig. 5 were taken as a basis. Thus, f(r) fulfills the most essential criterion for practical applications. The function can be calculated in advance for the material to be analysed. The measurement is performed as usual by solving the energy-dependent eqs. (19); however, in doing so, the product (vpA/~) or one of its factors is modified with the aid of eq. (23), which is also a function of the gamma-ray energies used. Because of the slight influence of Vp on f(r), a mean correction function f ( r ) may be employed in practice which is valid to a good approximation over the range of interest for %. In general, the errors arising from this procedure are negligible. A comprehensive discussion of this subject is included in ref. 12. Besides its dependence on the particle radius, f(r) depends also on the difference of the linear attenuation coefficients Aft =/% -/%m" Corresponding curves are shown in fig. 7. It should be pointed out that the absolute values of these coefficients do not enter.
H.L. The et al. / Particle size effects 1
454 i,o
662 keY--
~*,v
0.8
~ O.B
0.6
0.6 r'/,
o.&
0.2
I
I
COO[
manganese nodules in s e a - w a f e r 60keY: 0.748 0.196 662 keV: 0.122
in wafer
60k#4: 0.228 0.192
0.0854
662 keV: 0101
[cm-11 0.0
I
l
0,5
01085
p [cm -1 ]
1.0
1.5 --"
I
2.0 2.5 particle radius r [ c r n ]
I
1.0
0.5
1.5
2.0 porfic[e
1.0 ~
2.5
3.0
radiusr [ c r n l
1.0 ~
~
'
~
~
O.B
~
~
0.8
~
6
O.S 6%
0 '¢
0.4
dead rocks in wafer
sfeofite spheres in wafer 0.2
60keY 0.559
0.192
662keY: 0 179
0085
60keV
. . . .
[cr~- 1 ] o,o
1
[ 1.0
0.5
0 6 6 4 0192
662keV: 0.188 0085
1.5
2.0 2.5 particle radius r[crn]
o.o
3,0
i
~u [cm_ 11
0.5
1.0
I
1.5 ,
2.o 2.5 particle radius r [ c m )
3.
Fig. 6. ( a - d ) Correction f u n c t i o n / ( r ) vs. particle radius r with the part by v o l u m e ranging from 1 to 6% for some t w o - c o m p o n e n t mixtures. G a m m a - r a y energies 60 and 662 keV. The linear a t t e n u a t i o n coefficients used for the calculations are specified in the insets. 1.0
r 0.05 0.10
0.8
0.15 -0.20
v
:0.25
0.6
._.,..
"*---.,_.
0.4
Vp=5°lo
,0.30 '035 ~OJ,O :0,45 '0.50 ,0.55 ,060
AjJ [cm -1 ]
0.2
0.0 O.
1.0
2.
L.
=
2.5 pQrticie r a d i u s
3,0 r[crn]
Fig. 7. D e p e n d e n c e of the function f ( r ) on the particle radius r and the difference A/~ of the linear a t t e n u a t i o n coefficients/% and/L .... a s s u m i n g a p a r t by volume of 5%.
H.L. The et al. / Particle size effects 1
455
From eqs. (20) and (21e) the correction function /% -- btp ]'£rm
2r/)p
may be derived. Grain-size effects are considered by a procedure analogous to that described before. Again, the transmission length is eliminated and there is practically no dependence of h(r) on % (fig. 8). On the other hand, this function depends not only on A/~, but also in an explicit manner on /~rm which may prove to be a disadvantage. Therefore, the best choice between f(r) and h(r) is determined by the nature of the problem to be solved. Fig. 9 displays h(r) vs. the particle radius with A~t as a parameter, assuming /~rm = 0.192 cm ~ corresponding to water and Vp = 5%.
5.3. Transmission analysis of a three-component mixture The equations given in subsect. 5.2 apply to all multi-component systems containing one coarse-grained material provided that the remaining components can be treated as a unified homogeneous mixture. Formulae for special problems are easily derivable, although in most cases, additional assumptions have to be introduced. As an example, let the most important case of a three-component flow be discussed in some detail. Here, along with the percentage of the coarse-grained material, the individual parts by volume of the homogeneous components are of interest. Following the nomenclature of sects. 2 and 4, one obtains ~rm
l~/)p(~tq--~w)+~w ,
and thus
C= -2
I
(t~p-/~w)-
As a consequence, the functions F(r) and f(r) become dependent on the parts by volume in a rather complicated manner which requires cumbersome numerical solution procedures. For practical applications, therefore, a simplified treatment must be found in which the errors introduced remain negligible or, at least, tolerable. There are several possible approaches for solving this problem. The first procedure involves assigning mean values ~p(r) of the true linear attenuation coefficient/Xp(r, Vp, /)q), for the gamma-ray energies used, and, hence, mean values of the correction function h(r) over the expected ranges of variation of vp and Vq. The calculation of the data matrix is performed in advance for a proper set of particle radii. Extensive tables given in ref. 12 indicate that the resulting errors remain acceptable provided that the ranges of variation are not too large, which, for instance, is quite well valid for the transport of solids through pipes. A second rather efficient procedure may be derived by writing [cf. eq. (21d)] / ) p ( ~ p -- ~ r m ) *
q" ~ r m = /)p ~ p
=/)p
1- % - .w)*
- .w) + .w,
(25)
and correcting (t% -/~w) instead of (/~p -/~rm)" The pertinent correction function reads
f ( r ) = ln[~/)p{g'(r) - l} + 1] /)pC / r
with F ' ( r ) = 2 e C ' r ( C ' r - 1) + 1
and
C' = - 2 ( b t p - / x w ) .
(C'r) 2 Physically, this means that the problem is treated approximately within the framework of a mixed cell-sandwich model as illustrated in fig. 10. This can easily be shown by means of eq. (20). When applied
H.L. The et a L / Particle size effects 1
456
t- 6"/,
/
t~
- - t - -
.{ -
0 E
0
a~
7 manganese nodules in sea-wafer I
0.2 I
60key 0.748
662key o.~n
J
i
0.5
0196
I
o.o85~
t
o2 ~ -
J
60.~ 0.226 0.,92 662 keV: 0101
/J Icm-11 r
0085
]
,.C
1.5
2.0
particle
2.5
radius
o
5
i o
r[cm]
~
5
- - "
2,0
particle
radius
2.5
30
r[cm]
i.c
o
g
.{
-{ O.G .c
1 0.2
0.0
O2
05
I.O
1.5
2.0
---particle
2.5
radius
0.0
3,0
O.S
r[cm]
1.0
1.5 -
2.0 particle
radius
2.5
3.0
r[cm]
Fig. 8. (a-d) Correction function h(r) vs. particle radius r with the part by volume ranging from 1 to 6% for some two-component mixtures. Gamma-ray energies 60 and 662 keV. The linear attenuation coefficients used for the calculations are specified in the insets. 1.0
0.05 , 0.10
0.15 0.20 ..
0.8
0.25
0.30 0.35 O.t,O 0.45 050 0.55 0.60
ii
.~ o.~
Vp=5% 0.4
~,~u [cm-~]
0.2
0.0
1.0
1.5 -
2.0
parficle
2.5
radius
3.
r[cm
Fig. 9. Dependence of the function h(r) on the particle radius r and the difference A/~ of the linear attenuation coefficients/Xp and #rm assuming a part by volume of 5% and an absolute value of 0.192 c a - i for/~r,, corresponding to water.
H.L. The et al. / Particle size effects I
(£)
457
I. .: Ii..'
:
"
Fig. 10. Approximation of the pure cell model by a mixed cell-sandwich model.
to a mixture with the components p, w and a slab with component q, eq. (20) yields with
/)p q- 13w ~
1
/.tw(Vp + Vw) +f(r)Vp(~tp - btw)(Vp + Vw) + ~qL~q = ~tw(1 -- l)q) +f(r)vp(IXp -- ktw)(Vp + Vw) + #qVq
=f(r)vp(IXp - btw) + Vq(/Xq --/Zw) + ktw. The interpretation follows directly from comparison with relation (25). Again, detailed tables on the size of the resulting errors may be found in ref. 12. Concerning the dependence o f f ( r ) on Vp, we refer to subsect. 5.2. It is also shown in ref. 12 that employment of the maximum occurring value Vp max for calculatingf(r) may give the best results which further simplifies the evaluation. The parts by volume obtained from the preceding approximation method may also be used for recalculating the attenuation coefficients in an iteration procedure. In general, two steps already yield sufficient accuracy, even for larger particle sizes [12]. For small radii the product Cr becomes << 1. Then, the function F(r) can be expanded into a series: r(r)=
1 + ~ C r + (C-~2+ . . . .
It follows that f(r)=ln[1
+vpCr(l +~Cr)] l)pCr
and, hence, by expanding also the logarithmic expression
f ( r ) = (1 + 3 C r ) - ½vpCr(1 + 3Cr) 2 + . . . . The dependence on Vp occurs only in the terms of higher order. For sufficiently small values of r the second term may be neglected. The application of this approximation method in the case of fine-grained materials is, in particular, recommended if the evaluation is performed by means of microprocessors with a small word length, since rounding errors may possibly introduce inaccuracies when using the exact, but rather complex formulae.
5.4. Materials containing more than one coarse-grained constituent A stringent treatment of particle-size effects in materials with several coarse-grained components requires a theory based on complicated multinomial distributions. As a consequence, no closed analytical expressions can be derived. A more pragmatic approach, however, may be taken by using the mixed cell-sandwich model already utilized above with success. Consider first a three-component mixture with coarse-grained constituents p, q and a homogeneous residual constituent rm which in itself may be composed of several materials. Let the particles of components p and q be of different, but uniform size. Neglecting mutual obstruction of particles of
H.L. The et al. / Particle size effects I
458
different species and assuming %, Vq << 1, the transmission equations analogous to eqs. (20) and (2 ld) read
1 ln[~vp(Fp(rp)-1)+ _ ! 1 ln[(rp, rq)=P, rm_ 2-~p
1 ] - ~ r q1
ln[{vq(Fq(rq)_l)+l ]
1 and - 7 In t =/*r,n + VpA#~ + Vqa#q, with A g i = ~ , - - ~ r m (i = p, q) and corresponding substitution into the functions F,. Correction for particle size effects may thus be achieved by proper modification of the differences A/~i using separate functions f, (r,). Generalization to materials with n coarse-grained components yields 1 l n t - ( r 1 r2
,r,,)=/Zrm
~{
-
1
ln[~vi(Fi(r, )
1)+1]}
(26)
i=1
'
and - 7 In t = ~rm +
L V,A~a*, i=1
respectively. 6. Distributions of grain-sizes Since in many applications of the method the particle radii are not uniform, but follow more or less a distribution function, extension of the model to size distributions would be very attractive. One promising approach may be formulated as follows. By analogy to subsect. 5.2, consider a material with a coarse-grained component p and a homogeneous residual mixture. Let the particle radii obey a distribution function n(r). Then the associated normalized distribution function k(r) for the part by volume may be written in the sum representation as
1,(5) = n ( r, ) ri 3 i=1
and the differential part by volume is given by
vp(5) = k(5),
p0,
where %o is the total percentage of the constituent p. If each size interval is now thought of as a separate component, the problem becomes one of a multi-component mixture, which has already been discussed in subsect. 5.4. Since all particles belong to the same species, it follows that 1
t hii[k(r)l=.rm-- Z
{ 1
ln[3vpok(ri){F(r,) -
1)+ 11}
i=1
The rather cumbersome evaluation of this equation may be avoided if a representative particle radius can be found which leads to the same result as the stringent consideration of the size distribution. The proper choice depends on the distribution function presumed. A normal distribution of the grain radii hardly coincides with reality. Moreover, care must be taken that the condition r >/0 is fulfilled and, correspondingly, that the mean radius is not identical with the center of the distribution function. A more realistic description is provided by the Poisson distribution P(r), the asymmetric course of which is typical for many practical cases of application. Using Stirling's formula, P(r) may be written in the form P(F)--
l
e
-
~r
) .
H.L. The et aL / Particle size effects I
459
Fig. 1 la displays this function with the mean radius ? as a parameter. For comparison, in fig. 1 lb the associated distribution k(r) for the part by volume is shown. Here, rma~ denotes the grain radius at which the maximum in the distribution occurs. In search of a representative radius value on the basis of P(r) for the purpose given above, it is worthwhile comparing the correction function f(r), as obtained for the radius values ~ and rma ~, respectively, with that resulting from a stringent calculation on the basis of the complete distribution
0.20-
--£ c
0.15-
AG~ 0.05
*+°oz:
000 1.00
2.00
3.00 -
¢.00
5.00
parfide rcldius r [cm]
0. I0i... v
O.OB-
o
0.05
+_:'~
Yx~ E 0.02
0.00 .00
z. bo
3. bo
.
+. bo
parficte radius
5. oo
r(cm)
Fig. 11. (a) Poisson distribution n(r) = P ( r ) of the particle radius r with the mean value ~ varying from 0.28 to 1.14 cm. (b) Associated differential distribution k ( r ) of the part by volume.
H.L. The et aL / Particle size effects 1
460
,,o
>~
0.20 I
o.o~.~o
O. 6B
• parficte
J
O. 91
radius
]. I¢
O.Ot) O.O0
O. ~2
F [Era]
i . 26
• particle
I . 58
radius
2. I 0
rmax(cm )
Fig. 12. (a-b) Comparison of the correction function f ( r ) as obtained by using the grain radii ~ and rmax, respectively, with the function f [ k (r)] resulting from a stringent consideration of the distribution function k (r).
function k ( r ) . As is shown in fig. 12 for the case of steatite particles in water, there are severe discrepancies between f ( ~ ) and f [ k (r)], while the function f ( r ma, ) coincides quite well with f [ k ( r )] for both gamma-ray energies assumed. Thus, rma, turns out to be sufficiently suited as a representative value for simplified correction procedures. In the presence of several coarse-grained components with individual grain-radius distributions n~(r), n2(r ) .... and parts by volume distributions k l ( r ), k 2 ( r ) .... the problem becomes very complex. Quite formally, the algorithms given above can be generalized without difficulty. If ~ denotes the representative particle radius of component i, the grain-size dependent transmission equation reads [cf. eq. (26)] 1 ln t [ k l ( r ) , k 2 ( r ) . . . . . k n ( r ) ] : ~ t r m - i =~1{ 1 l n [ ~ v , 0 { F , ( ~ , ) _ l ) + l ] } However, it should be pointed out that this general formula is based on several approximations. Its application, therefore, requires a detailed error analysis in each case.
7. Conclusions
In the present paper a 'cell' model has been developed which allows the construction of a particle-size dependent attenuation law for gamma radiation passing through materials containing one or more coarse-grained constituents. From this law, correction functions can be derived that permit the application and evaluation of the 'conventional' transmission equations in gamma-ray absorptiometry, by properly modifying one of the physical quantities involved. Both particles of uniform size, and those foll6wing a Poisson size distribution can be considered. Although certain approximation procedures had to be introduced in the course of the calculations, proper handling of the algorithms described ,ensures sufficiently accurate results. Experimental confirmation will be given in part II of the paper. The theoretical model presented considerably extends the possible range of application of gamma-ray absorptiometry in non-destructive material assay, since particle-size effects can now be taken into account in a quite straightforward manner. Of course, the use of the established correction procedures requires information on the grain radius or its distribution. These data if unknown may be obtained either by taking a sample from the material under study or, preferably, in a non-destructive way by gamma-ray transmission analysis as well. Pertinent suggestions have already been made by the authors elsewhere [3,12]. For example, particle-size information
H.L. The et al. / Particle size effects I
461
may be derived from the depths or widths of the counting rate 'dips' from a transmission gate which occur during passage of particles through the gate. If higher parts by volume of the coarse-grained constituent and, accordingly, increased mutual obstruction effects preclude such a direct electronic evaluation, the resulting complicated structures have to be resolved by means of a computer program similar to those employed for spectrum analysis in nuclear spectroscopy. Other more involved instruments may derive benefit from a spatial reduction of the measuring volume, for instance, by using two crossed gates or a conical multislit collimator.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
Proc. Int. Conf. on Industrial application of radioisotopes and radiation technology, Grenoble, 1981 (IAEA, Vienna 1982). H.-U. Fanger, R. Pepelnik and W. Michaelis, Nuclear techniques and mineral resources (IAEA, Vienna, 1977) p. 539. R. Pepelnik, H.-U. Fanger, W. Michaelis and E. B6ssow, 3R International 17 (1978) 289. H.-U. Fanger, W. Michaelis, R. Pepelnik and H.L. The, Hydrotransport 5, Proc. Int. Conf. on the Hydraulic transport of solids in pipes (1978), Hannover, Paper G4. H.-U. Fanger, R. Pepelnik and W. Michaelis, Annual Report (1978) GKSS Research Centre Geesthacht, p. 37. H.-U. Fanger and R. Pepelnik, Meerestechnik 10 (1979) 189. R. Pepelnik, E. B6ssow and H.-U. Fanger, GKSS 78/E/32. Bergbau-Forschung GmbH, Essen, Untersuchungsbericht 2/82. P.F. Berry, TND-70-6-6, Boston College, Massachusetts (1970). K. Umiastowski, Rapport INT 86/1, IFiTJ Krakbw (1975). K. Umiastowski, M. Buniak, J. Gyurcsak and P. Maloszewski, Nucl. Instr. and Meth. 141 (1977) 347. H.L. The, Thesis, University of Hamburg (1982).