Energy-dependent scattering cross section of Plexiglass for thermal neutrons. Krzysztof Drozdowicz

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1 January Energy-dependent scattering cross section of Plexiglass for thermal neutrons Krzysztof Drozdowicz

2 CIH-RF-62 January 1989 Energy-dependent scattering cross section of Plexiglass for thennal neutrons Krzysztof Drozdowicz permanent address Institute of Nuclear Physics ul. Radzikowskiego 1S2, Kraköw, Poland Department of Reactor Physics Chalmers University of Technology, S Göteborg ISSN

3 Abstract The macroscopic total cross section of Plexiglass (poly-methylmethacrylate) for neutrons has been measured as a function of energy in the thermal region (from 15 up to 80 mev) in a transmission experiment. From this the macro and microscopic scattering cross sections have been derived. A semi-theoretical formula for the dependence of the scattering cross section upon the neutron energy is given. The measured and calculated values are compared and show a very good agreement.

4 1. Introduction. Poly-methylmethacrylate, (CJH 8 O 2 ) n, known as Plexiglass, is a hydrogeneous material which may be used as a good and convenient neutron moderator. It was applied in some experiments as a testing material for a theoretical investigation (CopiC et al. 1964) and as a moderator in measurements of the neutron properties of other materials (Yurova and Pankratenko (1975), Czubek et al. (1983)). Generally, the thermal neutron diffusion parameters of Plexiglass are close to those of water. Water and another hydrogeneous material, polyethylene, have been extensively studied and their properties are.cnown very weli. For Plexiglass, however, only the thermal neutron macroscopic parameters, i.e. the absorption cross section, the diffusion constant, and the diffusion cooling coefficient, have been measured (CopiC et al. (1964), Graffstein et al. (1966), Yurova and Pankratenko (1975), Drozdowicz and Wofnicka (1987)). No data seem to exist on the energy dependence of the scattering cross section and the average cosine of the scattering angle in the thermal region. And it is known that just there a strong dependence of the scattering upon the energy is observed for various hydrogeneous compounds. The diffusion constant, connected to the transport cross section, is a very useful but energy integrated parameter. The present experiment was based on a transmission measurement. The total cross section for Plexiglass was measured as a function of the thermal neutron energy. An arrangement for a crystal spectrometer at the R2 nuclear reactor in Studsvik was utilized to obtain a beam of well defined monoenergetic neutrons. The interval of available energies was quite narrow because the neutron flux intensity from this set was insufficient at very low or too high energies to perform measurements. However, the experiment has given results in the most interesting interval of thermal energies.

5 2. Measurement of the total cross section. The transmission measurement is based on the relation between the intensities NL and N of a collimated neutron beam before and after crossing a layer of the material investigated: -Ld l N = N o e, (1) where I is the total macroscopic cross section, and d is the thickness of the material. In practice, the intensities are measured with and without a sample placed between a neutron source and a neutron detector. The sample should be thin to avoid registration of the neutrons from multiple scatterings. The geometry of the measurement is shown in Fig.l. The monoenergetic neutrons were obtained from the neutron beam from the reactor by means of an arrangement for a nei n spectrometer based on Bragg reflection. The Cu(220) reflection was utilized in a double monochromator. However, there is also a contribution from half the wave length of the reflection from Cu(440). This was removed by using germanium in an additional monochromator where the Ge(lll) reflection was utilized. There is no reflection from Ge(222), so finally there was no contamination from half the wave length. The sample was placed close at the additional monochromator and the diameter of the neutron beam was of the order of 1 cm. A 3 He counter was used as neutron detector. The counts N and N~ (i.e. when the sample was placed in and out, respectively) were controlled by a BF. counter used as a monitor of neutron total counts. The distance from the sample to the neutron detector was 16 cm and the diameter of the collimation opening was 4 cm. It was chosen as the maximal one when no influence of a change of this size on the N/N Q ratio was still observed. Two Plexiglass samples of different thicknesses, d, were used: Sample 1, d = mm, Sample 2, d= mm. These average thicknesses in the middle of the sample irradiated by the neutrons were known with the accuracy of C(d) = mm. The neutron background was determined for each energy using a cadmium shield instead of the sample.

6 / 'A 7//////////////A NEUTRON MONITOR PLEXIGLASS SAMPLE R2 REACTOR Cu(220) Cu(220) % ^ V/. NEUTRON BEAM ADDITIONAL MONOCHROMATOR NEUTRON DETECTOR DOUBLE MONOCHROMATOR Fig.l. Geometry of the measurement (not in scale).

7 3. Data treatment. The total cross section for a given neutron energy from Eq.(l) is I t = Jln(^ (2) with the standard deviation (3) Each single value of N_ or N was determined from a series of 5 measurements. The procedure was repeated several times for each neutron energy. Individual Z. and o(l.) values were calculated from Eqs (2) and (3). The total cross section was determined as the mean value <«> The standard deviation, a, was calculated from the variance: o(z)= n-llo 2^)]* (5) and the estimator, a, from the experimental data: o(i) = -i L! L_, (6) 1 L n (n - 1) -I Finally, the estimate s of the standard deviation was assumed as

8 s(z t )= max{ a(2 t ), o(i t )} (7) A to avoid an underestimation of the measurement errors. (The relation between the c and G values varied due to some fluctuations in the operation of the system.) Sample 1 was the basic one during the measurements. Sample 2 was used for check-points at several energies and these results were also included. 4. Results. The macroscopic total cross section Z(E) for Plexiglass was measured at room temperature as a function of the neutron energy from 15 to 80 mev. The energy was defined with an accuracy better than 0.1 mev. The values obtained for both samples are listed in Table 1. The microscopic cross section, o(e), has been calculated using the Plexiglass density p = g cnr 3 and the molecular weight M = The macro- and microscopic scattering cross sections, (E) and o (E), can be calculated if the absorption cross section is known. Here there are two possibilities: one may utilize a value of the macroscopic cross section measured for the material or calculate the microscopic cross section for the Plexiglass molecule from nuclear data for the elements. A 1/v dependence of the absorption cross section has been assumed. Energy dependent values have been calculated both from the absorption rate ve measured by Drozdowicz and Wof nicka (1987) and from the microscopic cross sections o (v o =2200 m/s) taken from Mughabghab et al. (1981). The results were consistent and the standard deviations equal to zero at the accuracy level obtained from the measurement for the I values. The energy dependent total, absorption, and scattering cross sections are given in Table 2.

9 E [mcv] _ x Sample t I, [cm-i] s(i t ) Sample Mean * Relative standard deviation [*]

10 Table? t Macroscopic and microscopic cross sections for Plexiglass as a function of the thermal neutron energy. E s(z t ) r a r s s(i s ) t s(o fc ) a s [m ev] [cm" 1 ] [cm" 1 ] [cm" 1 ] [b] [to] [b] SS i , ,.127 < ,.111 i , , ,.993 i, , ,.866 <, , , ,830, ,,817, , ,704, ,691, , ,

11 5. Semi-theoretical dependence of the scattering cross section on energy. An effort was made to calculate theoretically the energy dependence of the scattering cross section for Plexiglass. A model was constructed in the following way. The formula for the macroscopic scattering cross section resulting from the free gas model (e.g. Williams 1966) is: I s (E) = 2 ^ [(2AW + 1) erf(/äw) VÄW e~ AW ], (8) where: I f - free scattering cross-section, A = M/m n, (9) M - nuclear mass, m - neutron mass, n W = E/E Q - relative neutron energy, (10) E Q = kt, (10a) k - Boltzmann constant, T - moderator temperature. Generally, Eq.(8) is not valid for hydrogeneous materials. However, the author modified it for hydrogen to the form: (11) 10

12 where an energy dependent coefficient a = a(e) (12) is introduced, and o f is the microscopic free scattering cross section for hydrogen. Next, use is made of the observation that Plexiglass contains the CH. and CH. groups in its structural formula: H CH 3 i-o A Values of the energy dependent total cross section for polyethylene, (CH 2 ) n, measured by Granada et al. (1987) were utilized to calculate the a(e) function: (13) where O^ 2(E) is given, and (14) contains the sought function in the second term according to Eq.(H). The formula for the carbon scattering cross section o^(e) follows exactly Eq.(8). 11

13 CHi The absorption cross section o 2 (E) was calculated from the microscopic cross sections for carbon and hydrogen from Mughabghab et al. (1981). The <x(e) function obtained from Eq.(13) with Eqs (14) and (11) was applied to calculate the total cross section o(e) for Plexiglass according to the elemental composition: o,(e) = o n (E) + (8o"(E) + 5c&E) + 2a (E)) (15) I II a S S (where o ( ) is the oxygen cross section after Eq.(8)). The results of the measurement and of the calculation are compared in Fig. 2. A surprisingly good agreement is observed. Therefore, the energy dependent total cross section was calculated for a much greater energy Literval and the result is plotted in Fig.3, where the experimental points are also marked and a line is drawn which shows the dependence calculated from the pure formula of the free gas model, i.e. when a(e) = 1 in Eq.(ll) for the hydrogen scattering cross section. 12

14 c L. 550 ro.q,_, U ^. \ N V n CD É l.... l ENERGY W = E/E0.... _ 3.5 Fig.2. Comparison of the measured and calculated results. EQ ev, the curve is calcjlated from Eq.(15). 13

15 ^ 800 C 600 LJ 400 p n> 280 E CD cn 0 **«' i^. i F, -x ENERGY N = E/E0 Fig.3. Dependence of the total cross section a t for Plexiglass upon thermal neutron energy. The curves are calculated from Eq.(lS), where O*j(E) for Curve 1 follows the free gas model Eq.(8) and for Curve 2 is given by the modified formula, Eq.(ll). Eo = 0.02S3 ev. The experimental points from the present measurement are marked (+). 14

16 Acknowledgments. I am indebted to prof. Nils G. Sjöstrand, head of the Dept. of Reactor Physics, who gave me the opportunity to work in this institute and arranged my contacts with the Studsvik laboratory. I thank Dr. Roland Tellgren and Eng. Håkan Rundlöf from the Institute of Chemistry of Uppsala University (Neutron Research Laboratory in Studsvik) for adaptation of their equipment at the R2 reactor for the present experiment and for their help during the measurements. The work has been supported by the Royal Society of Arts and Sciences in Göteborg and by the Swedish Natural Science Research Council. References. Copic M., Kalin T., Pregl G. and Zerdin F. (1964) Nucl.Sci.Eng. 12, 74. Czubek J.A., Drozdowicz K., Krynicka-Drozdowicz E., Igielski A. and Woznicka U. (1983) Int.J.Appl.Radiation Isotopes Drozdowicz K. and Woznicka U. (1987) J.Phys.D:Appl.Phys. 2Q, 985. Graffstein A., Rzeszot T. and Warda E. (1966) Nukleonika H, 277. Granada J.R., Dawidowski J., Mayer R.E. and Gillette V.H. (1987) Nucl.Instrum.Methods A Mughabghab S.F., Divadeenam M. and Holden N.E. (1981) Neutron Cross Sections, vol.1, Neutron Resonance Parameters and Thermal Cross Sections. Part A. Academic Press, New York. Williams M.M.R. (1966) The Slowing Down and Thermalization of Neutrons, North-Holland, Amsterdam. Yurova L.N. and Pankratenko D.A. (1975) Sov.Atom.Energy 28,120 (and Atomnaya Énergiya deposited paper No 768/7822 (in Russian)). 15

The Henryk Niewodniczański INSTITUTE OF NUCLEAR PHYSICS Polish Academy of Sciences ul. Radzikowskiego 152, Kraków, Poland.

The Henryk Niewodniczański INSTITUTE OF NUCLEAR PHYSICS Polish Academy of Sciences ul. Radzikowskiego 152, 31-342 Kraków, Poland. www.ifj.edu.pl/reports/23.html Kraków, listopad 23 Report No: 1933/PN Influence