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Design of microfocusing bent-crystal double monochromators
Nuclear Instruments and Methods in Physics Research North-Holland
A 338 (1994) 132-135
NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH Section A
Design of microfocusing bent-crystal double monochromators M. Popovici *, W.B. Yelon
Missouri University Research Reactor, Columbia, MO 65211, USA
Design relations are given for double bent-crystal monochromators ensuring both spatial microfocusing and focusing in reciprocal space in powder diffraction. The choice of crystals for monochromators for microfocus stress scanning in steels is discussed. Design computations for an instrument at the Missouri University Research Reactor (MURR, thermal neutron flux 10 14 n/cm 2 s) with small-mosaic Cu(200) crystals are presented. The spot size can be brought down to 2.5 x 4 mm, with a peak flux at focus approaching 10 8 n/cm 2 s.
1. Introduction
2. Theoretical background
We recently examined double bent-crystal monochromators [1-3]. The antiparallel (+, +) setting of two bent crystals turned out to have a remarkable property of allowing strong spatial focusing to be achieved easily, both with synchrotron radiation (SR) [1] and neutrons [2,3]. Such a setting is therefore of particular interest with small samples, as in stress scanning with neutrons [4]. The possibility of combining beam focusing onto the sample with focusing in scattering was confirmed experimentally [3]. Such a combination does not conflict with the Lionville theorem, because the phase space volume is not changed, but rather is optimally adjusted . It was found that flat mosaic crystals may also be used as first elements in microfocusing double monochromators [3] . In this paper the design of double monochromators with both crystals bent, ensuring simultaneous microfocusing at the sample position and focusing in scattering at the detector position, is further examined . The experience with stress scanning at MURR has shown [5] that the intrinsic linewidths of steel samples are quite large (slightly below 1°), so only a moderate instrument resolution is needed . The question of the maximal intensities that can be achieved in such a situation is addressed here . Copper plates of small mosaic spread (3' typically for as-grown monocrystals) appear to be the most promising. Results of design computations for a dedicated stress scanning instrument at MURR are presented.
Consideration of the Bragg reflection on two bent crystals has shown that a high reflection efficiency of large area, large angular divergence neutron beams is achieved when the focal lengths of the two crystals satisfy the conditions [6,2]:
* Corresponding author .
L2/fz =2-1 /a,,
0168-9002/94/$07 .00 © 1994 - Elsevier SSDI 0168-9002(93)E0791-P
Science
B.V. All
f, * = (L/2)/(l - l/a), f2 = (L/2)/(l -a),
(la) (lb)
where L is the distance between the crystals and a is the usual dispersion parameter, a = -tan 02/tan 0, (the indices i = 1, 2 specify the crystal), 0, and 02 being the Bragg angles (with signs, trigonometric convention) . In the symmetric reflection (Bragg) case, which will only be considered here, one has fi * = fi = (R,/2)lsin Oi l, Ri being the radius of curvature in the diffraction (horizontal for neutrons) plane. If the relations (la) and (lb) are satisfied, the entire extended source is imaged at a certain distance L, after the second crystal . By combining the two equations with the conventional optics relations [1,2], one finds for this distance : L2/fz* = 2(1 -a)/[1 - 2a + 1/(2/a - 1 +L/L,) ], where L, is the distance from source to first crystal . For L/L t << 1 it results:
(2a) L2/f2 = (2 - a)/(1 - a) . On the other hand, to have narrow powder diffraction lines at a detector angle 20, with a small sample, one has to satisfy a different condition involving the same ratio [2,3,7]:
rights reserved
133
M. Popovici, W.B. Yelon /Nucl. Instr. and Meth. in Phys . Res. A 338 (1994) 132-135
where a 5 is the dispersion parameter for powder diffraction, a 5 = -tan 0,/tan 02, taken with respect to the second crystal in the double monochromator . The conditions (2) and (3) are satisfied simultaneously if the two dispersion parameters obey the relation : a 5 = 1 - 1/a.
70 .
L,=6 m, L=0.8 m, Lz0.3 m, R20.85 m
50 .
(4)
Satisfying Eq . (4) automatically ensures minimal linewidths around 20 s when the sample is at the microfocus position . With a first crystal flat (mosaic), the corresponding relation is a 5 r 1 - 1/2a [3]. For identical crystals in (+, +) setting (a = -1), the value of a s must be close to 2. Therefore, the range of narrow diffraction lines will be around 90° if the take-off angle of the two (identical) crystals is close to 53° (cot 0 = 2) . The wavelength is then selected by choosing the crystal d-spacing. To put the bcc Fe(211) line, convenient for stress scanning (due to its high multiplicity), at 90°, the wavelength should be 0.166 nm, thus the crystal d-spacing should be close to 0.185 nm . The Cu(200) planes have the right spacing (d = 0.1808 nm). Other planes may be used as well, by allowing for different d-spacings (a z# -1), but the configuration with identical crystals seems to be the best . As discussed in more detail in ref. [3], the strong conditions (la) and (1b) may be softened by instead using a single condition referring to the SR point-source case [1,8,9], which turns out to work well in the vicinity of Eqs. (Ia) and (1b) . For this reason, the take-off angles are not tightly specified, both here and in ref. [3]. One can only say that the range of optimal take-off angles is 50-80° with identical crystals . A variety of situations occur if the crystals are allowed to be different. The necessary radii of curvatures are small and cannot be achieved by elastic bending at the required thicknesses . Therefore, plastically bent crystals, either perfect or of small mosaic spread, will have to be used . The peak reflectivity decreases steeply on bending [10,11], so that only strongly reflecting planes can be used . The low-index planes of copper satisfy all these requirements . The (00.2) plane of Be (d = 0.1797 nm) would be somewhat better than Cu(200), but beryllium crystals are expensive and their plastic bending might be a problem . Perfect copper crystals would also be better, but are not available.
0.0
025
0 .50
0.75
1 .00
1 .25
First crystal curvature 1/R, (11m)
1 .50
1 .75
-
2,
0
Fig. 1. Computed peak flux at focus vs curvature (inverse radius) of the first crystal in a microfocusing double monochromator with Cu(200) small-mosaic crystals . Neutron wavelength 0.166 nm, thermal source flux 10 14 n/cm 2/s. plates), of 7.5 cm diameter and 0.4 cm thickness. Plastic spherical bending, to the radii of curvature specified on the figures, was assumed. The peak flux at the focus position vs the first crystal curvature (inverse radius), with the curvature of the second crystal set to the value ensuring minimal linewidths at 90° detector angle, is shown in Fig. 1. It is seen to approach 5 X 107 n/cm 2/s at optimum (absorption in air neglected) . This figure can be doubled by optimal vertical focusing, quite feasible with plastic bending. Fig. 2 shows the beam width at the fixed focus position vs . the second crystal curvature (first at optimum) . The minimal spot size is computed to be 0.26 X 1 .21 cm (FWHM's) . With optimal vertical focusing, the second figure can be reduced to 0.38 cm . It is seen in Fig. 2 that the beam is narrow (below 6 mm) at any second crystal curvature. This is because a
L1 =6 m, L=0.8 m, L,=0.3 m, R,=0.93 m
3. Design computations Results of design computations for a microfocusing double monochromator for stress scanning at MURR are presented in Figs . 1-4. The computer program MAX) was described in ref. [12] . The computations refer to copper (200) plates of small mosaic spread (3' true, as measured recently at MURR for as-grown
00
D.0
0 .25
0.50
0 .75 1 .00 1 .25
1 .50 1 .75
Second crystal curvature 11R2 ('/M)
2 .00
Fig. 2. Computed beam width at focus (horizontally) vs curvature of the second crystal . III. APPLICATIONS
M. Popovici, W.B. Yelon I Nuel. Instr. and Meth. in Phys . Res. A 338 (1994) 132-135
134
second crystal in (+, +) setting always (at any curvature) selects a beam that goes into a small focus at some distance . However, to have both high intensity and a small spot, it is essential that the radii of curvature are set correctly. The necessary accuracy is seen (Figs. 1 and 2) to be not a problem. Fig. 3 shows the computed linewidths in powder diffraction vs detector angle. The double monochromator performance is compared to that of a single monochromator with a similar Cu(200) plate, spherically bent to the optimal radius for minimal linewidths at 90° detector angle. The resolution with the double monochromator is slightly worse, but the peak intensity in diffraction is still better by a factor of more than 7. Little can be done to improve the resolution in the double monochromator configuration with the given sample, because the large linewidth is related to the small distance L21 which is essential for microfocusing. An in-pile Soller collimator will have practically no effect, as a double (+, +) monochromator automatically back-selects an almost plane-parallel incoming beam . The resolution can be improved somewhat, at the expense of intensity, by limiting the beam horizontally before the first crystal. However, as always with bent crystal focusing [121, the resolution can be improved by thinning the crystals and the sample . With a smaller sample, hence a better spatial resolution on stress scanning, it is enough to make only the second crystal thinner. The situation for a sample cube of 1 mm edge and a second crystal 1.5 mm thin is shown in Fig. 4. The optimal radii are now closer to the theoretical values, relations (1a, b) . The linewidth is reduced from 0.9° to 0.65° (curve 1) . The
6
L,=6 m, L=Ö.8 m, L,=0 .3 m (or 1 .1 m with one crystal) 1 : Double R1 =0 .93 rn, mspherically R2-0 .85 Fe (211) peak: 750 n/sat -90° 2: Single Cu (200), R=3.15 m spherically I Fe (211) peak: 100 n/sat -90°
a.o 3.5
N N
m
L,=6 rn, L=0.8 m, L2--0 .3 rn R,=0.93 m, R2=0.88 rn spherically 1 : Mosaic (3' true) both, Fe (211) peak: 31 n/s 2: Perfectcrystals both, Fe (211) peak : 38 n/s
3.0
Ô 2.5 E 20
3
1 .5
120.
105,
90
-7560 Detector angle (degrees)
-
A5
30
Fig. 4. Resolution improvement in the focusing range on thinning the second crystal, for a smaller sample (cube of 1 mm edge). Curve 1: second crystal 1.5 mm thin, mosaic; curve 2: both crystals perfect (if), second 1.5 mm thin; the broken curve is the same as curve 1 in Fig. 3 .
spot size is reduced to 2 mm (horizontally) . Because of the 27 times smaller sample volume, the peak intensity is lower, naturally (by a factor of 24). If the crystals were perfect (curve 2), the linewidth would be further reduced (to 0.57°), with an even smaller (1 .8 mm) spot size, while the intensity would be only 20 times lower. The broken curve in Fig. 4 is the curve 1 in Fig. 3, shown for comparison . The absolute intensities indicated on figs . 3 and 4 for the Fe(211) line can be doubled in each case by optimal vertical focusing . An improvement in peak intensities by a factor of more than 100 over the recent stress scanning measurements at MURR [5] is anticipated with the microfocusing monochromator . References [11 M. Popovici and W.B . Yelon, Nucl . Instr. and Meth . A 319 (1992) 141 .
E 3
M. Popovici and W.B . Yelon, in Neutron Optical Devices and Applications, ed . C.F . Majkrzak and J.L . Wood, SPIE Proc . 1738 (1992) 422. [3] M. Popovici and W.B . Yelon, Z. Kristallografie (1993) in press. [4] A.D . Krawitz, in : Neutron Scattering for Materials Science, eds. S .M . Shapiro, S. Moss and J. Jorgensen, Mater. Res. Soc. Symp . Proc . 166 (1990) 281 . [51 A.D . Krawitz and R.A . Winholtz, in : Advances in X-Ray Analysis 37 (1994) in press. [6] M. Popovici, A.D . Stoica, B. Chalupa and P. Mikula, J. Appl . Cryst. 21 (1988) 258. [7] M. Popovici, A.D . Stoica and 1. lonilâ, to be published. [8] M. Popovici and W.B . Yelon, J. Appl . Cryst. 25 (1992) [2]
3
3.
a
2.
6.0
-150.
-120 .
-90.
-60.
~ -30 .
0 .0
30 .
Detector angle (degrees)
.. 60 .
9Ô .
120 .
, 150.
Fig. 3 . Linewidths in powder diffraction for the double monochromator considered and for a single monochromator with optimally bent Cu(200). The absolute peak intensities for the Fe(211) line of a small sample (cube of 3 mm edge) are indicated (uncorrected for absorption effects and detection efficiency).
331 .
M. Popovici, W.B . Yelon /Nucl. Instr. and Meth. in Phys. Res. A 338 (1994) 132-135 [9] F .N . Chukhovskii, M . Krisch and A .K . Freund, Rev . Sci . Instr. 63 (1992) 920 . [10] Z.H . Kalman and S . Weissmann, J . Appl . Cryst . 12 (1979) 209 . [11] J . Kulda, Acta Cryst . A 40 (1984) 120.
135
[12] M . Popovici, W .B . Yelon, R . Berliner, A.D . Stoica, I . Ionijâ and R . Law, these Proceedings (Workshop on Focusing Bragg Optics, Braunschweig, Germany, 1993), Nucl . Instr . and Meth . A 338 (1993) 99.
III. APPLICATIONS
A 338 (1994) 132-135
NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH Section A
Design of microfocusing bent-crystal double monochromators M. Popovici *, W.B. Yelon
Missouri University Research Reactor, Columbia, MO 65211, USA
Design relations are given for double bent-crystal monochromators ensuring both spatial microfocusing and focusing in reciprocal space in powder diffraction. The choice of crystals for monochromators for microfocus stress scanning in steels is discussed. Design computations for an instrument at the Missouri University Research Reactor (MURR, thermal neutron flux 10 14 n/cm 2 s) with small-mosaic Cu(200) crystals are presented. The spot size can be brought down to 2.5 x 4 mm, with a peak flux at focus approaching 10 8 n/cm 2 s.
1. Introduction
2. Theoretical background
We recently examined double bent-crystal monochromators [1-3]. The antiparallel (+, +) setting of two bent crystals turned out to have a remarkable property of allowing strong spatial focusing to be achieved easily, both with synchrotron radiation (SR) [1] and neutrons [2,3]. Such a setting is therefore of particular interest with small samples, as in stress scanning with neutrons [4]. The possibility of combining beam focusing onto the sample with focusing in scattering was confirmed experimentally [3]. Such a combination does not conflict with the Lionville theorem, because the phase space volume is not changed, but rather is optimally adjusted . It was found that flat mosaic crystals may also be used as first elements in microfocusing double monochromators [3] . In this paper the design of double monochromators with both crystals bent, ensuring simultaneous microfocusing at the sample position and focusing in scattering at the detector position, is further examined . The experience with stress scanning at MURR has shown [5] that the intrinsic linewidths of steel samples are quite large (slightly below 1°), so only a moderate instrument resolution is needed . The question of the maximal intensities that can be achieved in such a situation is addressed here . Copper plates of small mosaic spread (3' typically for as-grown monocrystals) appear to be the most promising. Results of design computations for a dedicated stress scanning instrument at MURR are presented.
Consideration of the Bragg reflection on two bent crystals has shown that a high reflection efficiency of large area, large angular divergence neutron beams is achieved when the focal lengths of the two crystals satisfy the conditions [6,2]:
* Corresponding author .
L2/fz =2-1 /a,,
0168-9002/94/$07 .00 © 1994 - Elsevier SSDI 0168-9002(93)E0791-P
Science
B.V. All
f, * = (L/2)/(l - l/a), f2 = (L/2)/(l -a),
(la) (lb)
where L is the distance between the crystals and a is the usual dispersion parameter, a = -tan 02/tan 0, (the indices i = 1, 2 specify the crystal), 0, and 02 being the Bragg angles (with signs, trigonometric convention) . In the symmetric reflection (Bragg) case, which will only be considered here, one has fi * = fi = (R,/2)lsin Oi l, Ri being the radius of curvature in the diffraction (horizontal for neutrons) plane. If the relations (la) and (lb) are satisfied, the entire extended source is imaged at a certain distance L, after the second crystal . By combining the two equations with the conventional optics relations [1,2], one finds for this distance : L2/fz* = 2(1 -a)/[1 - 2a + 1/(2/a - 1 +L/L,) ], where L, is the distance from source to first crystal . For L/L t << 1 it results:
(2a) L2/f2 = (2 - a)/(1 - a) . On the other hand, to have narrow powder diffraction lines at a detector angle 20, with a small sample, one has to satisfy a different condition involving the same ratio [2,3,7]:
rights reserved
133
M. Popovici, W.B. Yelon /Nucl. Instr. and Meth. in Phys . Res. A 338 (1994) 132-135
where a 5 is the dispersion parameter for powder diffraction, a 5 = -tan 0,/tan 02, taken with respect to the second crystal in the double monochromator . The conditions (2) and (3) are satisfied simultaneously if the two dispersion parameters obey the relation : a 5 = 1 - 1/a.
70 .
L,=6 m, L=0.8 m, Lz0.3 m, R20.85 m
50 .
(4)
Satisfying Eq . (4) automatically ensures minimal linewidths around 20 s when the sample is at the microfocus position . With a first crystal flat (mosaic), the corresponding relation is a 5 r 1 - 1/2a [3]. For identical crystals in (+, +) setting (a = -1), the value of a s must be close to 2. Therefore, the range of narrow diffraction lines will be around 90° if the take-off angle of the two (identical) crystals is close to 53° (cot 0 = 2) . The wavelength is then selected by choosing the crystal d-spacing. To put the bcc Fe(211) line, convenient for stress scanning (due to its high multiplicity), at 90°, the wavelength should be 0.166 nm, thus the crystal d-spacing should be close to 0.185 nm . The Cu(200) planes have the right spacing (d = 0.1808 nm). Other planes may be used as well, by allowing for different d-spacings (a z# -1), but the configuration with identical crystals seems to be the best . As discussed in more detail in ref. [3], the strong conditions (la) and (1b) may be softened by instead using a single condition referring to the SR point-source case [1,8,9], which turns out to work well in the vicinity of Eqs. (Ia) and (1b) . For this reason, the take-off angles are not tightly specified, both here and in ref. [3]. One can only say that the range of optimal take-off angles is 50-80° with identical crystals . A variety of situations occur if the crystals are allowed to be different. The necessary radii of curvatures are small and cannot be achieved by elastic bending at the required thicknesses . Therefore, plastically bent crystals, either perfect or of small mosaic spread, will have to be used . The peak reflectivity decreases steeply on bending [10,11], so that only strongly reflecting planes can be used . The low-index planes of copper satisfy all these requirements . The (00.2) plane of Be (d = 0.1797 nm) would be somewhat better than Cu(200), but beryllium crystals are expensive and their plastic bending might be a problem . Perfect copper crystals would also be better, but are not available.
0.0
025
0 .50
0.75
1 .00
1 .25
First crystal curvature 1/R, (11m)
1 .50
1 .75
-
2,
0
Fig. 1. Computed peak flux at focus vs curvature (inverse radius) of the first crystal in a microfocusing double monochromator with Cu(200) small-mosaic crystals . Neutron wavelength 0.166 nm, thermal source flux 10 14 n/cm 2/s. plates), of 7.5 cm diameter and 0.4 cm thickness. Plastic spherical bending, to the radii of curvature specified on the figures, was assumed. The peak flux at the focus position vs the first crystal curvature (inverse radius), with the curvature of the second crystal set to the value ensuring minimal linewidths at 90° detector angle, is shown in Fig. 1. It is seen to approach 5 X 107 n/cm 2/s at optimum (absorption in air neglected) . This figure can be doubled by optimal vertical focusing, quite feasible with plastic bending. Fig. 2 shows the beam width at the fixed focus position vs . the second crystal curvature (first at optimum) . The minimal spot size is computed to be 0.26 X 1 .21 cm (FWHM's) . With optimal vertical focusing, the second figure can be reduced to 0.38 cm . It is seen in Fig. 2 that the beam is narrow (below 6 mm) at any second crystal curvature. This is because a
L1 =6 m, L=0.8 m, L,=0.3 m, R,=0.93 m
3. Design computations Results of design computations for a microfocusing double monochromator for stress scanning at MURR are presented in Figs . 1-4. The computer program MAX) was described in ref. [12] . The computations refer to copper (200) plates of small mosaic spread (3' true, as measured recently at MURR for as-grown
00
D.0
0 .25
0.50
0 .75 1 .00 1 .25
1 .50 1 .75
Second crystal curvature 11R2 ('/M)
2 .00
Fig. 2. Computed beam width at focus (horizontally) vs curvature of the second crystal . III. APPLICATIONS
M. Popovici, W.B. Yelon I Nuel. Instr. and Meth. in Phys . Res. A 338 (1994) 132-135
134
second crystal in (+, +) setting always (at any curvature) selects a beam that goes into a small focus at some distance . However, to have both high intensity and a small spot, it is essential that the radii of curvature are set correctly. The necessary accuracy is seen (Figs. 1 and 2) to be not a problem. Fig. 3 shows the computed linewidths in powder diffraction vs detector angle. The double monochromator performance is compared to that of a single monochromator with a similar Cu(200) plate, spherically bent to the optimal radius for minimal linewidths at 90° detector angle. The resolution with the double monochromator is slightly worse, but the peak intensity in diffraction is still better by a factor of more than 7. Little can be done to improve the resolution in the double monochromator configuration with the given sample, because the large linewidth is related to the small distance L21 which is essential for microfocusing. An in-pile Soller collimator will have practically no effect, as a double (+, +) monochromator automatically back-selects an almost plane-parallel incoming beam . The resolution can be improved somewhat, at the expense of intensity, by limiting the beam horizontally before the first crystal. However, as always with bent crystal focusing [121, the resolution can be improved by thinning the crystals and the sample . With a smaller sample, hence a better spatial resolution on stress scanning, it is enough to make only the second crystal thinner. The situation for a sample cube of 1 mm edge and a second crystal 1.5 mm thin is shown in Fig. 4. The optimal radii are now closer to the theoretical values, relations (1a, b) . The linewidth is reduced from 0.9° to 0.65° (curve 1) . The
6
L,=6 m, L=Ö.8 m, L,=0 .3 m (or 1 .1 m with one crystal) 1 : Double R1 =0 .93 rn, mspherically R2-0 .85 Fe (211) peak: 750 n/sat -90° 2: Single Cu (200), R=3.15 m spherically I Fe (211) peak: 100 n/sat -90°
a.o 3.5
N N
m
L,=6 rn, L=0.8 m, L2--0 .3 rn R,=0.93 m, R2=0.88 rn spherically 1 : Mosaic (3' true) both, Fe (211) peak: 31 n/s 2: Perfectcrystals both, Fe (211) peak : 38 n/s
3.0
Ô 2.5 E 20
3
1 .5
120.
105,
90
-7560 Detector angle (degrees)
-
A5
30
Fig. 4. Resolution improvement in the focusing range on thinning the second crystal, for a smaller sample (cube of 1 mm edge). Curve 1: second crystal 1.5 mm thin, mosaic; curve 2: both crystals perfect (if), second 1.5 mm thin; the broken curve is the same as curve 1 in Fig. 3 .
spot size is reduced to 2 mm (horizontally) . Because of the 27 times smaller sample volume, the peak intensity is lower, naturally (by a factor of 24). If the crystals were perfect (curve 2), the linewidth would be further reduced (to 0.57°), with an even smaller (1 .8 mm) spot size, while the intensity would be only 20 times lower. The broken curve in Fig. 4 is the curve 1 in Fig. 3, shown for comparison . The absolute intensities indicated on figs . 3 and 4 for the Fe(211) line can be doubled in each case by optimal vertical focusing . An improvement in peak intensities by a factor of more than 100 over the recent stress scanning measurements at MURR [5] is anticipated with the microfocusing monochromator . References [11 M. Popovici and W.B . Yelon, Nucl . Instr. and Meth . A 319 (1992) 141 .
E 3
M. Popovici and W.B . Yelon, in Neutron Optical Devices and Applications, ed . C.F . Majkrzak and J.L . Wood, SPIE Proc . 1738 (1992) 422. [3] M. Popovici and W.B . Yelon, Z. Kristallografie (1993) in press. [4] A.D . Krawitz, in : Neutron Scattering for Materials Science, eds. S .M . Shapiro, S. Moss and J. Jorgensen, Mater. Res. Soc. Symp . Proc . 166 (1990) 281 . [51 A.D . Krawitz and R.A . Winholtz, in : Advances in X-Ray Analysis 37 (1994) in press. [6] M. Popovici, A.D . Stoica, B. Chalupa and P. Mikula, J. Appl . Cryst. 21 (1988) 258. [7] M. Popovici, A.D . Stoica and 1. lonilâ, to be published. [8] M. Popovici and W.B . Yelon, J. Appl . Cryst. 25 (1992) [2]
3
3.
a
2.
6.0
-150.
-120 .
-90.
-60.
~ -30 .
0 .0
30 .
Detector angle (degrees)
.. 60 .
9Ô .
120 .
, 150.
Fig. 3 . Linewidths in powder diffraction for the double monochromator considered and for a single monochromator with optimally bent Cu(200). The absolute peak intensities for the Fe(211) line of a small sample (cube of 3 mm edge) are indicated (uncorrected for absorption effects and detection efficiency).
331 .
M. Popovici, W.B . Yelon /Nucl. Instr. and Meth. in Phys. Res. A 338 (1994) 132-135 [9] F .N . Chukhovskii, M . Krisch and A .K . Freund, Rev . Sci . Instr. 63 (1992) 920 . [10] Z.H . Kalman and S . Weissmann, J . Appl . Cryst . 12 (1979) 209 . [11] J . Kulda, Acta Cryst . A 40 (1984) 120.
135
[12] M . Popovici, W .B . Yelon, R . Berliner, A.D . Stoica, I . Ionijâ and R . Law, these Proceedings (Workshop on Focusing Bragg Optics, Braunschweig, Germany, 1993), Nucl . Instr . and Meth . A 338 (1993) 99.
III. APPLICATIONS