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Neutron transmission study through simple bent conducting tube
NUCLEAR INSTRUMENTS AND METHODS
91 (1971) 429-437; ©
NORTH-HOLLAND PUBLISHING CO.
NEUTRON TRANSMISSION STUDY THROUGH SIMPLE BENT CONDUCTING TUBE A. BOBBIO and C. VACIAGO Is/III/to di Fisica Sperlllle/l/ale del Poli/ecnico, Torino, Ilaly
Received II August 1970 The neutron distribution at the exit window of a simple bent neutron conducting tube is studied using the Monte Carlo method. The method is also applied to obtain the neutron energy spectra and the transmission on exit from the tube, and to discuss
the optimal size ofa cold neutron source. A statistical law relating the average number ii of neutron reflections inside the tube to the wavelength .A. is obtained in a particular case.
1. Introduction
in the conducting tubes, and particularly on the distribution of fast and slow neutrons at the exit window; the average number of reflections versus the wavelength; the influence of the size of the neutron source, and therefore of the angular beam dispersion. on the neutron exit number and distribution, and on the tube transmission. This information is useful in order to calculate and design the neutron conducting tube, taking into account, with somewhat greater precision, its real physical properties.
The increasing interest in experimental techniques employing cold neutron scattering has shown the importance of neutron optical devices, especially of neutron conducting tubes l - 3 ). These devices allow a very low background of ')I rays and fast neutrons in the measurement area, and give a very high gain in the allowed solid angle and therefore in the transmitted neutron intensity. Neutron conducting tubes have been calculated, from the beginning, using the simplifying hypothesis that a nearly collimated neutron beam will enter the tube; that is, the absolute value of the wave number K is assumed approximately equal to its component K" along the tube axis. This assumption leads to some very easy relations between geometrical and physical parameters of the tube; however, it does not permit one to predict with sufficient accuracy the space and energy distributions of the neutrons at the exit window of the tube. To know such distributions is indeed extremely useful, mainly when one needs to analyse the results of neutron inelastic scattering experiments, or when the dimensiolls of the sample are smaller than the tube exit window. In order to handle such a problem in a realistic way, it is necessary to deal with all possible directions and energies ofthe neutrons entering the tu be, and therefore a statistical choice between them must of course be made: Monte Carlo techniques may be applied to solve our problem. The criteria by which statistically distributed neutrons were chosen, are discussed in section 2. Some of the computing program details are given 1n section 3. The discussion of the results concerning neutron flux and transmission at the exit window, average number of reflections, neutron energy and space distributions, and cold source dimensions will be found in section 4. Monte Carlo study gives results for neutron features
2. Statistical choice of neutrons Before analysing the computing program in detail let us briefly enumerate the information required by our experimental group, working with the cold neutron facilities at the Avogadro Swimming Pool Research Reactor at Saluggia: 1. The most probable space distribution of neutrons of various energies at the end of the tube. 2. The statistical law for the number of neutron reflections inside the tube as a function of the neutron wavelength. 3. The reliability of the calculations based on Alefeld's simplified hypothesis l ) especially concerning the neutron transmission through the tube.
429
z' y
z
Fig. 1. Reference system.
430
A. BOBBIO AND C. VACIAGO
RANDOM SELECTION OF THE NEUTRON ENTRY COORDINATES Xo"n Zo RANDOM SELECTION OF THE NEUTRON DIRECTION fOLLOWING THE LAW P :::
I\! cos 6'
UATION OF THE INITIAL TRAJECTORY
CALCULATION OF THE EffECTIVE INTERSECTION ON THE TUBE WALLS THE INTERSECTION IS
NUMBER AND COORDINATES IN EACH ROW AND COLUMN
Of
THE EXIT WINDOW
CALCULATION OF THE AVERAGE NUMBER Of REFLECTIONS CALCULATION Of TUE TRANSMISSION
Fig. 2. Flow-chart of the computing program.
431
NEUTRON TRANSMISSION STUDY THROUGH SIMPLE DENT CONDUCTING TUBE
4. The optimal size of the neutron source, in order to obtain the maximum neutron flux at the exit window. Neutrons emerging from a source having a crosssectional area of dimensions Z x Y, enter the tube through a window placed at a distance f from the source (fig. 1). The fl ux is supposed to be constant with respect to the local neutron coordinates in the source, while a distribution of the form P = Po cos 8 for the neutron directions is assumed; [) is the azimuthal angle of the neutron direction with respect to the x' axis. Then a neutron may be randomly chosen, by means of a random choice of its initial coordinates z", y" and direction angle 8, using four times the IBM subroutine RANDU*, and imposing only one condition, that such a neutron must really enter the tube; i.e., with the reference system defined in fig. 1, that the following conditions:
x = Of-lo !p-a/2 < Y < p+aj2 - bj2 < z <
+ bj2
transmission (ratio of neutrons at exit j neutrons at entry) and the average reflection number ii. The reflection losses due to failures in the reflecting properties of the tube walls (index ot reflection R == 0.98) were also taken into account. In order to answer point number 4 we divided the source area in seven zones. The central one has the same size as the sectional area of the tube; all the others were selected as having a similar shape and the same area. The neutron flux at the end of the tube, coming from each one of those zones, was then examined; it is interesting to observe the gain in neutron flux as additional zones are included. 4. Results We have applied our conclusions to the conducting tube which is being built at the Avogadro reactor at Saluggia. This tube was calculated, following Alefeld's suggestions, with the aim of having a cold neutron
should be satisfied. The energy E or wavelength A. of the neutron has been considered as a parameter, 'since it was our aim to study the energy dependence of the properties of the exit distribution. We have taken into consideration the following ten values of A.: 4, 6, 8, 10, 12, 14, 16, 20,25,30 A. For the various values of these wavelengths, we studied the trajectories of a well defined number of neutrons, that is a number proportional to the ordinate given by a normalized maxwellian distribution, at the temperature of the reactor core. However the computing program has been written so that any source temperature can be introduced. 3. Computing program
Tn the program we have constructed a mathematical model of the tube consisting basically of two concentric cylinders, limited by four planes, as can be seen in fig. 1. With the above reference system, all the surfaces have an extremely simple analytical expression. The general flow-chart of the program is shown in fig. 2. For every selected wavelength, the program COmputes the total number of neutrons at the exit window, their final coordinates and directions, the neutron • This subroutine is written for an IBM 360 system computer, and gives a uniform distribution of random 8 digit numbers included between 0 and 1.
I 1\
(A')
Fig. 3. Total spectrum of the neutrons emerging from the tube.
432
A. BOBBIO AND C. VACIAGO
flux with its maximum value at a wavelength of 10 A. For the tube we have: cross-sectional dimensions: a=3cm, b=IOcm; bending radius p=207.41m; AK of the reflecting walls AKN1 = 1.07 X 10- 2 A -1. 4.1.
neutrons coming from the outer zones of the source, only the slowest ones can be reflected on the tube walls with a reflecting angle smaller than its critical value.
TOTAL FLUX AND TRANSMISSION AT THE EXIT WINDOW
Fig. 3 shows the total number of neutrons emerging from the exit window as a function of the wavelength l. As may be seen, the neutron distribution reaches it maximum value at a wavelength of 12 A, while the guide tube had been calculated to give maximum flux at 10 A wavelength. The shift of the maximum value is obviously to be attributed to the dimensions of the neutron source which are greater than those ofthe tube; this gives rise to a beam divergence larger than that foreseen by the usual hypothesis 1,2) employed in calculations for such a tube. Actually, among the 1 I
, , I
",1
1 I I \
I
0--0--0
1 I
..
1
D-<>-O -.-.
,,
I
\
700
0---0--0
\,
•• .. ..
2 J 4
5
6
\ \ I
0--<>-0 _
1.0
\
\
500
400
ZONE .. 21
D--O--
••
3
~
..
0-0--0
..
4 5 6
~
__
..
••
7
0.9
\
\
"0
Fig. 5. Total neutron transmission through the tube.
r(A)
\
\
liD
7
Input Muwelhan
,,
600
10
ZONE: 1
\
\
\
\
0.8 \
\
0.7
\
\
\
\
\
\
0.6 \
\
\
300
,,
\
0.5
,,
0.4
,,
""
200
0.3
"
0.2 100
0.1
20
30
40
Fig. 4. Neutron exit spectra us a function of the wavelength.
Fig. 6. Neutron transmission through the tube from each source zone.
433
NEUTRON TRANSMISSION STUDY THROUGH SIMPLE BENT CONDUCTING TUBE
Fig. 4 shows the emerging distribution of neutrons issuing from the various source zones. We may observe the shifting of the maximum value versus the higher wavelengths, while passing from the central to the outer zones. Fig. 4a shows the distribution of neutrons emerging from the central zone of the source, the dimensions of which are equal to those of the cross section of the tube. As can be seen, the maximum value stands out very nearly at 10 A wavelength. In fig. 5 the total neutron transmission is plotted for each wavelength. Fig. 6 represents the single transmission curves of neutrons issuing from the various zones of the source. Diagram a) of fig. 6 representing the transmission from the central zone, coincides, within a good approximation, with Alefeld's theoretical deductions!), whereas the other diagrams, as well as the transmission shown in fig. 5 (as above explained) are strictly increasing, owing to the prevalence of very slow neutrons. As can be seen the numerical values of transmission are definitely lower than those obtained by theoretical deduction. This is due to our assumption about the angular divergence of the neutron beam emerging from the source.
along the tube depends, for a given tube geometry, on the wavelength, and on the initial coordinates and direction where Kx , K, and K: are the components of the wave number along the x, y and z axis (see fig. I). Fig. 7 is obtained by plotting the average number of reflections n for each wavelength J.. This curve shows a vertical asymptote for A. -+ O. This may be explained by observing that a fast neutron call only be transmitted through a very high number (increasing function of the neutron velocity) of "garland" reflections along the outer tube wall, at very low angles of incidence and reflection. The curve then decreases to a minimum value, at approximately the predicted optimum wavelength of 10 A, because higher incidence angles, and therefore longer paths between two successive reflections, are made possible. By increasing neutron wavelength - accompanied by higher transmission - ever greater reflection angles become possible this means that, as the percentage of "zig-zag" reflection is raised, the curve is bound to grow. At a very high wavelength the diagram leads to an horizontal asymptote.
4.2. AVERAGE NUMBER OF REFLECTIONS ALONG THE TUBE As was shown l ), the number n of neutron reflections
0---<>-<>
SOURCE: 10HE 1
+-+-+
1B
D--<>--
..
:)
..
5
..
7
111
o
o
14
'3
111
o
B
7
• L - - -__-----.m-----,3oi
111..-
.......
- - ,.....
-
•
/.. (A
Fig. 7. Average number of reflections.
Fig. 8. Average number of reflections for each source zone.
434
A. BODDIO AND C. VACIAGO
•
('!o)
o
8 A
c:
o 10 A __ TOTAL
:5
;0 ~ o 23 A
on
.
o
E c:
~
..
'x
gO
60
30
30
60
150
lBO
Fig. 9. Neutron gain as a function of the wavelength. COLUMNS
4A~
Fig. 10. Neutron intensity for each vertical column of the ex.it window.
NEUTRON TRANSMISSION STUDY THROUGH SIMPLE BENT CONDUCTING TUBE
435
1 2 3 4 1!5 Ei
Fig. II. Neutron intensity for each vertical column as a function of the wavelength.
While in the theory
lim T(A.) is equal to I [where ,l-looo
T(A.) is the transmission function] for any initial coordinates and direction, in reality this function cannot reach the limit value 1, because the asymptotic value of ii involves a decrease in transmission equal to RR at very high wavelengths. Fig. 8 shows the average reflection number for neutrons issuing from the various source zones. Obviously, neutrons coming from the outer zon~s undergo an increasing number of reflections, partIcularly at the highest wavelengths. That is one of the main reasons for the asymptotic shape of the neutron gain diagram, which is shown in fig. 9. 4.3. NEUTRON DISTRIBUTION AT THE EXIT WINDOW The exit window of the tube was divided into six vertical columns measuring 0.5 x 10 em, numbered 1 to 6 going from the concave to the convex wall of the tube. Fig. 10 shows the total neutron intensity predicted for each column. This diagram shows an increasing intensity the nearer one gets to the convex wall, which may be explained by observing that for
fast neutrons, only" garland" reflections are allowed, while slow neutrons may have "zig-zag" reflections as well. Further theoretical confirmation of this may be obtained by looking at figs. 11 and 12. Fig. 11 shows the intensity predicted for each column, as a function of wavelength. Fig. 12 shows the exit window distribution of neutrons coming from each of the source zones. The contribution of the central zone to the intensity of the neutrons in column 6 is 36.76% of the overall intensity in the column, whereas the contribution of the farthest zone only is 4.71 %; the same comparison, made for column I, gives a contribution for the central zone of 20.87%, while the farthest zone reaches 9.75%: this different behaviour may be explained once more by remembering the more uniform distribution of neutrons issuing from" zig-zag" reflections. Fig. 13 is obtained by dividing the exit window into 10 horizontal rows 0.5 x 3 cm large, and plotting the neutron intensity for each row. It can be seen that the distribution is very uniform with a standard deviation of 0.54%.
COLUMNS
ZONE 1
ZONE 3
ZONE 5
ZONE 7
~IT---'---,
ZONE 2
~~ ~~
ZONE 4
ZONE 6
~~ ~~ ~~
~~
Rows
1
R
3
4
Av.r.go valuo
IS
a
II
•
7
11
B 10
1467.50
Standard deviation 0.54 II.
Fig. 12. Distribution at the exit window of neutrons from each
zone.
Fig. 13. Neutron intensity lit the exit window for each horizontal row.
436
A. BOBBIO AND C. VACIAGO
4.4. COLD SOURCE DIMb""NSIONS We supposed the distribution of the incoming neutrons to be uniform throughout the cold source area: whereas the transmitted neutron distribution naturally decreases from the central to the outermost zone, as fig. 14 statistically confirms. The decreasing trend of the slow neutron rate is much lower than that of the fast neutrons (fig. 15). At a 30 A wavel ength the contribution of each zone is statistically the same. If we plot the total intensity of neutrons at the exit window as a function of the could source area, we find an increasing curve leading, approximately, to an horizontal aymptote as can be seen in fig. 9. It will be observed from the diagram that e.g. for a source area of 150 cm2 , no fewer than 92.6% of 10 A neutrons are transmitted. 5. Conclusion To sum up the calculations concerning the neutron space distribution at the exit window of the tube, ~4
it may be observed that the higher the neutron wavelength, the more uniform is the pattern, as shown by the diagrams. As has been said already, the calculations show that fast neutrons can only reach the exit window in a narrow zone adjacent to the convex wall. It could be pointed out that, where a very high neutron intensity is not required, but when a high energy resolution may be extremely useful, the column at the exit window adjacent to the convex wall can be shielded, so that the total energy distribution is abruptly shifted towards higher wavelengths. When the sample dimensions are smaller than those of the exit window of the tube, a pyramidal connection can then be prepared, so that the neutrons issuing from the columns nearest the convex wall (which are for the most part fast) do not reach the sample. As regards the dimensions of the cold source, it may be pointed out that each additional zone of the surface gives rise to a shift in the maximum value of the
A
_sA.
UllIlllDB A ~A. C)Ia A.
6
A
A
10
A
A
14
A
20
A
30
A
4
Po
8
12
IllIIIItM A. bD18 A
BIIIlIIlIiiIOA ~A.
~A
25
Fig. 14. Total intensity of the transmitted neutrons for each source zone.
A
Fig. 15. Intensity of the transmitted neutrons, for each source zone as a function of the wavelength.
NEUTRON TRANSMISSION STUDY THROUGH SIMPLE BENT CONDUCTING TUBE
neutron energy spectrum towards higher wavelengths, while the total intensity of the neutrons is asymptotically increased. We are very grateful to Prof. F. Demichelis constant encouragement and advice. We want thank Prof. H. Maier-Leibnitz for having discussed the results of our calculations with
for her also to kindly us.
437
References 1) B. Alefeld, J. Christ, D. Kukla, R. Scherm and W. Schmatz,
Jul. 294, NP (1965).
2) H. Maier-Leibnitz, Nukleonik 8 (1966) 61. 3) H. Maier-Leibnitz and T. Springer, J. Nucl. Energy
AlB 17 (1963) 211. 4) A. Chaudry and P. K. Bandopadhyay, Nucl. (nstr. and Meth. 68 ([969) 293. 5) A. Fllrnoux, B. Hennion and J. Fagot. Proc. Symp. Neutron inelastic scattering, vol. II, Copenhagen, 1968 (IAEA, Vienna, 1968) p. 353.
91 (1971) 429-437; ©
NORTH-HOLLAND PUBLISHING CO.
NEUTRON TRANSMISSION STUDY THROUGH SIMPLE BENT CONDUCTING TUBE A. BOBBIO and C. VACIAGO Is/III/to di Fisica Sperlllle/l/ale del Poli/ecnico, Torino, Ilaly
Received II August 1970 The neutron distribution at the exit window of a simple bent neutron conducting tube is studied using the Monte Carlo method. The method is also applied to obtain the neutron energy spectra and the transmission on exit from the tube, and to discuss
the optimal size ofa cold neutron source. A statistical law relating the average number ii of neutron reflections inside the tube to the wavelength .A. is obtained in a particular case.
1. Introduction
in the conducting tubes, and particularly on the distribution of fast and slow neutrons at the exit window; the average number of reflections versus the wavelength; the influence of the size of the neutron source, and therefore of the angular beam dispersion. on the neutron exit number and distribution, and on the tube transmission. This information is useful in order to calculate and design the neutron conducting tube, taking into account, with somewhat greater precision, its real physical properties.
The increasing interest in experimental techniques employing cold neutron scattering has shown the importance of neutron optical devices, especially of neutron conducting tubes l - 3 ). These devices allow a very low background of ')I rays and fast neutrons in the measurement area, and give a very high gain in the allowed solid angle and therefore in the transmitted neutron intensity. Neutron conducting tubes have been calculated, from the beginning, using the simplifying hypothesis that a nearly collimated neutron beam will enter the tube; that is, the absolute value of the wave number K is assumed approximately equal to its component K" along the tube axis. This assumption leads to some very easy relations between geometrical and physical parameters of the tube; however, it does not permit one to predict with sufficient accuracy the space and energy distributions of the neutrons at the exit window of the tube. To know such distributions is indeed extremely useful, mainly when one needs to analyse the results of neutron inelastic scattering experiments, or when the dimensiolls of the sample are smaller than the tube exit window. In order to handle such a problem in a realistic way, it is necessary to deal with all possible directions and energies ofthe neutrons entering the tu be, and therefore a statistical choice between them must of course be made: Monte Carlo techniques may be applied to solve our problem. The criteria by which statistically distributed neutrons were chosen, are discussed in section 2. Some of the computing program details are given 1n section 3. The discussion of the results concerning neutron flux and transmission at the exit window, average number of reflections, neutron energy and space distributions, and cold source dimensions will be found in section 4. Monte Carlo study gives results for neutron features
2. Statistical choice of neutrons Before analysing the computing program in detail let us briefly enumerate the information required by our experimental group, working with the cold neutron facilities at the Avogadro Swimming Pool Research Reactor at Saluggia: 1. The most probable space distribution of neutrons of various energies at the end of the tube. 2. The statistical law for the number of neutron reflections inside the tube as a function of the neutron wavelength. 3. The reliability of the calculations based on Alefeld's simplified hypothesis l ) especially concerning the neutron transmission through the tube.
429
z' y
z
Fig. 1. Reference system.
430
A. BOBBIO AND C. VACIAGO
RANDOM SELECTION OF THE NEUTRON ENTRY COORDINATES Xo"n Zo RANDOM SELECTION OF THE NEUTRON DIRECTION fOLLOWING THE LAW P :::
I\! cos 6'
UATION OF THE INITIAL TRAJECTORY
CALCULATION OF THE EffECTIVE INTERSECTION ON THE TUBE WALLS THE INTERSECTION IS
NUMBER AND COORDINATES IN EACH ROW AND COLUMN
Of
THE EXIT WINDOW
CALCULATION OF THE AVERAGE NUMBER Of REFLECTIONS CALCULATION Of TUE TRANSMISSION
Fig. 2. Flow-chart of the computing program.
431
NEUTRON TRANSMISSION STUDY THROUGH SIMPLE DENT CONDUCTING TUBE
4. The optimal size of the neutron source, in order to obtain the maximum neutron flux at the exit window. Neutrons emerging from a source having a crosssectional area of dimensions Z x Y, enter the tube through a window placed at a distance f from the source (fig. 1). The fl ux is supposed to be constant with respect to the local neutron coordinates in the source, while a distribution of the form P = Po cos 8 for the neutron directions is assumed; [) is the azimuthal angle of the neutron direction with respect to the x' axis. Then a neutron may be randomly chosen, by means of a random choice of its initial coordinates z", y" and direction angle 8, using four times the IBM subroutine RANDU*, and imposing only one condition, that such a neutron must really enter the tube; i.e., with the reference system defined in fig. 1, that the following conditions:
x = Of-lo !p-a/2 < Y < p+aj2 - bj2 < z <
+ bj2
transmission (ratio of neutrons at exit j neutrons at entry) and the average reflection number ii. The reflection losses due to failures in the reflecting properties of the tube walls (index ot reflection R == 0.98) were also taken into account. In order to answer point number 4 we divided the source area in seven zones. The central one has the same size as the sectional area of the tube; all the others were selected as having a similar shape and the same area. The neutron flux at the end of the tube, coming from each one of those zones, was then examined; it is interesting to observe the gain in neutron flux as additional zones are included. 4. Results We have applied our conclusions to the conducting tube which is being built at the Avogadro reactor at Saluggia. This tube was calculated, following Alefeld's suggestions, with the aim of having a cold neutron
should be satisfied. The energy E or wavelength A. of the neutron has been considered as a parameter, 'since it was our aim to study the energy dependence of the properties of the exit distribution. We have taken into consideration the following ten values of A.: 4, 6, 8, 10, 12, 14, 16, 20,25,30 A. For the various values of these wavelengths, we studied the trajectories of a well defined number of neutrons, that is a number proportional to the ordinate given by a normalized maxwellian distribution, at the temperature of the reactor core. However the computing program has been written so that any source temperature can be introduced. 3. Computing program
Tn the program we have constructed a mathematical model of the tube consisting basically of two concentric cylinders, limited by four planes, as can be seen in fig. 1. With the above reference system, all the surfaces have an extremely simple analytical expression. The general flow-chart of the program is shown in fig. 2. For every selected wavelength, the program COmputes the total number of neutrons at the exit window, their final coordinates and directions, the neutron • This subroutine is written for an IBM 360 system computer, and gives a uniform distribution of random 8 digit numbers included between 0 and 1.
I 1\
(A')
Fig. 3. Total spectrum of the neutrons emerging from the tube.
432
A. BOBBIO AND C. VACIAGO
flux with its maximum value at a wavelength of 10 A. For the tube we have: cross-sectional dimensions: a=3cm, b=IOcm; bending radius p=207.41m; AK of the reflecting walls AKN1 = 1.07 X 10- 2 A -1. 4.1.
neutrons coming from the outer zones of the source, only the slowest ones can be reflected on the tube walls with a reflecting angle smaller than its critical value.
TOTAL FLUX AND TRANSMISSION AT THE EXIT WINDOW
Fig. 3 shows the total number of neutrons emerging from the exit window as a function of the wavelength l. As may be seen, the neutron distribution reaches it maximum value at a wavelength of 12 A, while the guide tube had been calculated to give maximum flux at 10 A wavelength. The shift of the maximum value is obviously to be attributed to the dimensions of the neutron source which are greater than those ofthe tube; this gives rise to a beam divergence larger than that foreseen by the usual hypothesis 1,2) employed in calculations for such a tube. Actually, among the 1 I
, , I
",1
1 I I \
I
0--0--0
1 I
..
1
D-<>-O -.-.
,,
I
\
700
0---0--0
\,
•• .. ..
2 J 4
5
6
\ \ I
0--<>-0 _
1.0
\
\
500
400
ZONE .. 21
D--O--
••
3
~
..
0-0--0
..
4 5 6
~
__
..
••
7
0.9
\
\
"0
Fig. 5. Total neutron transmission through the tube.
r(A)
\
\
liD
7
Input Muwelhan
,,
600
10
ZONE: 1
\
\
\
\
0.8 \
\
0.7
\
\
\
\
\
\
0.6 \
\
\
300
,,
\
0.5
,,
0.4
,,
""
200
0.3
"
0.2 100
0.1
20
30
40
Fig. 4. Neutron exit spectra us a function of the wavelength.
Fig. 6. Neutron transmission through the tube from each source zone.
433
NEUTRON TRANSMISSION STUDY THROUGH SIMPLE BENT CONDUCTING TUBE
Fig. 4 shows the emerging distribution of neutrons issuing from the various source zones. We may observe the shifting of the maximum value versus the higher wavelengths, while passing from the central to the outer zones. Fig. 4a shows the distribution of neutrons emerging from the central zone of the source, the dimensions of which are equal to those of the cross section of the tube. As can be seen, the maximum value stands out very nearly at 10 A wavelength. In fig. 5 the total neutron transmission is plotted for each wavelength. Fig. 6 represents the single transmission curves of neutrons issuing from the various zones of the source. Diagram a) of fig. 6 representing the transmission from the central zone, coincides, within a good approximation, with Alefeld's theoretical deductions!), whereas the other diagrams, as well as the transmission shown in fig. 5 (as above explained) are strictly increasing, owing to the prevalence of very slow neutrons. As can be seen the numerical values of transmission are definitely lower than those obtained by theoretical deduction. This is due to our assumption about the angular divergence of the neutron beam emerging from the source.
along the tube depends, for a given tube geometry, on the wavelength, and on the initial coordinates and direction where Kx , K, and K: are the components of the wave number along the x, y and z axis (see fig. I). Fig. 7 is obtained by plotting the average number of reflections n for each wavelength J.. This curve shows a vertical asymptote for A. -+ O. This may be explained by observing that a fast neutron call only be transmitted through a very high number (increasing function of the neutron velocity) of "garland" reflections along the outer tube wall, at very low angles of incidence and reflection. The curve then decreases to a minimum value, at approximately the predicted optimum wavelength of 10 A, because higher incidence angles, and therefore longer paths between two successive reflections, are made possible. By increasing neutron wavelength - accompanied by higher transmission - ever greater reflection angles become possible this means that, as the percentage of "zig-zag" reflection is raised, the curve is bound to grow. At a very high wavelength the diagram leads to an horizontal asymptote.
4.2. AVERAGE NUMBER OF REFLECTIONS ALONG THE TUBE As was shown l ), the number n of neutron reflections
0---<>-<>
SOURCE: 10HE 1
+-+-+
1B
D--<>--
..
:)
..
5
..
7
111
o
o
14
'3
111
o
B
7
• L - - -__-----.m-----,3oi
111..-
.......
- - ,.....
-
•
/.. (A
Fig. 7. Average number of reflections.
Fig. 8. Average number of reflections for each source zone.
434
A. BODDIO AND C. VACIAGO
•
('!o)
o
8 A
c:
o 10 A __ TOTAL
:5
;0 ~ o 23 A
on
.
o
E c:
~
..
'x
gO
60
30
30
60
150
lBO
Fig. 9. Neutron gain as a function of the wavelength. COLUMNS
4A~
Fig. 10. Neutron intensity for each vertical column of the ex.it window.
NEUTRON TRANSMISSION STUDY THROUGH SIMPLE BENT CONDUCTING TUBE
435
1 2 3 4 1!5 Ei
Fig. II. Neutron intensity for each vertical column as a function of the wavelength.
While in the theory
lim T(A.) is equal to I [where ,l-looo
T(A.) is the transmission function] for any initial coordinates and direction, in reality this function cannot reach the limit value 1, because the asymptotic value of ii involves a decrease in transmission equal to RR at very high wavelengths. Fig. 8 shows the average reflection number for neutrons issuing from the various source zones. Obviously, neutrons coming from the outer zon~s undergo an increasing number of reflections, partIcularly at the highest wavelengths. That is one of the main reasons for the asymptotic shape of the neutron gain diagram, which is shown in fig. 9. 4.3. NEUTRON DISTRIBUTION AT THE EXIT WINDOW The exit window of the tube was divided into six vertical columns measuring 0.5 x 10 em, numbered 1 to 6 going from the concave to the convex wall of the tube. Fig. 10 shows the total neutron intensity predicted for each column. This diagram shows an increasing intensity the nearer one gets to the convex wall, which may be explained by observing that for
fast neutrons, only" garland" reflections are allowed, while slow neutrons may have "zig-zag" reflections as well. Further theoretical confirmation of this may be obtained by looking at figs. 11 and 12. Fig. 11 shows the intensity predicted for each column, as a function of wavelength. Fig. 12 shows the exit window distribution of neutrons coming from each of the source zones. The contribution of the central zone to the intensity of the neutrons in column 6 is 36.76% of the overall intensity in the column, whereas the contribution of the farthest zone only is 4.71 %; the same comparison, made for column I, gives a contribution for the central zone of 20.87%, while the farthest zone reaches 9.75%: this different behaviour may be explained once more by remembering the more uniform distribution of neutrons issuing from" zig-zag" reflections. Fig. 13 is obtained by dividing the exit window into 10 horizontal rows 0.5 x 3 cm large, and plotting the neutron intensity for each row. It can be seen that the distribution is very uniform with a standard deviation of 0.54%.
COLUMNS
ZONE 1
ZONE 3
ZONE 5
ZONE 7
~IT---'---,
ZONE 2
~~ ~~
ZONE 4
ZONE 6
~~ ~~ ~~
~~
Rows
1
R
3
4
Av.r.go valuo
IS
a
II
•
7
11
B 10
1467.50
Standard deviation 0.54 II.
Fig. 12. Distribution at the exit window of neutrons from each
zone.
Fig. 13. Neutron intensity lit the exit window for each horizontal row.
436
A. BOBBIO AND C. VACIAGO
4.4. COLD SOURCE DIMb""NSIONS We supposed the distribution of the incoming neutrons to be uniform throughout the cold source area: whereas the transmitted neutron distribution naturally decreases from the central to the outermost zone, as fig. 14 statistically confirms. The decreasing trend of the slow neutron rate is much lower than that of the fast neutrons (fig. 15). At a 30 A wavel ength the contribution of each zone is statistically the same. If we plot the total intensity of neutrons at the exit window as a function of the could source area, we find an increasing curve leading, approximately, to an horizontal aymptote as can be seen in fig. 9. It will be observed from the diagram that e.g. for a source area of 150 cm2 , no fewer than 92.6% of 10 A neutrons are transmitted. 5. Conclusion To sum up the calculations concerning the neutron space distribution at the exit window of the tube, ~4
it may be observed that the higher the neutron wavelength, the more uniform is the pattern, as shown by the diagrams. As has been said already, the calculations show that fast neutrons can only reach the exit window in a narrow zone adjacent to the convex wall. It could be pointed out that, where a very high neutron intensity is not required, but when a high energy resolution may be extremely useful, the column at the exit window adjacent to the convex wall can be shielded, so that the total energy distribution is abruptly shifted towards higher wavelengths. When the sample dimensions are smaller than those of the exit window of the tube, a pyramidal connection can then be prepared, so that the neutrons issuing from the columns nearest the convex wall (which are for the most part fast) do not reach the sample. As regards the dimensions of the cold source, it may be pointed out that each additional zone of the surface gives rise to a shift in the maximum value of the
A
_sA.
UllIlllDB A ~A. C)Ia A.
6
A
A
10
A
A
14
A
20
A
30
A
4
Po
8
12
IllIIIItM A. bD18 A
BIIIlIIlIiiIOA ~A.
~A
25
Fig. 14. Total intensity of the transmitted neutrons for each source zone.
A
Fig. 15. Intensity of the transmitted neutrons, for each source zone as a function of the wavelength.
NEUTRON TRANSMISSION STUDY THROUGH SIMPLE BENT CONDUCTING TUBE
neutron energy spectrum towards higher wavelengths, while the total intensity of the neutrons is asymptotically increased. We are very grateful to Prof. F. Demichelis constant encouragement and advice. We want thank Prof. H. Maier-Leibnitz for having discussed the results of our calculations with
for her also to kindly us.
437
References 1) B. Alefeld, J. Christ, D. Kukla, R. Scherm and W. Schmatz,
Jul. 294, NP (1965).
2) H. Maier-Leibnitz, Nukleonik 8 (1966) 61. 3) H. Maier-Leibnitz and T. Springer, J. Nucl. Energy
AlB 17 (1963) 211. 4) A. Chaudry and P. K. Bandopadhyay, Nucl. (nstr. and Meth. 68 ([969) 293. 5) A. Fllrnoux, B. Hennion and J. Fagot. Proc. Symp. Neutron inelastic scattering, vol. II, Copenhagen, 1968 (IAEA, Vienna, 1968) p. 353.