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Scattering and Absorption of Luminescent Light in Polycrystalline Materials
To determine the technical yield of luminescence from the known absolute yield, it is necessary to know what fraction of the luminescence excited within the substance reaches the surface of the layer and is emitted outward, since part of the luminescent light is absorbed in the lumi—
of the substance; and part of it is scattered on individual crystals of the substance.
Conversely, if, from the known technical yield, it is desired to determine the absolute yield of the luminescent substance, then it is necessary to determine both the absorbed part of the exciting light and the intensity of all the luminescence produced; the latter can be calculated from the technical yield only with knowledge of the scattering and absorption properties of the given layer of luminescent substance.
The author of the paper under review¹, together with other investigators, has previously studied the dependence of the intensity of luminescence emitted by a layer of luminescent substance on the thickness of this layer when excited by α-, β-, γ- and X-rays²˒³˒⁴. The experimental curves obtained varied both in passing from one kind of excitation to another for one and the same substance, and in passing from one substance to another under identical excitation. Working with thin layers, for which scattering and absorption are negligibly small, the indicated authors measured the absolute yield of the substances studied, particular difficulties being caused by the determination of the excitation energy absorbed in a thin layer under excitation by β- and X-rays, which considerably reduced the accuracy of the measurements. The best result can be achieved if one theoretically finds the dependence of the intensity of the emitted luminescence on the thickness of the layer of luminescent substance. In this case the intensity of all the excited luminescence for an arbitrary layer thickness can be calculated, while the measurement of the absorbed exciting energy is carried out more easily and more accurately for thicker layers of the luminescent substance. If, moreover, for a definite layer of the substance the absorption capacity with respect to the exciting radiation, the absolute yield, and the constants of scattering and absorption for the light of luminescence are known, then the technical yield for a layer of any thickness can be calculated.
The author of the paper under review derived, on the basis of a simple theoretical consideration, differential equations relating the absorption capacity of the luminescent substance, the absolute yield, the constants of absorption and scattering of the light of luminescence in the layer, and the thickness of this layer.
Integral coefficients of absorption \(A(x)\), reflection \(R(x)\), and transmission \(T(x)\) are introduced; that is, if radiation of intensity \(J_0\) is incident on the layer under study, then \(A(x)J_0\) is absorbed by the layer, \(R(x)J_0\) is scattered, and \(T(x)J_0\) passes through the layer, so that
\[ A(x) + R(x) + T(x) = 1. \]
Correspondingly, differential coefficients of absorption \(a\), reflection (scattering) \(r\), and transmission \(t\) are introduced, such that \(A(x)\) goes over into \(a\,dx\), \(R(x)\) goes over into \(r\,dx\), and \(T(x)\) goes over into \(t\,dx\) in passing from a layer of thickness \(x\) to a layer of thickness \(dx\).
In addition, a differential loss coefficient \(v\) is introduced; here \(v\,dx\) characterizes that part of the radiation incident on a layer of thickness \(dx\) which is lost as a result of reflection and absorption, so that
\[ v = a + r. \]
To characterize the optical properties of a layer of thickness \(dx\), it is sufficient to use any two coefficients, which may be chosen for convenience.
Considering a layer of thickness \(x\), the author obtains the following expressions:
\[ \frac{dJ}{dx}=E(x)[1+R(x)]-J[v-rR(x)], \tag{1} \]
\[ \frac{di}{dx}=E(x)T(x)+JrT(x), \tag{2} \]
where \(i\) is the intensity of the luminescence emitted by the layer from the side of excitation, \(J\) is the intensity of the luminescence emitted by the layer from the side opposite to the excitation side, and \(2E(x)\,dx\) is the intensity of the luminescence arising in a layer of thickness \(dx\). If we take into account that
\[ 2E(x)\,dx=\eta S(x)\,dx, \]
where \(\eta\) is the absolute yield of the luminescent substance and \(S(x)\) is the absorptive capacity of the luminescent substance with respect to the exciting radiation, then it is clear that equations (1) and (2) precisely relate the absolute yield, the absorptive capacity with respect to the exciting radiation, the scattering constant \(r\), and the absorption \(\alpha=-v-r\) of the luminescence light to the layer thickness \(x\); moreover, as calculation shows, the quantities \(R(x)\) and \(T(x)\) can be expressed in terms of \(v\) and \(r\) as follows:
\[ R(x)= \frac{-\dfrac{r}{p}\sin px} {\cos px+\dfrac{v}{p}\sin px}, \tag{3} \]
\[ T(x)= \frac{1} {\cos px+\dfrac{v}{p}\sin px}, \tag{4} \]
where
\[ p=\sqrt{v^2-r^2}. \]
In principle, the optical constants \(v\) and \(r\) can be determined from the indicated formulas by measuring the intensity of the luminescence emitted from both surfaces of the layer under investigation as a function of the thickness of this layer. In doing so, however, difficulties arise that can be circumvented if only a thin surface layer of the luminescent substance is excited to luminescence, which is especially convenient to accomplish with rays of low penetrating power.
The intensity of the luminescence emitted from the side of excitation (points in Fig. 1) and from the side opposite to the excitation side (crosses in Fig. 1) was measured with a photoelectron multiplier as a function of the layer thickness. It was assumed that the curve drops almost to its maximum as soon as the \(\alpha\)-rays pass only through the surface layer, and that scattering in this layer may be neglected. In this case the decay of the curve should be described by the function \(T(x)\). Formula (4) best (solid curve I in Fig. 1) described this course of the curve for ZnS-Cu phosphor at the values
\[ \left\{ \begin{aligned} v&=2.84\,p\\ r&=2.66\,p \end{aligned} \right\} \quad p=1.65\cdot 10^{-2}\ \frac{\mathrm{cm}^2}{\mathrm{mg}}. \]
(this means that a layer of \(1 \dfrac{mg}{cm^2}\) reflects \(r=4.68\%\) and absorbs \(v-r=0.3\%\) of the incident excitation energy).
Under the same conditions, the curve calculated by formula (3) must represent the arrangement of the experimental points when measuring the intensity of the luminescence emitted from the excitation side, which
Fig. 1. Luminescence intensity as a function of the layer thickness \(x\) under excitation by \(\alpha\)-particles.
is justified with a good degree of accuracy (solid curve II in Fig. 1).
Equation (1) is solved with respect to \(J\):
\[ J=\left\{\int_0^x E(x)\frac{(1+R)}{T}\,dx\right\}T. \tag{5} \]
If \(R(x)\) and \(T(x)\) are known, this expression is easily integrated for certain functions \(E(x)\). The case is especially simple when \(E(x)=E=\mathrm{const}\); then
\[ J=G(x)=\frac{E}{v-r}\{1-(R+T)\}. \tag{6} \]
This case occurs when the luminescent layer is excited by \(\gamma\)-rays, whose intensity practically did not decrease in passing through the layer, so that the whole substance is excited uniformly. The intensity of the luminescence emitted from the excitation side is here equal to the intensity of the luminescence emitted from the opposite side, which mathematically is expressed in the fact that, equating the right-hand sides of equations (1) and (2), one obtains expression (6).
The results of measurements under excitation by \(\gamma\)-rays are shown in Fig. 2. Only the intensity of the luminescence emitted in the direction opposite to the excitation was measured. The tangent at \(x=0\) serves as a measure of the value \(E\) and can be used for its determination. For a known \(E\), the function \(G(x)\) must, for the values found above,
at values of \(v, h, r\) convey the position of the experimental points, which, as is seen from Fig. 2 (solid curve), is justified with a good approximation. It is seen that for large values of \(x\) the curve asymptotically approaches a certain limiting value, which according to expression (6) is equal to
\[ \lim_{x\to\infty} G(x)=E\,\frac{1}{p}\left\{1+\frac{r}{v+p}\right\}. \tag{7} \]
Knowing \(E\) and carrying out measurements in thick layers of the luminescent substance, one can determine from this equality the ratio of \(v\) to \(r\). Conversely, knowing \(v\) and \(r\) and carrying out measurements in thin layers, one can calculate the maximum technical yield under excitation by \(\gamma\)-rays.
Fig. 2. Luminescence intensity as a function of the layer thickness \(x\) under excitation by \(\gamma\)-rays.
When exciting a luminescent layer by X-rays, it can no longer be assumed that the entire layer is excited uniformly (as under excitation by \(\gamma\)-rays), or that only the surface layer is excited (as under excitation by \(\alpha\)-particles). One might suppose that the absorbed energy of the X-rays falls off exponentially with increasing \(x\), i.e. represent \(E(x)\) in the form
\[ E(x)=Ce^{-\psi x}. \]
In this case equation (5) is also integrated elementarily; however, the exponential law, apparently, is strictly fulfilled only in exceptional cases.
But for X-rays, as in general for any kind of excitation, it is possible, without introducing any additional assumptions, to determine, according to equation (1), the form of \(E(x)\) from the curve measured for \(J\), by graphical integration or differentiation. Indeed, according to (1),
\[ E(x)=J\,\frac{v-rR}{1+R}+\frac{dJ}{dx}\,\frac{1}{1+R}. \tag{8} \]
or, respectively,
\[ \int_0^x E_i(x)\,dx=(v+r)\int_0^x J\,\frac{1-R}{(1+R)^2}\,dx+\frac{J}{1+R}. \tag{9} \]
All the quantities entering into these expressions can be measured and, consequently, \(E_i(x)\), or respectively
\[ \int_0^x E_i(x)\,dx \]
is determined.
The intensity of the luminescence emitted from the side opposite to the excitation was measured with a photomultiplier, as a function of the layer thickness (curve \(I\), Fig. 3). From the experimental curve \(I\), according to equations (8) and (9), the luminescence intensity per unit layer thickness, \(E(x)\) (curve \(II\), Fig. 3), and the intensity of all the luminescence excited in a layer of thickness \(x\),
\[ \int_0^x E(x)\,dx \]
(curve \(III\), Fig. 3), were found as functions of \(x\).
Fig. 3. Luminescence intensity as a function of the layer thickness \(x\) under excitation by X-rays.
According to curve \(II\), \(E(x)\) is not represented by an exponential law, but by some other law.
In conclusion, the author of the paper under review investigates the dependence of the optical constants on the size of the grains of the luminescent substance; moreover, as should be noted, in deriving the formulas cited above the granular structure of the substance was not taken into account. Curves are compared for the luminescence intensity emitted in the direction opposite to the excitation for two layers of identi...
of the same composition (ZnS–Cu), but with different grain sizes (mean ratio 1:2).
It turned out that the expected decrease in the values of \(v\) and \(r\) on going to coarser powders occurs in such a way that their ratio remains constant.
V. Bredel
References
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- I. Broser u. H. Kallmann, Zeits. Naturforsch. 2a, 440, 642 (1947).
- I. Broser, L. Herforth, H. Kallmann u. U. Martius, Zeits. Naturforsch. 3a, 6 (1948).
- I. Broser, H. Kallmann u. U. Martius, Zeits. Naturforsch. 4a, 204 (1949).