F hilips Res. Rep. 2, 55-67, 1947
R 35
RADIATION AND HEAT CONDUCTION IN
LIGHT-SCATTERING MATERIAL
by H. C. HAMAKER
536.24: 536.33
I. REFLECTION AND TRANSMISSION
Summary
On the basis of a set rif simultaneous differential equations originally
due to S ch u s ter the transmission and reflection of light in lightscattering layers is discussed. Formulae previously developed by
Kub elk a and M u n k are briefly recapitulated; they are extended
so as to describe the luminescence of fluorescent screens excited by
X-rays or electron bombardment. Likewise formulae are derived
that include temperature radiation.
1. Introduction
In 1905 Schuster
1) developed' a set of simultaneous differential
equations to describe the transmission of light through fog. Many years
later the same set of equations was applied hy Dreosti 2) in dealing with
transmission and reflection phenomena in opal' glass, and. by Kub elk a
and Munk 3) in discussing the coating properties of layers of paint.
The original purpose of the present investigations was to extend this
theory so as to include temperature radiation and the combined effect of
radiation and heat conduction. These more complicated problems, however,
will be reserved for a future paper. In the course ofthis work we came to the
conclusion that the equations of Kub elka and Munk could, with advantage, be written in a simpler form and that they could be 'extended
so as to apply to the radiation eurltted by a fluorescent screen irradiated
by X-rays *). These problems will form the main topic discussed below.
2: ,Basic' equations
Only a one-dimensional case is considered
flux is plit up into two parts, viz:
(.fig. 1) and the total radiant
J = the flux in the direction of the positive x axis and
= the flux in the direction of the negative x axis.
J
On passing through the infinitesimal layer dx a fraction a.I.dx of the
*) My attention was drawn to this point by Dr Klasens
of this laboratory. Practical applications of various equations developed below will be found elsewhere in
this issue.
.
56
H. C. HAMAKER
flux I will be absorbed and a fraction s.I.dx will be lost by scattering in a
.hackward direction. On the other hand a quantity s.J.dx will be added by
scattering from-the flux J, so that we obtain
dl
~ =-
(a
+ s) I + s.l ,
(la)
and by the same arguments *)
dJ
. -=
dx
(a
+ s) J -
sI .
(lb)
These are S ch uster's equations. Scattering sideways has been disregarded, it being supposed that any loss of radiant energy in a sideward
direction is compensated by an equal contribution from the neighbouring
parts of the layer; this implies that the dimensions of the area investigated
must be large compared with the thickness of the layer, a condition nearly
always fu]filled in practical cases. It will also be evident that observations
to which equations (la) and (lb) are applied must be made under completely diffuse illumination,
dl(
_X
48ZTI
Fig. 1. The radiant flux is split up intó two parts: -I in the direction of positive x and
the direction of negative x.
Jin
This last supposition is perhaps not' quite correct. It has been observed by
Spijkerboer 4) and Dreosti 2) that the inténsity distribution over various angles after
transmission through a light-scattering layer is independent of the intensity distribution
of the incident light and of the thickness of the layer, provided this thickness exceeds a
certain limit. The intensity distribution of the transmitted light for a thick layer may be
designated as the specific distribution of the material in question; it will mainly depend on
the ratio of absorption to scattering (aJs). The author inclines to the opinion that when the
.incident light happens to possess the specific intensity distribution then equations (1ar
and (lh) will strictly hold. If the distribution of the incident beam is different, it will be
*) It should be noted that equation (lb) can be deduced from (la) by interchanging I
and J and changing the sign of dx. This is a principle which also applies in more complicated cases (see for instance equations (28) and (32».
RADIATION
ÀND HEAT
CONDUCTION
IN LIGHT-SCATTERING
MATERIAL
I
57
gradually transformed into the specific distribution as the light penetrates the layer, so
that from a certain thickness onwards equations (la) and (lh) can he applied. For very
thin layers these equations may hreak down owing to the changes taking place in the
intensity distribution.
Whether we must preferably carry out our observations with completely diffuse light
or not depends on whether the specific distribution is near to the completely diffuse distribution or not. For instance, in the case of pure absorbtien the specific distrlbution is
evidently a parallel bundle incident at right angles, and equations (la) and (lh) apply
if s is put equal to zero.
To decide whcther these views are correct will require a theoretical investigation on the
basis of such integral equations as developed by Sch warzschild 5), King 6) and
Sp ijker b 0 er 4). This would appear to the author a profitable way ofattacking theproblem.
,
In his original paper Sc h u ster assumed isotropic scattering, the
fraction s being exactly one half of the total amount scattered. As pointed
out by Dreosti, this is an unnecessary restriction; whether scattering is
symmetric with respect to the y-z plane or not is immaterial, provided s
measures the fraction scattered backwards.
.
.'
To conclude this section it may he mentioned that according to a theoretical investigation by Schwarzschild
6) equations (la) and (lb) furnish a satisfactory approximation to actual conditions. This is corroborated
by Judd's7)
experimental verification ofthe equations of Kubelka
and
Munk:
Consequentlyequations
(la) and (lb) may be adopted as a satisfactory
basis for the discuesion of such phenomena as we have in view.
3•. The general solution
Equations
(la) and (lb) are solved by putting
I
= Cl . e"
+C
2 •
e-Gx; J
only two of the four constants Cl ....
solution reduced to its simplest form is
= Ca . e"
+C
4 •
C4 being arbitrary.
+ B(l + (Jo) e-a,x,
J = A(l + (Jo) ea,x + B(l- (Jo) e-a,x ,
1= A(l-
(Jo) ea,,;
,----+ 2s) ,
where
0'0
= 1 a(a
{Jo
=
e-a,; ,
(2)
The complete
(3a)
(3h)
(4)
/ a
1, (a + 2s) = a + 2s ,
0'0
(5)
both roots being taken with the positive sign *). In these equations A and B
are two constants determined by the boundary conditions .
• ) This is of course a pure convention. It is only essential that both roots should have the
same sign:
The suffix 0 added to a and {J might have been omitted in the present case, but has
been added in view of extensions which will be developed in a later paper.
58
x
H.C.HAMAKER
For instance in the simplest case, a layer infinitely thick stretching from
to x =
00 with an incident beam of intensity 10, we have
=
°
+
I
I
=
10 for x
= J=
°
=
°
ann
+
for x =
(6)
00
from which we- obtain
A =
°
and B
10
= --I
+ Po
(7)
The intensity of the reflected beam is then
J(O)
=
B(1 -
Po)
'Po
+ Po
1-
= 10 --_
1
and the reflectivity is repesented by
J(O)
Reo = -10
=
1 - {Jo
1
+ {Jo
,
(8)
an equation which may be of service in estimating the value ofthe constant
{Jo from observations or estimates of the reflectivity of an infinitely thick
layer.
From (3) and (7) we find
so that 0"0 measures the rate of decay of the incident beam as we penetrate
into the interior of the layer; I is reduced to about 1 % of its original value
when 0"0 x = 4'5. Consequently if we observe that certain specimens of '.
porcelain are visibly transparent when 0'2 cm thick we may conclude
that 0"0 (for visible light at least) must be ofthe order of 4'5/0'2 = 22·5 cm-I;
likewise for a paint which in a layer of 0'01 cm completely covers the underlayer, 0"0 must be of the order of 4'5/,001 = 450 cm-I. In such a way 0"0 may
be estimated, when, as in most cases, no data are available from which it
can be calculated with accuracy.
When considering a layer of finite thickness D applied to an underlayer
ofreflectivity r (fits. 2) the boundary conditions wiJl be:
I = 10 when x = 0,
J= r.I whenx
=
+ D.
(9)
It has been found, however, that practically
all our formulae are considerably simplified by introducing instead of r the related quantity
l-r
(1=
l+r
(10)
RADIATION
AND HEAT
CONDUCTION
IN LIGHT-SCATTERING
MATERIAL
I
59
Then
T
l-e
l+e
(11)
= --,
and the boundary conditions (9) become
I = 10 when x = 0,
e) I when x =
+ e) J = (1-
(1
(12)
+ D.
Whenever similar situations arise, the boundary conditions will consistently
be used in this form.
After inserting equations (3) in (12) and solving for A and B we 'obtain
- 10. (e - (Jo) e-aoD
A = -(I-+----=-(J'o...,...)---:-(-e+-P-Co) eaDD-_ (i':___(J-o)-(-e-(J-o-)
10· (e
B _
-
(1
+ Po)
(e + (Jo)
ë
and consequently for the reflectivity
R
= J(O) = A(1
+ (Jo)
+ (Jo)
aD
• -
ea,D
(1 - (Jo) (e -
•
(Jo) e-aoD
(13a)
(13b)
of the layer
+ B(I-
(Jo) =
10
10
.
(1 - (Jo) (e + Po) eUoD - (1 + (Jo) (e - (Jo)
= (1
(Jo) (e + (Jo) eUoD - (1 - (Jo) (e - (Jo)
+
e-~-DD
e-aDD
(14)
e-aDD •
This is Kubelka
and Munk's equation, which has been experimenta11y
verified by J udd and colloborators.
When D is large equation (14) reduces to (8), as we sh~uld expect. The
same simplification is introduced if the underlayer possesses the same
reflectivity as an infinite layer of the light-scattering material. This is
Fig. 2. A light-scattering layer (thickness D) applied to a reflecting underlayer (reflectivity r).
60
H. C. HA1tL\KER
mathematically realized by putting (J = {Jo (compare equations (8) and
(17», which again reduces (14) to (8).
In order to simplify our equations as much as possible the constants a
and s figuring in the differential equations (I) have above been replaced by
Go and {Jo. K uh elk a and Munk used Ra:> and s and these two were
computed by Ju d d from his ohservations. The three sets (a, s), ({Jo, Go)'
or (Ra:>' s) are completely equivalent and the mutual relations between
them are covered by equations (4), (5) and (8). In addition we note that
the following relations, which are easily derived from the equations just
mentioned, may sometimes he 'useful in calculating one constant from
another:
Go =
. {Jo
=
t
S
1
(Ra:> -
,)
Ra:> '_
I-Ra:>
1 Ra:> '
+
(IS)
(16)
e;
s
2
~= (1 - Ra:> )2
•
(17)
By these relations and (11) equation (14) can readily be transformed into
the form given by Kubelka
and Mun k.
A still more general case occurs when the light-scattering layer is enclosed between two transparent plates as indicated in fig. 3, for instance
when we investigate a powder compressed between two glass plates. Let
rl he the reflectivity and tI the transmissivity ofthe front plate (on the side
ofnegative x), the same quantities for the back being r2 and t2• Then ofthe
incident intensity ID the fraction tI' ID will enter the light-scattering material and the boundary conditions will he
a
1
x.D
-x
"'Ill
Fig. 3. A light-scattering layer (thickness D) between two transparent plates (1 and 2)
10 is the radiation incident from the left; J, is the amount reflected and It the amount
transmitted,
RADIATION
AND HEAT CONDUCTION
IN LIGHT
SCATTERING
MATERIAL
I
61
when x = 0, and
(18b)
when x = D, (2t and
e2
being defined by equation
(10).
Inserting (3) in (18);A and B may be computed, but it is not essential
to give their expressions in detail. From them we obtain
which expresses the radiant energy incident from the right on the front
plate 1. The total amount reflected will be
When there is no plate in front we may put Tl = 0, el = 1, ti = 1 and the
expressions (19) or (20) simplify to (14), as we would expect.
In the same manner the intensity of the transmitted light can be computed, with the result:
-
an expression symmetric with respect to the suffixes 1 and 2, as it ought
to be .
. 4 Radiation excited by X-rays
In X-ray photography a common arrangement is that ofthe intensifying
back screen; the action of the X-rays on the photographic emulsion is
strongly intensified by the light excited in a :fluorescent screen pressed
against the film at the back (see fig. 4).
Now let No be the intensity of the X-ray beam at x = 0, that is after
passing the photo-sensitive layer. On passing through the :fluorescent
material the X-ray intensity will decrease as
N= No.e-p",
where f-l is the total coefficient of X-ray absorption. The amount of Röntgen
radiation absorbed between the plane x = X and x = X
dX is
+
-dN
= f1 •
No . e-Px dX
and an equivalent amount of :fluorescent light will be excited in this' infinitesimallayer,
viz.
62
H. C.HAMAKER
di = k • fl . No . e-Px dX
(21)
the constant k measuring the efficiency of the fluorescence.
Transmission of thisvisible light through the fluorescent layer will take
place according to equations (la, b), and the resulting intensity in the plane
x = 0, that is the light incident on the photographic plate, will now he
computed.
Fig: 4. A layer of fluorescent material (thickness D) applied to a piece of cardboard, used
. as an intensifying back screen in Xvray photography.
Let u~ suppose an amount of light di to be created in the layer dx at
other half to the left.
To find the resulting distribution of light intensities the layer must he
. divided into two parts, viz: 0 < x <'X and X < x < D. Assuming
x = X, half of which is emitted to the right, the
+ Bl(1 + (Jo)
= Al(1 + (JO) ea,,, + BI (1 - (Jo)
(Jo) ea•x
1= 11 = Al(lJ
when 0
= Jl
< x < X,
whenX
=
(22a)
e-a",
(22b)
and
+ B (1 + (Jo) e-a,x
+ Po) ea,x + B (1 - (Ju) e-a",
I = 12 = A2(1- (Jo) ca",
J
e-a,x
J2 = A2(1
2
(22c)
2
(22d)
< X < D,
the boundary conditions will he
+
11= Tl . Jl or (1
el) 11J1 = J2 t di, 12 = 11
J2 = T2 • 12 or (1
(2) J2 -
+
+
Tl' el referring to the film and
+
(1 - el) Jl = 0
t di
(1 - t.?2) 12 = 0
T2, e2
to the cardboard.
forx = 0
forx= X
forx = D
!
(23)
RADIATION
AND HEAT
CONDUCTION IN LIGHT-SCATTERING
MATERIAL
63
I
By combining (22) and (23) Al' B1, A2, and B2 may be calculated, and with
these we subsequently obtain
J1(0} = A1(1
1 .
=
1fdL
(1
+ {Jo} + B
+ el)
{Jo} =
1(1(e2
{JO} eu.(D-X)
(el
+
+ {Jo}
(e2
+ {Jo} eU'
(e2 - {JO} e-u.(D-Xj
D-
-
--
---
-
(el - {Jo} (e2 -- {Jo} e-G·
D
(24)
and similarly
12(D) = A2(l - {Jo} eu•D
+ B2(1 + {Jo} e-G,D
{J
}
-'<::1
u
e •
=
-=-
+ po __eU'x
= -}di (1 + e )
2 (el + {Jo} (e2 + {Jo} D (n
(
{J } e-u•x
(11 -
0 -
(el - {Jo} (e2 -
•
flo)
(25)
e-G·D
for the radiation incident from the right on the photographic plate and
from the left on the cardboard underlayer respectively. The latter quantity
is not of essential interest in our present problem hut has been computed
for the sake of completeness.
.
To find the resultant intensities under X-ray irradiation we have to
replace di in (24) and ,(25) by the expression (21) and to integrate from
X = P to X = D, with the result:
According to circumstances these formulae
For instance when D is large the terms e-u,D
pect to the term eu•D • We shall not consider
for practical applications we may refer to
An alternative method to treat the same
differential equations
dl
<Ix = - (a
dj
- = (a
dx
can sometimes be simplified.
can be disregarded with resthese questions in detail here;
a paper by Dr Kl.a sen s.
problem is to start from the
.
+ s) I + sj + .~.
C e-
+ s) J -
PX
(28)
sI -
i
C e-PX
which are deduced from equations (I) by adding the amount of radiation
excited by the X-rays, half ofwhich contributes to I and the other halfto J.
64
H. C. HAMAKER
By common principles the general solution of (28) is found to he
(92)
A and B must now be determined by the boundary conditions:
+ lh) J (1 + (12) J -
(1
(1 - (h) J
(1 -
(12)
=
0
I= 0
whenx = 0,
whenx=
D.
(30)
That the expressions (26) and (27) result, may readily he verified.
It should be mentioned that equations (28) are practically identical with
the equations used by D reo s ti 2) in discussing the scattering of light
when the layer is illuminated by parallel rays of visible light. Owing to
scattering and absorption the incident beam will decrease in intensity
as C . e-kx from left to right. When a parallel beam of intensity H passes
the layer dx a fraction p.H.dx will he scattered in the forward direction
(positive x) and a fraction q.lI.dx ill the backward direction (negative x).
If to this scattered light we apply the principles adopted throughout this
paper we must have
dl
+ s) I + sJ + qCe.
-- = -
(a
dJ
+ s) J -
dx
dx
kx
,
(31)
= (a
sI - pCe-
kx
,
the only difference from (28) being that the coefficients pand q are unequal. The solution of (31) ~ ill describe the amount of diffuse light emanating
from both sides of the layer. According to Dreosti's
observations this
theory is in good accord with experiments.
5. Temperosure radiatiun
As a last problem we will consider a light-scattering layer heated to a
uniform temperature and investigate the resulting radiation.
.
From the total radiant flux I in the direction of positive x a fraction
a.I.dx is absorbed in the layer dx (fig. I). Consequently to conform to
Kirchhoff's
law this layer dx will contribute by radiation the amount
a.Eo.dx in the direction of negative x, where Eo designates the black-body
radiation at the temperature and wavelength in question. The same
amount will 'of course be radiated in the opposite direction. By adding
these quantities to equations (1) we are led to
RADIATION
AND HEAT CONDUCTION
dl
- - = - (a
dx 'dj
- = (a
dx
IN LIGHT·SCATTERING
MATERIAL
I
65
+ s) I + sj + aEo
+ s)
(32)
J - - sI - aEo
which are solved by
+ B(1 + Po} e-a,,, + Eo
J = A(1 + Po} ea?"+ 8(1- Po} e-a,,, + Eo ..
1=
A(1 - Po) ea,,,
(33)
If the light-scattering material is applied as a coating on an underlayer of
reflectivity r the boundary conditions will read:
1=0
J
=
r.I
+ (1-·r)
+ e) J -
Eo or (1
forx
=
0
(1- e) 1 = 2eEo for x
=
IJ,
(34)
the term (1- r) Eo representing the radiation contributed by the underlayer according to Kirchhoff's
rule. After computing A and B we find
for the temperature radiation
where R is the reflectivity expressed by formula (14). This is again in
keeping with Kirchhoff's
law.
Other cases may be worked out in the same manner.
6. The constants
a, s, etc.
To conclude this paper we shall briefly recapitulate the data available
concerning the magnitude of the constants a and s and the derived quantities ao and Reo. The most extensive observations were made by Judd
and collaborators 7) who checked equation (14) for various ,materials.
They computed Reo and s, from which, however, a and a can easily be
calculated with the aid of equations (15) and (17). A few characteristic
data have. been collected in table 1. The dental silicate cements, having the
lowest value of ao, are most transparent; next come the vitreous enamels,
whereas the cold-water paints are very opaque; a coating of these of 0'01
to 0'02 cm will suffice to mask the colour of the underlayer almost completely.
.
Some further data have been deduced by Dreosti
for opal glasses
and mast~ emulsions. His results are reproduced in table Il. The opal
glass has optical properties of the same order as the dental silicate cements
of J udd, though they are somewhat more opaque. The mastix emulsions
66
H. C. HAUAKER
TABLE I
R""
a and s according to observations by Judd
s
R",
cm-1
and collaborators
s
a
cm-1
a
I. Vitreous enamels
100
90
64
56
0·89
0·90
0·89
0·82
u.
I
0·68
0·50
0·435
0·875
I
147
180
147
64
1I·7
9·5
7·5
9·9
Cold-water paints (white)
0·88
0·86
0·93
900
720
650
122
88
380
lIS
7·35
8·20
1·72
109
47·5
UI. Dental silicate cements
6·2
3·6
3·8
2·1
1·7
0·76
0·50
0·40
0·59
0·16
White
White
Light yellow
Light yellowgrey
Dark grey
26·4
4·0
2·2
7"0
0·45
1·72
2·70
4·00
1·16
5·15
0·235
0·90
1·71
0·30
3·74
-
TABLE II
R""
a and s observed by Dreosti,
s
cm-1
R",
À = 5370
a
cm-1
A
• Go
s
cm-1
a
1·69
5·33
3·57
4·95
43
24
29·5
5·7
1. Opal glasses·, various samples
7·8
18·3
13·6
8·0
0·807
0·750
0·771
0·56
u.
Go
G1
G2
Ko
Kl
K2
I
I
0·39
0·091
0·020
0·338
0·176
0·030
I
I
I
0·18
0·76
0·46
1·49
Mastix emulsions
0·31
0·22
0·06
0·147
0·915
1·12
0·336
1·11
2·21
0·66
0·60
0·20
0·42
1·15
3·00
0·85
1·64
3·19
.
2·1
0·24
0·03
1·6
0·52
0·06
Emulsions K and G both contained 5 g mastix per litre. The average particle diameter
was 0·67 [J. for G and 0·49 [J. for K. In the sequence Ko, Kl' K2 acid fuchsine has been added
to the intermicellar fluid in increasing quantities. The exact concentrations have not been
stated, and the description of the emulsions is, on the whole, rather incomplete.
RADIATION
AND HEAT CONDUCTION
IN LIGHT SCATTERING
MATERIAL
I
67
are much more transparent. Unfortunately the description of these emulsions is incomplete, but it is of interest to note that adding a dye to the
Huid not only increases the absorption a but at the same time diminishes
the coefficient of scattering s. This clearly shows how closely these two
constants are connected;' they are not independent of one another.
Eindhoven, October 1942.
1)
2)
3)
4)
5)
6)
7)
A.
G.
P.
J.
K.
L.
D.
REFERENCES
Schuster, Astrophys. J. 21, 1, 1905.
M. Dr eos t i, Thesis, Utrecht 1930.
Kubelka & F. Munk, Z. tech. Phys. 12, 593, 1931.
Spijkerboer,
Thesis, Utrecht 1917.
Schwarzschild,
Berl. Ber. 1914, p. lI83.
V. King, Phil. Trans. Roy. Soc. London A 212, 375, 1912.
B. Judd & collaborators, Bur. Stand. J. Res. 19, 317, 1937.
ABSTRACTS'
(Continued
1696:
from page 54)
J. F. H. Custers
and J. C. Riemersma,
The texture of straightrolled and of cross-rolled molybdenum, Physica 's-Grav.12, 195-208,
1946.
.
With the aid of pole figures the textures of straight-rolled and of crossrolled molybdenum are determined. The pole figures thus obtained show
that earlier authors described these textures in a too simple way; they are
at least twofold.
For example, after cross-rolling, there is found besides the so-called (100)
[HO] tecture ((~OO)parallel to the rolling plane, and [110] parallel to the
rolling direction) a second texture, which is rotary symmetrical around the
normal to the rolling plane, and which has a (111) plane parallel to this plane.
The texture of straight-rolled molybdenum turns out to be in good
agreement with the texture of straight-rolled iron, as determined by
K urdjumow
and Sachs; the texture of cross-rolled iron is not known.
1697: N. G. de Bruijn,
A combinatorial problem, Proc. Kon. Ned.
Akad. Wetenschappen Amsterdam 49, 758-764, 1946.
A Pn-cycle is defined as an ordered cycle of 2n digits 0 or 1 (i.e. a series
of such digits placed on the circumference of a circle), such that the 2n
possible sets of n consecutiv~ digits of that cycle are all different '(as a
consequence, any ordered set of n digits 0 or 1 occurs exactly once in that
cycle). Posthumus
studied these cycles in connection with a practical
problem of telecommunication, and was led to the conjecture, that the
number of Pn-cycles be equal to 2N where N = 2n-I_n. This conjecture
is proved to be correct, as follows from a theorem concerning a special
type of networks. Another application of this theorem is mentioned too.
(Continued on page 79)