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The Kiel University fast-chopper neutron time-of-flight spectrometer: Monte Carlo model and experimental test
148
Nuclear Instruments and Methods in Physics Research A249 (1986) 148-154 North-Holland, Amsterdam
THE KIEL UNIVERSITY FAST-CHOPPER NEUTRON TIME-OF-FLIGHT SPECTROMETER: MONTE CARLO MODEL AND EXPERIMENTAL TEST Bernd ASMUSSEN and Hans Georg PRIESMEYER Institut für Reine und Angewandte Kernphysik der Universität Kiel, c/o GKSS Forschungszentrum GmbH, Max Planck Str. 1, D-2054 Geesthacht, FRG
Received 25 February 1986 The Kiel University fast-chopper has been simulated by a Monte Carlo model taking into account the geometric outlines of the system, neutron leakage through rotor edges and neutron beam divergence . The calculated results have been verified by experimental investigations . Special emphasis was laid on the very sensitive dependence of the measured neutron time-of-flight spectra on the rotor displacement from the collimator axis, which is also well reproduced by the calculations. Therefore the model is regarded to be authentic to derive a better estimate for the resolution function of the spectrometer, which is needed for measurement tasks like neutron resonance spectroscopy and shape analysis . The FORTRAN programs permit achievement of the best spectrometer adjustment to a given experimental problem. They may be modified to calculate different system geometries .
1. Introduction Since the first experiments of Fermi, chopper designs with a vertical axis of rotation and slit systems grooved horizontally into the rotor material have proven to fulfil the demands of high background suppression and effective neutron beam chopping. When used on a powerful steady state neutron source, these types of choppers are still competitive with other pulsed neutron sources below about 100 eV . Their importance will increase when they are used on a spallation neutron source to shorten the neutron pulse length for epithermal neutron work. Many of the early papers describe the design and operation considerations of the so-called "fast chopper" [11 for neutron energies between 1 eV and 2 keV. Among the properties of a fast chopper, the cutoff or transmission function and the energy resolution are the most important parameters. These properties can, in principle, be calculated easily only when well collimated parallel neutron beams and perfect neutron-absorbing properties of the rotor material are assumed. A fast chopper time-of-flight system consists of a number of static collimators with the rotor in between. In order to achieve transmissions as a function of time-of-flight, or energy, which are equal to unity over a wide range, the slit geometry in the "rotating collimator" must fit the neutron trajectories in the moving system . These can very well be approximated by assuming parabolas. For epithermal neutrons this leads to "cigar-shaped" slits. Early calculations [2] show that an ideal slit of parabolic form has a rather broad energy range of 100` transmission, but differs from the realis0168-9002/86/$03.50 C Elsevier Science Publishers B.V . (North-Holland Physics Publishing Division)
tic case, which exhibits a peaked behaviour. In this early work, the transmission function was calculated without taking into consideration the influence of the collimators . It is necessary to investigate the realistic spectrometer system under the assumption of beam collimation, beam divergence and "grey" edges of the chopping rotor in order to understand such empirical findings as the shaping of the neutron time-of-flight spectrum as a function of the displacement of the rotor relative to the position of the collimators . Finally it is necessary to arrive at an energy resolution function, which needs to be known for the purpose of neutron resonance shape analysis from cross section measurements. In order to check the calculations by experiments, the Kiel University fast-chopper neutron spectrometer performance and geometrical outlines have been taken for a Monte Carlo simulation, which allows to study the influence of several parameters - omitted in earlier publications - on the neutron spectrum, the cutoff function and on the energy resolution . The method and its results are discussed and compared to measurements . 2. The numerical modeling of the TOF spectrometer The influence of the rotor-collimator system on the neutron beam intensity is shown by Monte Carlo calculations. In order to be able to check the theoretical results by experiments, the geometry of the Kiel University neu-
B . Asmussen, H. G. Priesmeyer / Fast-chopper neutron TOFspectrometer COLLIMATOR I
ROTOR
COLLIMATOR II
149
COLLIMATOR III
*b NEUTRON TRAJECTO RY®y=ax
--- do=
"CHOPPING POINT"
Fig. 1 . Diagram of the chopper TOF system (without the detector), top view. tron chopper spectrometer has been assumed to derive the necessary input parameters for the calculations. The neutron trajectory y = ax + b (cf. fig. 1) is determined by the parameters a, b and $o . a and b define the slope and intercept of the neutron path in the laboratory system, and S6o is the position of the rotor at the time instant when the neutron passes the flight path zero or so-called "chopping point" of the system . For the calculation these parameters are taken from a set of equally distributed random numbers. In this way a continuous neutron beam which completely covers the whole cross section area of the input collimator isotropically is represented . Every neutron path determined has its own transmission W, which depends on the distance s traversed in the bulk material of rotor and collimators having a macroscopic cross section T, W(E) = exp( - -Y i (E)s) . The transmission W is used as a weight or probabil-
ity factor for the specific neutron trajectory. The neutron beam intensity at the exit of the spectrometer is then represented by the sum of the weights as a function of the x and y coordinates. In the following we assume a clockwise rotating coordinate system with angular frequency w. The center of this rotation is the rotor axis . Fig. 2 gives the actual geometric outlines of the rotor-collimator system, which were used as input for the calculations. It is a horizontal cut through the middle slit of the system, which has a total of seven identical parallel slits. The slit boundaries in the collimators and in the marginal parts of the rotor are made of nickel, the center part of the rotor consists of depleted uranium. The microsopic total cross sections of U and Ni have been used for the calculations . They were taken from ref. [3]. The slit boundaries within the rotor have been described by ten piece-wise linear portions as given in table 1, defining an approximate cigar-shaped geometry.
y'=cx'+d COLLIMATOR I
ROTOR COLLIMATOR II
COLLIMATOR III
forxi
(2)
The collimators may be displaced laterally which may be considered in the calculation . We have introduced the following approximations : Table 1 Geometrical parameters of the Kiel chopper rotor used as input for the calculation Region
10
ROTOR I detailed)
02
Fig. 2. Geometrical outlines of the chopper system (rotor and collimator I, 11 and III, in mm) .
I II I1I IV V
XI XII XIII XIV XV
x, [nun]
xz [mm]
-158 -150 -80 80 150
-150 -80 80 150 158
0 -0 .00571 0 0.00571 0
-158 -150 -80 80 150
-150 -80 80 150 158
0 0.00571 0 -0 .00571 0
c
d [mm]
-0.1 -0.957 -0.5 -0.957 -0.1 0.1 0.957 0.5 0.957 0.1
15 0
B. Asmussen, H. G. Priesmeyer / Fast-chopper neutron TOFspectrometer
a) if the weight factor is lower than 10 -3 , it is set to zero ; b) the angles of divergence a are small so that cos a =1 and sin a = a; c) sin¢ =,o for small rotor displacements 0. The second assumption is a consequence of the collimator geometries ; the correctness of the third assumption is later verified by the calculation itself. The calculation of the point of intersection of the neutron trajectory with the slit boundaries inside the rotor is simplified by considering a rotor-fixed coordinate system . The laboratory and rotor frames are connected by the following transformations : y= -x'¢+y',
x = x'+Y'I>,
(3)
where the primed quantities refer to the rotating system. The slit boundaries within the rotor are described by ten piece-wise linear portions of the form y'=ex'+d for X1 :!_-X ' :!~- X2, with the numerical values given in table 1. A neutron of velocity v on a trajectory y = ax + b will cross the ordinate y in the laboratory system at a defined rotor angular position 00 . The position for the neutron being at an arbitrary point x is then given by W
where w is the rotor angular velocity . Inserting eq. (4) into the transformation eqs. (3) and using eq . (5) one gets the slit-boundary equation in the laboratory system : 2
1
W J(d02 1 + Cq)p + C-q0 V
v
_)x2 W_
U
-<
x -<
x2 .
- cdOo +dq)o +d, J
(6)
The values of x12 may be found by transforming the boundaries (x,, y, (x,)) i =1, 2 into XI
w =x ;+Yk(xr)((po + -xr), v
i=1,2 .
(7)
Here the approximation = 1~o+ V x = Po+
V x,
(7a)
was used. If the neutron trajectory intersects the slit boundary, we can write in the laboratory frame the following: Yk =ax+b.
W
Q= - (2doo-bc)-a+c-tpo(l+ac),
R=duo-bcoo -b+d . Eq. (8) has two solutions for Q2
Q_
XS,
2P
x52
4P 2
Q2
>-
4PR :
R - p,
= - Q + 2P
A point of intersection exists if the following conditions hold : Q2 >-4PR and xl _x 51 -x 2 , (10) Q2 >- 4PR and x1 5 x 52 -< x2 . It can be shown that the expression for P always has a nonzero value, as follows. A neutron can only pass a rotor slit if its velocity v is large compared to the tangential velocity of the rotor circumference V T: v T/v«1 .
r v
v2
v
since d « r. On the other hand, I ac I < 10 -4 , so that d
2
vw2
«
Û (1 + ac),
(11)
and P =# 0. For reasons of clarity the calculations are carried out in the rotating (x',y') system, using the back-transformations x' = x - Yo, Y'=x$+Y .
+(2d¢ow-cdw+c-gpo )x v v
for x1
W2
P=dz - = (1+ac), v
Using v = wr one finds
$ = ~5o + v x,
Yk=
We introduce the abbreviations
Using the approximation (7a) one gets
Y I=
WZ
-)X W
b z + -lx v 1 v
+00(1+26`=')x'+ b v b and d are always of the same order of magnitude so that from eq . (11) we get b-+`-'>0 . V V2
The trajectories of neutrons in the right-hand rotating system of the considered case may therefore be approximated by parabolas with positive bends. The bends decrease with increasing energy (see fig. 3) . A FORTRAN program has been written to calculate
B. Asmussen, H.G. Priesmeyer / Fast-chopper neutron TOFspectrometer El E2 >El
Fig. 3. Neutron trajectories through the rotor. points of intersection of neutron trajectories randomly distributed in an interval of parameters defined by the given geometry - in this case the outlines of the Kiel fast-chopper TOF spectrometer. 3. Neutron transmission and cutoff behaviour of a chopper system The most important quantity of a time-of-flight spectrometer system is its energy resolution function, which is difficult to determine experimentally. A very intense, monoenergetic neutron source, the energy width of which is very small as compared to the resolution width of the spectrometer, would be needed for such an experimental investigation . Therefore determinations of the resolution function rely on calculations under certain assumptions. On the other hand, the model used in this work yields results which may be directly compared to experiments. These are the chopper transmission and the cutoff function of the system and their dependence on quantities like the neutron energy, rotor speed and relative position of the static collimators to the rotor slits. The transmission of the chopper is defined as the ratio of the neutron intensity at given energy E to the intensity of the high-energy neutron component, which traverses the rotor slits undisturbed : T(E) = I(E)/I(EmaJ
(12)
The transmission depends on the speed of rotation of
Z
T(E)=0, T(E) = e_T2-8T1 +'; T,, T(E)=1- ;T12,
Ero
(13)
T(E) = 1 ;
m n Wr2 _ st 2E s, + 2s,
with the definitions used : cutoff energy : E~ = 2m W2r4 > z ( 32 + sl) transition energy : E,r =
8m W2r4
(23 2 - sl ) critical energy :
z
(15)
2m W2r4 Ecr = ( ( 16 ) z' 32-31) and the neutron mass m , the slit width at the rotor edge sl, the central slit width s2 , the rotor radius r, the angular velocity of the rotor W.
i
ln
N Z Q F-
E
0.8-
_o
N
the chopper and is therefore calculated for a fixed value of this quantity. For fast neutrons the transmission is equal to 1, but declines from the "critical" energy EST via a transition region downwards to the "cutoff" energy E., below which the transmitted intensity is zero . Earlier calculations of transmission and cutoff functions have considered parallel neutron beams and complete neutron absorption of the rotor and collimator materials [4,51. The work presented here includes beam divergence and incompletely absorbing ("grey") rotor and collimators. In the ideal case of totally absorbing material, the .utoff function of a rotor with parabolic slits in a parallel neutron beam may be described by the following expressions [6] :
0.4 1
0
f
1.6
3.2
NEUTRONENERGY
4.8
64
[eV]
8.0
9.6
11 .2
12 .8
14 .4
Fig. 4. Chopper transmission as a function of neutron energy : (1) realistic geometry, "black" system ; (2) parabolic slits, "black" system.
152
B . Asmussen, H. G. Prlesmeyer / Fast-chopper neutron TOFspectrometer
5 4 3
0
200
400
600
800
1000
2000
2600
CHANNEL NO
0
ui 0
0
160
320
(ROTORSPEED ) 2 .10 1
480
6
640
800
960
1120
min -2 1
Fig. 5 . Dependence of the cutoff energy on rotor speed. (1) "black" system, (2) "grey" system . The Monte Carlo model was first tested against the functional relationships (13) . The result is shown in fig. 4. For the given geometry of the Kiel chopper: s 1 = 0.2 mm, s 2 =1 .0 mm, r =158.0 mm, at a rotor speed of 5000 rpm corresponding to an angular velocity of w = 523.6 s-1 , one finds Eco = 2.48 eV, E« = 4.41 eV, Ecr _ 5.59 eV . The deviation of the Monte Carlo result from the exact formula is due to the fact that the rotor slits are not parabolic, but composed of piece-wise linear parts. When the neutron absorption properties of the rotor and collimator materials are included into the calculation, the cutoff energy is shifted to lower values . Fig. 5 shows its dependence on the rotor speed for both the "black" and the "grey" system . The linear regression of the calculated points for the "black" system yields Eco [eV] = 9 .89
x 10-8U2
(U in rpm),
w z z x
s z
ô U
8000 6400
3200 1600 0 CHANNEL NO
Fig. 6. Time-of-flight spectra for different rotor-collimator displacements. 12
z 08
F N z
in good agreement to the ideally absorbing parabola approximation E,, [eV] = 9 .92 E~o[eV] = 4.61
x 10 -8 U2 ,
from the Monte Carlo results. The relative rotor-to-collimator position d 0 can be adjusted for the Kiel chopper with a precision of 0.01 mm . Neutron time-of-flight spectra have been calculated and measured as a function of the rotor displacement relative to the collimators. It is well known that the form of the spectrum depends in a very sensitive way on d0. The Monte Carlo calculation is a good means to study this dependence . In fig. 6 three open-beam time-of-flight spectra for different displacements d0 are displayed. They were measured with 1 inch thick Lib glass scintillator detectors, type NE912, for which the energy-dependent ef-
0
12
x 10 -8 U2 ;
whereas for the realistic "grey" case one gets
04
ô
08
1
a 04
Za a
0
2
0
4
2
d o =0mm
6
8
1
12
ô
12
14
_
3, 08
10
[ eV 1
NEUTRONENERGY
2
04
r
do
0 0
2
4
NEUTRONENERGY
8
6 [ eV
1
10
0.20 Mm 12
14
Fig. 7. Cutoff function at 5000 rpm for different rotor-collimator displacements. (1) Measurement, (2) Monte Carlo calculation, (3) ideal chopper system.
153
B. Asmussen, H. G. Priesmeyer / Fast-chopper neutron TOFspectrometer
ficiency in this range may be described by e(E)=1- exp { -7 .48/ 7 .481E[eV] ) . The primary neutron spectrum O(E) is proportional to 1/E. If the measured intensity I*(E) is weighted by e(E) and O(E):
+0.20 +0.100-
2
i
a
I(E) = I e(E)E the transmission may be found from eq. (12). Without loss of accuracy I(E= 15 eV) was taken as denominator in eq. (12). Fig. 7 shows the cutoff functions of a "grey" rotor-collimator system at 5000 rpm rotor speed for do = +0.17 mm, d o = 0.00 mm and do = -0 .20 mm. Measurements (curves (1)) are well approximated by the calculations (curves (2)), but approach the ideal case (curves (3)) only for the rotor-collimator slit axis difference d o = -0.20 mm. (d o = 0 is the position where the maximum integral fast neutron and gamma-ray count rate is measured; positive values of do describe a displacement of the rotor in the direction indicated in fig. 1 .) 4. The energy resolution of the chopper The total energy resolution capability of a TOF spectrometer is determined by flight-path and flight-time uncertainties; in practice there may also be time-of-flight delays caused by neutron scattering from boundaries of the flight path. We will restrict ourselves here to consider the form of the neutron burst produced by the system . In a FORTRAN program we calculate for given neutron energy E and rotor speed U the time dependence of the transmitted, divergent neutron beam at the chopping point, which is identical to the rotor center in this case. The collimator slit axes may in general be
-- -0.2
ww
-0,4
w ââJ f ,_ 1 aN -0.6 00 -0.7
Fig . 8. Optimal rotor-collimator displacement as a function of neutron energy.
-.9 - 0.20
-020 -0.16 -012 -0.08 -0 .04 0 +0 .04 do f mm 1 Fig. 9. Time-zero to dependence on the relative rotor-collimator position for a rotor speed of 5000 rpm : (1) collimator I and rotor, (2) complete system as shown in fig . 1 . chosen arbitrarily, but in these calculations collinearity has been assumed . The position of the center of gravity of the transmitted neutron spatial intensity distribution on the y-axis is called d.. We define the case when d, coincides with the collimator axes do as the "optimal position" . Experimentally it may be determined by first aligning the collimators and then shifting the rotor until a maximum neutron intensity for a given energy E is attained . It turns out that the "optimal" displacement dc is energy-dependent in a way shown in fig. 8 for U = 5000 rpm. The calculated distributions show an interesting feature : they may all be fitted by Gaussians, but only one of them is symmetric with respect to to = 0. This is the case when do = d.. In all other cases there is a time shift to which may lead to a systematic error in the energy determination according to 4 E = 0.00065E3/210 . Fig. 9 shows how to depends on the relative rotor-collimator position. Curve (1) shows the result for a system consisting of collimator I and rotor, whereas curve (2) shows how this is modified by taking into account the additional collimators II and III . For the ideal case of a parabola-shaped rotor the fwhm of the neutron time burst Gaussian - generally accepted as the resolution width - may be calculated wr' ,dt=1 .272 so that for the case of the Kiel TOF spectrometer one gets 4 t [ /As] = 7676/U [rpm] . The Monte Carlo result for "black" material is in good agreement 4t [ ps] = 7600/U [rpm]
154
B. Asm ssen, H. G. Pnesmeyer / Fast-chopper neutron TOF spectrometer 1.6 1.2
w a 1w z 0
ô
0 .8
0.4
1-1
r
i
w
0 Im 101 10 100 10 2 3 feV1 NEUTRONENERGY
Fig. 10 . Energy resolution of the Kiel University fast-chopper as a function of neutron energy .
and for the realistic "grey" material one finds At [ Its] = 8850/U [rpm] .
We have also shown that the vertical divergence of the neutron beam, which may in general contribute to the energy resolution, has no influence in our case and can therefore be omitted. Fig. 10 shows the optimal energy resolution for the Kiel University TOF spectrometer . The linear part up to about 30 eV is due to the fact that in this range the rotor speed can be increased until the maximum speed of 12000 rpm is reached ; above 30 eV the E3/2 dependence sets on .
5. Conclusion It has been shown by Monte Carlo calculations and subsequent experimental verifications that a fast-chopper time-of-flight spectrometer system with collimated
neutron beams can be well modeled, so that important quantities like the cutoff function and resolution width may be studied in their dependence on system geometry, neutron energy and rotor speed. We feel encouraged from the results of this work to regard the time width At[Its] = 8850/U[rpm] as a very good approximation to the true resolution width of the chopper system. The resolution function is well represented by a Gaussian . The facility is used to investigate neutron resonances, especially in highly radioactive material which in favorable cases may be analysed by the resonanceshape method for Breit-Wigner parameter determination . This method of analysis requires a good knowledge of the spectrometer resolution, which we think has been achieved by the results of the present work . Detailed information, including the FORTRAN program listings, is contained in ref. [7]. The programs may be adapted to slit geometries other than those of the Kiel fast chopper. This was provided for the possibility to calculate choppers for epithermal neutron beam tailoring at future spallation sources [8]. References
[2] [3] [4]
[6] [7] [8]
R.E. Chnien, in : Neutron Physics and Nuclear Data in Science and Technology vol. 2, ed., S. Cierjacks (Pergamon, New York, 1983). L.M . Bollinger, R.E . Cote and G.E . Thomas, 2nd Conf. on Peaceful Uses of Atomic Energy, Geneva 15 (1958) p. 127. BNL 325, Neutron Cross Sections, 3rd ed., vol. 2, curves. V .I . Mostovoi, M.I . Pevzner and A.P . Tsitovich, Geneva Conf. (1955) p. 640. K.E. Larsson, U. Dahlborg, S. Holmroyd, K. Otnes and R. Stedman, Ark. Fys . 16 (1959) 194 . W. Biel, Kiel, private communication (1965). B. Asmussen, Diplomarbeit, Kiel (1985) and report GKSS 85/F/49. G.S. Bauer, Atomkernenergie 41 (1982) 234.
Nuclear Instruments and Methods in Physics Research A249 (1986) 148-154 North-Holland, Amsterdam
THE KIEL UNIVERSITY FAST-CHOPPER NEUTRON TIME-OF-FLIGHT SPECTROMETER: MONTE CARLO MODEL AND EXPERIMENTAL TEST Bernd ASMUSSEN and Hans Georg PRIESMEYER Institut für Reine und Angewandte Kernphysik der Universität Kiel, c/o GKSS Forschungszentrum GmbH, Max Planck Str. 1, D-2054 Geesthacht, FRG
Received 25 February 1986 The Kiel University fast-chopper has been simulated by a Monte Carlo model taking into account the geometric outlines of the system, neutron leakage through rotor edges and neutron beam divergence . The calculated results have been verified by experimental investigations . Special emphasis was laid on the very sensitive dependence of the measured neutron time-of-flight spectra on the rotor displacement from the collimator axis, which is also well reproduced by the calculations. Therefore the model is regarded to be authentic to derive a better estimate for the resolution function of the spectrometer, which is needed for measurement tasks like neutron resonance spectroscopy and shape analysis . The FORTRAN programs permit achievement of the best spectrometer adjustment to a given experimental problem. They may be modified to calculate different system geometries .
1. Introduction Since the first experiments of Fermi, chopper designs with a vertical axis of rotation and slit systems grooved horizontally into the rotor material have proven to fulfil the demands of high background suppression and effective neutron beam chopping. When used on a powerful steady state neutron source, these types of choppers are still competitive with other pulsed neutron sources below about 100 eV . Their importance will increase when they are used on a spallation neutron source to shorten the neutron pulse length for epithermal neutron work. Many of the early papers describe the design and operation considerations of the so-called "fast chopper" [11 for neutron energies between 1 eV and 2 keV. Among the properties of a fast chopper, the cutoff or transmission function and the energy resolution are the most important parameters. These properties can, in principle, be calculated easily only when well collimated parallel neutron beams and perfect neutron-absorbing properties of the rotor material are assumed. A fast chopper time-of-flight system consists of a number of static collimators with the rotor in between. In order to achieve transmissions as a function of time-of-flight, or energy, which are equal to unity over a wide range, the slit geometry in the "rotating collimator" must fit the neutron trajectories in the moving system . These can very well be approximated by assuming parabolas. For epithermal neutrons this leads to "cigar-shaped" slits. Early calculations [2] show that an ideal slit of parabolic form has a rather broad energy range of 100` transmission, but differs from the realis0168-9002/86/$03.50 C Elsevier Science Publishers B.V . (North-Holland Physics Publishing Division)
tic case, which exhibits a peaked behaviour. In this early work, the transmission function was calculated without taking into consideration the influence of the collimators . It is necessary to investigate the realistic spectrometer system under the assumption of beam collimation, beam divergence and "grey" edges of the chopping rotor in order to understand such empirical findings as the shaping of the neutron time-of-flight spectrum as a function of the displacement of the rotor relative to the position of the collimators . Finally it is necessary to arrive at an energy resolution function, which needs to be known for the purpose of neutron resonance shape analysis from cross section measurements. In order to check the calculations by experiments, the Kiel University fast-chopper neutron spectrometer performance and geometrical outlines have been taken for a Monte Carlo simulation, which allows to study the influence of several parameters - omitted in earlier publications - on the neutron spectrum, the cutoff function and on the energy resolution . The method and its results are discussed and compared to measurements . 2. The numerical modeling of the TOF spectrometer The influence of the rotor-collimator system on the neutron beam intensity is shown by Monte Carlo calculations. In order to be able to check the theoretical results by experiments, the geometry of the Kiel University neu-
B . Asmussen, H. G. Priesmeyer / Fast-chopper neutron TOFspectrometer COLLIMATOR I
ROTOR
COLLIMATOR II
149
COLLIMATOR III
*b NEUTRON TRAJECTO RY®y=ax
--- do=
"CHOPPING POINT"
Fig. 1 . Diagram of the chopper TOF system (without the detector), top view. tron chopper spectrometer has been assumed to derive the necessary input parameters for the calculations. The neutron trajectory y = ax + b (cf. fig. 1) is determined by the parameters a, b and $o . a and b define the slope and intercept of the neutron path in the laboratory system, and S6o is the position of the rotor at the time instant when the neutron passes the flight path zero or so-called "chopping point" of the system . For the calculation these parameters are taken from a set of equally distributed random numbers. In this way a continuous neutron beam which completely covers the whole cross section area of the input collimator isotropically is represented . Every neutron path determined has its own transmission W, which depends on the distance s traversed in the bulk material of rotor and collimators having a macroscopic cross section T, W(E) = exp( - -Y i (E)s) . The transmission W is used as a weight or probabil-
ity factor for the specific neutron trajectory. The neutron beam intensity at the exit of the spectrometer is then represented by the sum of the weights as a function of the x and y coordinates. In the following we assume a clockwise rotating coordinate system with angular frequency w. The center of this rotation is the rotor axis . Fig. 2 gives the actual geometric outlines of the rotor-collimator system, which were used as input for the calculations. It is a horizontal cut through the middle slit of the system, which has a total of seven identical parallel slits. The slit boundaries in the collimators and in the marginal parts of the rotor are made of nickel, the center part of the rotor consists of depleted uranium. The microsopic total cross sections of U and Ni have been used for the calculations . They were taken from ref. [3]. The slit boundaries within the rotor have been described by ten piece-wise linear portions as given in table 1, defining an approximate cigar-shaped geometry.
y'=cx'+d COLLIMATOR I
ROTOR COLLIMATOR II
COLLIMATOR III
forxi
(2)
The collimators may be displaced laterally which may be considered in the calculation . We have introduced the following approximations : Table 1 Geometrical parameters of the Kiel chopper rotor used as input for the calculation Region
10
ROTOR I detailed)
02
Fig. 2. Geometrical outlines of the chopper system (rotor and collimator I, 11 and III, in mm) .
I II I1I IV V
XI XII XIII XIV XV
x, [nun]
xz [mm]
-158 -150 -80 80 150
-150 -80 80 150 158
0 -0 .00571 0 0.00571 0
-158 -150 -80 80 150
-150 -80 80 150 158
0 0.00571 0 -0 .00571 0
c
d [mm]
-0.1 -0.957 -0.5 -0.957 -0.1 0.1 0.957 0.5 0.957 0.1
15 0
B. Asmussen, H. G. Priesmeyer / Fast-chopper neutron TOFspectrometer
a) if the weight factor is lower than 10 -3 , it is set to zero ; b) the angles of divergence a are small so that cos a =1 and sin a = a; c) sin¢ =,o for small rotor displacements 0. The second assumption is a consequence of the collimator geometries ; the correctness of the third assumption is later verified by the calculation itself. The calculation of the point of intersection of the neutron trajectory with the slit boundaries inside the rotor is simplified by considering a rotor-fixed coordinate system . The laboratory and rotor frames are connected by the following transformations : y= -x'¢+y',
x = x'+Y'I>,
(3)
where the primed quantities refer to the rotating system. The slit boundaries within the rotor are described by ten piece-wise linear portions of the form y'=ex'+d for X1 :!_-X ' :!~- X2, with the numerical values given in table 1. A neutron of velocity v on a trajectory y = ax + b will cross the ordinate y in the laboratory system at a defined rotor angular position 00 . The position for the neutron being at an arbitrary point x is then given by W
where w is the rotor angular velocity . Inserting eq. (4) into the transformation eqs. (3) and using eq . (5) one gets the slit-boundary equation in the laboratory system : 2
1
W J(d02 1 + Cq)p + C-q0 V
v
_)x2 W_
U
-<
x -<
x2 .
- cdOo +dq)o +d, J
(6)
The values of x12 may be found by transforming the boundaries (x,, y, (x,)) i =1, 2 into XI
w =x ;+Yk(xr)((po + -xr), v
i=1,2 .
(7)
Here the approximation = 1~o+ V x = Po+
V x,
(7a)
was used. If the neutron trajectory intersects the slit boundary, we can write in the laboratory frame the following: Yk =ax+b.
W
Q= - (2doo-bc)-a+c-tpo(l+ac),
R=duo-bcoo -b+d . Eq. (8) has two solutions for Q2
Q_
XS,
2P
x52
4P 2
Q2
>-
4PR :
R - p,
= - Q + 2P
A point of intersection exists if the following conditions hold : Q2 >-4PR and xl _x 51 -x 2 , (10) Q2 >- 4PR and x1 5 x 52 -< x2 . It can be shown that the expression for P always has a nonzero value, as follows. A neutron can only pass a rotor slit if its velocity v is large compared to the tangential velocity of the rotor circumference V T: v T/v«1 .
r v
v2
v
since d « r. On the other hand, I ac I < 10 -4 , so that d
2
vw2
«
Û (1 + ac),
(11)
and P =# 0. For reasons of clarity the calculations are carried out in the rotating (x',y') system, using the back-transformations x' = x - Yo, Y'=x$+Y .
+(2d¢ow-cdw+c-gpo )x v v
for x1
W2
P=dz - = (1+ac), v
Using v = wr one finds
$ = ~5o + v x,
Yk=
We introduce the abbreviations
Using the approximation (7a) one gets
Y I=
WZ
-)X W
b z + -lx v 1 v
+00(1+26`=')x'+ b v b and d are always of the same order of magnitude so that from eq . (11) we get b-+`-'>0 . V V2
The trajectories of neutrons in the right-hand rotating system of the considered case may therefore be approximated by parabolas with positive bends. The bends decrease with increasing energy (see fig. 3) . A FORTRAN program has been written to calculate
B. Asmussen, H.G. Priesmeyer / Fast-chopper neutron TOFspectrometer El E2 >El
Fig. 3. Neutron trajectories through the rotor. points of intersection of neutron trajectories randomly distributed in an interval of parameters defined by the given geometry - in this case the outlines of the Kiel fast-chopper TOF spectrometer. 3. Neutron transmission and cutoff behaviour of a chopper system The most important quantity of a time-of-flight spectrometer system is its energy resolution function, which is difficult to determine experimentally. A very intense, monoenergetic neutron source, the energy width of which is very small as compared to the resolution width of the spectrometer, would be needed for such an experimental investigation . Therefore determinations of the resolution function rely on calculations under certain assumptions. On the other hand, the model used in this work yields results which may be directly compared to experiments. These are the chopper transmission and the cutoff function of the system and their dependence on quantities like the neutron energy, rotor speed and relative position of the static collimators to the rotor slits. The transmission of the chopper is defined as the ratio of the neutron intensity at given energy E to the intensity of the high-energy neutron component, which traverses the rotor slits undisturbed : T(E) = I(E)/I(EmaJ
(12)
The transmission depends on the speed of rotation of
Z
T(E)=0, T(E) = e_T2-8T1 +'; T,, T(E)=1- ;T12,
Ero
(13)
T(E) = 1 ;
m n Wr2 _ st 2E s, + 2s,
with the definitions used : cutoff energy : E~ = 2m W2r4 > z ( 32 + sl) transition energy : E,r =
8m W2r4
(23 2 - sl ) critical energy :
z
(15)
2m W2r4 Ecr = ( ( 16 ) z' 32-31) and the neutron mass m , the slit width at the rotor edge sl, the central slit width s2 , the rotor radius r, the angular velocity of the rotor W.
i
ln
N Z Q F-
E
0.8-
_o
N
the chopper and is therefore calculated for a fixed value of this quantity. For fast neutrons the transmission is equal to 1, but declines from the "critical" energy EST via a transition region downwards to the "cutoff" energy E., below which the transmitted intensity is zero . Earlier calculations of transmission and cutoff functions have considered parallel neutron beams and complete neutron absorption of the rotor and collimator materials [4,51. The work presented here includes beam divergence and incompletely absorbing ("grey") rotor and collimators. In the ideal case of totally absorbing material, the .utoff function of a rotor with parabolic slits in a parallel neutron beam may be described by the following expressions [6] :
0.4 1
0
f
1.6
3.2
NEUTRONENERGY
4.8
64
[eV]
8.0
9.6
11 .2
12 .8
14 .4
Fig. 4. Chopper transmission as a function of neutron energy : (1) realistic geometry, "black" system ; (2) parabolic slits, "black" system.
152
B . Asmussen, H. G. Prlesmeyer / Fast-chopper neutron TOFspectrometer
5 4 3
0
200
400
600
800
1000
2000
2600
CHANNEL NO
0
ui 0
0
160
320
(ROTORSPEED ) 2 .10 1
480
6
640
800
960
1120
min -2 1
Fig. 5 . Dependence of the cutoff energy on rotor speed. (1) "black" system, (2) "grey" system . The Monte Carlo model was first tested against the functional relationships (13) . The result is shown in fig. 4. For the given geometry of the Kiel chopper: s 1 = 0.2 mm, s 2 =1 .0 mm, r =158.0 mm, at a rotor speed of 5000 rpm corresponding to an angular velocity of w = 523.6 s-1 , one finds Eco = 2.48 eV, E« = 4.41 eV, Ecr _ 5.59 eV . The deviation of the Monte Carlo result from the exact formula is due to the fact that the rotor slits are not parabolic, but composed of piece-wise linear parts. When the neutron absorption properties of the rotor and collimator materials are included into the calculation, the cutoff energy is shifted to lower values . Fig. 5 shows its dependence on the rotor speed for both the "black" and the "grey" system . The linear regression of the calculated points for the "black" system yields Eco [eV] = 9 .89
x 10-8U2
(U in rpm),
w z z x
s z
ô U
8000 6400
3200 1600 0 CHANNEL NO
Fig. 6. Time-of-flight spectra for different rotor-collimator displacements. 12
z 08
F N z
in good agreement to the ideally absorbing parabola approximation E,, [eV] = 9 .92 E~o[eV] = 4.61
x 10 -8 U2 ,
from the Monte Carlo results. The relative rotor-to-collimator position d 0 can be adjusted for the Kiel chopper with a precision of 0.01 mm . Neutron time-of-flight spectra have been calculated and measured as a function of the rotor displacement relative to the collimators. It is well known that the form of the spectrum depends in a very sensitive way on d0. The Monte Carlo calculation is a good means to study this dependence . In fig. 6 three open-beam time-of-flight spectra for different displacements d0 are displayed. They were measured with 1 inch thick Lib glass scintillator detectors, type NE912, for which the energy-dependent ef-
0
12
x 10 -8 U2 ;
whereas for the realistic "grey" case one gets
04
ô
08
1
a 04
Za a
0
2
0
4
2
d o =0mm
6
8
1
12
ô
12
14
_
3, 08
10
[ eV 1
NEUTRONENERGY
2
04
r
do
0 0
2
4
NEUTRONENERGY
8
6 [ eV
1
10
0.20 Mm 12
14
Fig. 7. Cutoff function at 5000 rpm for different rotor-collimator displacements. (1) Measurement, (2) Monte Carlo calculation, (3) ideal chopper system.
153
B. Asmussen, H. G. Priesmeyer / Fast-chopper neutron TOFspectrometer
ficiency in this range may be described by e(E)=1- exp { -7 .48/ 7 .481E[eV] ) . The primary neutron spectrum O(E) is proportional to 1/E. If the measured intensity I*(E) is weighted by e(E) and O(E):
+0.20 +0.100-
2
i
a
I(E) = I e(E)E the transmission may be found from eq. (12). Without loss of accuracy I(E= 15 eV) was taken as denominator in eq. (12). Fig. 7 shows the cutoff functions of a "grey" rotor-collimator system at 5000 rpm rotor speed for do = +0.17 mm, d o = 0.00 mm and do = -0 .20 mm. Measurements (curves (1)) are well approximated by the calculations (curves (2)), but approach the ideal case (curves (3)) only for the rotor-collimator slit axis difference d o = -0.20 mm. (d o = 0 is the position where the maximum integral fast neutron and gamma-ray count rate is measured; positive values of do describe a displacement of the rotor in the direction indicated in fig. 1 .) 4. The energy resolution of the chopper The total energy resolution capability of a TOF spectrometer is determined by flight-path and flight-time uncertainties; in practice there may also be time-of-flight delays caused by neutron scattering from boundaries of the flight path. We will restrict ourselves here to consider the form of the neutron burst produced by the system . In a FORTRAN program we calculate for given neutron energy E and rotor speed U the time dependence of the transmitted, divergent neutron beam at the chopping point, which is identical to the rotor center in this case. The collimator slit axes may in general be
-- -0.2
ww
-0,4
w ââJ f ,_ 1 aN -0.6 00 -0.7
Fig . 8. Optimal rotor-collimator displacement as a function of neutron energy.
-.9 - 0.20
-020 -0.16 -012 -0.08 -0 .04 0 +0 .04 do f mm 1 Fig. 9. Time-zero to dependence on the relative rotor-collimator position for a rotor speed of 5000 rpm : (1) collimator I and rotor, (2) complete system as shown in fig . 1 . chosen arbitrarily, but in these calculations collinearity has been assumed . The position of the center of gravity of the transmitted neutron spatial intensity distribution on the y-axis is called d.. We define the case when d, coincides with the collimator axes do as the "optimal position" . Experimentally it may be determined by first aligning the collimators and then shifting the rotor until a maximum neutron intensity for a given energy E is attained . It turns out that the "optimal" displacement dc is energy-dependent in a way shown in fig. 8 for U = 5000 rpm. The calculated distributions show an interesting feature : they may all be fitted by Gaussians, but only one of them is symmetric with respect to to = 0. This is the case when do = d.. In all other cases there is a time shift to which may lead to a systematic error in the energy determination according to 4 E = 0.00065E3/210 . Fig. 9 shows how to depends on the relative rotor-collimator position. Curve (1) shows the result for a system consisting of collimator I and rotor, whereas curve (2) shows how this is modified by taking into account the additional collimators II and III . For the ideal case of a parabola-shaped rotor the fwhm of the neutron time burst Gaussian - generally accepted as the resolution width - may be calculated wr' ,dt=1 .272 so that for the case of the Kiel TOF spectrometer one gets 4 t [ /As] = 7676/U [rpm] . The Monte Carlo result for "black" material is in good agreement 4t [ ps] = 7600/U [rpm]
154
B. Asm ssen, H. G. Pnesmeyer / Fast-chopper neutron TOF spectrometer 1.6 1.2
w a 1w z 0
ô
0 .8
0.4
1-1
r
i
w
0 Im 101 10 100 10 2 3 feV1 NEUTRONENERGY
Fig. 10 . Energy resolution of the Kiel University fast-chopper as a function of neutron energy .
and for the realistic "grey" material one finds At [ Its] = 8850/U [rpm] .
We have also shown that the vertical divergence of the neutron beam, which may in general contribute to the energy resolution, has no influence in our case and can therefore be omitted. Fig. 10 shows the optimal energy resolution for the Kiel University TOF spectrometer . The linear part up to about 30 eV is due to the fact that in this range the rotor speed can be increased until the maximum speed of 12000 rpm is reached ; above 30 eV the E3/2 dependence sets on .
5. Conclusion It has been shown by Monte Carlo calculations and subsequent experimental verifications that a fast-chopper time-of-flight spectrometer system with collimated
neutron beams can be well modeled, so that important quantities like the cutoff function and resolution width may be studied in their dependence on system geometry, neutron energy and rotor speed. We feel encouraged from the results of this work to regard the time width At[Its] = 8850/U[rpm] as a very good approximation to the true resolution width of the chopper system. The resolution function is well represented by a Gaussian . The facility is used to investigate neutron resonances, especially in highly radioactive material which in favorable cases may be analysed by the resonanceshape method for Breit-Wigner parameter determination . This method of analysis requires a good knowledge of the spectrometer resolution, which we think has been achieved by the results of the present work . Detailed information, including the FORTRAN program listings, is contained in ref. [7]. The programs may be adapted to slit geometries other than those of the Kiel fast chopper. This was provided for the possibility to calculate choppers for epithermal neutron beam tailoring at future spallation sources [8]. References
[2] [3] [4]
[6] [7] [8]
R.E. Chnien, in : Neutron Physics and Nuclear Data in Science and Technology vol. 2, ed., S. Cierjacks (Pergamon, New York, 1983). L.M . Bollinger, R.E . Cote and G.E . Thomas, 2nd Conf. on Peaceful Uses of Atomic Energy, Geneva 15 (1958) p. 127. BNL 325, Neutron Cross Sections, 3rd ed., vol. 2, curves. V .I . Mostovoi, M.I . Pevzner and A.P . Tsitovich, Geneva Conf. (1955) p. 640. K.E. Larsson, U. Dahlborg, S. Holmroyd, K. Otnes and R. Stedman, Ark. Fys . 16 (1959) 194 . W. Biel, Kiel, private communication (1965). B. Asmussen, Diplomarbeit, Kiel (1985) and report GKSS 85/F/49. G.S. Bauer, Atomkernenergie 41 (1982) 234.