## ABSTRACT

All the existing methods for determining the diameter of red cells by the observation of the diffraction patterns which they produce are concerned with the determination of mean diameter only, and rest upon expressions which relate the position of the various maxima and minima to the diameter of a number of circular disks of uniform size (Pijper, 1919, 1923, 1925; Millar, 1926; Allen and Ponder, 1928). As is well known, the red cells which produce the diffraction patterns are not of uniform diameter, but it is assumed in the use of these methods that the diameter given by the formulae is the mean diameter of the red cell population.

This paper is concerned with the properties of the diffraction patterns which arise from a population of cells of various diameters, and with the relation of the relative intensities of the maxima and minima to the diversity of size in the population.

## 1. METHOD OF CALCULATION

*N*cells of radius

*R*, scattered in a haphazard fashion on the surface of a slide. In the figure, let

*OA*be the axis of symmetry,

*p*a point in a plane normal to this axis, and θ the diffraction angle corresponding to

*p*the illumination at

*p*will be where

*m = (2*π

*R*.sin θ)/λ, and where

*k*is a constant depending on λ, the wave-length of the light employed, but which may be put equal to unity for the purposes of this problem. At the points where

*J*

_{1}(

*m*) = 0 the illumination is zero and we have the various diffraction minima, while at the points where

*J*is at a maximum we have the illuminated diffraction maxima.

_{1}(m)/m*N*cells is heterogeneous with respect to diameter (or radius), many cells being of mean radius, and the numbers falling away on either side of the mean until we have only a very few large cells and a very few small ones ; the number

*N*will now be composed of a series of groups of,

*N*

_{1},

*N*

_{2},…

*N*

_{n}cells of radius R

_{1},

*R*

_{2},…

*R*

_{n}, and these numbers will be normally distributed according to the expression where

*N*

_{0}is the number of cells of mean radius

*R*

_{0}, where

*x*is the difference between the radius of the cells of any group and the mean radius, and where

*a*is the standard deviation of the population. Such a distribution applies to the red cells of all mammals which have been hitherto examined. The illumination at

*p*will now be the sum of the illumination produced by the various groups of cells,

*i*.

*e*. that produced by the group of

*N*

_{1}cells of radius, plus that produced by the group of

*N*

_{2}cells of radius

*R*

_{2},…and so on. The individual illuminations can be calculated from (1), in which the proper value of

*R*and of

*N*, obtained from (2), is inserted. The illumination function for the diffraction patterns will thus differ from the illumination function for the diffraction pattern produced by

*N*cells of the same size, for if a minimum occurs at the point

*p*when the cells are uniform in size, that minimum will be illuminated, in the case of a heterogeneous population, by light coming from cells whose diameter is greater or less than the mean diameter. Zero illumination cannot occur at the minima, accordingly, when the cells are heterogeneous with respect to size, and a little consideration will show that in such a case the maxima will be less illuminated, just as the minima are more illuminated, than in the case where the diameter of the cells is uniform.

The values of *I* at various points corresponding to various diffraction angles are found by carrying out the following calculations in order.

(i) The value of is tabulated from θ

*=*0 to θ = 15°, by intervals of 0.2° for a series of values of*R*ranging from 3.0*μ*, to 5.0*μ*,. Such a range covers the first three maxima and minima on the one hand, and on the other all values for the diameter of dried human erythrocytes which are likely to occur in practice (6.0*μ*, to 10.0*μ*). Values of*J*_{1}(*m*) are to be found in Gray, Mathews, and McRobert’s*Bessel Functions*, from which the value of*I*can be calculated with great ease. In evaluating*m*, the wave-length used throughout was 5461,*i*.*e*. the wave-length of the intense green line of the Hg. arc; all figures in this paper are therefore to be considered with reference to this wave-length only.(ii) In order to obtain

*N*_{n}in (2) I have selected a frequency distribution of cells with a mean of 7.8*μ*(the average diameter of the normal dried human erythrocyte), assigned the value 0.2*μ*to or,*σ*and calculated the number of cells in successive groups of 0.1*μ*. In this particular case such grouping is perhaps rather coarse, for it exceeds σ /4, but in the other calculations for the larger values of σ (see (iv) below) it is as fine as is ordinarily employed, and the results of the computations show that over 98 per cent, of the total light is accounted for. The result of this calculation is to give the number of cells in various groups, the diameter of successive groups differing by 0.1*μ*.(iii) The value of (1) can now be calculated from steps (i) and (ii), and the results supply as many illumination functions, from θ = 0° to θ = 15 °, as there are groups in the frequency distribution. These functions vary both in amplitude, for the illumination supplied by the many cells of mean size exceeds that produced by the smaller number of cells in groups above and below the mean, and alsoun that their maxima and minima do not coincide; the maxima for the smaller cells correspond, of course, to larger values of θ than do those for the larger cells. The total illumination at any point

*p*, corresponding to any diffraction angle θ may now be found by summing the ordinates at*p*of all the individual illumination functions; this process gives a graph as in Fig. 2, in wh ch the illumination functions for the heterogeneous population corresponds very closely with the position of the first minimum and of the first maximum for the homogeneous population, but that (b) the ratio of the greatest illumination in the first maximum to the least illumination in the first minimum is infinity in the latter case, but about 9 in the former.(iv) The calculations described in (ii) and (iii) are repeated for distributions with the same mean (7 · 8

*μ*) but with standard deviation of 0·4*μ*, 0·6*μ*, and 0·8*μ*. Three new series of illumination functions are obtained by continuing the calculation as in (iii), and by summing the components of each series three illumination functions for the three frequency distributions can be found. When these are plotted together with the two functions referred to in (iii), it will be seen that the illumination of the first minimum becomes progressively greater, and that of the first maximum progressively less, as the value of*a*increases.

*μ*:

The values for the ratio are expressed in round numbers only, although the first decimal place is justified by the method of calculation, for the experimental procedure to be described below is capable of measuring the ratio of the intensities to the unit figure only; this, however, is no great disadvantage, for values of σ are rarely reproducible to more than the first place of decimals^{1}. The values for σ = 0·3, 0·5, and 0·7 have been obtained by interpolation.

## 2. EXPERIMENTAL METHOD

If the diffractometer designed by Millar (1926) is used for the study of the intensity of the illumination in various parts of the diffraction patterns, a hitherto unobserved difficulty is encountered. The pinhole of the diffractometer, being a circular aperture, produces a diffraction pattern of its own, and this pattern appears superimposed upon that produced by the red cells. As a result, the illumination immediately surrounding the image of the pinhole is greater than it would be were it due to diffraction from the cells alone, and at the same time the intensity of illumination in the first minimum is considerably increased ; the contrast in brightness between the first minimum and the first maximum is accordingly greatly reduced, and the minimum appears less “black” than it should. If accurate values for the illumination at various distances from the axis of symmetry are to be obtained, it is accordingly necessary to reduce the light diffracted by the pinhole to the smallest possible amount. This may be accomplished by very careful levelling of the various parts of the optical system, so that the beam of monochromatic light falling on the prism is perfectly parallel, by reducing the cross-sectional area of this beam with the iris diaphragm of the collimator, and by shutting off the periphery of the field lens of the telescope with a similar diaphragm. The proper adjustments are by no means easy to make, but result in the pinhole being surrounded by only a small illuminated area in which a very large number of maxima and minima appear as exceedingly narrow concentric rings. The edge of this illuminated area should not extend more than 3° from the central image, and if the telescope is set at a greater angle, the field should appear unilluminated. Since the first minimum of the diffraction pattern produced by the red cells occurs at nearly 5° from the axis of symmetry, these adjustments remove the diffraction pattern of the pinhole so far as the intensity of the illumination in the first maximum and the first minimum of the red cell diffraction pattern is concerned, and the removal of the diffracted light from the pinhole results in a very much greater contrast between the first maximum and the first minimum.

The diffractometer is now set so that the crosslines in the eyepiece intersect on the outer edge of the dark band which represents the first minimum; the first minimum and the first maximum are thus seen in the central part of the field. The telescope is now clamped firmly in position and a camera attached to the eyepiece ; I use a small “Mikam” camera made by Leitz, which has a side telescope through which the field can be observed and which is very convenient in every way. Owing to the low intensity of the diffracted light very rapid plates require to be used ; the best results are to be obtained with a fast emulsion sensitised to green, but Hammer Process plates (700 H and D), or Gaevert Sensima plates (600 H and D), are quite satisfactory, and both have a considerable latitude. These require an exposure of from one to four hours according to the brightness of the illuminant, the aperture of the iris diaphragm on the collimator and the magnification of the telescope. The illuminant cannot be varied to any great extent and it is usually necessary to close the iris diaphragm on the collimator until the aperture of the lens is reduced to about one-half ; reduction of exposure must, accordingly, be effected principally by using a low magnification in the telescope and a small camera extension. The extension of the “Mikam” is about 8 inches; an eyepiece of the lowest magnification obtainable should accordingly be used in the telescope.

The plate is developed at a fixed temperature for a time which requires to be determined beforehand (see below) and is masked off except for two small circular areas about 2 mm. in diameter, one situated over the darkest part of the first minimum and the other over the brightest part of the first maximum. The density of the plate in these two areas is then determined in the usual way by means of a potassium cell, a constant light source, and a galvanometer. In order to convert the densities in the two areas of the plate into values for the relative illumination of the maximum and minimum respectively, it is necessary for the characteristic curve of the plate, obtained by plotting the logarithm of the exposure (or intensity) against the plate density, to be known. This curve shows a more or less extensive linear portion within which the density is proportional to the logarithm of the exposure or intensity, and two curved terminal parts in which this simple relation does not hold, and which corresponds to “over-exposure” and “under-exposure” respectively. The slope of the linear part of the curve is determined by the degree of development and may vary in practice from about 20° to 45 °. The characteristic curve for the particular emulsion used in the experiment must accordingly be determined beforehand, and it is convenient to obtain a number of curves for different degrees of development. This is not by any means a difficult matter, for all that is required is a standard light source, a method for making exposures of various length, care in producing standard development at a fixed temperature and a method for measuring density (potassium cell and galvanometer). From the series of characteristic curves obtained for the emulsion to be used, one is selected and the conditions of development which determine the slope of this curve are rigorously adhered to in the development of the image of the diffraction pattern; it is advisable to select a characteristic curve which rises at a fairly steep angle. From this curve the plate densities in the two small areas in the maximum and minimum respectively can be converted into relative values for illumination. It is of course important that the densities in the two small areas of the plate shall both lie on the linear part of the characteristic curve. This condition may not be fulfilled in the first few trials owing, usually, to under-exposure.

Once the ratio of the illumination in the first maximum to that in the first minimum is obtained, it is necessary to apply a small correction in all cases where the mean size of the cells is other than 7·8*μ*, for the table gives results for cells of this mean size only. Provided that the mean diameter of the sample does not vary greatly (more than ± 1·o*μ*. from 7·8*μ*.), it is sufficient to correct the figure for the ratio by multiplying it by *μ*_{2}/7 · 8, where *μ*_{2} is the mean size of the sample under consideration, for the standard deviation in terms of the mean, rather than the standard deviation itself, is the quantity which regulates the ratio between the illumination of the maximum and the minimum respectively.

In all cases it is extremely important to examine the slide to be used for production of the diffraction pattern in order to be certain that the cells are not greatly distorted and that they are not collected together in clumps or over-lapping groups. The diffraction pattern observed is produced by all the cells upon which the beam of parallel light falls, and entirely erroneous values for the standard deviation will be obtained if many of these are distorted or clumped together. The fact that such distortion occurs and that it influences the illumination in the diffraction pattern constitutes a serious difficulty in connection with the method, for photographic measurements of the cells are based on circular cells only, distorted cells being excluded ; the measurement of scatter accordingly often tends to be a little higher when the diffraction method is used than when cells are measured by the photographic method. The fact that the iris diaphragm of the collimator is closed down, however, makes it easier to obtain the diffraction pattern from an area containing undistorted cells only, for the illuminated area of the slide is smaller; the loss in intensity which is brought about by the closing of the diaphragm is accordingly compensated for by the purity of the spectrum.

## 3. RESULTS

The following table shows the mean size and standard deviation of the dried cells of six samples of normal human blood obtained *(a)* by photography, *(b)* by diffraction.

These, being figures for normal dried human erythrocytes, show very little variation. The following table refers to red cells obtained from three cases of pernicious anaemia in which the mean size of the cells is somewhat increased and the standard deviation much greater than normal.

The increased standard deviation results in a very poor contrast between the first maximum and the first minimum, the former being only about three times as bright as the latter, and the difference in density on the plate being correspondingly reduced. Pijper specially comments on this point, observing that the spectra are always much “paler” *(i*.*e*. show less contrast) in cases of pernicious anaemia than in normal individuals.

The following figures refer to rabbit cells. The first three results were obtained from normal rabbit blood and the remaining three from blood films of rabbits injected with ricin. The effect of this substance is to produce considerable ani-socytosis.

The figures in these tables show that the standard deviation may be found approximately by the diffraction method, and that the results obtained thereby are very similar to those obtained by direct photography and measurement. It is only right, however, to mention in conclusion that the former method requires far greater technical skill than does the latter, and that, although it may be carried out rapidly when all the preliminary calibration is completed, it contains a greater number of possible sources of error. The diffraction method of measuring scatter, for these reasons, is accordingly not suitable, in its present form, for routine laboratory use.

## SUMMARY

A method is described whereby the mean diameter and the degree of scatter of a red cell population can be found from the characteristics of the diffraction pattern produced. The mean diameter may be found from the position of the various maxima and minima, and the scatter from the ratio of the illumination intensities in the first maximum and the first minimum respectively.

## ACKNOWLEDGEMENTS

The expenses of this investigation were in part defrayed by a grant from the Government Grants Committee of the Royal Society.

## REFERENCES

*Journ. Phys*

*Bessel Functions*

*Proc. Roy. Soc*. B

*S. Afr. Med. Journ*

*Afr. Med. Journ*

*Afr. Med. Journ*

^{1}

The reason for this is not that differences in the second place of the value of *a* are not statistically significant, but that such differences are not experimentally significant where populations of dried cells are concerned.