The interpretation of diffuse reflectance spectra
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JOUR NAL OF RESEA RC H of t h e Nationa l Burea u of Standa rdsA. Physics and Chemistry Vol. 80A, No. 4, J ul y August 1976
The Interpretation of Diffuse Reflectance Spectra
Harry G. Hecht
Department of Chemistry, South Dakota State Un iversity, Brookings, South Dakota 57006
(May 26, 1976)
Numerous treatments of the diffuse r eflecting properties of scattering media have been d escribed. Many theories give an adequate account of the refl ectance for a specific set of
conditions for which the model was constructed and the soluti on t ested experimentally. Only those models which are considered to be fairl y general are considered her e.
It is conveni ent to di vide the theories into those based upon continuum models and
those based upon s tatistical models. The continuum m odels typica ll y describ e the scatterin g a nd absorbin g properti es of a given m edium in t erm s of two phenomenological constants.
These models may all be regarded as varying levels of approxima t e soluti on to the general equation of radi ative tra nsfer. This provides a convenient basis for compariso n of the various
theories. The statistical models a re based upon a summ a tion of transmittances a nd r efl ectances
fr om individual l ayers or parti cles. Thus, ~ome ass um ptions must be m ade abo ut the n ature
of the fund amenta l units, and the validity of the ultim a te result will depend up on how closely these assumptions correspond with r eality. Only the statistical models lead to ex pressions
fr om which absolute absorptivities and sca ttering coefficients can be calculat ed and r elated to the actual particle characteristics.
The relationship between the various models will be discussed and the features which typify the ab~orptivity and scatterin g coeffici ent accordin g to each will be compared an d related to the available experimental data. This leads t o a consideration of the charac teristics
of appropri ate model systems and standards.
Key words: Absolute absorptivities; continuum models; diffuse reflectance; r adiative transfer; reflectance spectra; scatt ering coefficients ; statistical models.
List of Principal Symbols Used
H(iJ.)
(Note: Where a given letter is used in both capital [ and lower case form (e.g., r,R and t,T), the capital ['
letter refers to the macroscopic observable and the
lower case letter to the corresponding variable for an J
individual particle or layer of the material. A bar J
over a given symbol means the average value for k
that variable).
K
a
total absorption of a single par ticle
(layer); also S~K (equation (18))
Gaussian weighting function (equation
(33) ) L
}R ozenberg constants (equation (23)ff.) m
(a2 _1)1 /2 (equation (21))
M
[ ;/[1 polarization of radiant beam (equa
tion (24»)
fraction of radiation reflected (equation
(75) )
fraction of radiation transmitted (equa
tion (76»)
R ozenberg constant (equation (29))
Hintegral of Chandrasekhar (equation
(35) ) radiant intensity component of source function for selfradiation (equation (5)) scattering function (equation (2)) source func tion (equation (4)) absorptivity KubelkaMunk absorption constant (equations (16) and (17)) Attenuation constants in Gurevic layer model (equations (10) and (11)) r.article diameter or layer thickness 'free" path length of AntonovRomanov
sky (equation (58)) Gurevic constant=~K22 _K12 mean number of reflections
transmittance of a particle for a single
pass
radiant flux density
phase function (equation (2))
Legendre polynomial Rozenberg constant (equation (29))
Rozenberg multiple reflection constant
(equation (26))
567
214427 0  76  2
r
total reflectance of a single particle
(layer)
simple reflectance of a single particle
(layer) surface
r*
mean external reflection coefficient for
side scatter
reflectance of a particle (layer) for
externally incident radiation
reflectance of a particle (layer) for
internally incident radiation
R
reflectance of a (macroscopic) layer
s!s!'"
reflectance of an infinitely thick layer effective path length (equation (58»
KubelkaMunk scattering constant
(equations (16) and (17»
Rozenberg weak scatter constant (equa
tion (29»
total transmittance of a single particle
(layer)
T
transmittance of a (macroscopic) layer
u
radiation emerging from a particle per
unit solid angle
shading factor (equation (70»
fraction of radiation emerging from a
particle in a downward direction
fraction of radiation emerging from a
particle in a sideways direction
fraction of radiation emerging from a
particle in an upward direction
x
length
y
Johnson multiple reflection factor (equa
tion (51»
a
absorption coefficient
(3
a/a (equation (23»
'Y
FasslerStodolski constant (equation (82»
~
FasslerStodolski constant (equation (82»
()
polar angle
K
attenuation coefficient=a+a (equation
(1»
wavelength
cos ()
one of the roots of the Legendre poly
nominal P n(J..l.)
effective hole cross section (equation
(100»
p
density
a
scattering coefficient (equation (2»
T
optical thickness (equation (3»
rP
azimuthal angle
q,.
radiant flux
Wo
albedo of single scatter (equation (30»
I. Introduction
It is now recognized that diffuse reflectance spectroscopy is a very useful companion technique to transmission spectroscopy. Not only can it provide absorption data in some cases where transmission measurements fail, but for many industrial and research applications, it may in fact be the preferred technique.
Our discussion will be concerned with the behavior of radiation within a scattering medium. For simplicity we assume the scattering centers, whieh may also absorb radiation, are imbedded in a mpdium which neither scatters nor absorbs. The medium may usually be taken as air, although there are many other cases of interest in which the refractive index of the medium is much greater than unity. We will not deal specifically with these cases in the present review, nor with those processes which alter the frequency of the radiation, such as luminescence and fluorescence.
Scattering takes place under a wide variety of conditions. One may be concerned with the glowing photosphere of the sun, which is surrounded by a cloud of electrons that reradiate the direct sunlight incident on them. Sunlight is also reradiated by cosmic dust, which accounts for the outer part of the corona.
In a more downtoearth situation one may be dealing in the laboratory with the spectroscopy of a powder, paper, opal glass, photographic emulsion, etc. As so often happens, concepts developed in one area of science are slow to find their way into another. We will attempt to show that there is a close relationship between the astrophysical solutions, which are based largely on radiative transfer theory, and the various models which are more familial' to the spectroscopist.
Since we will not be following a historical development, it may be useful to point out some relationships between the early studies. The first attempt to account for transmission and reflection of a layered material was carried out by Stokes in about 1860 [1]/ and led to some very useful relationships which have also been derived by other workers (vide infra). Lord Rayleigh [2] and Mie [3] developed the theory of single scatter to a high degree, but Schuster [4] was the first to consider multiple scatter. He was concerned with the cloudy atmospheres of stars, and developed a planeparallel plate model in which the radiation field was divided into forward and backward components. This same model was used much later by Kubelka and Munk [5], whose names are usually attached 1,0 it by spectroscopists. Schwarzschild [6] showed that the radiation field should be characterized by a complete angular distribution, and if one integrates over the forward and backward hemispheres, the Schuster model is obtained as a first approximation. A further generalization of the Schwarzschild formulation leads to an integrodifferential equation known as the equation of radiative transfer, which is very general in concept, but can be solved exactly in only a few cases.
The radiative transfer theory and various models stemming from it are referred to as Continuum Models. They have in common the characterization of the scattering and absorbing properties of the medium through phenomenological constants, usually two in number. These theories will be considered in section II. A completely satisfactory theory must of course relate the measurable quantities to
1 Figures in brackets indicate the literature references at the end of this paper.
568

fundamental properties of the scattering material, such as particle size, refractive index, and absorptivity. This is the goal of the statistical theories. Many have been proposed; those which show the most promise are discussed in section III. In section IV we discuss some of the strengths and weaknesses of the various models and some relationships between them. Section V considers the meaning of the absorption and scattering coefficients often used to interpret reflectance data, as well as the characteristics required of appropriate model systems.
We willlimit our discussion to ideal, homogeneous dispersions. There have been interesting developments in the theory of nonhomogeneous media, mixtures, luminescing materials, diffusing media in which photochemical reactions are taking place, and reflectance for materials dispersed in a highly refracting matrix, but these topics are considered to be outside the scope of the present review. Many of these topics are covered in the books which have been written on the subject [7,8,9J . Further information has been presented at various symposia which have been h eld, and the proceedings of at least two of them are available [10,11]. These sources, together with various reviews devoted to applications in particular fields, have made workers aware of the power of the technique. Thus we will confine ourselves to certain theoretical aspects.
The present paper may be regarded as an update to a former review of the subject by the author [12]. Most of the theoretical work of the intervening years has been in the development of more refined statistical models, and in showing the relationship between the various theories. At present there is some reason to feel optimistic that out of a morass of apparently divergent and unrelated theory, some order is beginning to emerge.
II. Continuum Theory
As is often the case with applications of physical theory, the real media with which diffuse reflectance spectroscopy is concerned are intermediate between two extreme cases, each of which is well understood. The one limiting case involves the propagation of radiation through quasihomogeneous matter, where the radiation field is characterized by smoothly and slowly varying functions of the coordinates. The other extreme involves the emission, absorption, and scattering of radiation by single particles in a homogeneous medium. The local transformation of the radiation field at the boundaries of quasihomogeneous media is a special case which has likewise been characterized for a long time, and leads to such wellknown phenomena as reflection, refraction, etc.
In dense scattering media, it is important to consider the manner in which the radiation fields from the various scattering centers interact with one another. Rozenberg [13, 14, 15] has pointed out that the interaction can be treated as a sum of two parts,
one of which is coherent and the other incoherent.
The coherent part is largely due to nearest neighbor interactions and gives rise to dispersion effects; i.e., is involved with changes of refractive index. The incoherent part is that with which we are primarily concerned in the present discussion. It involves multiple scattering as a sum of interactions throughout the entire medium. In most treatments the multiple scattering, because of its complexity, is separated rath€r arbitrarily from dispersive effects. Such a separation reduces the problem to one involving geomrtrical optics, and allows one to write a radiative transfer equation in which the absorbing and scattering properties of the medium are treated as phenomenological constants (spoken of as "external parameters" by Stepanov[16J).
The concept that each volume element is irradiated by scattering from every other volume element of the medium (The Principle of SelfIllumination), is a basic concept of radiative transfer theory which clearly pervades the writings of such early workers as Schuster [4, 17], King [18], and Schwarz3child [6]. The equation of radiative transfer can in fact be regarded as a simple statement of the law of conservation of energy. The change in intensity of a beam along its direction of propagation, dI, is equal to the radiation which is lost through absorption and scattering KpIdx, plus that which is scattered in this direction from all other directions, jpdx:
dI= KpIdx+jpdx
(1)
Here p is the density, K is the attenuation coefficient, j is the scattering function , and dx is the element of path length. The scattering function can be written as
j(e,
rr CP)=47r
J{o" J(o2>rpee,
cp;
e',
cp')I(e',
cp')
sin
e'de'dcp'
(2)
where rr is the scattering coefficient and p(O,cp; e' ,cp') is the phase function which defines the
probability that radiation which travels initially in
the direction (0' ,cp') is scattered into the direction (e,cp). Equation (1) is usually solved in terms of
the optical thickness
(3)
Defining 0 as the angle with respect to the inward surface normal gives
J.L dI( Td,TJ.L, cp)
I( T,
J.L,
cp ) 
J( T,
J.L,
A.)
'I'
(4)
where w=cos e and J=j/rr is the source function.
The equation of radiative transfer can be gene~al ized to include dispersion effects as well as scattenng [14, 15]. In terms of the components of the Stokes
vector, I,
569
d1i~; ¢) tt { Kii j(O, ¢)
+~
471"
J(o"
J(o2" p(O,
¢;
0',
¢')lj (O',
¢')
sin
0' do' d¢'
}
+1/(0, cf».
(5)
The term l/ (0 ¢) accounts for possible self radin,tion
of the volume' element, which is of importance in the infrared spectral region or in ~uminescing medi.a. As mentioned above, this term WIll be suppressed m
the present discussion.
We generally deal with the case where Ti=T/ and Ri=R/; i.e., where the reflectance an~ tra?smittance of a layer do not. depend on the dIre~t~on
of incidence. This is qUIte a general condItIOn
which applies even to the case of nonhomogeneous
sheets [19]. With these substitutions eqs (6) and (7)
reduce to
TiTj
(8)
lRi Rj
+ T/Rj
R Hj=R i 1_RR
(9)
, .1
A. The Layer Model
A model in which the scattering medium is ap
proximated by planeparallel layers has been used
by numerous workers as a basis for reflectance
theory [7, 8]. We consider a pair of adjacent layers,
wmiittthanRci~Rs
j
an for
d r
Ti,T adiat
j i
as on
the reflectances and transincident in one direction,
maintdtaRnI~eRsJ
and for
TI, TJ as the reflectances and transradiation incident in the other
direction.
If the incident beam is of unit intensity, then the
portion Ri is reflected and the portion Ti transmit~ed
by the first layer. At the .second l::yer the portIOn TiR} is reflected and T/!'j IS transmI~ted. The beam TiRj strikes the undersIde o~ layer'/, where 'fiRjT/
is transmitted, while T iR jR/ IS reflected. Contmumg
this process indefinitely (see fig. 1) we find that. the
transmittance and reflectance of the combmed
layers are given by
Tt+j=TiT/l +R/Rj+RiRl+ ... )
Rt+j=Ri+Ti T/R/l +R/Rj+R/Rl+
Summing the geometric series gives
.. )
TiTj
T i+j =I_R/Rj
(6)
We now make the assumption that the layers are homogeneous 3;nd thick comp.are~ ~it~ individual particle dimenSIOns so that no mt~msIc mhomogeneities are evident. We can then wnte that the reflectance of a given layer is proportional to its thickness [20],
(10)
Similarly we write
T(dx) = 1K 2dx.
(11 )
Here KJ and K2 are phenomenological constants which characterize the medium. We assume that K2 '2K J, where the inequality applies t~ absorbi~g media and the equality to nonabsorbmg' medIa. We substitute R(x) and T(x) for Ri and Ti m eqs (8) and (9), with R(dx) and T(dx) from eq~ ~1O) and (11) being substituted for R j and T j. Wntmg R Hj and T i+j as R(x+dx) and T(x+dx) allows eqs (8) and (9) to be expanded in series to give
dR K JT 2dx and
dT= T(K2KJR)dx.
Integrating this set of coupled di~~rential equations subject to the boundary condItIOns R(O) =0, T(O) =1 gives
R
i
+
j=
R
i
+
TiT/R j lR/R
(7)
j
(12)
T=(IR ~) 1~oLo2xe 2LX'
(13)
Xi
T
.L
~~~~~":":T
R is the reflectance of an infinitely thick layer, and it 'is related to KJ and K2 through
R =K2L=K2.,JK22_KJ2
(14)
co
KJ
KJ
while L is given by
(15)
FIGURE 1. Reflectance and transmittance oj a pair oj inhomogeneous layers (Kubelka [19]).
Equations (12) and (13) were derived long ago by Stokes [1] for planeparallel plates, with similar su~: sequent derivations by Schuster [4, 17] and GurevIc
570
[20] for lightscattering layers. These ar e now known to be a special case of the more general KubelkaMunk theory [21], which we will now consider.
B. The KubelkaMunk Theory
The KubelkaMunk theory [5] is based upon a model in which the r adiation field is approximated by
two fluxes, the one, h, traveling from the illuminated
sample surface, and the other, L, traveling toward the illuminated surface (see fig. 2). As radiation travels from the surface, its intensity is decreased by scattering and absorption processes, both assumed to be proportional to the thickness of the medium traversed . This is partially offset by scattering from the other b eam, so we have
(16 )
R (RgRx,)/R oo Roo (R gI /R oo ) oxp [SX(1 /R oo Roo) ] RgRoo (Rgl/R ro ) exp [SX(1 /RooRro)]
(20)
Rg is th.e reflectan~e of t he b ackground (see fig . 2) and R oo IS once agam the reflectance of a layer which is so thick that fur ther increase in thickness does not alter the reflectan ce. If eq (19) is integrated over the limits x=O to x = ro, a simple formula results [5]
R ro= lim R=a(a21)1 /2== ab
(2 1)
X"'" 00
Usin g eq (18) this can be rearranged to give the wellknown KubelkaMunk function F(R ro )
(22)
The component travelling toward the illuminated surface is similarly described:
d1_ = (S+K)LdxS1+dx.
( 17)
We note in passing that this result follows directly from eq (12) in the limi t x+ro [23], which once again
shows the close relationship of the Gurevic and KubelkaMunk models.
The constants which we h ave introduced h er e are once again phenomenological constants which describe scattering (S) and absorption (K) within the medium. If we make the following definition,
S+K a==S
( 18)
we can write
d1+ Sdx = a1++L
~~~=aL+1+
C. Rozenberg Solutions
For a homogeneous semiinfinite medium, a good approximate treatment of the reflectance for strong absorbers has been given by Rozenberg [1 315]. The solu tion involves summing various successive contI' butions to the reflecting power by scattering of different degrees of multiplicity, and is based upon concepts developed by Kuznetsov for problems of visibility [23]. To the nth degree of approximation , the reflected intensity in an isotropic medium is given by
l'ef(e A.)=JO~ au(e, c/J).
(23)
t , ' I ' I ~ (1+,6)1
which can be combined into a single differential equation,
(19)
where R== L /1+. Equation (19) can be easily integrated over the entire thickness x of the scat tering medium to obtain [5]
l '. '.'  X~X
In this case e and ¢ define the direction of obser
vation, 11° is th e inciden t beam intensity, /3=:: is the (J
ratio of the absorption and scattering coefficients, and the index i runs over the four components of the radiation field (see eq (5)) . The a tl coefficients in eq (23) are given by
m where the C](eo,¢o) == n define the polarization
of the incident beam, and the atil (e, c/J) are coeffi
cients which depend only on the angles of incidence (eo, c/Jo) and observation (e,¢), and on the form of the scattering indicatrix.
The reflectance of the medium is given by
FIGURE 2. Model for the ICub elkaMunk analysis of refl ectance
(24)
and transmittance of a scattering medium [5J.
571
where Gi = [ti [1 is the polarization of the reflected
radiation. It will be observed that as (3 increases, scattering of higher multiplicities becomes less important. Ambartsumian [24] has shown that the
+ mean multiplicity of scatter in the case of reflection
from a semiinfinite turbid medium is .JI 1/(3. Thus when (32:: 1/3, a fairly accurate solution is obtained by inclusion of terms up to second or third degree. It is further assumed that the scatter is independent of (3, which should be a good approximation for mixtures of polydispersive media with different a and IJ'; i.e., as with the addition of dye to a suspension. The re
sulting equation is
D. Exact Solutions
In problems of spectroscopy we often assume isotropic scatter. We know that in no case is single scatter actually isotropic [3), although the random distribution of anisotropic particles and scatter apparently tends toward an isotropic result [15]. Problems in highly anisotropic scattering media have been considered by some workers [2934]. A detailed discussion of these solutions, often obtained by numerical computer methods, will not be discussed here.
For the simple case of isotropic scatter, the phase function (see eq (2)) can be written
(25)
p«(J,cp;(J'Cp')= wo=I+J' .
(30)
IJ' a
where Ro is the reflectance of the medium itself (when (3=(30), and
(26)
Thus Q is a quantity which defines the relative contribution of higher multiplicities of scattering when (3=(30' Both it and Ro are constants which are independent of the nature and concentration of the colorant.
Equation (25) may be regarded a '> a generalization of Lambert's law to the case of colored media. It has been derived in a somewhat different form by Chekalinskaia [25] from scattering theory. In terms of the reflection (r), forward scatter (t), and absorption (a) constants of a single scattering layer( a+r
+t= 1) used by Chekalinskaia, the Rozenberg con
stants can be written
Here Wo is known as the albedo of single scatter. It represents the fraction of the radiation lost by
scattering in a medium where both absorption (a)
and scattering (IJ') take place. With this assumption, the equation of radiative transfer for a plane
parallel semiinfinite medium becomes independent of the azimuthal angle cp and we have
f +1 p. df(drr, p.)
f( r, p.) 21' Wo
1
f
( r,
')d'
iJ. iJ..
(31)
The integral occurring in eq (31) may be approximated by a Gaussian quadrature, in which case a set of coupled linear differential equations is obtained,
(32)
The constants a] are Gaussian weighting functions given by
Q = l +t ·
(27)
(33)
r+t
Il'ina and Rozenberg [26] have demonstrated the
> validity of eq (25) in several instances. Obviously
in a highly absorbing medium where (3) 1, eq (25) further simplifies to
(28)
and iJ.] is one of the zeros of the Legendre polynomial, Pn(iJ.).
Passing to the limit n7a:l gives an exact so1ution which Chandrasekhar [35] has shown to be of the form,
(34)
In the other limit; i.e., where (3 « 1, Rozenberg [15, 27] has shown that the reflectance can be written as an exponential of the form ,
Here fr(iJ.) is the reflected intensity in the direction iJ. from a collimated incident beam in the direction iJ.o, whose flux per unit area normal to the beam is 1I'.(iJ.o) . The Hintegrals are defined by
s where h(p., p.o), (p., p.o), and q are quantities which
depend on the form of the scattering indicatrix. Romanova [28] has determined these quantities by exact solution of the radiative transfer equation.
and tables of them have been given by Chandrasekhar [35].
572
Giovanelli [36] has given explicit expressions for several cases of interest. The total reflectance for light incident in the direction J.l.o is
solutions are in fact a good approximation for real scattering media.
(36)
III. Statistical Theory
while that for diffused incident radiation is
Tables of the first moment of the H  integral, which occurs in eq (37), have also been given by Chandrasekhar [35] .
Exact solutions to the equation of radiative transfer can be derived for other phase functions as well. In general, the phase function may be expanded as a series of Legendre polynomials
p (cos e) = ~ WI PI (cos 8)
/=0
where axial symmetry is assumed. Terms higher than first degree contribute very little [36, 37], and thus the approximate phase function.
p(cos 8)=wo(1+x cos 8) (0::::: x::::: 1)
is sometimes used. Exact solutions are available for scatter according to this phase function also [36].
Equations (36) and (37) for isotropic scatter can be readily applied using tables given by Giovanelli [36], and they of course are of considerable theoretical interest since they represent exact solutions to which the various approximate theories can be compared. We do not expect all media to scatter isotropically, but we might expect the range of applicability of the equation s to be extended if an appropriate average scattering coefficient were used. If we consider a diffuser in which each scattering center scatters light symmetrically about the direction of incidence, we may write [38]
U eU= (1 p.)u
(38)
Continuum models, as we have seen, are somewhat limited. They involve the use of phenomenological constants with no obvious relationship in general to the fundamental constants with which we are familiar (molar absorptivity, refractive index, particle size and shape, etc.). Statistical theories, on the other hand, involve the con truction of an appropriate model and the success of the theory depends on just how closely the model approximates real sample conditions.
It appears certain that one of the most severe limitations of continuum models is the assumption that they remain valid even when infinitesimal thicknesses are considered. This is in fact contrary to the assumption of homogeneous layers previously invoked (see section IIA), and it is this contradiction which is largely responsible for limiting the range of applicability of continuum models, as our subsequent discussion will show.
Let us return Lo eqs (8) and (9) and assume that we are now dealing with thin layers whose thickness is that of the individual particles. If we take layer i to be the first layer and layer j to be the combination of all the other layers of an nlayer sample, we have
and
+ t =1'
rl.2.3.... n I
I2r2.3.4• .. . n
1 r lr2.3.4•... n
Passing to the limit n~ 00 , we write
(40) (41)
1'1.2.3.... n=r2.3. 4••.• n=R.,
Equation (41) then becomes
where
J.I.
(39)
In this approach it is assumed that the same isotropic solutions (eqs (36) and (37)) may be used for arbitrary angular distribu tions of scatter, so long as the scattering is averaged according to eqs (38) and (39). Blevin and Brown [38] have shown that the reflectance curves are essentially the same for isotropic scatter or for scatter according to the
phase functions 1+P1 (J.I.) , 1+P2 (J.I.) , and 1+P3 (J.I.) .
This suggests that the reflectance is not a sensitive
function of the scattering indicatrix, and the isotropic
_rRt2R
R.,=r+ 1
(42)
co
where we have assumed that all layers are the same so the subscripts on l' and t can be dropped. Equation (42) can be solved for Roo to give an expression for the reflectance of an infinitely thick sample in terms of the reflectance and transmittance of a single layer.
The result is
(43)
This equation is fundamental to essentially all statistical theories, the only difference being in the
method used to calculate l' and t.
573
We have seen that the KubelkaMunk theory leads to a solution of the form,
When this is solved for Roo, we get
plot of F(R "') versus K deviates from linearity for high values of K [79], and it appears that eq (48) can be used to explain the deviations in part. It should be recognized that the deviations at high values of K are probably a result of anomalous dispersion effects also, but eq (48) does represent an improvement in the range of validity and shows the need to consider the particulate nature of scattermg media in developing a more precise theory by which absolute absorptivites can be determined.
which is not of the same form as eq (43). We assume with Simmons [39] that the plane
parallel layers of the KubelkaMunk model cannot be made infinitesimally small, but are restricted to layers of finite thickness l, where l may be interpreted as the mean particle diameter of the sample. Then the fundamental differential equations of the KubelkaMunk theory (eqs (16) and (1 7)) are replaced by the finite difference equations:
d1+"'(/+);+1(/+) i
dx
l
A. The Bodo Model
Bod6 [40] used a procedure similar to that used to derive eq (43) for the derivation of l' and t. We will denote the simple reflectance of the layer surface by 1'0, the absorptivity (defined through 1= 10 exp (  kx)) by k, and the layer thickness (equivalent to the mean particle diameter) by I. Then according to figure 3, the reflectance and transmittance of a single layer are given by
1'=1'0+ (11'o)2d 2kl+ (11'o)21'03e4kt
dL",(L);+I(LL (K+8) (L);+18(1+);
(IX
l
(46)
where the subscripts i and (i+ 1) refer to the ith and (i+ l)st sample layers, respectively. Now for an infinitely thick sample.
R =(L)I=(L);+1
00 (1+)1 (/+)1+1
and Eqs (45) and (46) can be solved to give
R",=
2 ( 8 + K  K l 8  K2l / 2)
v'4(8+K  Kl8 K21 /2)2482 28 (47)
Equations (43) and (47) are identical if we make the following identifications:
+ (1 1'oFro5e6kl+ . ..
t= (11'o)2ekl+ (1ro)21'o2e3kl+ (11'o)2r04e5kl
+ (1 1'o)21'06e7kl+ ....
Summing these series gives
1'o[1+(12ro) exp (2kl)]
1'= "':1~1'co2;;e'x"'p'("';;2:'k"I)~
(49)
t (11'0)2 exp (kl) .
11'02 exp (2kl)
(50)
Equations (49) and (50) together with eq (43) constitute the Bodo formulation, which is in fact equivalent to that of Stokes [1] and Girin and Stepanov [41]. Bodo [40] obtained good results with these formulae for powdered glass samples using the arbitrary assumption that 1'0=0.10. Karvaly [42] has shown that this was at least in part due to a particularly favorable position for the absorption band
8=1'/1
K=(l1't) / I=a/1
where a is the fraction of the incident radiation which
ro
is absorbed by the layer.
The difference between the traditional and modi
fied KubelkaMunk solutions may be seen by writing
eq (47) in the form,
(1 ) F(R )=K KK2= 1 a 1  a2. (48)
00
8
28
l'
21'
It will be recognized that the difference lies in the FIGURE 3. Refl6Ctance and transmittance of a single layer of
addition of the last two terms. It is well known that a
thickness I according to Bod6 [40].
574
chosen for study, but in the gen eral case, Roo is a very sensitive function of To.
Bauer [43] showed that in some cases the layers should be considered to have rou gh surfaces where total internal refiection can take place, and he has derived expressions analogous to eqs (49) and (50)
for this case.
B. The Johnson Mod e l
Johnson [44] h as carried out the summation somewhat differently than B od6, but with quite similar results (see fig 4). It is assumed that there are p layers and that the mean number of attenuating refiections which the rays undergo in the 2p traversals is yp, so that the r efiectance is given by
Rp=To+2ToL::; (1To)V1' exp ( 2kpl ). (5 1) P
Thus y can be regarded as an adjustable parameter which gives a semiempirical account of multiple r efl ections as well as scattering losses. The sum for an infmite number of layers is
R
"
,

To
+
2
To
exp [y In (1To) 2kl] 1
(52)
1exp [y n (1To)2kl]
It will b e observed from figure 4 that y should be se t equal to 4 for the case of no multiple refiections. In such a case only one h alf of the incident light is reflected , however. Johnson h as suggested that y can
be estimated from eq (55) by setting R oo = 1 for
k = O. This gives
(l To)V l 1+1 . .J. 1 2 1 3
Y
241ol·6To+STo+... ·
(56)
from which y = 2 is seen to be a satisfactory approximation for refr active indices smaller than 1.5. A smaller value of y is required for larger refractive indices. Companion and Winslow [43] have used a model similar to Johnson's, bu t which includes all multiple reflections. The summation was carried ou t by compu ter and no explicit expression for the reflectance was given by these workers.
Johnson [441 also suggested that To be equated to 1.5 times the normal Fresnel reflectance. This is
meant to account for the random distribution of particle surfaces and corresponds with an average incidence angle of approximately 30°. It was shown [44] that eq (55) yields absorption coefficien ts for KCl :Tl, KBr :TI, and didymium glass which agree satisfactorily with those obtained by tr ansmission measurements of the same materials.
which is equivalent to eq (43) with
T=To[l+ (l  To)V exp (2k l )]
(53)
t =(1To)v/2 exp (k l ).
(54)
The denominator of equation (52) can be expanded to !rive
R ", To [ 2(1kTo)V exp (2kl )+ 1] .
(55)
2 l  y l n ( 1  To)
This result could also be obtained directly by in tegration of eq (5 1), which suggests that it may in fact be a more realistic representation of a real sample whose particles actually have a range of diameters.
. . . " ... ,'. . ..'.., .. .
~
I
T
C. The Antonov Romanov s ky Model
AntonovRomanovsky [22] has developed express~ons which. can be used to calculate the true absorptIOn coefficlent from reflectance measurements by connecting the KubelkaMunk and B od6 th eories . AntonovRomanovsky treats two limiting cases of r egul arly shaped sample particles, spher es and parallelepipeds (see fig. 5) .
For spherical particles the radiation impinges on the surface from within at th e same angle that i t entered the particle, since the angles which the cord of a circle makes with respect to th e surface normal must be the same. Therefore, total in ternal r eflection
is impossible, and l will approximate the p article
diameter, l , actually being somewhat smaller. In the case of t he parallelepiped, some total
internal reflection is possible, but most radiation probably exits through an opposite face without
further reflection. In this case also it is obvious that l
. ' . .. . . ', . . ' .. ' .
. ", ... .; ", . , ' "
FIGURE 4. Model for Johnson analysis of refl ectance from a scattering medium [44].
FIGURE 5. Th e AntonovRomanovsk y model for regular spheres and parallelepi peds [22].
575
is approximately of the same magnitude as l, but
somewhat larger.
.
Thus for regularly ~haped particles it is assumed
that l=l and
k=~ln (lro)22ro(l2ro)F(R,,).
(57)
2l
(lro)22roF(Roo)
< This is an approximate form of the Bodo model which
is valid for media in which ro2< l. For irregularly shaped particles it is assumed that
the emerging radiation meets the surface of the particle with an equal probability for all angles. This will be the case if the mean number of reflections in
the layer m, is large:
This condition requires that
where l, is defined as the "free" path length. The effective path length is then
(58)
and it is assumed by AntonovRomanovsky that
(59)
The assumption that the emerging radiation is independent of angle allows us to divide it equally between the two sides of the layer whose thickness is
l, the mean particle diameter. We write
rro=t
(60)
and from the law of conservation of energy,
(lro)(1exp( sk))=Irt.
(61)
Equations (60) and (61) can be solved to give
2ro+(1ro) exp (mkl/2)
r
2
(62)
(lro) exp (Titkl/2) (63)
2
Using these in eq (43) gives
k=2 ln (lro)2(1 ro)F(Roo).
(64)
Titl
(lro)22roF(R..,)
tive derivation of .Karvaly [47], which we follow ~ere (see .fig. 6). A smg;le .par~icle of the surface layer
IS sho;yn m th.e figure; It IS shIfted laterally with each reflectIOn t~ Illustrate the path of the radiation.
Of the dIffuse radiation which is incident on a
given where
surface particle, the subscript e
itsheusferadcttioonin2duicr~ties
rreefflleecctteiod~
of externally incident radiation. Here u is defined as the ~adiation ~merging per unit solid angle from a
partIcle. 47rU IS the solid angle which would be
observed from the second layer if a particle were
removed (see fig. 7).
O~ the radiant flux (l2ure) which enters the
partIcle, the part (12ur e)ut contributes to the re
flectance from the first internal reflection and the par:t Ao= (l2ur ~)(Iu)t strikes the underlyi~g layers,
whIch are conSIdered to form an infinitely thick
powder mass of reflectance Roo. The transmission
of an individual particle is represented by the
symbol t. The part AoRoo is reflected back into the layer,_ ~here AoRoo (11' e). enters the particle and
AoRoore IS reflecte:i back mto the underlying layers.
,9f that pa.rt whIch entered the particle, AoRoo (1
+ :'e)ut con.tnbutes to the reflectance and AoRoo(l
r ej (lu)t IS reflected downward again. This combines
wIth th':.t externally reflected to give AoR",re AoR", (l re)( l u)t AoR", [r e+ (lu)t]=AoR", Q as the
flux reflected downward of which AoR",2Q returns
to the layer, etc.
Ao AoR.
A oR . Q
FIGURE 6. The Melamed Model for powder reflectance as viewed by Karvaly [47].
'\ '\
,'\ '\
D. The Melamed Model
. In contrast with the st~tistical theories previously dIscussed, Melamed earned out a summation over the reflectance and transmittance of individual particles rather than layers [46] . Some features of thIs model are better understood using an alterna
FIGURE 7. Illustration of the meaning of the radiant intensity factor u [47].
576
The Interpretation of Diffuse Reflectance Spectra
Harry G. Hecht
Department of Chemistry, South Dakota State Un iversity, Brookings, South Dakota 57006
(May 26, 1976)
Numerous treatments of the diffuse r eflecting properties of scattering media have been d escribed. Many theories give an adequate account of the refl ectance for a specific set of
conditions for which the model was constructed and the soluti on t ested experimentally. Only those models which are considered to be fairl y general are considered her e.
It is conveni ent to di vide the theories into those based upon continuum models and
those based upon s tatistical models. The continuum m odels typica ll y describ e the scatterin g a nd absorbin g properti es of a given m edium in t erm s of two phenomenological constants.
These models may all be regarded as varying levels of approxima t e soluti on to the general equation of radi ative tra nsfer. This provides a convenient basis for compariso n of the various
theories. The statistical models a re based upon a summ a tion of transmittances a nd r efl ectances
fr om individual l ayers or parti cles. Thus, ~ome ass um ptions must be m ade abo ut the n ature
of the fund amenta l units, and the validity of the ultim a te result will depend up on how closely these assumptions correspond with r eality. Only the statistical models lead to ex pressions
fr om which absolute absorptivities and sca ttering coefficients can be calculat ed and r elated to the actual particle characteristics.
The relationship between the various models will be discussed and the features which typify the ab~orptivity and scatterin g coeffici ent accordin g to each will be compared an d related to the available experimental data. This leads t o a consideration of the charac teristics
of appropri ate model systems and standards.
Key words: Absolute absorptivities; continuum models; diffuse reflectance; r adiative transfer; reflectance spectra; scatt ering coefficients ; statistical models.
List of Principal Symbols Used
H(iJ.)
(Note: Where a given letter is used in both capital [ and lower case form (e.g., r,R and t,T), the capital ['
letter refers to the macroscopic observable and the
lower case letter to the corresponding variable for an J
individual particle or layer of the material. A bar J
over a given symbol means the average value for k
that variable).
K
a
total absorption of a single par ticle
(layer); also S~K (equation (18))
Gaussian weighting function (equation
(33) ) L
}R ozenberg constants (equation (23)ff.) m
(a2 _1)1 /2 (equation (21))
M
[ ;/[1 polarization of radiant beam (equa
tion (24»)
fraction of radiation reflected (equation
(75) )
fraction of radiation transmitted (equa
tion (76»)
R ozenberg constant (equation (29))
Hintegral of Chandrasekhar (equation
(35) ) radiant intensity component of source function for selfradiation (equation (5)) scattering function (equation (2)) source func tion (equation (4)) absorptivity KubelkaMunk absorption constant (equations (16) and (17)) Attenuation constants in Gurevic layer model (equations (10) and (11)) r.article diameter or layer thickness 'free" path length of AntonovRomanov
sky (equation (58)) Gurevic constant=~K22 _K12 mean number of reflections
transmittance of a particle for a single
pass
radiant flux density
phase function (equation (2))
Legendre polynomial Rozenberg constant (equation (29))
Rozenberg multiple reflection constant
(equation (26))
567
214427 0  76  2
r
total reflectance of a single particle
(layer)
simple reflectance of a single particle
(layer) surface
r*
mean external reflection coefficient for
side scatter
reflectance of a particle (layer) for
externally incident radiation
reflectance of a particle (layer) for
internally incident radiation
R
reflectance of a (macroscopic) layer
s!s!'"
reflectance of an infinitely thick layer effective path length (equation (58»
KubelkaMunk scattering constant
(equations (16) and (17»
Rozenberg weak scatter constant (equa
tion (29»
total transmittance of a single particle
(layer)
T
transmittance of a (macroscopic) layer
u
radiation emerging from a particle per
unit solid angle
shading factor (equation (70»
fraction of radiation emerging from a
particle in a downward direction
fraction of radiation emerging from a
particle in a sideways direction
fraction of radiation emerging from a
particle in an upward direction
x
length
y
Johnson multiple reflection factor (equa
tion (51»
a
absorption coefficient
(3
a/a (equation (23»
'Y
FasslerStodolski constant (equation (82»
~
FasslerStodolski constant (equation (82»
()
polar angle
K
attenuation coefficient=a+a (equation
(1»
wavelength
cos ()
one of the roots of the Legendre poly
nominal P n(J..l.)
effective hole cross section (equation
(100»
p
density
a
scattering coefficient (equation (2»
T
optical thickness (equation (3»
rP
azimuthal angle
q,.
radiant flux
Wo
albedo of single scatter (equation (30»
I. Introduction
It is now recognized that diffuse reflectance spectroscopy is a very useful companion technique to transmission spectroscopy. Not only can it provide absorption data in some cases where transmission measurements fail, but for many industrial and research applications, it may in fact be the preferred technique.
Our discussion will be concerned with the behavior of radiation within a scattering medium. For simplicity we assume the scattering centers, whieh may also absorb radiation, are imbedded in a mpdium which neither scatters nor absorbs. The medium may usually be taken as air, although there are many other cases of interest in which the refractive index of the medium is much greater than unity. We will not deal specifically with these cases in the present review, nor with those processes which alter the frequency of the radiation, such as luminescence and fluorescence.
Scattering takes place under a wide variety of conditions. One may be concerned with the glowing photosphere of the sun, which is surrounded by a cloud of electrons that reradiate the direct sunlight incident on them. Sunlight is also reradiated by cosmic dust, which accounts for the outer part of the corona.
In a more downtoearth situation one may be dealing in the laboratory with the spectroscopy of a powder, paper, opal glass, photographic emulsion, etc. As so often happens, concepts developed in one area of science are slow to find their way into another. We will attempt to show that there is a close relationship between the astrophysical solutions, which are based largely on radiative transfer theory, and the various models which are more familial' to the spectroscopist.
Since we will not be following a historical development, it may be useful to point out some relationships between the early studies. The first attempt to account for transmission and reflection of a layered material was carried out by Stokes in about 1860 [1]/ and led to some very useful relationships which have also been derived by other workers (vide infra). Lord Rayleigh [2] and Mie [3] developed the theory of single scatter to a high degree, but Schuster [4] was the first to consider multiple scatter. He was concerned with the cloudy atmospheres of stars, and developed a planeparallel plate model in which the radiation field was divided into forward and backward components. This same model was used much later by Kubelka and Munk [5], whose names are usually attached 1,0 it by spectroscopists. Schwarzschild [6] showed that the radiation field should be characterized by a complete angular distribution, and if one integrates over the forward and backward hemispheres, the Schuster model is obtained as a first approximation. A further generalization of the Schwarzschild formulation leads to an integrodifferential equation known as the equation of radiative transfer, which is very general in concept, but can be solved exactly in only a few cases.
The radiative transfer theory and various models stemming from it are referred to as Continuum Models. They have in common the characterization of the scattering and absorbing properties of the medium through phenomenological constants, usually two in number. These theories will be considered in section II. A completely satisfactory theory must of course relate the measurable quantities to
1 Figures in brackets indicate the literature references at the end of this paper.
568

fundamental properties of the scattering material, such as particle size, refractive index, and absorptivity. This is the goal of the statistical theories. Many have been proposed; those which show the most promise are discussed in section III. In section IV we discuss some of the strengths and weaknesses of the various models and some relationships between them. Section V considers the meaning of the absorption and scattering coefficients often used to interpret reflectance data, as well as the characteristics required of appropriate model systems.
We willlimit our discussion to ideal, homogeneous dispersions. There have been interesting developments in the theory of nonhomogeneous media, mixtures, luminescing materials, diffusing media in which photochemical reactions are taking place, and reflectance for materials dispersed in a highly refracting matrix, but these topics are considered to be outside the scope of the present review. Many of these topics are covered in the books which have been written on the subject [7,8,9J . Further information has been presented at various symposia which have been h eld, and the proceedings of at least two of them are available [10,11]. These sources, together with various reviews devoted to applications in particular fields, have made workers aware of the power of the technique. Thus we will confine ourselves to certain theoretical aspects.
The present paper may be regarded as an update to a former review of the subject by the author [12]. Most of the theoretical work of the intervening years has been in the development of more refined statistical models, and in showing the relationship between the various theories. At present there is some reason to feel optimistic that out of a morass of apparently divergent and unrelated theory, some order is beginning to emerge.
II. Continuum Theory
As is often the case with applications of physical theory, the real media with which diffuse reflectance spectroscopy is concerned are intermediate between two extreme cases, each of which is well understood. The one limiting case involves the propagation of radiation through quasihomogeneous matter, where the radiation field is characterized by smoothly and slowly varying functions of the coordinates. The other extreme involves the emission, absorption, and scattering of radiation by single particles in a homogeneous medium. The local transformation of the radiation field at the boundaries of quasihomogeneous media is a special case which has likewise been characterized for a long time, and leads to such wellknown phenomena as reflection, refraction, etc.
In dense scattering media, it is important to consider the manner in which the radiation fields from the various scattering centers interact with one another. Rozenberg [13, 14, 15] has pointed out that the interaction can be treated as a sum of two parts,
one of which is coherent and the other incoherent.
The coherent part is largely due to nearest neighbor interactions and gives rise to dispersion effects; i.e., is involved with changes of refractive index. The incoherent part is that with which we are primarily concerned in the present discussion. It involves multiple scattering as a sum of interactions throughout the entire medium. In most treatments the multiple scattering, because of its complexity, is separated rath€r arbitrarily from dispersive effects. Such a separation reduces the problem to one involving geomrtrical optics, and allows one to write a radiative transfer equation in which the absorbing and scattering properties of the medium are treated as phenomenological constants (spoken of as "external parameters" by Stepanov[16J).
The concept that each volume element is irradiated by scattering from every other volume element of the medium (The Principle of SelfIllumination), is a basic concept of radiative transfer theory which clearly pervades the writings of such early workers as Schuster [4, 17], King [18], and Schwarz3child [6]. The equation of radiative transfer can in fact be regarded as a simple statement of the law of conservation of energy. The change in intensity of a beam along its direction of propagation, dI, is equal to the radiation which is lost through absorption and scattering KpIdx, plus that which is scattered in this direction from all other directions, jpdx:
dI= KpIdx+jpdx
(1)
Here p is the density, K is the attenuation coefficient, j is the scattering function , and dx is the element of path length. The scattering function can be written as
j(e,
rr CP)=47r
J{o" J(o2>rpee,
cp;
e',
cp')I(e',
cp')
sin
e'de'dcp'
(2)
where rr is the scattering coefficient and p(O,cp; e' ,cp') is the phase function which defines the
probability that radiation which travels initially in
the direction (0' ,cp') is scattered into the direction (e,cp). Equation (1) is usually solved in terms of
the optical thickness
(3)
Defining 0 as the angle with respect to the inward surface normal gives
J.L dI( Td,TJ.L, cp)
I( T,
J.L,
cp ) 
J( T,
J.L,
A.)
'I'
(4)
where w=cos e and J=j/rr is the source function.
The equation of radiative transfer can be gene~al ized to include dispersion effects as well as scattenng [14, 15]. In terms of the components of the Stokes
vector, I,
569
d1i~; ¢) tt { Kii j(O, ¢)
+~
471"
J(o"
J(o2" p(O,
¢;
0',
¢')lj (O',
¢')
sin
0' do' d¢'
}
+1/(0, cf».
(5)
The term l/ (0 ¢) accounts for possible self radin,tion
of the volume' element, which is of importance in the infrared spectral region or in ~uminescing medi.a. As mentioned above, this term WIll be suppressed m
the present discussion.
We generally deal with the case where Ti=T/ and Ri=R/; i.e., where the reflectance an~ tra?smittance of a layer do not. depend on the dIre~t~on
of incidence. This is qUIte a general condItIOn
which applies even to the case of nonhomogeneous
sheets [19]. With these substitutions eqs (6) and (7)
reduce to
TiTj
(8)
lRi Rj
+ T/Rj
R Hj=R i 1_RR
(9)
, .1
A. The Layer Model
A model in which the scattering medium is ap
proximated by planeparallel layers has been used
by numerous workers as a basis for reflectance
theory [7, 8]. We consider a pair of adjacent layers,
wmiittthanRci~Rs
j
an for
d r
Ti,T adiat
j i
as on
the reflectances and transincident in one direction,
maintdtaRnI~eRsJ
and for
TI, TJ as the reflectances and transradiation incident in the other
direction.
If the incident beam is of unit intensity, then the
portion Ri is reflected and the portion Ti transmit~ed
by the first layer. At the .second l::yer the portIOn TiR} is reflected and T/!'j IS transmI~ted. The beam TiRj strikes the undersIde o~ layer'/, where 'fiRjT/
is transmitted, while T iR jR/ IS reflected. Contmumg
this process indefinitely (see fig. 1) we find that. the
transmittance and reflectance of the combmed
layers are given by
Tt+j=TiT/l +R/Rj+RiRl+ ... )
Rt+j=Ri+Ti T/R/l +R/Rj+R/Rl+
Summing the geometric series gives
.. )
TiTj
T i+j =I_R/Rj
(6)
We now make the assumption that the layers are homogeneous 3;nd thick comp.are~ ~it~ individual particle dimenSIOns so that no mt~msIc mhomogeneities are evident. We can then wnte that the reflectance of a given layer is proportional to its thickness [20],
(10)
Similarly we write
T(dx) = 1K 2dx.
(11 )
Here KJ and K2 are phenomenological constants which characterize the medium. We assume that K2 '2K J, where the inequality applies t~ absorbi~g media and the equality to nonabsorbmg' medIa. We substitute R(x) and T(x) for Ri and Ti m eqs (8) and (9), with R(dx) and T(dx) from eq~ ~1O) and (11) being substituted for R j and T j. Wntmg R Hj and T i+j as R(x+dx) and T(x+dx) allows eqs (8) and (9) to be expanded in series to give
dR K JT 2dx and
dT= T(K2KJR)dx.
Integrating this set of coupled di~~rential equations subject to the boundary condItIOns R(O) =0, T(O) =1 gives
R
i
+
j=
R
i
+
TiT/R j lR/R
(7)
j
(12)
T=(IR ~) 1~oLo2xe 2LX'
(13)
Xi
T
.L
~~~~~":":T
R is the reflectance of an infinitely thick layer, and it 'is related to KJ and K2 through
R =K2L=K2.,JK22_KJ2
(14)
co
KJ
KJ
while L is given by
(15)
FIGURE 1. Reflectance and transmittance oj a pair oj inhomogeneous layers (Kubelka [19]).
Equations (12) and (13) were derived long ago by Stokes [1] for planeparallel plates, with similar su~: sequent derivations by Schuster [4, 17] and GurevIc
570
[20] for lightscattering layers. These ar e now known to be a special case of the more general KubelkaMunk theory [21], which we will now consider.
B. The KubelkaMunk Theory
The KubelkaMunk theory [5] is based upon a model in which the r adiation field is approximated by
two fluxes, the one, h, traveling from the illuminated
sample surface, and the other, L, traveling toward the illuminated surface (see fig. 2). As radiation travels from the surface, its intensity is decreased by scattering and absorption processes, both assumed to be proportional to the thickness of the medium traversed . This is partially offset by scattering from the other b eam, so we have
(16 )
R (RgRx,)/R oo Roo (R gI /R oo ) oxp [SX(1 /R oo Roo) ] RgRoo (Rgl/R ro ) exp [SX(1 /RooRro)]
(20)
Rg is th.e reflectan~e of t he b ackground (see fig . 2) and R oo IS once agam the reflectance of a layer which is so thick that fur ther increase in thickness does not alter the reflectan ce. If eq (19) is integrated over the limits x=O to x = ro, a simple formula results [5]
R ro= lim R=a(a21)1 /2== ab
(2 1)
X"'" 00
Usin g eq (18) this can be rearranged to give the wellknown KubelkaMunk function F(R ro )
(22)
The component travelling toward the illuminated surface is similarly described:
d1_ = (S+K)LdxS1+dx.
( 17)
We note in passing that this result follows directly from eq (12) in the limi t x+ro [23], which once again
shows the close relationship of the Gurevic and KubelkaMunk models.
The constants which we h ave introduced h er e are once again phenomenological constants which describe scattering (S) and absorption (K) within the medium. If we make the following definition,
S+K a==S
( 18)
we can write
d1+ Sdx = a1++L
~~~=aL+1+
C. Rozenberg Solutions
For a homogeneous semiinfinite medium, a good approximate treatment of the reflectance for strong absorbers has been given by Rozenberg [1 315]. The solu tion involves summing various successive contI' butions to the reflecting power by scattering of different degrees of multiplicity, and is based upon concepts developed by Kuznetsov for problems of visibility [23]. To the nth degree of approximation , the reflected intensity in an isotropic medium is given by
l'ef(e A.)=JO~ au(e, c/J).
(23)
t , ' I ' I ~ (1+,6)1
which can be combined into a single differential equation,
(19)
where R== L /1+. Equation (19) can be easily integrated over the entire thickness x of the scat tering medium to obtain [5]
l '. '.'  X~X
In this case e and ¢ define the direction of obser
vation, 11° is th e inciden t beam intensity, /3=:: is the (J
ratio of the absorption and scattering coefficients, and the index i runs over the four components of the radiation field (see eq (5)) . The a tl coefficients in eq (23) are given by
m where the C](eo,¢o) == n define the polarization
of the incident beam, and the atil (e, c/J) are coeffi
cients which depend only on the angles of incidence (eo, c/Jo) and observation (e,¢), and on the form of the scattering indicatrix.
The reflectance of the medium is given by
FIGURE 2. Model for the ICub elkaMunk analysis of refl ectance
(24)
and transmittance of a scattering medium [5J.
571
where Gi = [ti [1 is the polarization of the reflected
radiation. It will be observed that as (3 increases, scattering of higher multiplicities becomes less important. Ambartsumian [24] has shown that the
+ mean multiplicity of scatter in the case of reflection
from a semiinfinite turbid medium is .JI 1/(3. Thus when (32:: 1/3, a fairly accurate solution is obtained by inclusion of terms up to second or third degree. It is further assumed that the scatter is independent of (3, which should be a good approximation for mixtures of polydispersive media with different a and IJ'; i.e., as with the addition of dye to a suspension. The re
sulting equation is
D. Exact Solutions
In problems of spectroscopy we often assume isotropic scatter. We know that in no case is single scatter actually isotropic [3), although the random distribution of anisotropic particles and scatter apparently tends toward an isotropic result [15]. Problems in highly anisotropic scattering media have been considered by some workers [2934]. A detailed discussion of these solutions, often obtained by numerical computer methods, will not be discussed here.
For the simple case of isotropic scatter, the phase function (see eq (2)) can be written
(25)
p«(J,cp;(J'Cp')= wo=I+J' .
(30)
IJ' a
where Ro is the reflectance of the medium itself (when (3=(30), and
(26)
Thus Q is a quantity which defines the relative contribution of higher multiplicities of scattering when (3=(30' Both it and Ro are constants which are independent of the nature and concentration of the colorant.
Equation (25) may be regarded a '> a generalization of Lambert's law to the case of colored media. It has been derived in a somewhat different form by Chekalinskaia [25] from scattering theory. In terms of the reflection (r), forward scatter (t), and absorption (a) constants of a single scattering layer( a+r
+t= 1) used by Chekalinskaia, the Rozenberg con
stants can be written
Here Wo is known as the albedo of single scatter. It represents the fraction of the radiation lost by
scattering in a medium where both absorption (a)
and scattering (IJ') take place. With this assumption, the equation of radiative transfer for a plane
parallel semiinfinite medium becomes independent of the azimuthal angle cp and we have
f +1 p. df(drr, p.)
f( r, p.) 21' Wo
1
f
( r,
')d'
iJ. iJ..
(31)
The integral occurring in eq (31) may be approximated by a Gaussian quadrature, in which case a set of coupled linear differential equations is obtained,
(32)
The constants a] are Gaussian weighting functions given by
Q = l +t ·
(27)
(33)
r+t
Il'ina and Rozenberg [26] have demonstrated the
> validity of eq (25) in several instances. Obviously
in a highly absorbing medium where (3) 1, eq (25) further simplifies to
(28)
and iJ.] is one of the zeros of the Legendre polynomial, Pn(iJ.).
Passing to the limit n7a:l gives an exact so1ution which Chandrasekhar [35] has shown to be of the form,
(34)
In the other limit; i.e., where (3 « 1, Rozenberg [15, 27] has shown that the reflectance can be written as an exponential of the form ,
Here fr(iJ.) is the reflected intensity in the direction iJ. from a collimated incident beam in the direction iJ.o, whose flux per unit area normal to the beam is 1I'.(iJ.o) . The Hintegrals are defined by
s where h(p., p.o), (p., p.o), and q are quantities which
depend on the form of the scattering indicatrix. Romanova [28] has determined these quantities by exact solution of the radiative transfer equation.
and tables of them have been given by Chandrasekhar [35].
572
Giovanelli [36] has given explicit expressions for several cases of interest. The total reflectance for light incident in the direction J.l.o is
solutions are in fact a good approximation for real scattering media.
(36)
III. Statistical Theory
while that for diffused incident radiation is
Tables of the first moment of the H  integral, which occurs in eq (37), have also been given by Chandrasekhar [35] .
Exact solutions to the equation of radiative transfer can be derived for other phase functions as well. In general, the phase function may be expanded as a series of Legendre polynomials
p (cos e) = ~ WI PI (cos 8)
/=0
where axial symmetry is assumed. Terms higher than first degree contribute very little [36, 37], and thus the approximate phase function.
p(cos 8)=wo(1+x cos 8) (0::::: x::::: 1)
is sometimes used. Exact solutions are available for scatter according to this phase function also [36].
Equations (36) and (37) for isotropic scatter can be readily applied using tables given by Giovanelli [36], and they of course are of considerable theoretical interest since they represent exact solutions to which the various approximate theories can be compared. We do not expect all media to scatter isotropically, but we might expect the range of applicability of the equation s to be extended if an appropriate average scattering coefficient were used. If we consider a diffuser in which each scattering center scatters light symmetrically about the direction of incidence, we may write [38]
U eU= (1 p.)u
(38)
Continuum models, as we have seen, are somewhat limited. They involve the use of phenomenological constants with no obvious relationship in general to the fundamental constants with which we are familiar (molar absorptivity, refractive index, particle size and shape, etc.). Statistical theories, on the other hand, involve the con truction of an appropriate model and the success of the theory depends on just how closely the model approximates real sample conditions.
It appears certain that one of the most severe limitations of continuum models is the assumption that they remain valid even when infinitesimal thicknesses are considered. This is in fact contrary to the assumption of homogeneous layers previously invoked (see section IIA), and it is this contradiction which is largely responsible for limiting the range of applicability of continuum models, as our subsequent discussion will show.
Let us return Lo eqs (8) and (9) and assume that we are now dealing with thin layers whose thickness is that of the individual particles. If we take layer i to be the first layer and layer j to be the combination of all the other layers of an nlayer sample, we have
and
+ t =1'
rl.2.3.... n I
I2r2.3.4• .. . n
1 r lr2.3.4•... n
Passing to the limit n~ 00 , we write
(40) (41)
1'1.2.3.... n=r2.3. 4••.• n=R.,
Equation (41) then becomes
where
J.I.
(39)
In this approach it is assumed that the same isotropic solutions (eqs (36) and (37)) may be used for arbitrary angular distribu tions of scatter, so long as the scattering is averaged according to eqs (38) and (39). Blevin and Brown [38] have shown that the reflectance curves are essentially the same for isotropic scatter or for scatter according to the
phase functions 1+P1 (J.I.) , 1+P2 (J.I.) , and 1+P3 (J.I.) .
This suggests that the reflectance is not a sensitive
function of the scattering indicatrix, and the isotropic
_rRt2R
R.,=r+ 1
(42)
co
where we have assumed that all layers are the same so the subscripts on l' and t can be dropped. Equation (42) can be solved for Roo to give an expression for the reflectance of an infinitely thick sample in terms of the reflectance and transmittance of a single layer.
The result is
(43)
This equation is fundamental to essentially all statistical theories, the only difference being in the
method used to calculate l' and t.
573
We have seen that the KubelkaMunk theory leads to a solution of the form,
When this is solved for Roo, we get
plot of F(R "') versus K deviates from linearity for high values of K [79], and it appears that eq (48) can be used to explain the deviations in part. It should be recognized that the deviations at high values of K are probably a result of anomalous dispersion effects also, but eq (48) does represent an improvement in the range of validity and shows the need to consider the particulate nature of scattermg media in developing a more precise theory by which absolute absorptivites can be determined.
which is not of the same form as eq (43). We assume with Simmons [39] that the plane
parallel layers of the KubelkaMunk model cannot be made infinitesimally small, but are restricted to layers of finite thickness l, where l may be interpreted as the mean particle diameter of the sample. Then the fundamental differential equations of the KubelkaMunk theory (eqs (16) and (1 7)) are replaced by the finite difference equations:
d1+"'(/+);+1(/+) i
dx
l
A. The Bodo Model
Bod6 [40] used a procedure similar to that used to derive eq (43) for the derivation of l' and t. We will denote the simple reflectance of the layer surface by 1'0, the absorptivity (defined through 1= 10 exp (  kx)) by k, and the layer thickness (equivalent to the mean particle diameter) by I. Then according to figure 3, the reflectance and transmittance of a single layer are given by
1'=1'0+ (11'o)2d 2kl+ (11'o)21'03e4kt
dL",(L);+I(LL (K+8) (L);+18(1+);
(IX
l
(46)
where the subscripts i and (i+ 1) refer to the ith and (i+ l)st sample layers, respectively. Now for an infinitely thick sample.
R =(L)I=(L);+1
00 (1+)1 (/+)1+1
and Eqs (45) and (46) can be solved to give
R",=
2 ( 8 + K  K l 8  K2l / 2)
v'4(8+K  Kl8 K21 /2)2482 28 (47)
Equations (43) and (47) are identical if we make the following identifications:
+ (1 1'oFro5e6kl+ . ..
t= (11'o)2ekl+ (1ro)21'o2e3kl+ (11'o)2r04e5kl
+ (1 1'o)21'06e7kl+ ....
Summing these series gives
1'o[1+(12ro) exp (2kl)]
1'= "':1~1'co2;;e'x"'p'("';;2:'k"I)~
(49)
t (11'0)2 exp (kl) .
11'02 exp (2kl)
(50)
Equations (49) and (50) together with eq (43) constitute the Bodo formulation, which is in fact equivalent to that of Stokes [1] and Girin and Stepanov [41]. Bodo [40] obtained good results with these formulae for powdered glass samples using the arbitrary assumption that 1'0=0.10. Karvaly [42] has shown that this was at least in part due to a particularly favorable position for the absorption band
8=1'/1
K=(l1't) / I=a/1
where a is the fraction of the incident radiation which
ro
is absorbed by the layer.
The difference between the traditional and modi
fied KubelkaMunk solutions may be seen by writing
eq (47) in the form,
(1 ) F(R )=K KK2= 1 a 1  a2. (48)
00
8
28
l'
21'
It will be recognized that the difference lies in the FIGURE 3. Refl6Ctance and transmittance of a single layer of
addition of the last two terms. It is well known that a
thickness I according to Bod6 [40].
574
chosen for study, but in the gen eral case, Roo is a very sensitive function of To.
Bauer [43] showed that in some cases the layers should be considered to have rou gh surfaces where total internal refiection can take place, and he has derived expressions analogous to eqs (49) and (50)
for this case.
B. The Johnson Mod e l
Johnson [44] h as carried out the summation somewhat differently than B od6, but with quite similar results (see fig 4). It is assumed that there are p layers and that the mean number of attenuating refiections which the rays undergo in the 2p traversals is yp, so that the r efiectance is given by
Rp=To+2ToL::; (1To)V1' exp ( 2kpl ). (5 1) P
Thus y can be regarded as an adjustable parameter which gives a semiempirical account of multiple r efl ections as well as scattering losses. The sum for an infmite number of layers is
R
"
,

To
+
2
To
exp [y In (1To) 2kl] 1
(52)
1exp [y n (1To)2kl]
It will b e observed from figure 4 that y should be se t equal to 4 for the case of no multiple refiections. In such a case only one h alf of the incident light is reflected , however. Johnson h as suggested that y can
be estimated from eq (55) by setting R oo = 1 for
k = O. This gives
(l To)V l 1+1 . .J. 1 2 1 3
Y
241ol·6To+STo+... ·
(56)
from which y = 2 is seen to be a satisfactory approximation for refr active indices smaller than 1.5. A smaller value of y is required for larger refractive indices. Companion and Winslow [43] have used a model similar to Johnson's, bu t which includes all multiple reflections. The summation was carried ou t by compu ter and no explicit expression for the reflectance was given by these workers.
Johnson [441 also suggested that To be equated to 1.5 times the normal Fresnel reflectance. This is
meant to account for the random distribution of particle surfaces and corresponds with an average incidence angle of approximately 30°. It was shown [44] that eq (55) yields absorption coefficien ts for KCl :Tl, KBr :TI, and didymium glass which agree satisfactorily with those obtained by tr ansmission measurements of the same materials.
which is equivalent to eq (43) with
T=To[l+ (l  To)V exp (2k l )]
(53)
t =(1To)v/2 exp (k l ).
(54)
The denominator of equation (52) can be expanded to !rive
R ", To [ 2(1kTo)V exp (2kl )+ 1] .
(55)
2 l  y l n ( 1  To)
This result could also be obtained directly by in tegration of eq (5 1), which suggests that it may in fact be a more realistic representation of a real sample whose particles actually have a range of diameters.
. . . " ... ,'. . ..'.., .. .
~
I
T
C. The Antonov Romanov s ky Model
AntonovRomanovsky [22] has developed express~ons which. can be used to calculate the true absorptIOn coefficlent from reflectance measurements by connecting the KubelkaMunk and B od6 th eories . AntonovRomanovsky treats two limiting cases of r egul arly shaped sample particles, spher es and parallelepipeds (see fig. 5) .
For spherical particles the radiation impinges on the surface from within at th e same angle that i t entered the particle, since the angles which the cord of a circle makes with respect to th e surface normal must be the same. Therefore, total in ternal r eflection
is impossible, and l will approximate the p article
diameter, l , actually being somewhat smaller. In the case of t he parallelepiped, some total
internal reflection is possible, but most radiation probably exits through an opposite face without
further reflection. In this case also it is obvious that l
. ' . .. . . ', . . ' .. ' .
. ", ... .; ", . , ' "
FIGURE 4. Model for Johnson analysis of refl ectance from a scattering medium [44].
FIGURE 5. Th e AntonovRomanovsk y model for regular spheres and parallelepi peds [22].
575
is approximately of the same magnitude as l, but
somewhat larger.
.
Thus for regularly ~haped particles it is assumed
that l=l and
k=~ln (lro)22ro(l2ro)F(R,,).
(57)
2l
(lro)22roF(Roo)
< This is an approximate form of the Bodo model which
is valid for media in which ro2< l. For irregularly shaped particles it is assumed that
the emerging radiation meets the surface of the particle with an equal probability for all angles. This will be the case if the mean number of reflections in
the layer m, is large:
This condition requires that
where l, is defined as the "free" path length. The effective path length is then
(58)
and it is assumed by AntonovRomanovsky that
(59)
The assumption that the emerging radiation is independent of angle allows us to divide it equally between the two sides of the layer whose thickness is
l, the mean particle diameter. We write
rro=t
(60)
and from the law of conservation of energy,
(lro)(1exp( sk))=Irt.
(61)
Equations (60) and (61) can be solved to give
2ro+(1ro) exp (mkl/2)
r
2
(62)
(lro) exp (Titkl/2) (63)
2
Using these in eq (43) gives
k=2 ln (lro)2(1 ro)F(Roo).
(64)
Titl
(lro)22roF(R..,)
tive derivation of .Karvaly [47], which we follow ~ere (see .fig. 6). A smg;le .par~icle of the surface layer
IS sho;yn m th.e figure; It IS shIfted laterally with each reflectIOn t~ Illustrate the path of the radiation.
Of the dIffuse radiation which is incident on a
given where
surface particle, the subscript e
itsheusferadcttioonin2duicr~ties
rreefflleecctteiod~
of externally incident radiation. Here u is defined as the ~adiation ~merging per unit solid angle from a
partIcle. 47rU IS the solid angle which would be
observed from the second layer if a particle were
removed (see fig. 7).
O~ the radiant flux (l2ure) which enters the
partIcle, the part (12ur e)ut contributes to the re
flectance from the first internal reflection and the par:t Ao= (l2ur ~)(Iu)t strikes the underlyi~g layers,
whIch are conSIdered to form an infinitely thick
powder mass of reflectance Roo. The transmission
of an individual particle is represented by the
symbol t. The part AoRoo is reflected back into the layer,_ ~here AoRoo (11' e). enters the particle and
AoRoore IS reflecte:i back mto the underlying layers.
,9f that pa.rt whIch entered the particle, AoRoo (1
+ :'e)ut con.tnbutes to the reflectance and AoRoo(l
r ej (lu)t IS reflected downward again. This combines
wIth th':.t externally reflected to give AoR",re AoR", (l re)( l u)t AoR", [r e+ (lu)t]=AoR", Q as the
flux reflected downward of which AoR",2Q returns
to the layer, etc.
Ao AoR.
A oR . Q
FIGURE 6. The Melamed Model for powder reflectance as viewed by Karvaly [47].
'\ '\
,'\ '\
D. The Melamed Model
. In contrast with the st~tistical theories previously dIscussed, Melamed earned out a summation over the reflectance and transmittance of individual particles rather than layers [46] . Some features of thIs model are better understood using an alterna
FIGURE 7. Illustration of the meaning of the radiant intensity factor u [47].
576
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