Method for determining the average radius of gyration of particles with a size of less than or equal to 1 micron in a suspension, and device for carrying out the method

Abstract

The invention relates to a method for determining the average radius of gyration (r.sub.g) of particles with a size of 1 m in a suspension, and to a device for carrying out the method according to the invention. The method is based on the scattering of linearly polarised electromagnetic radiation on nanoparticles, which, suspended in a solution, are moved through a through-flow cell. The irradiation is carried out perpendicular to the movement direction, wherein the scattering intensity is measured via at least four detectors that are arranged in a defined plane at defined angles. Alternatively, at least one mirror can be used in the position of at least one of the detectors, which deflects the radiation to at least one detector. Based on the scattering intensities, both the average radius of gyration (r.sub.g) of the particles as well as the concentration thereof in the suspension can be determined.

Claims

1. A method for determining the average radius of gyration r.sub.g of particles with a size of 200 nm, in a suspension, comprising the steps: a) moving a suspension comprising particles of a size of 200 nm through a flow cell at a specified flow rate along a movement straight line; b) irradiating the suspension in the flow cell with linearly polarised electromagnetic radiation of a wavelength of 800 nm via a radiation source along an irradiation straight line, the irradiation straight line intersecting the movement straight line at a scattering point; and c) detecting at least two first scattering intensities of the electromagnetic radiation via at least one first and at least one second detector, the at least one first detector being disposed, relative to the irradiation straight line, at a first angle .sub.1 and the at least one second detector, relative to the irradiation straight line, at a second angle 180.sub.1; d) detecting at least two further scattering intensities of the electromagnetic radiation via at least one third detector and at least one fourth detector, the at least one third detector being disposed, relative to the irradiation straight line, at a third angle .sub.2 and the at least one fourth detector, relative to the irradiation straight line, at a fourth angle 180.sub.2, wherein .sub.2.sub.1; e) repeating steps a) to d) with a reference liquid comprising no particles instead of the suspension comprising particles; f) forming at least four differences of scattering intensities, respectively the scattering intensities of the reference liquid without particles being subtracted from the scattering intensities of the suspension with particles; and g) calculating the average radius of gyration r.sub.g of the particles from the at least four differences of scattering intensities, wherein the at least one first detector, the at least one second detector, the at least one third detector, and the at least one fourth detector are disposed in a plane which is parallel to the irradiation straight line and perpendicular to the movement straight line, at least one mirror for deflecting the electromagnetic radiation to the at least one detector being utilized alternatively at the position of at least one of the detectors; wherein the average radius of gyration r.sub.g of the particles is calculated by a method comprising the steps of: (i) calculating a value Z.sub.1 from the at least two first scattering intensities according to the formula $Z_1=\frac{I_{{}_1}}{I_{180-{}_1}}=\frac{P_{{}_1}}{P_{180-{}_1}}\left[1-\frac{2A_2{M}_p{c}_P}{\left(1+2A_2{M}_p{c}_P{P\left(q\right)}_{}\right)}\left(P_{{}_1}-P_{180-{}_1}\right)\right]$ wherein I.sub.1: scattering intensity with the first angle .sub.1; I.sub.1801: scattering intensity with the second angle 180.sub.1; P(.sub.1)/P(180.sub.1): quotient of the shape factors which is produced from the shape and size of the particles; M.sub.p: average molecular mass of the particles; c.sub.p: concentration, A.sub.2: second virial coefficient; and (ii) calculating a value Z.sub.2 from the at least two further scattering intensities according to the formula $Z_2=\frac{I_{{}_2}}{I_{180-{}_2}}=\frac{P_{{}_2}}{P_{180-{}_2}}\left[1-\frac{2A_2{M}_p{c}_P}{\left(1+2A_2{M}_p{c}_P{P\left(q\right)}_{}\right)}\left(P_{{}_2}-P_{180-{}_2}\right)\right]$ wherein I.sub.2: scattering intensity with the third angle .sub.2; I.sub.1802: scattering intensity with the fourth angle 180.sub.2; P(.sub.2)/P(180.sub.2): quotient of the shape factors which is produced from the shape and size of the particles; M.sub.p: average molecular mass of the particles; c.sub.p: concentration, A.sub.2: second virial coefficient; and (iii) numerically determining the average radius of gyration r.sub.g by means of a calibration curve calculated for the particles according to the following formula:
F(r.sub.g)=0.5.Math.[Z.sub.1Z.sub.2+(Z.sub.1/Z.sub.21)].

2. The method according to claim 1, wherein the first angle .sub.1 is from 45 to 55, relative to the beam direction, and/or .sub.14.sub.2.sub.120 applies.

3. The method according to claim 1, wherein the number of detection angles between 40 and 90, relative to the beam direction, is 2.

4. The method according to claim 1, wherein the pair of the at least one first and at least one second, and/or the pair of the at least one third and at least one fourth detector are disposed symmetrically, relative to the 9 axis.

5. The method according to claim 1, wherein the detectors comprise respectively an input surface or input opening which is disposed, at least in regions, on a flow cell surface or abuts or is identical, at least in regions, to the latter, the input surface or input opening being a flat cavity which is transparent for the electromagnetic radiation, optionally an air space, or a window area which is transparent for electromagnetic radiation.

6. The method according to claim 1, wherein the flow cell a) has a volume of 1 to 450 l; b) has a cross-section perpendicular to the flow direction of the suspension, a circular, rectangular or polygonal configuration; c) comprises glass and/or a polymer; and/or d) has, on a contact surface with the suspension, a refractive index which corresponds essentially to the refractive index of the suspension which deviates from the refractive index of the suspension by 0.1.

7. The method according to claim 1, wherein the suspension is moved at a flow rate of 0.02 to 2 ml/min through the flow cell.

8. The method according to claim 1, wherein, before detecting the scattering intensity, the detectors are calibrated, the deviation of the actual position of the detectors from the ideal position of the detectors being compensated for.

9. The method according to claim 1, wherein a radiation path from the scattering point to a detector is isolated optically relative to a radiation path from the scattering point to an adjacent detector, by a screen, which is not transparent for the electromagnetic radiation, along the radiation path, the screen having an anti-reflection surface.

10. The method according to claim 1, which comprises: a) moving the suspension with the particles through the flow cell; b) detecting the scattering intensities of the electromagnetic radiation of the suspension in the flow cell; c) moving air or another suitable separation medium through the flow cell for spatial separation of the suspension and the reference solution; d) moving a reference solution which consists of the suspension without particles through the flow cell; e) detecting the scattering intensities of the electromagnetic radiation of the reference solution in the flow cell; and f) subtracting the scattering intensity measured for the reference solution from the scattering intensity determined for the suspension.

11. The method according to claim 1, wherein the detectors are Si detectors and/or CCD sensors.

12. The method according to claim 1, wherein the detected scattering intensity is corrected with an electronic data filter which corrects the scattering intensity.

13. The method according to claim 1, wherein the electromagnetic radiation a) has a wavelength in the range of 250 nm to 800 nm; b) is monochromatic; c) is linearly polarised parallel or perpendicular to the movement straight line; or d) is produced via a laser light source.

14. The method according to claim 1, wherein, prior to step a), a suspension comprising a liquid and particles of a size of 200 nm and particles of a size of >200 nm is filtered through a membrane, whereby the particles of a size of >200 nm are separated from the suspension.

15. The method according to claim 1, wherein the radiation source i) is disposed along the irradiation straight line; or ii) the irradiation source is disposed at an angle relative to the irradiation straight line and the electromagnetic radiation is deflected via at least one mirror onto the irradiation straight line.

16. A method for determining the average radius of gyration r.sub.g of particles with a size of 200 nm, in a suspension, comprising the steps: a) moving a suspension comprising particles of a size of 200 nm through a flow cell at a specified flow rate along a movement straight line; b) irradiating the suspension in the flow cell with linearly polarised electromagnetic radiation of a wavelength of 800 nm via a radiation source along an irradiation straight line, the irradiation straight line intersecting the movement straight line at a scattering point; and c) detecting at least two first scattering intensities of the electromagnetic radiation via at least one first and at least one second detector, the at least one first detector being disposed, relative to the irradiation straight line, at a first angle .sub.1 and the at least one second detector, relative to the irradiation straight line, at a second angle 180.sub.1; d) detecting at least two further scattering intensities of the electromagnetic radiation via at least one third detector and at least one fourth detector, the at least one third detector being disposed, relative to the irradiation straight line, at a third angle .sub.2 and the at least one fourth detector, relative to the irradiation straight line, at a fourth angle 180.sub.2, wherein .sub.2.sub.1; e) repeating steps a) to d) with a reference liquid comprising no particles instead of the suspension comprising particles; f) forming at least four differences of scattering intensities, respectively the scattering intensities of the reference liquid without particles being subtracted from the scattering intensities of the suspension with particles; and g) calculating the average radius of gyration r.sub.g of the particles from the at least four differences of scattering intensities, wherein the at least one first detector, the at least one second detector, the at least one third detector, and the at least one fourth detector are disposed in a plane which is parallel to the irradiation straight line and perpendicular to the movement straight line, at least one mirror for deflecting the electromagnetic radiation to the at least one detector being utilized alternatively at the position of at least one of the detectors; wherein, in a further step, the concentration of the particles in the suspension is determined by a method comprising the steps of: (i) detecting a scattering intensity, 90 relative to the irradiation direction of the electromagnetic radiation, via a fifth detector and detecting the irradiation intensity I.sub.0 via a sixth detector, the fifth detector being disposed, relative to the irradiation straight line, at a fifth angle of 90 and the sixth detector, relative to the irradiation straight line, at a sixth angle of 180; (ii) calculating R(.sub.1), R(.sub.m) of any pair from the five scattering intensities I.sub.l, I.sub.m at the irradiation intensity I.sub.0 wherein $R\left({}_i\right)=\frac{I_{{}_i}}{I_0}\frac{r^2}{V\cos {}_i}:$ Rayleigh ratio at angle .sub.i; V: scattering volume; and r.sup.2: square of the spacing between the center of the scattering volume and the corresponding detector; (iii) combining, in pairs, the scattering intensities detected at least with the first, second, third, fourth and fifth angle with each other and calculating at least 10 independent concentrations c.sub.i according to the formula
c.sub.i=1/(M.sub.p*K)*[1/P(.sub.1)1/P(.sub.m)]/[1/R(.sub.1)1/R(.sub.m)] wherein M.sub.p: molecular weight of the particles; K: contrast factor relative to material and concentration for characterisation of the scattering capacity of the particles; P(.sub.1), P(.sub.m): shape factors for scattering signals of any pair from the five scattering intensities; R(.sub.1), R(.sub.m): Rayleigh ratios, dependent upon material and particle size, of any pair from the five scattering intensities; and (iv) calculating the average of the at least 10 independent concentrations c.sub.i and calculating the standard deviation.

17. A device for determining the average radius of gyration r.sub.g of particles with a size of 200 nm in a suspension, comprising a) a flow cell, optionally with a pump device for moving a suspension comprising particles of a size of 200 nm at a specific flow rate along a movement straight line; b) a radiation source for irradiating the suspension in the flow cell with linearly polarised electromagnetic radiation of a wavelength of 800 nm along an irradiation straight line, the irradiation straight line intersecting the movement straight line at a scattering point; c) at least one first and second detector for detecting at least two first scattering intensities of the electromagnetic radiation, the at least one first detector being disposed, relative to the irradiation straight line, at a first angle .sub.1 and the at least one second detector, relative to the irradiation straight line, at a second angle 180.sub.1; d) at least one third and fourth detector for detecting at least two further scattering intensities of the electromagnetic radiation, the at least one third detector being disposed, relative to the irradiation straight line, at a third angle .sub.2 and the at least one fourth detector, relative to the irradiation straight line, at a fourth angle 180.sub.2; and e) a computing unit which is configured to subtract at least the scattering intensities of a reference solution without particles, detected at the first, second, third and fourth detector, from the scattering intensities, detected there respectively, of a suspension comprising particles, and calculating the average radius of gyration r.sub.g of the particles from the at least four differences, wherein the at least one first detector, the at least one second detector, the at least one third detector and the at least one fourth detector are disposed in a plane which is parallel to the irradiation straight line and perpendicular to the movement straight line, at least one mirror for deflecting the electromagnetic radiation to the at least one detector is disposed alternatively at the position of at least one of the detectors; wherein the computing unit is set to determine the average radius of gyration r.sub.g of the particles via the following steps: (i) calculating a value Z.sub.1 from the two first scattering intensities according to the formula $Z_1=\frac{I_{{}_1}}{I_{180-{}_1}}=\frac{P_{{}_1}}{P_{180-{}_1}}\left[1-\frac{2A_2{M}_p{c}_P}{\left(1+2A_2{M}_p{c}_P{P\left(q\right)}_{}\right)}\left(P_{{}_1}-P_{180-{}_1}\right)\right]$ wherein I.sub.1: scattering intensity with the first angle .sub.1; I.sub.1801: scattering intensity with the second angle 180.sub.1; P(.sub.1)/P(180.sub.1): quotient of the shape factors which is produced from the shape and size of the particles; M.sub.p: average molecular mass of the particles; c.sub.p: concentration, A.sub.2: second virial coefficient; and (ii) calculating a value Z.sub.2 from the two further scattering intensities according to the formula $Z_2=\frac{I_{{}_2}}{I_{180-{}_2}}=\frac{P_{{}_2}}{P_{180-{}_2}}\left[1-\frac{2A_2{M}_p{c}_P}{\left(1+2A_2{M}_p{c}_P{P\left(q\right)}_{}\right)}\left(P_{{}_2}-P_{180-{}_2}\right)\right]$ wherein I.sub.2: scattering intensity with the third angle .sub.2; I.sub.1802: scattering intensity with the fourth angle 180.sub.2; P(.sub.2)/P(180.sub.2): quotient of the shape factors which is produced from the shape and size of the particles; M.sub.p: average molecular mass of the particles; c.sub.p: concentration, A.sub.2: second virial coefficient; and (iii) numerically determining the average radius of gyration r.sub.g by means of a calibration curve calculated for a concrete type of particles according to the following formula:
F(r.sub.g)=0.5.Math.[Z.sub.1Z.sub.2+(Z.sub.1/Z.sub.21)].

18. The device according to claim 17, wherein the first angle .sub.1 is from 45 to 55 relative to the beam direction, and/or .sub.14.sub.2.sub.120 applies.

19. The device according to claim 17, wherein the number of detection angles between 400 and 90, relative to the beam direction, is 2.

20. A device for determining the average radius of gyration r.sub.g of particles with a size of 200 nm in a suspension, comprising a) a flow cell, optionally with a pump device for moving a suspension comprising particles of a size of 200 nm at a specific flow rate along a movement straight line; b) a radiation source for irradiating the suspension in the flow cell with linearly polarised electromagnetic radiation of a wavelength of 800 nm along an irradiation straight line, the irradiation straight line intersecting the movement straight line at a scattering point; c) at least one first and second detector for detecting at least two first scattering intensities of the electromagnetic radiation, the at least one first detector being disposed, relative to the irradiation straight line, at a first angle .sub.1 and the at least one second detector, relative to the irradiation straight line, at a second angle 180.sub.1; d) at least one third and fourth detector for detecting at least two further scattering intensities of the electromagnetic radiation, the at least one third detector being disposed, relative to the irradiation straight line, at a third angle .sub.2 and the at least one fourth detector, relative to the irradiation straight line, at a fourth angle 180.sub.2; and e) a computing unit which is configured to subtract at least the scattering intensities of a reference solution without particles, detected at the first, second, third and fourth detector, from the scattering intensities, detected there respectively, of a suspension comprising particles, and calculating the average radius of gyration r.sub.g of the particles from the at least four differences, wherein the at least one first detector, the at least one second detector, the at least one third detector and the at least one fourth detector are disposed in a plane which is parallel to the irradiation straight line and perpendicular to the movement straight line, at least one mirror for deflecting the electromagnetic radiation to the at least one detector is disposed alternatively at the position of at least one of the detectors; wherein the computing unit is set to determine the concentration of the particles in the suspension, the determination of the concentration comprising the following steps: (i) detection of two further scattering intensities of the electromagnetic radiation, via a fifth detector and sixth detector, the fifth detector being disposed, relative to the irradiation straight line, at a fifth angle of 90 and the sixth detector, relative to the irradiation straight line, at a sixth angle of 180; (ii) calculation of R(1), R(m): Rayleigh ratios, dependent upon material and particle size, of any pair from the five scattering intensities I.sub.l, I.sub.m at the irradiation intensity I.sub.O wherein $R\left({}_i\right)=\frac{I_{{}_i}}{I_0}\frac{r^2}{V\cos {}_i}:$ Rayleigh ratio at the angle .sub.i; V: scattering volume and r.sup.2: square of the spacing between the center of the scattering volume and the corresponding detector; (iii) combination, in pairs, of the scattering intensities detected at least with the first, second, third, fourth and fifth angle with each other and calculation of at least 10 independent concentrations c.sub.i according to the formula
c.sub.i=1/(M.sub.p*K)*[1/P(.sub.1)1/P(.sub.m)]/[1/R(.sub.1)1/R(.sub.m)] wherein Mp: molecular weight of the particles; K: contrast factor relative to material and concentration for characterisation of the scattering capacity of the particles; P(.sub.1), P(.sub.m): shape factors for scattering signals of any pair from the five scattering intensities; R(.sub.1), R(.sub.m): Rayleigh ratios, dependent upon material and particle size, of any pair from the five scattering intensities; and (iv) calculation of the average of the at least 10 independent concentrations ci and calculation of the standard deviation.

Description

(1) The subject according to the invention is intended to be explained in more detail with reference to the subsequent Figures and examples without wishing to restrict said subject to the specific embodiments represented here.

(2) FIG. 1 shows a schematic illustration of the geometric arrangement of a device according to the invention. There emerges from the illustration, the flow cell 1, the radiation source 3 and the individual detectors D1, D2, D3, D4, D5, D6 which are used in the method according to the invention or in the device according to the invention. Via a radiation source 3, linearly polarised electromagnetic radiation is directed along an irradiation straight line 4 onto a flow cell 1. Through the flow cell, a suspension with the particles to be measured is conducted along a movement straight line 2, the movement straight line 2 being perpendicular to the irradiation straight line 4. In a plane about the flow cell 1 which is parallel to the irradiation straight line 4 and perpendicular to the movement straight line 2, a first detector D1, a second detector D2, a third detector D3, a fourth detector D4, a fifth detector D5 and a sixth detector D6 are disposed, which detectors respectively measure the scattering intensities at a first, second, third, fourth, fifth angle and also the irradiation intensity at a sixth angle. From the scattering intensities at the first to fourth angle, the average radius of gyration r.sub.g is determined according to the invention. For determination of the concentration of the particles in the suspension, in addition, the scattering intensity at the fifth angle (90 relative to the irradiation straight line 4), and also in addition irradiation intensity at the sixth angle (180 relative to the irradiation straight line 4) is taken into account. FIG. 1 also depicts mirrors ML M2, M3, M4, M5, and M6 which can be used to deflect radiation onto detectors DL D2, D3, D4, D5, and D6, respectively. FIG. 1 also depicts screens SL, S2, S3, S4, S5, and S6 which are not transparent for electromagnetic radiation and which optically isolate detectors D1-D6, respectively.

(3) FIG. 2 shows a schematic illustration of the geometric arrangement of a further device according to the invention. The arrangement of the flow cell 1, of the radiation source 3 and of the individual, here 20, detectors D1, D2, D3, D4, D5, D6, D7, D8, D9, D10, D11, D12, D13, D14, D15, D16, D17, D18, D19, D20 is illustrated. The detectors D1 and D2, D3 and D4, D5 and D6, D7 and D8, D9 and D10, D11 and D12, D13 and D14, D15 and D16, D17 and D18, D19 and D20 form respectively pairs of values for a determined angle .sub.1 and its associated angle 180.sub.1. Via a radiation source 3, linearly polarised electromagnetic radiation is directed along an irradiation straight line 4 onto a flow cell 1. Through the flow cell, a suspension is conducted here also with the particles to be measured along a movement straight line 2, the movement straight line 2 being perpendicular to the irradiation straight line 4. In a plane about the flow cell 1 which is parallel to the irradiation straight line 4 and perpendicular to the movement straight line 2, the in total 20 detectors D1, D2, D3, D4, D5, D6, D7, D8, D9, D10, D11, D12, D13, D14, D15, D16, D17, D18, D19, D20 are disposed, which detectors measure respectively the scattering intensities at a first to twentieth angle. A twenty-first detector D(0) measures the irradiation intensity and a twenty-second detector D(90) measures the 90 scattering intensity. From the scattering intensities at the first to twentieth angle, the average radius of gyration r.sub.g is determined. For determination of the concentration of the particles in the suspension, in addition the scattering intensity at the twenty-second angle (90 relative to the irradiation straight line 4), and also in addition irradiation intensity at the twenty-first angle (180 relative to the irradiation straight line 4), is taken into account. FIG. 2 also depicts mirrors M1, M2, M3, M4, M5, M6, M7, M8, M9, M10, M11, M12, M13, M14, M15, M16, M17, M18, M19, and M20, which can be used to deflect radiation onto detectors D1, D2, D3, D4, D5, D6, D7, D8, D9, D10, D11, D12, D13, D14, D15, D16, D17, D18, D19, and D20, respectively. FIG. 2 also depicts screens S1, S2, S3, S4, S5, S6, S7, S8, S9, S10, S11, S12, S13, S14, S15, S16, S17, S18, S19, and S20, which are not transparent for electromagnetic radiation and which optically isolate detectors D1-D20, respectively.

EXAMPLE 1

Calculation of the Average Radius of Gyration r.SUB.g.

(4) The average radius of gyration r.sub.g is calculated from the averaged and, on larger particles, corrected intensity values corresponding to the dissymmetry method from the data for two corresponding measurement angles set symmetrically relative to the movement straight line.

(5) There applies:
Z.sub.i=I.sub.i/I.sub.180i=P(.sub.i)/P(180.sub.i),

(6) wherein

(7) I.sub.i/I.sub.180i: quotient of the intensities measured at the corresponding angles;

(8) P(.sub.i)/P(180.sub.i): a quotient of the shape factors, corresponding to the angles i and 180.sub.I, which depends upon the shape of the particles and their size and from which the average radius of gyration can be calculated.

(9) Calculation of the average radius of gyration functions by means of P(.sub.i)/P(180.sub.i), if the interaction between the particles and water and also amongst the particles themselves is negligibly small.

(10) Otherwise there applies:
I.sub.i/I.sub.180i=P(.sub.i)/P(180.sub.i).Math.[1f(A.sub.2,c,M.sub.p).Math.(P(.sub.i)/P(180.sub.i))],

(11) wherein

(12) f(A.sub.2, c, M.sub.p)>0;

(13) A.sub.2: second virial coefficient (responsible for the interaction);

(14) c: concentration of the particles;

(15) M.sub.p: molecular weight of the particles.

(16) Normally, this is calculated according to the Zimm Method. According to the invention, the interaction is calculated from the angle-dependent measurement of the quotient of .sub.i and 180.sub.i for two pairs of angles Z.sub.1 and Z.sub.2. From Z.sub.1 and Z.sub.2 the difference (Z.sub.1Z.sub.2), the value (Z.sub.1/Z.sub.21) and the average of Z.sub.1 and Z.sub.2 is determined.

(17) For small particle sizes (d/<0.25, wherein d: particle diameter and : wavelength) there applies:
Z.sub.1=P(.sub.1)/P(180.sub.1).Math.[1f(A.sub.2,c,M.sub.p).Math.(P(.sub.1)/P(180.sub.1))];
Z.sub.2=P(.sub.2)/P(180.sub.2).Math.[1f(A.sub.2,c,M.sub.p).Math.(P(.sub.2)/P(180.sub.2))];
Z=P(.sub.1)/P(180.sub.1)P(.sub.2)/P(180.sub.2)f(A.sub.2,c,M.sub.p).Math.([(P(.sub.1)/P(180.sub.1)](P(.sub.2)/P(180.sub.2))]);

(18) In the case where .sub.1 and .sub.2 are relatively close together, there applies:
[P(.sub.1)P(180.sub.1)][P(.sub.2)P(180.sub.2)]<<P(.sub.1)P(180.sub.1); and
[P(.sub.1)P(180.sub.1)][P(.sub.2)P(180.sub.2)]<<P(.sub.2)P(180.sub.2); and hence
ZP(.sub.1)/P(180.sub.1)P(.sub.2)/P(180.sub.2)difference of the quotients in the case of an interaction of the particles amongst each other.fwdarw.0).

(19) There applies then for the quotient:
Z.sub.1/Z.sub.2=P(.sub.1)/P(.sub.2).Math.P(180.sub.2)/P(180.sub.1).Math.[(1f(A.sub.2,c,M.sub.p).Math.(P(.sub.1)P(180.sub.1))]/[(1f(A.sub.2,c,M.sub.p).Math.(P(.sub.2)/P(180.sub.2))]1;
wherein
(1f(A.sub.2,c,M.sub.p).Math.(P(.sub.1)P(180.sub.1))(1f(A.sub.2,c,M.sub.p).Math.(P(.sub.2)/P(180.sub.2)); and
Z.sub.1/Z.sub.21=P(.sub.1)/P(.sub.2).Math.P(180.sub.2)/P(180.sub.1)1<1, with the ratio of the quotients Z.sub.1 and Z.sub.2 in the case of an interaction of the particles amongst each other.fwdarw.0).

(20) There applies hence for the ideal case and d/<0.25:
ZZ.sub.1/Z.sub.21;

(21) Since the influence of f(A.sub.2, c, M.sub.p) is reduced by the formation of Z and (Z.sub.1/Z.sub.21) to a negligibly small value, the particle size can be calculated from the average of both measured Z. This is effected according to the following correlation:
[Z+Z.sub.1/Z.sub.21]/2

EXAMPLE 2

Calculation of the Particle Concentration

(22) The concentration can then be calculated from the angle-dependent measured intensities. In order to compensate for manufacturing and positioning errors, the intensity measured with at least five (optionally also six) scattering angles is used.

(23) There applies:
K/R(.sub.1)=1/(M.sub.p.Math.P(.sub.1).Math.c)+2A2.sub.1/P/V;
K/R(.sub.2)=1/(M.sub.p.Math.P(.sub.2).Math.c)+2A2.sub.2/P/V;

(24) wherein

(25) K=const.;

(26) P=(1+cos.sup.2 .sub.i)/2r.sup.2;

(27) r=spacing of detection cell to the detector or mirror;

(28) R(.sub.i)=[(I.sub.AnalystI.sub.Ref)/I.sub.0)].sub.I;

(29) V=scattering volume.

(30) Furthermore, there applies:
K/R(.sub.1)K/R(.sub.2)=1/(M.sub.p.Math.c).Math.[1/P(.sub.1)1/P(.sub.2)];
c=1/(M.sub.p.Math.K).Math.[1/P(.sub.1)1/P(.sub.2)]/[1/R(.sub.1)1/R(.sub.2)]; and
2A.sub.2=(K/R(90)(1/(c.Math.M.sub.p.Math.P(90)));

(31) There result then from the five angles, ten values for the concentration which are averaged and subsequently the standard deviation is calculated. The standard deviation provides a measurement of the accuracy of the method, a conclusion being able to be drawn hence with respect to the system-related inaccuracy of the device according to the invention (e.g. a construction-conditioned, slight deviation of the detectors or mirror from the ideal angle) which can be compensated for via a corresponding calibration.