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Straggling effects in a proton recoil neutron spectrometer
Nuclear Instruments and Methods in Physics Research 228 (1985) 417-424 North-Holland, Amsterdam
417
STRAGGLING EFFECTS IN A PROTON RECOIL NEUTRON SPECTROMETER Klaes Hakan BEIMER Department of Reactor Physics, Chalmers University of Technology, S-412 96 Gbteborg, Sweden Received 27 July 1984
The influence of proton straggling on the efficiency and resolution of a combined proton recoil and time-of-flight neutron spectrometer, TANSY, is studied. The spectrometer is designed for 14 MeV neutron diagnostics at JET (Joint European Torus). Energy and angular distributions for 14 MeV protons slowing down in polyethylene foils with various thicknesses are presented. The results indicate that for a foil thickness of 1.24 mg cm -Z about 10% of the protons lose more than twice the average energy loss. Furthermore, the angular straggling leads to a coincidence loss of 4.6 t0.4%.
1 . Introduction This paper is a study of the effects of straggling on protons in thin polyethylene foils used as scattering material in a spectrometer designed for JET (Joint European Torus). The energy loss and the angular spread of the proton distribution for different foil thicknesses are presented and a calculation of the coincidence losses due to angular straggling is made. The calculation is based on the dimensions of the reference spectrometer, TANSY [1] . The proton recoil method [2-5] has been used as a tool for fast neutron spectrometry since the mid-fifties . The use of (n, p) scattering has the advantage over other scattering processes that the cross section is relatively large and its variation with energy is well known. Furthermore, (n, p) scattering gives the highest possible energy transfer from the neutron to the target nucleus . The energy of the recoil proton is proportional to the initial neutron energy for a certain scattering angle . By measuring the recoil proton energy with a detector positioned in a certain direction, the energy of the neutron can be determined . The energy resolution obtained using proton energy measurement is usually a few percent . Its actual size depends on the variation of the angle between the initial neutrons and the recoil proton paths, and on the slowing down of the protons in the scattering material. Resolution can be increased by decreasing the slowing down paths of the protons in the scattering material, but this leads to decreased efficiency. The recoil protons created in the scattering foil lose energy during the passage out of the foil. The energy loss is a quantum mechanical slowing down process and results in a broad proton spectrum. The energy spread is mainly caused by slowing down from different positions in the scattering foil. However, the slowing down process has two side effects, namely energy straggling and angular straggling. The energy straggling, which is proportional to the thickness of the foil, causes a spread of the proton energy. The angular straggling causes a loss of protons which will be compensated for by other protons scattered into the detector. However, this leads to a worse angular resolution which contributes to a further spread of the proton energy. The straggling effect is a well known concept and has often been studied before [6-9] . However, to my knowledge there is no study of the resulting decrease in the resolution of a spectrometer based on the proton recoil method . The reason may be that the resolution is mainly limited by the variation in the proton recoil angles, and so the effects of straggling need not be considered. The angular variation can be compensated for by measurement of the energy of the scattered neutron . The present method is based on a combination of the proton recoil method and the time-of-flight method, and is used in the design study for the spectrometer TANSY. The purpose of TANSY is to measure the energy distribution of neutrons 0168-9002/85/$03.00 © Elsevier Science Publishers B.V . (North-Holland Physics Publishing Division)
418
K.H. Beimer / Straggling effects in a spectrometer
emerging from deuterium-tritium plasmas. Accurate measurements of the neutron spectrum are important in verifying that the neutrons are of thermonuclear origin . From this point of view it is valuable to study the effects of straggling on the resolution and efficiency of TANSY. It is found for the recoil protons that the mean energy loss in a polyethylene foil (1.24 mg cm -z) is 24 keV and that the maximum energy loss is 70 keV. A direct calculation, without considering the energy straggling phenomenon, gives a mean energy loss of 23 keV and a maximum energy loss of 46 keV. The angular straggling in TANSY leads to a coincidental loss of 4.6 ± 0.4% if all angles defined by the proton detector are used . 2. The spectrometer The main parts of the spectrometer are the annular foil, the proton detector and the annularly-situated neutron detector (see fig. 1). The scattering process takes place in the foil and the energy of the incoming neutron is distributed on the scattered neutron and the recoil proton . The basic idea behind this spectrometer is that it can make a simultaneous measurement of the energies of the scattered neutron and the recoil proton . The proton energy is measured directly by a silicon detector. The neutron energy is derived from the time difference between the signals from the proton detector and a neutron detector . Then the sum of the energies of the scattered neutron and the recoil proton determines
Fig. 1 . Geometrical definitions of TANSY . R 0 = 9, R, = 7, D = 30 and R a =1 cm.
K.H. Betmer / Straggling effects in a spectrometer
41 9
the energy of the initial neutron . The accuracy of the measurement depends on the resolution and efficiency of the spectrometer. The resolution is defined as the full width at half maximum of a recorded spectrum created by a monoenergetic neutron source . The efficiency is defined as the number of recorded events per second per neutron flux density unit at the scattering foil. The efficiency and the resolution are functions of several parameters, which depend on the design. The aim of the design study was to obtain a resolution of 100 keV or lower and an efficiency of 10-6 or better . It would have been possible to approach this in various ways, but since it is difficult to derive an expression for the resolution as a function of efficiency the trial and error method was used. For most of the parameters, e.g. the foil thickness, increased efficiency leads to decreased resolution. Both the efficiency and the energy spread of the protons are proportional to the thickness of the foil. However, the loss of coincidences due to angular straggling decreases the proportionality constantly . One set of parameters which determine the limits of efficiency and resolution defines a reference design, TANSY [1] (see fig . 1). It has an annular polyethylene foil with a mean radius of 8 cm, a width of 2 cm and a thickness of 1.24 mg cm -2. Thus, the inner diameter is 7 cm and the outer diameter is 9 cm. The area is 100 cm2 . The proton detector has a radius of 1 cm and is placed on the central axis, 30 cm from the foil. 32 plastic neutron detectors form a ring around the central axis. The mean design angles are 15° for the proton path and 75 for the neutron path. TANSY has an efficiency of 10 -6 counts per neutron per cm?, and its resolution is about 100 keV .
°
3. Slowing down The energy deposited by charged particles when slowing down is dependent on the Coulomb interaction with the atomic electrons and scattering against the atomic nuclei . The elastic nuclear collision is a scattering process which leads to a large energy loss and deflection in each interaction . The role of nuclear collision in the slowing down process has been discussed extensively by Bohr [6] . More recently, Bohr's ideas have been developed and refined by Lindhard, Scharf and Schiott [10]. They have derived a nuclear stopping power curve for heavy ions. For protons this curve shows a maximum value at about 10 keV, and nuclear slowing down is comparable with electronic slowing down below 25 keV. Perkins and Cullen [11] discussed the nuclear elastic scattering process, including its interference with the Coulomb interaction . Their results show that the slowing down due to nuclear large angle interaction for 14 MeV protons is less than 1%o of the electronic slowing down. Because several processes contribute to the slowing down it is convenient to study slowing down at three different energy intervals [12]. The low energy region for protons is below 100 keV and the high region is above 940 MeV . Nuclear interactions occur relatively rarely, but their large effect on the energy of the particles is of importance in the low and high energy regions. The large deflection in each collision means that the coincidence criterion is not normally fulfilled for protons . Therefore, the scattering leads to a sensitivity loss. However, at intermediate energies the charged particles slow down mainly because of Coulomb interaction with the atomic electrons . The stopping formula for this energy region, based on the quantum mechanical model of interaction, is derived in detail by Fano [8] . His model contains corrections for the structure of the inner shells and for high ionisation density. Both effects are small at 14 MeV . Thus, they are neglected in the present study. The average energy loss per unit path length is called the stopping power and is written -
dx PE
N°emZ _~ In( 1mep2 ) -ß2 - ln(I) [
I,
where p = density, No = Avogadro's number, Z = atomic number of the stopping material (for polyethylene Z = 8), M = mol weight of the stopping material, ß = velocity of the proton relative to the velocity of light, I = average excitation potential per electron (for polyethylene I = 51), and m e = rest mass of the electron in energy units.
420
K.H. Beimer / Straggling effects in a spectrometer
The variation of the stopping power in the range 12-16 MeV is small . Therefore, a second order power expansion based on numerical values from eq. (1) gives sufficient accuracy in this range : - dE = 37 .6 - 2 .28(E-14) + 0.146(E-14)2 . pdx Here, E is given in MeV, x in cm and p in mg cm -3. 4. Energy straggling The stopping formulae, eqs . (1) and (2), determine the average energy loss, E°, for a number of protons which traverse a material of thickness x. The energy loss varies around the average value due to statistical fluctuations of the energy loss process . The protons participate in a large number of collisions during the slowing down and lose a certain energy in every collision. The number of collisions and the energy loss per collision are not constant . This results in a distribution of energies, known as energy straggling. The shape of the energy distribution is determined by a parameter related to the mean energy loss, E°, in a material of thickness x and by the maximum energy transfer to an atomic electron in one collision . It has been found by Vavilov [7] that for large values of the parameter the energy straggling is closely approximated by a Gaussian distribution . The energy distribution then takes the form: P(E' E°) dE
2 2 exp(
z E- E°)2 ( 29
where q2 is the variance of the energy straggling distribution, first calculated by Bohr [6] as q2 = 0.157 Zx/M, where E° is the average energy loss in MeV calculated from eq. (1) and x is the penetrated distance. The energy loss distribution of a monoenergetic beam of protons traversing a foil of thickness x is determined by the Gaussian distribution eq. (3) . However, in our case the protons are created inside the foil with a probability independent of the distance between the point where the scattering process takes place and the surface of the foil. Therefore, the energy distribution of protons from the foil is a folding of distributions with penetration distances varying from zero to the thickness of the foil ,JE)=KJxP(E, E° )dEdx, °
(4)
where K is a normalization constant. 5. Angular straggling Another effect of the Coulomb interaction is the deflection of protons traversing a material. The repeated small angle deflection spread along a parallel beam of protons is of importance even for very thin foils. This effect is neglected in spectrometers using only the protons, because the protons lost at one angle interval are compensated for by other protons scattered into the interval . This is also true in a neutron-proton coincidence instrument, as long as the angle is far from the angular boundaries defined by the recoil proton detector and the neutron detectors . However, in the neighbourhood of the boundaries, the coincidence condition causes losses. The recoil protons which are scattered in the direction of the proton detector but are deflected through multiple scattering do not give signals in the proton detector. The protons deflected from outside to inside do not generate corresponding signals in the neutron detectors . Multiple scattering of particles has been carefully studied and different, closely related theories have
421
KH Beimer / Straggling effects in a spectrometer
been published [9,14,15]. In this paper the Molière theory [9], studied by Bethe [13], is used . The distribution function, the relative number of protons scattered to an angle element dB in a cone with the top angle 29 after traversing a thickness x, is given by f(0,
x)0d0=vdv[ f o (v)+B-lfl ( v)+B -Zf 2 (v)] .
The variable v is a function of the distance x, and is given by -1/2 v = 0(0.178 X 10 -6B(1 - ß 2 )Z(Z+ I)x)/(ß4M)] and the coefficient B is evaluated from the transcendental equation B - ln(B) =1n{ [6680(Z + 1) Z113 x] /[ ß 2M(1 + 3 .34a 2 )] ),
. The three functions are: where a = Z/137ß fo(v) =2e -"
O fl(v)= ƒ ( v)l-(v2_1) (1)+Jol[(1_x)eX°Z_1_x(v2_1)jdx~ a
f2(v)=f°(v)l( 2
_2v2+1)(q,2(2)+t'(2))+2 ƒ1[ln lxx _ (2)J
X[(1-x)2ex°Z_1-(v2-2)x -(
2a
-2v2+1)x21d 3 ~, x
where *(1) = 0.4228, ~P(2) = 0.9228 and `y'(2) = 0.3949. The angular distribution of protons with initial direction * in the foil is obtained by the integration of eq. (5) over all distances P,,(0, P)=KfXf( 0, x) 0 dB dx.
(6)
0
The calculation of the losses in TANSY involves a calculation of the losses for all possible angles and scattering positions. The probability that a proton which is scattered towards the proton detector with the angle IP will miss the detector, due to multiple scattering in the foil, is PIP=
fo
~Pas(B,
fde,
where 0 = arctan
Rá p +(
Z
D
2 +rf +rd
rp
cos
4~
l)
2
.
-rp - rp cos ~P l
-2rf rd sinof -(R d _rd)sin
(see figs. 2 and 3). The angular distribution of the initially scattered protons in the foil is given by Psp -cos'Psin~P,
where * = arccos
D(D2
+
rf2
+ rd2
2 rfrd sin Of)
1/2 .
The total number of protons which miss the detector is then Pl = f f Psp(~)Pas(0, ~)dA d dAf de, A f Ad where Af is the foil area and Ad is the detector area.
(8)
42 2
K.H. Beimer / Straggling effects in a spectrometer
6. Numerical method Eq. (8) was solved numerically with the integrals approximated with sums . Changing to dimensionless variables the relative number of protons which miss the detector are P, =
E E E E Lr \ Pspl ,,(Pas)mrdàrd 27rrfdrfdOk4t4em
1
A,A,,P,-21r a
f
where _
P
o
sp -
D Psp
Y . f f k
p rfdrf4k
A point on the detector surface is defined by the coordinates (rd , 0) and a point on the foil by the coordinates (rf, 0k) . Because of the symmetry between the centrally situated detector and the annular foil the proton flux on the detector surface is independent of 0. Therefore, the calculation is only made for = 90° . For each point on the detector a number of directions T from different points on the foil could be defined (see fig. 2). The detector surface seen by the protons with direction * is the projection of the detector surface on to the plane perpendicular to the direction 'k. To make the calculation easier the resulting ellipse was approximated to a circle with the radius equal to the smaller semi-axis of the ellipse (see fig. 3). The relative number of protons which miss the projected detector in a certain angle 0, is Y_(Pas m
A0,,A0l/21T .
In fig. 3 this number of protons is symbolized by the dashed area between the inner
and outer circle. The proton angular distribution was cut off at 3° and the outer circle defines this limit. The computer program for the calculation was written in FORTRAN and was run on an IBM-computer for different values on N, O, P and Q. The determination of the proton angular distribution, Pas , was made on a HP desk computer and the results were transferred to the IBM-computer.
(a)
Foil
(b)
Projected Detector
Fig . 2. Sketch showing the quantities of importance for determining the proton angular distribution in the direction ~P towards the detector. The protons are deflected from their initial scattering direction ~P due to angular straggling in the foil. For clarity, the sketch is not to scale. R o = 9, R, = 7, D = 30 and R d =1 cm . Fig . 3 . The foil and the projection of the detector surface on to a plane perpendicular to a direction ~P . (a) The foil seen by the detector . R o = 9 and R, = 7 cm. (b) The detector seen by protons with direction ~P. The dashed area between the inner and outer circle defines those protons which will miss the detector . R dp is the radius of the projected circle and rp is the projection of r .
K.H. Be :mer / Straggling effects in a spectrometer
423
7. Results Proton energy distributions for different foil thicknesses are shown in fig. 4. The effect of energy straggling determines the slope of the curves . On the assumption that the scattering process takes place with the same probability everywhere in the foil, the proton distribution, disregarding straggling, is rectangular . For example, the energy distribution for 1 .24 mg cm -2 has a cut-off at 47 keV, if straggling is neglected . However, the figure indicates that the protons may lose as much as 70 keV . The conclusion for TANSY is that the mean energy loss is 24 keV, the minimum is 0 keV and the maximum is 70 keV . About 10% of the protons lose more than 47 keV . Proton angular distributions for different foil thicknesses are presented in figs. 5 and 6 . The distributions are determined for ~P = 0. About half of the protons produced in a foil with a thickness of 1 .24 mg cm -2 are deflected less than 0.1° when they leave the foil. The calculation of the coincidence losses, in accordance with eq. (9), gives 4.6 ± 0.4% . The reliability of the results depends on the accuracy of the approximations connected with the method . The proton angular distribution for 1 .28 mg cm -2 was used for every direction ~P. However, the distribution is a function of the average slowing down distance of the protons, and for different directions 5 .0 d
á v N c O
á
1000
4 .5
- Energy Distribution - - Energy Distribution Energy Distribution - - Energy Distribution
4 .0 3 .5
for for for for
.75 mg/cW 1 .24 mgicmz 1 .50 mgicmz 2 .00 mgicm2
900
m
800
y a
700
v
3 .0 2 .5
O V 0
2 .0
á 0
0 .0
z 0
10
20
30
40
50
60
70
80
90
Energy Loss,keV
100 110 120
Iii111111rtT7 Angle - - - Angle Angle - - Angle
Distribution Distribution Distribution Distribution
for for for for
.75 mgicmz 1 .24 mg/cW 1 .50 mgicmz 2 .00 mg/cW
=
600 500 400 300 200 100 0 0 .0 Spatial flngle,degrees
Fig. 4. Energy straggling for 14 MeV protons in polyethylene foils of different thicknesses. The average energy loss according to eq . (2) is 28, 47, 56 and 75 keV, respectively . Fig . 5 . Angular straggling for 14 MeV protons 1n polyethylene foils of different thicknesses . The angle 9 is defined in fig. 2 . 100 90 N C O
t ó
m
E z v d V N
árn m c
80 70 60 50 40
Angle Angle Angle Angle
Distribution Distribution Distribution Distribution
for for for for
.75 mgicmz 1 .24 mg/cW 1 .50 mg/cm' : 2 .00 mg/cm
30 20 10 0
Fiiiiliiiiliiii il 0 .0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 Spatial Rngle,degrees
iiiliiir 0 .7
0 .8
Fig. 6 . Integrated angular straggling. The figure shows the relative number of protons inside the angle B, defined in fig. 2 .
424
K.H. Beimer / Straggling effects :n a spectrometer
from the foil the distances vary between 1.26 and 1 .31 mg cm -2. Calculations with the two foil thicknesses showed that the results differed by about 14%. Another approximation connected with the method is that the coincidence losses were calculated with the projected radius equal to the small semi-axis of the ellipse. This means that the result of the calculation gives an upper limit to the correct value. In order to estimate a lower limit, the losses were calculated with the projected radius equal to the radius of the detector . The results from the two calculations differed by less than 3 .5%. Furthermore, the accuracy of the summations in eq. (9) affect the result. For N = 100, O = 5, P = 9, Q = 50 and R = 400 the summations lead to a truncation error of less than 1% . Finally, the errors connected with the determination of the proton angular distribution are negligible compared to the errors mentioned above. The truncation error and the 3° limit for deflection together affect the result by less than 1%. I want to thank Prof . Nils Gbran Sjöstrand, head of the department, and all my colleagues at the department for valuable discussions and help during this work . Discussions with Tony Elmroth at the Department of Mathematics are greatly appreciated. Finally, I thank John McDonald at the Main Library and Linda Schenck at the Department of the English Language, University of Göteborg, for revising the English text. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
G . Grosshoeg et a] ., Combined proton-recoil and neutron time-of-flight spectrometer for 14 MeV neutrons, ISSN 99-0358116-5, JET-JB2-9008 (Chalmers University of Technology Goteborg Sweden, 1983) . J .B . Marion and J .L. Fowler, Fast neutron physics part 1 (Interscience New York, 1960). G . Grosshoeg, Nucl. Instr . and Meth. 162 (1979) 539. G .F . Knoll, Radiation detection and measurement (Wiley, New York, 1979) . N . Tsoulfanidis, Measurement and detection of radiation (Hemisphere Publishing Corporation, 1983) . N . Bohr, Dan . Vid . Selsk . Mat. Fys . Medd . 1 8 (1948) no . 8 . P.V . Vavilov, Ionization losses of high-energy heavy particles, Soviet Physics JETP 5 (4) (1957) English translation . U . Fano, Ann . Rev . Nucl . Sci . 1 3 (1963) 1 . G. Molière, Z . Naturforschg . 3a (1948) 78. J . Lindhard, M . Scharff and H .E. Schiott, Dan. Vid . Selsk . Mat . Fys . Medd. 3 3 (1963) no. 4 . S .T. Perkins and D.E. Cullen, Nucl . Sci. Eng. 77 (1981) 20 . H. Bichsel, Charged particle interactions, in: Radiation dosimetry, vol . 1, 2nd ed ., eds., F.H . Attix et al . (Academic Press, New York 1968). H.A. Bethe, Phys . Rev . 89 (6) (1953). H. Snyder and W.T. Scott, Phys . Rev. 76 (1949) 220. H W . Lewis, Phys . Rev. 78 (1950) 526.
417
STRAGGLING EFFECTS IN A PROTON RECOIL NEUTRON SPECTROMETER Klaes Hakan BEIMER Department of Reactor Physics, Chalmers University of Technology, S-412 96 Gbteborg, Sweden Received 27 July 1984
The influence of proton straggling on the efficiency and resolution of a combined proton recoil and time-of-flight neutron spectrometer, TANSY, is studied. The spectrometer is designed for 14 MeV neutron diagnostics at JET (Joint European Torus). Energy and angular distributions for 14 MeV protons slowing down in polyethylene foils with various thicknesses are presented. The results indicate that for a foil thickness of 1.24 mg cm -Z about 10% of the protons lose more than twice the average energy loss. Furthermore, the angular straggling leads to a coincidence loss of 4.6 t0.4%.
1 . Introduction This paper is a study of the effects of straggling on protons in thin polyethylene foils used as scattering material in a spectrometer designed for JET (Joint European Torus). The energy loss and the angular spread of the proton distribution for different foil thicknesses are presented and a calculation of the coincidence losses due to angular straggling is made. The calculation is based on the dimensions of the reference spectrometer, TANSY [1] . The proton recoil method [2-5] has been used as a tool for fast neutron spectrometry since the mid-fifties . The use of (n, p) scattering has the advantage over other scattering processes that the cross section is relatively large and its variation with energy is well known. Furthermore, (n, p) scattering gives the highest possible energy transfer from the neutron to the target nucleus . The energy of the recoil proton is proportional to the initial neutron energy for a certain scattering angle . By measuring the recoil proton energy with a detector positioned in a certain direction, the energy of the neutron can be determined . The energy resolution obtained using proton energy measurement is usually a few percent . Its actual size depends on the variation of the angle between the initial neutrons and the recoil proton paths, and on the slowing down of the protons in the scattering material. Resolution can be increased by decreasing the slowing down paths of the protons in the scattering material, but this leads to decreased efficiency. The recoil protons created in the scattering foil lose energy during the passage out of the foil. The energy loss is a quantum mechanical slowing down process and results in a broad proton spectrum. The energy spread is mainly caused by slowing down from different positions in the scattering foil. However, the slowing down process has two side effects, namely energy straggling and angular straggling. The energy straggling, which is proportional to the thickness of the foil, causes a spread of the proton energy. The angular straggling causes a loss of protons which will be compensated for by other protons scattered into the detector. However, this leads to a worse angular resolution which contributes to a further spread of the proton energy. The straggling effect is a well known concept and has often been studied before [6-9] . However, to my knowledge there is no study of the resulting decrease in the resolution of a spectrometer based on the proton recoil method . The reason may be that the resolution is mainly limited by the variation in the proton recoil angles, and so the effects of straggling need not be considered. The angular variation can be compensated for by measurement of the energy of the scattered neutron . The present method is based on a combination of the proton recoil method and the time-of-flight method, and is used in the design study for the spectrometer TANSY. The purpose of TANSY is to measure the energy distribution of neutrons 0168-9002/85/$03.00 © Elsevier Science Publishers B.V . (North-Holland Physics Publishing Division)
418
K.H. Beimer / Straggling effects in a spectrometer
emerging from deuterium-tritium plasmas. Accurate measurements of the neutron spectrum are important in verifying that the neutrons are of thermonuclear origin . From this point of view it is valuable to study the effects of straggling on the resolution and efficiency of TANSY. It is found for the recoil protons that the mean energy loss in a polyethylene foil (1.24 mg cm -z) is 24 keV and that the maximum energy loss is 70 keV. A direct calculation, without considering the energy straggling phenomenon, gives a mean energy loss of 23 keV and a maximum energy loss of 46 keV. The angular straggling in TANSY leads to a coincidental loss of 4.6 ± 0.4% if all angles defined by the proton detector are used . 2. The spectrometer The main parts of the spectrometer are the annular foil, the proton detector and the annularly-situated neutron detector (see fig. 1). The scattering process takes place in the foil and the energy of the incoming neutron is distributed on the scattered neutron and the recoil proton . The basic idea behind this spectrometer is that it can make a simultaneous measurement of the energies of the scattered neutron and the recoil proton . The proton energy is measured directly by a silicon detector. The neutron energy is derived from the time difference between the signals from the proton detector and a neutron detector . Then the sum of the energies of the scattered neutron and the recoil proton determines
Fig. 1 . Geometrical definitions of TANSY . R 0 = 9, R, = 7, D = 30 and R a =1 cm.
K.H. Betmer / Straggling effects in a spectrometer
41 9
the energy of the initial neutron . The accuracy of the measurement depends on the resolution and efficiency of the spectrometer. The resolution is defined as the full width at half maximum of a recorded spectrum created by a monoenergetic neutron source . The efficiency is defined as the number of recorded events per second per neutron flux density unit at the scattering foil. The efficiency and the resolution are functions of several parameters, which depend on the design. The aim of the design study was to obtain a resolution of 100 keV or lower and an efficiency of 10-6 or better . It would have been possible to approach this in various ways, but since it is difficult to derive an expression for the resolution as a function of efficiency the trial and error method was used. For most of the parameters, e.g. the foil thickness, increased efficiency leads to decreased resolution. Both the efficiency and the energy spread of the protons are proportional to the thickness of the foil. However, the loss of coincidences due to angular straggling decreases the proportionality constantly . One set of parameters which determine the limits of efficiency and resolution defines a reference design, TANSY [1] (see fig . 1). It has an annular polyethylene foil with a mean radius of 8 cm, a width of 2 cm and a thickness of 1.24 mg cm -2. Thus, the inner diameter is 7 cm and the outer diameter is 9 cm. The area is 100 cm2 . The proton detector has a radius of 1 cm and is placed on the central axis, 30 cm from the foil. 32 plastic neutron detectors form a ring around the central axis. The mean design angles are 15° for the proton path and 75 for the neutron path. TANSY has an efficiency of 10 -6 counts per neutron per cm?, and its resolution is about 100 keV .
°
3. Slowing down The energy deposited by charged particles when slowing down is dependent on the Coulomb interaction with the atomic electrons and scattering against the atomic nuclei . The elastic nuclear collision is a scattering process which leads to a large energy loss and deflection in each interaction . The role of nuclear collision in the slowing down process has been discussed extensively by Bohr [6] . More recently, Bohr's ideas have been developed and refined by Lindhard, Scharf and Schiott [10]. They have derived a nuclear stopping power curve for heavy ions. For protons this curve shows a maximum value at about 10 keV, and nuclear slowing down is comparable with electronic slowing down below 25 keV. Perkins and Cullen [11] discussed the nuclear elastic scattering process, including its interference with the Coulomb interaction . Their results show that the slowing down due to nuclear large angle interaction for 14 MeV protons is less than 1%o of the electronic slowing down. Because several processes contribute to the slowing down it is convenient to study slowing down at three different energy intervals [12]. The low energy region for protons is below 100 keV and the high region is above 940 MeV . Nuclear interactions occur relatively rarely, but their large effect on the energy of the particles is of importance in the low and high energy regions. The large deflection in each collision means that the coincidence criterion is not normally fulfilled for protons . Therefore, the scattering leads to a sensitivity loss. However, at intermediate energies the charged particles slow down mainly because of Coulomb interaction with the atomic electrons . The stopping formula for this energy region, based on the quantum mechanical model of interaction, is derived in detail by Fano [8] . His model contains corrections for the structure of the inner shells and for high ionisation density. Both effects are small at 14 MeV . Thus, they are neglected in the present study. The average energy loss per unit path length is called the stopping power and is written -
dx PE
N°emZ _~ In( 1mep2 ) -ß2 - ln(I) [
I,
where p = density, No = Avogadro's number, Z = atomic number of the stopping material (for polyethylene Z = 8), M = mol weight of the stopping material, ß = velocity of the proton relative to the velocity of light, I = average excitation potential per electron (for polyethylene I = 51), and m e = rest mass of the electron in energy units.
420
K.H. Beimer / Straggling effects in a spectrometer
The variation of the stopping power in the range 12-16 MeV is small . Therefore, a second order power expansion based on numerical values from eq. (1) gives sufficient accuracy in this range : - dE = 37 .6 - 2 .28(E-14) + 0.146(E-14)2 . pdx Here, E is given in MeV, x in cm and p in mg cm -3. 4. Energy straggling The stopping formulae, eqs . (1) and (2), determine the average energy loss, E°, for a number of protons which traverse a material of thickness x. The energy loss varies around the average value due to statistical fluctuations of the energy loss process . The protons participate in a large number of collisions during the slowing down and lose a certain energy in every collision. The number of collisions and the energy loss per collision are not constant . This results in a distribution of energies, known as energy straggling. The shape of the energy distribution is determined by a parameter related to the mean energy loss, E°, in a material of thickness x and by the maximum energy transfer to an atomic electron in one collision . It has been found by Vavilov [7] that for large values of the parameter the energy straggling is closely approximated by a Gaussian distribution . The energy distribution then takes the form: P(E' E°) dE
2 2 exp(
z E- E°)2 ( 29
where q2 is the variance of the energy straggling distribution, first calculated by Bohr [6] as q2 = 0.157 Zx/M, where E° is the average energy loss in MeV calculated from eq. (1) and x is the penetrated distance. The energy loss distribution of a monoenergetic beam of protons traversing a foil of thickness x is determined by the Gaussian distribution eq. (3) . However, in our case the protons are created inside the foil with a probability independent of the distance between the point where the scattering process takes place and the surface of the foil. Therefore, the energy distribution of protons from the foil is a folding of distributions with penetration distances varying from zero to the thickness of the foil ,JE)=KJxP(E, E° )dEdx, °
(4)
where K is a normalization constant. 5. Angular straggling Another effect of the Coulomb interaction is the deflection of protons traversing a material. The repeated small angle deflection spread along a parallel beam of protons is of importance even for very thin foils. This effect is neglected in spectrometers using only the protons, because the protons lost at one angle interval are compensated for by other protons scattered into the interval . This is also true in a neutron-proton coincidence instrument, as long as the angle is far from the angular boundaries defined by the recoil proton detector and the neutron detectors . However, in the neighbourhood of the boundaries, the coincidence condition causes losses. The recoil protons which are scattered in the direction of the proton detector but are deflected through multiple scattering do not give signals in the proton detector. The protons deflected from outside to inside do not generate corresponding signals in the neutron detectors . Multiple scattering of particles has been carefully studied and different, closely related theories have
421
KH Beimer / Straggling effects in a spectrometer
been published [9,14,15]. In this paper the Molière theory [9], studied by Bethe [13], is used . The distribution function, the relative number of protons scattered to an angle element dB in a cone with the top angle 29 after traversing a thickness x, is given by f(0,
x)0d0=vdv[ f o (v)+B-lfl ( v)+B -Zf 2 (v)] .
The variable v is a function of the distance x, and is given by -1/2 v = 0(0.178 X 10 -6B(1 - ß 2 )Z(Z+ I)x)/(ß4M)] and the coefficient B is evaluated from the transcendental equation B - ln(B) =1n{ [6680(Z + 1) Z113 x] /[ ß 2M(1 + 3 .34a 2 )] ),
. The three functions are: where a = Z/137ß fo(v) =2e -"
O fl(v)= ƒ ( v)l-(v2_1) (1)+Jol[(1_x)eX°Z_1_x(v2_1)jdx~ a
f2(v)=f°(v)l( 2
_2v2+1)(q,2(2)+t'(2))+2 ƒ1[ln lxx _ (2)J
X[(1-x)2ex°Z_1-(v2-2)x -(
2a
-2v2+1)x21d 3 ~, x
where *(1) = 0.4228, ~P(2) = 0.9228 and `y'(2) = 0.3949. The angular distribution of protons with initial direction * in the foil is obtained by the integration of eq. (5) over all distances P,,(0, P)=KfXf( 0, x) 0 dB dx.
(6)
0
The calculation of the losses in TANSY involves a calculation of the losses for all possible angles and scattering positions. The probability that a proton which is scattered towards the proton detector with the angle IP will miss the detector, due to multiple scattering in the foil, is PIP=
fo
~Pas(B,
fde,
where 0 = arctan
Rá p +(
Z
D
2 +rf +rd
rp
cos
4~
l)
2
.
-rp - rp cos ~P l
-2rf rd sinof -(R d _rd)sin
(see figs. 2 and 3). The angular distribution of the initially scattered protons in the foil is given by Psp -cos'Psin~P,
where * = arccos
D(D2
+
rf2
+ rd2
2 rfrd sin Of)
1/2 .
The total number of protons which miss the detector is then Pl = f f Psp(~)Pas(0, ~)dA d dAf de, A f Ad where Af is the foil area and Ad is the detector area.
(8)
42 2
K.H. Beimer / Straggling effects in a spectrometer
6. Numerical method Eq. (8) was solved numerically with the integrals approximated with sums . Changing to dimensionless variables the relative number of protons which miss the detector are P, =
E E E E Lr \ Pspl ,,(Pas)mrdàrd 27rrfdrfdOk4t4em
1
A,A,,P,-21r a
f
where _
P
o
sp -
D Psp
Y . f f k
p rfdrf4k
A point on the detector surface is defined by the coordinates (rd , 0) and a point on the foil by the coordinates (rf, 0k) . Because of the symmetry between the centrally situated detector and the annular foil the proton flux on the detector surface is independent of 0. Therefore, the calculation is only made for = 90° . For each point on the detector a number of directions T from different points on the foil could be defined (see fig. 2). The detector surface seen by the protons with direction * is the projection of the detector surface on to the plane perpendicular to the direction 'k. To make the calculation easier the resulting ellipse was approximated to a circle with the radius equal to the smaller semi-axis of the ellipse (see fig. 3). The relative number of protons which miss the projected detector in a certain angle 0, is Y_(Pas m
A0,,A0l/21T .
In fig. 3 this number of protons is symbolized by the dashed area between the inner
and outer circle. The proton angular distribution was cut off at 3° and the outer circle defines this limit. The computer program for the calculation was written in FORTRAN and was run on an IBM-computer for different values on N, O, P and Q. The determination of the proton angular distribution, Pas , was made on a HP desk computer and the results were transferred to the IBM-computer.
(a)
Foil
(b)
Projected Detector
Fig . 2. Sketch showing the quantities of importance for determining the proton angular distribution in the direction ~P towards the detector. The protons are deflected from their initial scattering direction ~P due to angular straggling in the foil. For clarity, the sketch is not to scale. R o = 9, R, = 7, D = 30 and R d =1 cm . Fig . 3 . The foil and the projection of the detector surface on to a plane perpendicular to a direction ~P . (a) The foil seen by the detector . R o = 9 and R, = 7 cm. (b) The detector seen by protons with direction ~P. The dashed area between the inner and outer circle defines those protons which will miss the detector . R dp is the radius of the projected circle and rp is the projection of r .
K.H. Be :mer / Straggling effects in a spectrometer
423
7. Results Proton energy distributions for different foil thicknesses are shown in fig. 4. The effect of energy straggling determines the slope of the curves . On the assumption that the scattering process takes place with the same probability everywhere in the foil, the proton distribution, disregarding straggling, is rectangular . For example, the energy distribution for 1 .24 mg cm -2 has a cut-off at 47 keV, if straggling is neglected . However, the figure indicates that the protons may lose as much as 70 keV . The conclusion for TANSY is that the mean energy loss is 24 keV, the minimum is 0 keV and the maximum is 70 keV . About 10% of the protons lose more than 47 keV . Proton angular distributions for different foil thicknesses are presented in figs. 5 and 6 . The distributions are determined for ~P = 0. About half of the protons produced in a foil with a thickness of 1 .24 mg cm -2 are deflected less than 0.1° when they leave the foil. The calculation of the coincidence losses, in accordance with eq. (9), gives 4.6 ± 0.4% . The reliability of the results depends on the accuracy of the approximations connected with the method . The proton angular distribution for 1 .28 mg cm -2 was used for every direction ~P. However, the distribution is a function of the average slowing down distance of the protons, and for different directions 5 .0 d
á v N c O
á
1000
4 .5
- Energy Distribution - - Energy Distribution Energy Distribution - - Energy Distribution
4 .0 3 .5
for for for for
.75 mg/cW 1 .24 mgicmz 1 .50 mgicmz 2 .00 mgicm2
900
m
800
y a
700
v
3 .0 2 .5
O V 0
2 .0
á 0
0 .0
z 0
10
20
30
40
50
60
70
80
90
Energy Loss,keV
100 110 120
Iii111111rtT7 Angle - - - Angle Angle - - Angle
Distribution Distribution Distribution Distribution
for for for for
.75 mgicmz 1 .24 mg/cW 1 .50 mgicmz 2 .00 mg/cW
=
600 500 400 300 200 100 0 0 .0 Spatial flngle,degrees
Fig. 4. Energy straggling for 14 MeV protons in polyethylene foils of different thicknesses. The average energy loss according to eq . (2) is 28, 47, 56 and 75 keV, respectively . Fig . 5 . Angular straggling for 14 MeV protons 1n polyethylene foils of different thicknesses . The angle 9 is defined in fig. 2 . 100 90 N C O
t ó
m
E z v d V N
árn m c
80 70 60 50 40
Angle Angle Angle Angle
Distribution Distribution Distribution Distribution
for for for for
.75 mgicmz 1 .24 mg/cW 1 .50 mg/cm' : 2 .00 mg/cm
30 20 10 0
Fiiiiliiiiliiii il 0 .0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 Spatial Rngle,degrees
iiiliiir 0 .7
0 .8
Fig. 6 . Integrated angular straggling. The figure shows the relative number of protons inside the angle B, defined in fig. 2 .
424
K.H. Beimer / Straggling effects :n a spectrometer
from the foil the distances vary between 1.26 and 1 .31 mg cm -2. Calculations with the two foil thicknesses showed that the results differed by about 14%. Another approximation connected with the method is that the coincidence losses were calculated with the projected radius equal to the small semi-axis of the ellipse. This means that the result of the calculation gives an upper limit to the correct value. In order to estimate a lower limit, the losses were calculated with the projected radius equal to the radius of the detector . The results from the two calculations differed by less than 3 .5%. Furthermore, the accuracy of the summations in eq. (9) affect the result. For N = 100, O = 5, P = 9, Q = 50 and R = 400 the summations lead to a truncation error of less than 1% . Finally, the errors connected with the determination of the proton angular distribution are negligible compared to the errors mentioned above. The truncation error and the 3° limit for deflection together affect the result by less than 1%. I want to thank Prof . Nils Gbran Sjöstrand, head of the department, and all my colleagues at the department for valuable discussions and help during this work . Discussions with Tony Elmroth at the Department of Mathematics are greatly appreciated. Finally, I thank John McDonald at the Main Library and Linda Schenck at the Department of the English Language, University of Göteborg, for revising the English text. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
G . Grosshoeg et a] ., Combined proton-recoil and neutron time-of-flight spectrometer for 14 MeV neutrons, ISSN 99-0358116-5, JET-JB2-9008 (Chalmers University of Technology Goteborg Sweden, 1983) . J .B . Marion and J .L. Fowler, Fast neutron physics part 1 (Interscience New York, 1960). G . Grosshoeg, Nucl. Instr . and Meth. 162 (1979) 539. G .F . Knoll, Radiation detection and measurement (Wiley, New York, 1979) . N . Tsoulfanidis, Measurement and detection of radiation (Hemisphere Publishing Corporation, 1983) . N . Bohr, Dan . Vid . Selsk . Mat. Fys . Medd . 1 8 (1948) no . 8 . P.V . Vavilov, Ionization losses of high-energy heavy particles, Soviet Physics JETP 5 (4) (1957) English translation . U . Fano, Ann . Rev . Nucl . Sci . 1 3 (1963) 1 . G. Molière, Z . Naturforschg . 3a (1948) 78. J . Lindhard, M . Scharff and H .E. Schiott, Dan. Vid . Selsk . Mat . Fys . Medd. 3 3 (1963) no. 4 . S .T. Perkins and D.E. Cullen, Nucl . Sci. Eng. 77 (1981) 20 . H. Bichsel, Charged particle interactions, in: Radiation dosimetry, vol . 1, 2nd ed ., eds., F.H . Attix et al . (Academic Press, New York 1968). H.A. Bethe, Phys . Rev . 89 (6) (1953). H. Snyder and W.T. Scott, Phys . Rev. 76 (1949) 220. H W . Lewis, Phys . Rev. 78 (1950) 526.