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Tungsten moderator of Venetian blinds- and honeycomb-type for the slow positron source on hard synchrotron radiation of SPring-8 storage ring
Nuclear Instruments and Methods in Physics Research A 470 (2001) 44–49
Tungsten moderator of Venetian blinds- and honeycomb-type for the slow positron source on hard synchrotron radiation of SPring-8 storage ring V.V. Plokhoi*, Ya.Z. Kandiev, S.I. Samarin, G.N. Malyshkin, G.V. Baidin, I.A. Litvinenko, V.P. Nikitin Russian Federal Nuclear Center, All-Russian Science and Research Institute of Technical Physics, 456770 Snezhinsk, Russia
Abstract The paper considers designs of moderators where fast positron stopping medium consists of very fine tungsten strips separated by vacuum gaps and the strips are arranged into Venetian blinds- or honeycomb-type structures. Moderator efficiency is evaluated through Monte-Carlo simulations. According to the maximal estimate, the efficiency of conversion of fast positrons into slow ones in the Venetian blinds and honeycomb-type moderators is B5 10@3 for the reasonable thickness of the tungsten foil. If such moderator is used, the intensity of slow positron source on the hard synchrotron of SPring-8 storage ring can reach the level of B5 1010 e+/s. r 2001 Elsevier Science B.V. All rights reserved. PACS: 25.30.h; 61.43.b; 07.85.q Keywords: Positron; Moderator; Synchrotron radiation; Monte-carlo; SPring-8
1. Introduction The paper [1,2] provides results of calculations of efficiency of the mutliwire tungsten moderator proposed for conversion of hard synchrotron radiation from SPring-8 into slow positrons. In this paper, consideration is given to one more, alternative, design of a moderator where the fast positrons stopping medium consists of strips of very thin tungsten foil separated by vacuum gaps, sufficient for drawing out the reemitted positrons from the whole volume of a moderator. Two options for positioning the foil strips are reviewed: *Corresponding author. Fax: +7-35172-30979. E-mail address: [email protected] (V.V. Plokhoi).
parallel, in the form of Venetian blinds, and perpendicular, in the form of a lattice with a honeycomb (cellular) structure. For the time being, quite a number of moderator designs have been published, both of moderators used, or proposed for use in the projects on creation of new positron sources [3–10]. In particular, Venetian blinds-type moderators are used in slow positron sources based on linear accelerators with the energy of accelerated electrons B100 MeV in the United States (LLNL) and Japan (KEK) [11]. In spite of this fact, we are providing results of our investigations on the efficiency of such type of moderators for the slow positron source on hard synchrotron radiation, because efficiency and design of a moderator for every slow positron
0168-9002/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 1 ) 0 1 0 0 6 - 3
V.V. Plokhoi et al. / Nuclear Instruments and Methods in Physics Research A 470 (2001) 44–49
source is determined, to a very large extent, by the spatial and energetic characteristics of the source of fast positrons that are slowed down in a moderator.
2. Target Modeling of the processes of synchrotron radiation interacting with the target, transport of positrons and electrons in the target, was performed through Monte-Carlo simulation using the PRIZMA baseline code that was built to solve the problems of joint transport of photons, neutrons, and charged particles in a medium [12–15]. 2.1. Fast positron source In all calculations presented in this paper we will use the target shown in Fig. 1 as a converter of
synchrotron radiation into the fast positrons beam to evaluate the efficiency of the moderators under consideration. All calculations used the spectrum of synchrotron radiation corresponding to the wiggler magnetic field of 10 T [1,2]. As was shown in Refs. [2,3], the maximal efficiency for the targets considered is reached for the thin Pb-target, oriented at the angle of 5 mrad relative to the direction photon beam flow. We use a similar target 6 10@3 cm thick of tungsten due to high heating (Fig. 1). The transversal size of the target is 10 20 cm2, taking into account the decrease of synchrotron radiation horizontal divergence angle down to 2.8 mrad [1]. Our calculations yield the following characteristics of the fast positron source, normalized with respect to a photon with energy above 1.022 MeV: *
*
*
Fig. 1. Technique for generation of fast positrons by irradiating W-target by hard synchrotron radiation of 10 T wiggler.
45
total yield of positrons from the target – (0.8870.009) 10@2 e+/photon; yield through the top surface (see Fig. 1) above (0.570.004) 10@2 e+/photon; mean positron energy 0.67 MeV.
The positron spectrum obtained in our calculations is shown in Ref. [2]. Fig. 2 shows the angle distribution of fast positrons, emitted from the upper side of tungsten target irradiated by hard synchrotron radiation of 10 T wiggler. Angle distribution of fast positrons, emitted from the lower side of tungsten target is the same.
Fig. 2. Angle distribution of fast positrons, emitted from upper side of tungsten target irradiated by hard synchrotron radiation of 10 T wiggler.
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V.V. Plokhoi et al. / Nuclear Instruments and Methods in Physics Research A 470 (2001) 44–49
3. Moderator 3.1. Moderator design The moderator investigated in this work is a structure comprising several hundreds of strips of tungsten monocrystalline foil B10@3 cm thick situated parallel to each other at a distance of B10@1 cm perpendicular to the surface of the target on both sides (Fig. 3). The overall structure is assembled in a vacuum chamber. Width of the strips is chosen to be not more than B5 10@1 cm in order to efficiently draw reemitted positrons out of the moderator by the electric field. A moderator of such a design is usually called a Venetian blinds-type moderator. Further advancement of this design is to join two
Fig. 3. Venetian blinds-type moderator for slow positrons source on hard synchrotron radiation, and one of the techniques for drawing reemitted positrons out of the moderator using electric field that pushes positrons out of the moderator.
such moderators with the, respectively, perpendicular position of the strips (Fig. 4) in order to increase the efficiency of slowing down fast positrons, while preserving the same thickness and width of the strips. Such a design makes it possible to increase the distance between the strips, while the share of positrons implanted into the moderator remains the same, to achieve more efficient output of reemitted positrons from the moderator. A moderator of such a design has a cellular structure and later on we will be calling it a honeycomb-type (or cell-type) moderator. We shall also observe that in a real design the shape of cells in the honeycomb structure can be arbitrary, depending on the technology of manufacturing such cellular structures. Monocrystalline thin strips can be obtained using the directional recrystallization technique. Apart from this, the strips can be coated by a brush of filament-like crystals (bristles) formed by gaseous phase deposition method, in order to increase the reemitting surface of the strips and to improve its quality. Regular positioning of the strips is necessary in order to achieve high efficiency of the moderator using relatively simple solutions for the problem of drawing positrons reemitted from the surface of the strips out of the moderator by an electric field applied across the target and the moderator. The thickness of strips in the moderator design that we used in the subsequent calculations is 10@3 cm, width of the stripsF5 10@1 cm, distance between the stripsF1 10@1 and 2 10@1 cm. Size of the cells in the cellular-type moderator was taken as equal to 0.1 0.1, 0.2 0.2, and 0.3 0.3 cm2. 3.2. Calculation of the profile of implanted positrons in the moderator
Fig. 4. Honeycomb-type moderator for slow positrons source on hard synchrotron radiation, and one of the techniques for drawing reemitted positrons out of the moderator using electric field that pushes positrons out of the moderator.
In order to choose the efficient thickness of the Venetian blinds- and cellular-type moderators, we have performed a Monte-Carlo simulation of the share of positrons implanted into the moderator, and a distribution of the implanted positrons, along the height of the moderator matching the width of the strips. Modification of the PRIZMA code was used for these calculations. PRIZMA code was modified using the model of generalized
47
V.V. Plokhoi et al. / Nuclear Instruments and Methods in Physics Research A 470 (2001) 44–49 Table 1 Moderator type
Venetian blinds of strips 0.5 20 cm2, 1 mm spacing
Venetian blinds of strips 0.5 20 cm2, 2 mm spacing
Cells of strips 0.5 20 cm2 and 0.5 10 cm2, cell size 1 1 mm2
Cells of strips 0.5 20 cm2 and 0.5 10 cm2, cell size 2 2 mm2
Cells of strips 0.5 20 cm2 and 0.5 10 cm2, cell size 3 3 mm2
Number of implanted positrons (%)
29.1
14.4
50.0
31.0
21.2
Number of positrons reflected from the moderator (%)
26.5
22.5
31.5
27.3
24.4
Number of positrons that escaped through the back side (%)
40.0
57.9
16.3
38.5
50.2
forces of oscillator [16], the model is based on optical data that enable the tracking of history of a positron in tungsten up to the energy of B10 eV. Table 1 presents the results of the calculation of shares of positrons implanted into the moderator, reflected from the moderator, and those escaped through the back (with respect to the target) and lateral sides of the moderator, expressed as percentage of the number of fast positrons flying out of the target. The calculation was performed for various gaps between the strips in Venetian blinds-type moderator, and for various sizes of cells in the honeycomb-type moderator. Fig. 5 shows the results of calculation of the density of distribution of implanted positrons with respect to the height of the upper half (relative to the figure) for both types of moderators. Distance is counted from the edge of the strips that is closest to the target. Results of the implanted positrons distribution density are normalized by one photon from the synchrotron radiation source with the full spectrum. Analysis of these results indicates that the largest number of positrons flying out of the target can be implanted into the honeycomb-type moderator with the cell size 0.1 0.1 cm2. In the cases of a Venetian blinds-type moderator with 0.1 cm step, and of honeycomb-type moderator with cell size 0.2 0.2 cm2, the share of implanted positrons is close to the share of implanted positrons in
Fig. 5. Density of distribution of implanted positrons with respect to the height of Venetian blinds-type tungsten moderator with various steps, and of honeycomb-type moderator with varying cell sizes. The distribution is normalized by one photon from the synchrotron radiation source of the full spectrum.
the previously considered multiwire moderator consisting of 100 layers of tungsten wire 10@5 mm thick [2]. Similar results are obtained for the lower half. It is visible from the presented distributions that considerable increase in the share of implanted positrons can be reached by substantial increase in the height of the moderator (width of strips), but this is undesirable due to the increasing difficulty of drawing reemitted positrons out of the moderator.
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V.V. Plokhoi et al. / Nuclear Instruments and Methods in Physics Research A 470 (2001) 44–49
4. Yield of reemitted positrons into vacuum gaps of the moderator Uniform distribution of thermalized positrons through the depth in the surface layer of the moderator’s strips, combined with smallness of diffusion path of positrons with respect to the thickness of the strips, makes it possible, as it was in Ref. [2], to use the solution of the diffusion problem for the case of planar layer with uniform source and zero condition for positron density on the surface. As earlier, we will assume that the share of reemitted positrons among those that reached the strips’ surface is, approximately, Z ¼ 13 [7,17–19]; therefore, to get an estimate of the current of reemitted positrons, one must multiply the diffusion current by this coefficient. The diffusion path, as earlier, will be estimated through the lifetime of a positron in the moderator’s medium tþ B10@10 s and diffusion coefficient at the temperature T ¼p 300 K that is D+(300) B1.26 cm2/s ffiffiffiffiffiffiffiffiffiffiffiffi [2], i.e. Lþ ¼ Dþ tþ ¼ 1:12 10@5 cm. Thus, the share of thermalized positrons exiting from a strip into the vacuum gap of the moderator is k ¼ 2ZLþ =d, where d is the thickness of a strip. For a strip 10@3 cm thick, this value is equal to k=0.75 10@2. The estimate we obtained holds true for every strip in the moderator and, therefore, for the moderator as a whole. It is easy to see that when the thickness of tungsten wires is equal to the diameter of the wires in the multiwire moderator [2], the share of positrons reemitted into the vacuum gaps (with respect to those implanted into the moderator) in the multiwire moderator is a factor of two higher than in the Venetian blinds- or honeycomb-type moderators.
5. Slow positrons dynamics in the moderator vacuum gaps We are considering a situation when positrons reemitted into vacuum gaps between tungsten strips are moving from tungsten foil to the exterior boundaries of the moderator. Such consideration relies on the model that corresponds to the foil
with non-ideal planar surface or the foil with bristles grown on its surface; therefore, the angular distribution for reemitted and reflected positrons was taken as isotropic in the exterior semi-sphere. The energy of reemitted electrons was taken as equal to tungsten work function for a positron of 2.75 eV [17–19], and scattering on the surface potential was assumed to be elastic. Calculation of the positron dynamics was done numerically, using the differential scheme of the second order of accuracy. Calculations in this section have a model nature and are used to do a comparative estimate of the efficiency of positrons’ extraction from the space between the strips in the Venetian blinds-type moderator, and from the space of vacuum cells in the honeycomb-type moderator where the extraction is performed by an electric field. Venetian blinds almost touch the target by their edges. Injection of the simulated positrons was performed from the foil’s surface at various distances from the target’s surface. A 3D code numerically solving the Poisson equation was used to calculate the electrical fields. The calculated fields were then used as external stationary fields acting upon positrons’ movement during the simulation of the positrons dynamics. After the onset of its movement, if a positron had a collision with the foil surface, that collision was assumed to be random. In the calculations performed for various modifications of the Venetian blinds-type moderator, we determined the parameter nFmean number of positron’s collisions with foil surface until the positron does not fly out of the moderator. The fewer the collisions, the smaller the number of reemitted positrons that would be lost due to scattering on moderator walls and the higher the moderator efficiency. To draw positrons out of the moderator of the mentioned types it must be recommended to have low values of the target’s positive potential which is slightly above the initial energy of the positron. The Venetian blinds-type design enables the efficient extraction of positrons from the moderator and has the lowest probability of positrons loss (the mean number of collisions with the surface of
V.V. Plokhoi et al. / Nuclear Instruments and Methods in Physics Research A 470 (2001) 44–49
the strips until the positron escapes from the moderator is n ¼ 3212 depending on the point of departure of the reemitted positron). For the honeycomb-type design, the mean number of collisions with the cell’s walls where the cell size is 0.1 0.1 cm2 until the positron escapes from the moderator grows substantially up to n ¼ 11240 depending on the point of departure of the reemitted positron.
6. Estimate of the moderator efficiency For a Venetian blinds-type moderator where electric field is used to draw out positrons, and the assumption of small probability of positron’s capture in collision with moderator’s strips (we will call such an estimate as the maximal one), the efficiency of fast-to-slow positrons conversion can reach the value kef ¼ 0:2 10@2 . Intensity of slow positron source on hard synchrotron radiation of SPring-8 will reach the value of 5 1010 e+/s. For a honeycomb-type moderator with the cell size of 0.1 0.1 cm2 a similar maximal estimate yields kef ¼ 0:3 10@2 that approaches the earlier obtained estimate for the multiwire moderator.
8. Conclusion Honeycomb-type moderator is preferable to the Venetian blinds-type moderator: according to the maximal estimate it can reach a fairly high efficiency of conversion of fast positrons into slow positrons B0.4 10@2. If such a moderator is used, the intensity of slow positrons source on hard synchrotron radiation from SPring-8 can achieve the level of B5 1010 e+/s.
49
References [1] M.G. Fedurin, G.N. Kulipanov, N.A. Mezentsev, V.A. Shkaruba, Nucl. Instr. and Meth. A 448 (1–2) (2000) 51. [2] V. Plokhoi, Ya. Kandiev, S. Samarin, G. Malyshkin, G. Baydin, I. Litvinenko, V. Nikitin, Nucl. Instr. and Meth. A 448 (1-2) (2000) 94. [3] A. Ando, V.N. Belyaev, S. Date, H. Kamitsubo, A.V. Kiselev, G.N. Kulipanov, N. Kumagai, N.A. Mezentsev, et al., J. Synchrotr. Radiat. 3 (1996) 201. [4] P.J. Shultz, K.G. Lin, Rev. Mod. Phys. 60 (1988) 701. [5] M.J. Pushka, R.M. Nieminen, Rev. Mod. Phys. 66 (1994) 841. [6] R. Suzuki, Y. Kobayashi, T. Mikado, H. Ohgaki, M. Chiwaki, T. Yamazaki, Hyperfine Interact. 84 (1994) 345. [7] A.P. Mills Jr., E.M. Gullikson, Appl. Phys. Lett. 49 (1986) 1121. [8] S. Okada, H. Sunaga, Nucl. Instr. and Meth. B 56/57 (1991) 604. [9] S. Okada, I! . Kaneko, Appl. Surf. Sci. 85 (1995) 149. [10] D. Taqqu, Helv. Phys. Acta 63 (1990) 442. [11] A. Asami, A. Enomoto, H. Kobayashi, T. Kurihara, K. Nakahara, T. Shidara, Mater. Sci. Forum 105-110 (1992) 1833. [12] Ya.Z. Kandiev, V.B. Krjuchenkov, V.V. Plokhoi, Phys. Plasma 5 (1979) 171. [13] V.V. Plokhoi, Ya.Z. Kandiev, P.L. Volegov, D.D. Batkaev, A.G. Plotnikov, IET 36 (2) (1993) 209. [14] Ya.Z. Kandiev, V.V. Plokhoi, VANT, Mathematical Simulation of Physical Processes Series, issue 2, 1993 p. 70. [15] V.V. Plokhoi, Ya.Z. Kandiev, IET 40 (5) (1997) 25. [16] J.M. Fernandez-Varea, R. Mayol, D. Liljequist, F. Salvat, J. Phys. Condens. Matter 5 (1993) 3593. [17] R. Suzuki, T. Ohdaira, A. Uedono, Y.K. Cho, S. Yoshida, T. Ohshima, H. Itoh, M. Chiwaki, T. Mikado, T. Yamazaki, S. Tanigawa, Investigation of moderators for intense slow positron beams, Jpn. J. Appl. Phys. 37 (1998) 4636. [18] D.M. Chen, K.G. Lynn, R. Pareja, B. Nielsen, Phys. Rev. B 31 (1985) 4123. [19] A. Goodyear, A.P. Knights, P.G. Coleman, J. Phys. Condens. Matter 6 (1994) 9601.
Tungsten moderator of Venetian blinds- and honeycomb-type for the slow positron source on hard synchrotron radiation of SPring-8 storage ring V.V. Plokhoi*, Ya.Z. Kandiev, S.I. Samarin, G.N. Malyshkin, G.V. Baidin, I.A. Litvinenko, V.P. Nikitin Russian Federal Nuclear Center, All-Russian Science and Research Institute of Technical Physics, 456770 Snezhinsk, Russia
Abstract The paper considers designs of moderators where fast positron stopping medium consists of very fine tungsten strips separated by vacuum gaps and the strips are arranged into Venetian blinds- or honeycomb-type structures. Moderator efficiency is evaluated through Monte-Carlo simulations. According to the maximal estimate, the efficiency of conversion of fast positrons into slow ones in the Venetian blinds and honeycomb-type moderators is B5 10@3 for the reasonable thickness of the tungsten foil. If such moderator is used, the intensity of slow positron source on the hard synchrotron of SPring-8 storage ring can reach the level of B5 1010 e+/s. r 2001 Elsevier Science B.V. All rights reserved. PACS: 25.30.h; 61.43.b; 07.85.q Keywords: Positron; Moderator; Synchrotron radiation; Monte-carlo; SPring-8
1. Introduction The paper [1,2] provides results of calculations of efficiency of the mutliwire tungsten moderator proposed for conversion of hard synchrotron radiation from SPring-8 into slow positrons. In this paper, consideration is given to one more, alternative, design of a moderator where the fast positrons stopping medium consists of strips of very thin tungsten foil separated by vacuum gaps, sufficient for drawing out the reemitted positrons from the whole volume of a moderator. Two options for positioning the foil strips are reviewed: *Corresponding author. Fax: +7-35172-30979. E-mail address: [email protected] (V.V. Plokhoi).
parallel, in the form of Venetian blinds, and perpendicular, in the form of a lattice with a honeycomb (cellular) structure. For the time being, quite a number of moderator designs have been published, both of moderators used, or proposed for use in the projects on creation of new positron sources [3–10]. In particular, Venetian blinds-type moderators are used in slow positron sources based on linear accelerators with the energy of accelerated electrons B100 MeV in the United States (LLNL) and Japan (KEK) [11]. In spite of this fact, we are providing results of our investigations on the efficiency of such type of moderators for the slow positron source on hard synchrotron radiation, because efficiency and design of a moderator for every slow positron
0168-9002/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 1 ) 0 1 0 0 6 - 3
V.V. Plokhoi et al. / Nuclear Instruments and Methods in Physics Research A 470 (2001) 44–49
source is determined, to a very large extent, by the spatial and energetic characteristics of the source of fast positrons that are slowed down in a moderator.
2. Target Modeling of the processes of synchrotron radiation interacting with the target, transport of positrons and electrons in the target, was performed through Monte-Carlo simulation using the PRIZMA baseline code that was built to solve the problems of joint transport of photons, neutrons, and charged particles in a medium [12–15]. 2.1. Fast positron source In all calculations presented in this paper we will use the target shown in Fig. 1 as a converter of
synchrotron radiation into the fast positrons beam to evaluate the efficiency of the moderators under consideration. All calculations used the spectrum of synchrotron radiation corresponding to the wiggler magnetic field of 10 T [1,2]. As was shown in Refs. [2,3], the maximal efficiency for the targets considered is reached for the thin Pb-target, oriented at the angle of 5 mrad relative to the direction photon beam flow. We use a similar target 6 10@3 cm thick of tungsten due to high heating (Fig. 1). The transversal size of the target is 10 20 cm2, taking into account the decrease of synchrotron radiation horizontal divergence angle down to 2.8 mrad [1]. Our calculations yield the following characteristics of the fast positron source, normalized with respect to a photon with energy above 1.022 MeV: *
*
*
Fig. 1. Technique for generation of fast positrons by irradiating W-target by hard synchrotron radiation of 10 T wiggler.
45
total yield of positrons from the target – (0.8870.009) 10@2 e+/photon; yield through the top surface (see Fig. 1) above (0.570.004) 10@2 e+/photon; mean positron energy 0.67 MeV.
The positron spectrum obtained in our calculations is shown in Ref. [2]. Fig. 2 shows the angle distribution of fast positrons, emitted from the upper side of tungsten target irradiated by hard synchrotron radiation of 10 T wiggler. Angle distribution of fast positrons, emitted from the lower side of tungsten target is the same.
Fig. 2. Angle distribution of fast positrons, emitted from upper side of tungsten target irradiated by hard synchrotron radiation of 10 T wiggler.
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V.V. Plokhoi et al. / Nuclear Instruments and Methods in Physics Research A 470 (2001) 44–49
3. Moderator 3.1. Moderator design The moderator investigated in this work is a structure comprising several hundreds of strips of tungsten monocrystalline foil B10@3 cm thick situated parallel to each other at a distance of B10@1 cm perpendicular to the surface of the target on both sides (Fig. 3). The overall structure is assembled in a vacuum chamber. Width of the strips is chosen to be not more than B5 10@1 cm in order to efficiently draw reemitted positrons out of the moderator by the electric field. A moderator of such a design is usually called a Venetian blinds-type moderator. Further advancement of this design is to join two
Fig. 3. Venetian blinds-type moderator for slow positrons source on hard synchrotron radiation, and one of the techniques for drawing reemitted positrons out of the moderator using electric field that pushes positrons out of the moderator.
such moderators with the, respectively, perpendicular position of the strips (Fig. 4) in order to increase the efficiency of slowing down fast positrons, while preserving the same thickness and width of the strips. Such a design makes it possible to increase the distance between the strips, while the share of positrons implanted into the moderator remains the same, to achieve more efficient output of reemitted positrons from the moderator. A moderator of such a design has a cellular structure and later on we will be calling it a honeycomb-type (or cell-type) moderator. We shall also observe that in a real design the shape of cells in the honeycomb structure can be arbitrary, depending on the technology of manufacturing such cellular structures. Monocrystalline thin strips can be obtained using the directional recrystallization technique. Apart from this, the strips can be coated by a brush of filament-like crystals (bristles) formed by gaseous phase deposition method, in order to increase the reemitting surface of the strips and to improve its quality. Regular positioning of the strips is necessary in order to achieve high efficiency of the moderator using relatively simple solutions for the problem of drawing positrons reemitted from the surface of the strips out of the moderator by an electric field applied across the target and the moderator. The thickness of strips in the moderator design that we used in the subsequent calculations is 10@3 cm, width of the stripsF5 10@1 cm, distance between the stripsF1 10@1 and 2 10@1 cm. Size of the cells in the cellular-type moderator was taken as equal to 0.1 0.1, 0.2 0.2, and 0.3 0.3 cm2. 3.2. Calculation of the profile of implanted positrons in the moderator
Fig. 4. Honeycomb-type moderator for slow positrons source on hard synchrotron radiation, and one of the techniques for drawing reemitted positrons out of the moderator using electric field that pushes positrons out of the moderator.
In order to choose the efficient thickness of the Venetian blinds- and cellular-type moderators, we have performed a Monte-Carlo simulation of the share of positrons implanted into the moderator, and a distribution of the implanted positrons, along the height of the moderator matching the width of the strips. Modification of the PRIZMA code was used for these calculations. PRIZMA code was modified using the model of generalized
47
V.V. Plokhoi et al. / Nuclear Instruments and Methods in Physics Research A 470 (2001) 44–49 Table 1 Moderator type
Venetian blinds of strips 0.5 20 cm2, 1 mm spacing
Venetian blinds of strips 0.5 20 cm2, 2 mm spacing
Cells of strips 0.5 20 cm2 and 0.5 10 cm2, cell size 1 1 mm2
Cells of strips 0.5 20 cm2 and 0.5 10 cm2, cell size 2 2 mm2
Cells of strips 0.5 20 cm2 and 0.5 10 cm2, cell size 3 3 mm2
Number of implanted positrons (%)
29.1
14.4
50.0
31.0
21.2
Number of positrons reflected from the moderator (%)
26.5
22.5
31.5
27.3
24.4
Number of positrons that escaped through the back side (%)
40.0
57.9
16.3
38.5
50.2
forces of oscillator [16], the model is based on optical data that enable the tracking of history of a positron in tungsten up to the energy of B10 eV. Table 1 presents the results of the calculation of shares of positrons implanted into the moderator, reflected from the moderator, and those escaped through the back (with respect to the target) and lateral sides of the moderator, expressed as percentage of the number of fast positrons flying out of the target. The calculation was performed for various gaps between the strips in Venetian blinds-type moderator, and for various sizes of cells in the honeycomb-type moderator. Fig. 5 shows the results of calculation of the density of distribution of implanted positrons with respect to the height of the upper half (relative to the figure) for both types of moderators. Distance is counted from the edge of the strips that is closest to the target. Results of the implanted positrons distribution density are normalized by one photon from the synchrotron radiation source with the full spectrum. Analysis of these results indicates that the largest number of positrons flying out of the target can be implanted into the honeycomb-type moderator with the cell size 0.1 0.1 cm2. In the cases of a Venetian blinds-type moderator with 0.1 cm step, and of honeycomb-type moderator with cell size 0.2 0.2 cm2, the share of implanted positrons is close to the share of implanted positrons in
Fig. 5. Density of distribution of implanted positrons with respect to the height of Venetian blinds-type tungsten moderator with various steps, and of honeycomb-type moderator with varying cell sizes. The distribution is normalized by one photon from the synchrotron radiation source of the full spectrum.
the previously considered multiwire moderator consisting of 100 layers of tungsten wire 10@5 mm thick [2]. Similar results are obtained for the lower half. It is visible from the presented distributions that considerable increase in the share of implanted positrons can be reached by substantial increase in the height of the moderator (width of strips), but this is undesirable due to the increasing difficulty of drawing reemitted positrons out of the moderator.
48
V.V. Plokhoi et al. / Nuclear Instruments and Methods in Physics Research A 470 (2001) 44–49
4. Yield of reemitted positrons into vacuum gaps of the moderator Uniform distribution of thermalized positrons through the depth in the surface layer of the moderator’s strips, combined with smallness of diffusion path of positrons with respect to the thickness of the strips, makes it possible, as it was in Ref. [2], to use the solution of the diffusion problem for the case of planar layer with uniform source and zero condition for positron density on the surface. As earlier, we will assume that the share of reemitted positrons among those that reached the strips’ surface is, approximately, Z ¼ 13 [7,17–19]; therefore, to get an estimate of the current of reemitted positrons, one must multiply the diffusion current by this coefficient. The diffusion path, as earlier, will be estimated through the lifetime of a positron in the moderator’s medium tþ B10@10 s and diffusion coefficient at the temperature T ¼p 300 K that is D+(300) B1.26 cm2/s ffiffiffiffiffiffiffiffiffiffiffiffi [2], i.e. Lþ ¼ Dþ tþ ¼ 1:12 10@5 cm. Thus, the share of thermalized positrons exiting from a strip into the vacuum gap of the moderator is k ¼ 2ZLþ =d, where d is the thickness of a strip. For a strip 10@3 cm thick, this value is equal to k=0.75 10@2. The estimate we obtained holds true for every strip in the moderator and, therefore, for the moderator as a whole. It is easy to see that when the thickness of tungsten wires is equal to the diameter of the wires in the multiwire moderator [2], the share of positrons reemitted into the vacuum gaps (with respect to those implanted into the moderator) in the multiwire moderator is a factor of two higher than in the Venetian blinds- or honeycomb-type moderators.
5. Slow positrons dynamics in the moderator vacuum gaps We are considering a situation when positrons reemitted into vacuum gaps between tungsten strips are moving from tungsten foil to the exterior boundaries of the moderator. Such consideration relies on the model that corresponds to the foil
with non-ideal planar surface or the foil with bristles grown on its surface; therefore, the angular distribution for reemitted and reflected positrons was taken as isotropic in the exterior semi-sphere. The energy of reemitted electrons was taken as equal to tungsten work function for a positron of 2.75 eV [17–19], and scattering on the surface potential was assumed to be elastic. Calculation of the positron dynamics was done numerically, using the differential scheme of the second order of accuracy. Calculations in this section have a model nature and are used to do a comparative estimate of the efficiency of positrons’ extraction from the space between the strips in the Venetian blinds-type moderator, and from the space of vacuum cells in the honeycomb-type moderator where the extraction is performed by an electric field. Venetian blinds almost touch the target by their edges. Injection of the simulated positrons was performed from the foil’s surface at various distances from the target’s surface. A 3D code numerically solving the Poisson equation was used to calculate the electrical fields. The calculated fields were then used as external stationary fields acting upon positrons’ movement during the simulation of the positrons dynamics. After the onset of its movement, if a positron had a collision with the foil surface, that collision was assumed to be random. In the calculations performed for various modifications of the Venetian blinds-type moderator, we determined the parameter nFmean number of positron’s collisions with foil surface until the positron does not fly out of the moderator. The fewer the collisions, the smaller the number of reemitted positrons that would be lost due to scattering on moderator walls and the higher the moderator efficiency. To draw positrons out of the moderator of the mentioned types it must be recommended to have low values of the target’s positive potential which is slightly above the initial energy of the positron. The Venetian blinds-type design enables the efficient extraction of positrons from the moderator and has the lowest probability of positrons loss (the mean number of collisions with the surface of
V.V. Plokhoi et al. / Nuclear Instruments and Methods in Physics Research A 470 (2001) 44–49
the strips until the positron escapes from the moderator is n ¼ 3212 depending on the point of departure of the reemitted positron). For the honeycomb-type design, the mean number of collisions with the cell’s walls where the cell size is 0.1 0.1 cm2 until the positron escapes from the moderator grows substantially up to n ¼ 11240 depending on the point of departure of the reemitted positron.
6. Estimate of the moderator efficiency For a Venetian blinds-type moderator where electric field is used to draw out positrons, and the assumption of small probability of positron’s capture in collision with moderator’s strips (we will call such an estimate as the maximal one), the efficiency of fast-to-slow positrons conversion can reach the value kef ¼ 0:2 10@2 . Intensity of slow positron source on hard synchrotron radiation of SPring-8 will reach the value of 5 1010 e+/s. For a honeycomb-type moderator with the cell size of 0.1 0.1 cm2 a similar maximal estimate yields kef ¼ 0:3 10@2 that approaches the earlier obtained estimate for the multiwire moderator.
8. Conclusion Honeycomb-type moderator is preferable to the Venetian blinds-type moderator: according to the maximal estimate it can reach a fairly high efficiency of conversion of fast positrons into slow positrons B0.4 10@2. If such a moderator is used, the intensity of slow positrons source on hard synchrotron radiation from SPring-8 can achieve the level of B5 1010 e+/s.
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