Two types of wide-angle fan analyzers for neutron beams

Nuclear Instruments and Methods in Physics Research A 634 (2011) S117–S121

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Two types of wide-angle fan analyzers for neutron beams N.K. Pleshanov n, A.F. Schebetov Petersburg Nuclear Physics Institute, 188300 Gatchina, St. Petersburg, Russia

a r t i c l e in fo

abstract

Available online 7 July 2010

Fan analyzers are devices with tapered channels formed by polarizing neutron (super) mirrors. Two types of wide-angle fan analyzers for the polarization of neutrons scattered by the sample are considered. Fan analyzers with the geometry of straight channel walls (type I) have been produced in PNPI (Gatchina) and tested in JINR (Dubna) and GKSS (Geesthacht). A more consecutive and simple approach to their design is now proposed. An approach to the correction of the neutron data for the analyzer transmission is discussed. Some aspects concerning the design and the use of such analyzers in neutron reflectometers are taken into account. The theory of a fan analyzer with curved channel walls (type II) is presented. It is pointed out that it can also be used to design focusing polarizers. & 2010 Published by Elsevier B.V.

Keywords: Neutron analyzer Fan analyzer Polarizing supermirror Focusing polarizer

1. Introduction To achieve a reasonable resolution, polarized neutron beams with relatively small divergence are required and the polarization of scattered neutrons is analyzed with numerous channels of small angular aperture. Then neutron optical polarizers [1,2] may turn out to be more appropriate than 3He spin filters [3,4] and their development remains to be an important task. The design of the analyzer often differs from that of the polarizer, because the apertures of the detector and the slits forming the primary beam are quite different. The difference is especially evident when position-sensitive detectors are used. In this case the supermirror analyzer should consist of numerous channels, each of which analyzes neutrons scattered by the sample into a restricted angular range. A wide-angle analyzer can be built with the use of stacks of thin substrates coated on both sides by polarizing supermirrors with a characteristic wavelength lc. Such stacks, of the supermirrors of length L0 ¼ 4wlc =lmin

ð1Þ

separated by spacers of thickness w and bent into a circumference with radius R ¼ L20 ð8wÞ1 ¼ ðlc =lmin ÞL0 =2

ð2Þ

to exclude the traversal of neutrons without being reflected at least once from the channel walls, are used [5] as multichannel polarizers and analyzers. This geometry ensures that neutrons n

Corresponding author.Tel.: + 7 81371 46973; fax: + 7 81371 39053. E-mail address: [email protected] (N.K. Pleshanov).

0168-9002/$ - see front matter & 2010 Published by Elsevier B.V. doi:10.1016/j.nima.2010.06.359

with wavelengths l Z lmin , entering the channel window of width w at the right angle, easily traverse the channel. Such devices ¨ (Scharpf benders) have been successfully used [6] at many instruments. In a wide-angle analyzer the numerous benders are turned with respect to each other to achieve the maximum transmission for neutrons scattered by the sample into different angles [1,7]. However, one cannot focus all channels in a bender onto the sample, so the number of its channels is restricted and they are not equivalent in transmission. In addition, ‘‘dead stripes’’ on the detector cannot be avoided, because of the necessity to use wedge spacers, made of neutron absorbing materials, at the joints between the neighboring benders. The mentioned shortcomings are removed using analyzers with tapered channels (fan analyzers). The first such analyzer with curved channel walls was designed (using a Monte Carlo program), produced and tested by Krist et al. [8]. A similar geometry was used earlier [9] to design a focusing polarizer, which remarkably increased the luminosity of the DNS instrument, now at FRM-II (Garching, Germany). The idea was quite successfully used [7] to build a focusing polarizer at the spectrometer D7 in ILL (Grenoble, France). The first fan analyzer with straight channel walls was designed, produced and installed by PNPI (Gatchina, Russia) at the reflectometer REMUR in JINR (Dubna, Russia) [10]. To exclude the direct view of the sample through the channels, the analyzer is turned by a definite angle. The analyzer transmittance is about 50% and its polarizing efficiency is 95–97% for the white beam. A wide-angle (90  160 mm2) analyzer was produced in PNPI also for the neutron reflectometer NERO in GKSS (Geesthacht, Germany). Analyzer transmittance of 58% and the polarizing efficiency of 98% for a l ¼0.435 nm neutron beam were achieved [11].

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N.K. Pleshanov, A.F. Schebetov / Nuclear Instruments and Methods in Physics Research A 634 (2011) S117–S121

Section 2 describes a simple approach, relative to that described in Ref. [10], to the design of fan analyzers with straight channel walls (type I). The approach takes into account the special features concerning the use of fan analyzers in neutron reflectometers. The exact theory of a fan analyzer with curved channel walls (type II) is presented in Section 3 and the Appendix. Previously, one could evaluate design parameters of such analyzers by very simplified approaches.

To find the width v of the exit channel window, we require, in addition, that the ray from the sample center S cross the nearest point (E) of the front wall of the neighboring channel at the same designed angle a. In this case the channel taper e (the angle between the two adjacent walls forming the channel) is equal to the angle d between the rays SA and SE:

e ¼ d ffi ðwþ uÞ=fa

ð5Þ

where u is the thickness of the channel walls. Knowing e, we find the channel exit window width: 2. Design of fan analyzer with straight channel walls (type I) The formulas for building a fan analyzer with straight channel walls are derived from the geometry and simple physical considerations. In Fig. 1 the origin of the reference frame is at the sample center, the x-axis is along the central ray of the neutron beam and the y-axis is also in the plane perpendicular to the channels of the fan analyzer. Transmission of a neutron through the channel does not depend on the z-component of its velocity. In most cases we may assume in our considerations that the sample cross-section is a circle with radius r, i.e. the sample is in the form of a sphere, a cylinder, etc. To achieve high transmittance of the channels, the maximum glancing angle amax should not exceed the critical angle of the neutron supermirror for analyzed neutrons with a minimum wavelength lmin, i.e. myNi lmin , where m is the angular acceptance of the supermirror in units of the critical angle of Ni (yNi ffi 1.7 ˚ Therefore, the designed turn mrad for neutrons of wavelength 1 A). angle for the channel is

a ¼ amax rmax =fa ¼ myNi lmin rmax =fa :

ð3Þ

Here we assume that facrmax, i.e. the distance fa from the sample to the analyzer by far exceeds the maximum possible sample radius rmax. Note that the undervalueing of rmax would restrict the possibilities of the instrument, whereas its overstating is fraught with designing an analyzer with diminished transmittance. For thermal and cold neutrons the angle a is a small quantity. To provide the analysis of polarization of scattered neutrons and maximum transmittance, the entrance channel window width should be the maximum, such that the condition of the absence of the direct view between the sample and the detector is still fulfilled. Assuming that the distances from the sample to the neighboring channels are almost the same (9SC9ffi 9SA9), we find the required width for the entrance channel window as w ffiLa bmin ffi aLa 

aLa þ rmax fa =La þ1

¼

afa rmax fa =La þ 1

ð4Þ

where La is the channel length.

v ¼ w þ La sin e ffiw þ ðw þ uÞLa =fa :

The designing of the fan analyzer can be started by fixing the entrance channel window width w. Then the channel length is determined from Eq. (4) as   r þ w 1 : ð7Þ La ffi w a max fa The other equations remain unchanged. They can be used iteratively, e.g., when the sample–detector distance is fixed and distance fa depends on La. Building the channels in the manner described above, one can design a fan analyzer that can, in principle, cover any angular range. If the distances from the sample to different channels differ but slightly, all channels can be made geometrically equivalent. Then a is the designed turn angle of the analyzer. The transmittance of the channel is determined by the reflectivity of its upper and lower walls. Solving the corresponding geometric problem with rmax { fa in the approximation of small glancing angles, we can find the portion of neutrons scattered at a point rs of the sample into the given channel and then reflected from its walls no less than n times: 8 > < 0, G o0, aðe þ w=La Þðn1Þ þ wðrs Þ Kn ðr s Þ ¼ G, 0 r G r1, G ¼ ðn ¼ 1, 2, :::Þ: > wð1=La þ 1=fa Þ : 1, G 41, ð8Þ In deriving of this formula it was taken into account that when

rmax { fa , the angle w(rs) subtended by the projection of the vector rs onto the plane (x,y) is practically the same for all points of the channel entrance window, viz. wðr s Þ ffir? =fa , where r? is the length of the projection of rs onto the axis, which is in the plane (x,y) and perpendicular to the ray SA (Fig. 1). The portion of neutrons traversing the channel with n reflections is accordingly ( ðn ¼ 0Þ 1K1 ðr s Þ ð9Þ kn ðrs Þ ¼ Kn ðr s ÞKn þ 1 ðrs Þ ðn ¼ 1, 2, :::Þ Hence the average number of collisions inside the channel for neutrons scattered at the point rs is X X nðr s Þ ¼ Kn ðr s Þ ¼ nkn ðr s Þ: ð10Þ n

Fig. 1. Geometry of a single channel of the fan analyzer with straight channel walls.

ð6Þ

n

By definition, k0 ðr s Þ is the portion of neutrons that pass the channel without being reflected. The condition of the absence of the direct view should be fulfilled for any sample point, i.e. k0 ðrs Þ ¼ 0. The respective portions of neutrons scattered by the entire sample depend not only on the geometry (w, La, e) and the turn angle of the channels (a), but also on the incident beam divergence and the illumination of the sample by the beam, the shape and dimensions of the sample, the angular dependence of the scattering cross-section and the absorption in the sample. Therefore, for the correction of the neutron data it may not be sufficient just to calibrate the transmission coefficients of different channels and use them in all measurements.

N.K. Pleshanov, A.F. Schebetov / Nuclear Instruments and Methods in Physics Research A 634 (2011) S117–S121

Correction of the neutron data obtained with the use of the fan analyzer can be made with formulas (8) and(9). In the special case where reflectivity R + (y,l) of the supermirror coating on the channel walls may be approximated by a linear function of the glancing angle y, the sample does not absorb, is uniformly illuminated by the incident beam and scatters isotropically into the channel, the transmission coefficient is þ ðlÞ ¼ V 1 tch

R

dr s

P

n

kn ðrs Þ

n 1 Y

R þ ðgn ðr s Þje, lÞ

j¼0

gn ðrs Þ ¼ ðKn1 ðrs Þ þKn ðrs Þ þ 2n1Þ

w þ e þ wðr s Þ 2La

ð11Þ

where V is the sample volume. It is taken into account that the glancing angle at each subsequent reflection inside a tapered channel is reduced by e. Let us define the quantities that depend only on the geometry and reflecting properties of the coating and which can be easily measured: transmittance and polarizing efficiency. The transmittance Tch(l,g) of a channel is the transmission coefficient for a strongly collimated and spatially uniform beam, entering the channel at a certain angle g. The beam uniformity can be imitated by scanning the channel window with a narrow collimated beam. The polarizing efficiency of the channel is defined by the 7 of neutrons with opposite spins ( 7): transmittances Tch þ  þ  ðl, gÞTch ðl, gÞ½Tch ðl, gÞ þ Tch ðl, gÞ1 Pch ðl, gÞ ¼ ½Tch

ð12Þ

The transmittance of the analyzer is given by Ta ðl, gÞ ¼ Tg T ch ðl, gÞ,

ð13Þ

where T ch ðl, gÞ is the average transmittance of the channels for the beam entering at an angle g and Tg ¼ w=ðw þ uÞ,

ð14Þ

the geometric transmission factor characterizing the reduction of the transmittance of the multichannel device due to absorption of neutrons in the substrates of thickness u. When designing the analyzer, one should take into account the number of collisions experienced by neutrons, scattered at the most distant point of the sample, in the channels: nmax ¼ nðrmax Þ. Examples of the design of the fan analyzers are given in Table 1. The transition from the coating with m¼2 to the coating with m ¼3 allows diminishing the length of the analyzer or shifting it closer to the sample. Yet, an excessive reduction of distance fa to 1000 mm increases the number nmax up to 4.2, i.e. the number of collisions with walls may reach 5. The transmittance of the channels will be noticeably decreased. A decrease in channel width w leads to a drop in geometric transmission factor of the analyzer. When the distance fa is diminished to 780 mm, transmittance of the channels drops to 0, since the calculated width of the entrance window is equal to 0. When the distance from the sample to the analyzer is too small and/or the sample size is too large, one should use the fan analyzers of type II, with curved channels (see Section 3). Table 1 Analyzers designed for neutrons with wavelengths l Z lmin ¼ 0:5 nm on the basis of substrates of thickness u¼ 0.4 mm (rmax ¼ 10 mm). Analyzer design parameters

a (deg) w (mm) v (mm) Tg

nmax

m ¼2, m ¼3, m ¼3, m ¼3, m ¼3, m ¼3,

0.70 1.19 1.19 1.09 0.90

2.34 1.62 1.62 2.03 4.2

fa ¼2000 mm, La ¼ 200 mm fa ¼2000 mm, La ¼ 200 mm fa ¼2000 mm, La ¼ 100 mm fa ¼1500 mm, La ¼ 200 mm fa ¼1000 mm, La ¼ 200 mm fa ¼780 mm, La ¼ 200 mm

1.30 2.86 1.50 2.19 0.96 0

1.47 3.19 1.59 2.54 1.23

0.77 0.88 0.79 0.85 0.705 0

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The use of a fan analyzer in neutron reflectometers is a special case. Then the sample is a plate with a layered structure on its flat surface (Fig. 2). The collimated beam is incident at the glancing angle y onto the sample with a length Ls. The specularly reflected neutrons yield information about depth profiles (laterally averaged) of the nuclear and magnetic potentials, whereas offspecular scattering is sensitive to variation of structural and magnetic properties of the layered structure along the sample surface. The sample surface defines the ‘‘horizon plane’’. For a channel located above the horizon plane, the maximum glancing angle under reflection from the frontal wall will be for the trajectory TA (Ls{fa ¼ 9SA9):

amax ffi a þ

Ls Ls sinðfyÞ ¼ a þ sin Z 2fa 2fa

ð15Þ

where Z ¼ j  y is the angular elevation of the channel above the horizon. As in the previous case, we assume that amax should not exceed the critical angle of the coating for neutrons with a minimum wavelength lmin. To define the designed turn angle a, the elevation of the channel above the ‘‘zero’’ horizon should be , the taken for y ¼0 (Z ¼ j) and Ls should be substituted by Lmax s maximum possible length of the sample:   a ¼ myNi lmin  Lmax =2fa sin f: ð16Þ s Thus, we may infer that for calculation of the channel geometry it is sufficient to substitute   rmax ðfÞ ¼ Lmax =2 sin f ð17Þ s into Eqs. (3)–(7). The essential dependence of effective maximum radius rmax on position of the channel above the level of the ‘zero’ horizon is a special feature. As a consequence, all channels should, generally speaking, have a different geometry. Different approaches to the design of the fan analyzer are possible. Fixing the length of the channels, one can build an analyzer with channel entrance windows with different widths and taper angles (Table 2). One may fix the width of the entrance windows and change the length and the taper of the channels, building an analyzer with equidistant channels. Using the length

Fig. 2. Geometry of a single channel of the fan analyzer for the neutron reflectometer.

Table 2 Fan analyzer channels with a fixed length La ¼200 mm and different elevations fa sin Z above the horizon (design parameters: lmin ¼ 0:5 nm, Lmax ¼ 100mm, m ¼2, s u¼0.4 mm, fa ¼2000 mm). Elevation above the horizon (mm)

a (deg) w (mm) v (mm) Tg

nmax

0 50 100 200 400

0.98 0.95 0.92 0.86 0.74

1 1.06 1.14 1.33 1.98

3.12 2.92 2.72 2.32 1.56

3.47 3.25 3.03 2.59 1.75

0.89 0.88 0.87 0.85 0.80

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N.K. Pleshanov, A.F. Schebetov / Nuclear Instruments and Methods in Physics Research A 634 (2011) S117–S121

and the width of the channel with a minimum angle a for all channels, one can build an analyzer with equivalent channels. Anyway, transmission conditions for different channels will be different at a given glancing angle y. Transmission conditions for each channel will be different at different y, unless the analyzer is rotated by y (y: y mode). The y: y mode can be recommended for reflectometers. In addition, it is to be noted that the use of the correction formulas (9)–(11) is quite simple for mirror samples.

3. Design of fan analyzer with curved channel walls (type II) The use of curved channels in a fan analyzer (Fig. 3) excludes the direct view, independently of the sample dimensions and the distance from the sample to the analyzer. Thus, the analyzer with curved channel walls is more efficient for short distances from the sample and large sample dimensions. The radius of curvature R is found from Eq. (2) for the coating with lc ¼(myNi)  1 and the direct-view channel length L0. The widths of the entrance (w) and exit (v) windows of the channel at a distance fa from the sample center can be found using the condition of the absence of the direct view (Appendix A) and the condition of the focusing of the channels onto the sample center. When u{(fa,L0){R, one may use the approximate solution yt fa u, xt þ fa pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xt ffi fa ðfa þL0 Þ þ 2Rufa ,

wffi

yt ðLa þfa Þ u xt þ fa ð2fa þ L0 Þxt fa L0 yt ffi 2R

In conclusion, note that lately the interest in wide-angle neutron analyzers has noticeably increased. Results obtained in this paper may help in designing wide-angle fan analyzers for neutron instruments. It is to be noted that the analyzer with straight channel walls is noticeably shorter (and cheaper) than the analyzer with curved channel walls. When the angular dimensions of samples as seen from the analyzer are significantly less than the critical angle of the coating, fan analyzers with straight channel walls are preferable. Otherwise, it is better to use fan analyzers with curved channel walls. In addition, it should be noted that the geometry with curved channel walls can be used to design focusing polarizers [9]. Indeed, when the flux in Fig. 3 is reversed, neutrons traverse numerous channels coated with polarizing supermirrors and focused onto the sample. Then the polarized neutron flux density at the sample will be proportional to angular acceptance m of the supermirror coating and to the number of channels. Using a focusing polarizer and the world’s largest solidangle analyzer with improved polarizing supermirrors, the luminosity of the polarized neutron spectrometer D7 [12] had been increased by almost two orders (!) of magnitude [7]. It seems to be one of the best illustrations of the potential of neutron optics combined with great collaborative efforts and expertise.

Appendix A. Tapered channels with curved walls

vffi

ð18Þ

The quantity La in the expression for v is the length of supermirrors, which is made 5–15% larger than L0 to rule out the possibility for neutrons to traverse the channels without being reflected due to imperfections of their geometry. A preliminary design of such an analyzer was made for a restricted geometry of the diffractometer E4 (HMI, Berlin), where the sample–detector distance is 850 mm. With the design parameters lmin ¼ 0:2nm, m¼2.5, L0 ¼340 mm, La ¼400 mm, u ¼0.3 mm, fa ¼435 mm, the widths w¼0.45 mm and v¼ 1.15 mm were obtained. The analyzer transmission coefficients T + for the spin-up neutrons with a working wavelength l ¼0.23 nm were calculated on the assumption that cylindrical samples with different diameters ds scatter isotropically. The calculated transmission coefficient T + is 0.50 for ds ¼ 1 mm, 0.49 for ds ¼5 mm, 0.39 for ds ¼10 mm, 0.33 for ds ¼12 mm and 0.27 for ds ¼15 mm. With the m¼3 supermirrors, the transmission coefficients are increased by about 5%. It is also of interest that for wavelengths l 4 0.5 nm the transmission coefficients for this analyzer would exceed 0.5, approaching Tg ¼ 0.6.

Fig. 3. Geometry of the fan analyzer with curved channel walls.

Conclusion

In order to avoid direct view through a channel with walls of length L0, bent into a circumference with a radius R, and with the exit window wider than the entrance window, it is sufficient that the concave wall rest on a tangent to the convex wall surface (Fig. A-1). For a tapered channel the extensions of the chords AB and CD subtending the arcs formed by the bottom surfaces of the two walls intersect at a point F at a distance f from the convex wall. In Fig. A-1 the origin of the reference frame is at point C, the x-axis is along the chord CD and the y-axis is also in the plane of the figure. Then the equation of a straight line traversing point F may be written as y ¼ tanðeÞðx þf Þ

ðA:1Þ

where e is its inclination angle. The equation of the circumference constituting a part of the convex wall surface is ðxxc Þ2 þðyyc Þ2 ¼ R2

ðA:2Þ

with its center coordinates xc ¼ R sin gc ,

yc ¼ R cos gc þ u

ðA:3Þ

where u is the wall thickness and gc ¼L0/(2R). It follows from (A.2) that qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y ¼ yc þ R2 ðxxc Þ2 ðA:4Þ

Fig. A-1. Condition sufficient to avoid the direct view through a tapered channel with curved walls.

N.K. Pleshanov, A.F. Schebetov / Nuclear Instruments and Methods in Physics Research A 634 (2011) S117–S121

so the direction of the tangent at a point (x,y) of the circumference is defined by yu ¼

dy xxc : ¼ yyc dx

xc f yc yt , xc þf

P ¼ yc

yt ¼ P þ

ðxc þ f Þ2 x2c ðxc þ f Þ2 þ y2c

,

Q¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi P2 þ Q

ðxc þf Þ2 ðR2 y2c Þx4c

ðA:6Þ

ðxc þ f Þ2 þ y2c

It follows also from the geometry that   yt : e ¼ arctan xt þ f

ðA:7Þ

The exact formulas (A.6) and (A.7) can be used to design fan analyzers with channel walls stacked in the manner described. Particularly, if the edge of the concave wall is also at the y-axis (Fig. A-1), the direct view through the channel with length L0 will be avoided with entrance and exit window widths. Accordingly w ¼ f sin eu,

is small). By substituting f ffi fa (see the main text) and assuming additionally that u{(fa,L0){R, we find the approximate solution (18).

ðA:5Þ

Solving the system of two Eqs. (A.1) and (A.2) with yu ¼ tanðeÞ, we find the coordinates (xt,yt) of the contact of the tangent (drawn from the point F) with the convex wall: xt ¼

S121

v ffiðL0 þf Þsin eu

ðA:8Þ

The formula for v was obtained on the assumption that the lengths of the chord and the respective arc are equal (the angle gc

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