- Email: [email protected]
Monochromators for Synchrotron radiation based on SiGe gradient crystals
Nuclear Instruments and Methods in Physics Research A 448 (2000) 152}157
Monochromators for Synchrotron radiation based on SiGe gradient crystals E.V. Shulakov *, V.Sh. Shekhtman, S.S. Khasanov, I.A. Smirnova Institute of Microelectronics Technology, RAS, 142432 Chernogolovka, Moscow Region, Russia Institute of Solid State Physics, RAS, 142432 Chernogolovka, Moscow Region, Russia
Abstract This work is devoted to the investigation of monochromatization of the divergent synchrotron beam at the radiation di!raction by single crystals of silicon}germanium solid solutions with the given gradient of the lattice parameter along the sample surface. It has been shown that such crystals provide a high spectral resolution and increasing luminosity at a given radiation wavelength. The concentration dependence of the lattice parameter of SiGe crystals for compositions from 1 to 12 at % of Ge has been re"ned. The real structure of crystals has been analyzed by methods of two-crystal spectrometer and section topography. The description of the dynamic problem on the Bragg symmetric di!raction of the divergent beam of the X-ray radiation is represented for the case when the scattering vector is perpendicular to the direction of the lattice parameter gradient of the crystal. 2000 Elsevier Science B.V. All rights reserved. PACS: 07.85.-m; 07.85-Qe Keywords: Si-Ge gradient crystals; X-ray di!raction; Synchrotron radiation; X-ray monochromator
1. Introduction Crystals with the composition concentration gradient along the sample surface are of great interest for a new generation of synchrotrons with the character small beam cross-section (several tens of mkm) and divergence of an order of angular minute. The basic e!ect of monochromatization of the divergent beam is providing the variation of the lattice parameter in the crystal which allows one to select the di!racted beam with the given
* Corresponding author. Tel.: #7-095-962-8074; fax: #7095-962-8047. E-mail address: [email protected] (E.V. Shulakov).
wavelength from the polychromatic radiation. The present work is devoted to the investigation of SiGe crystals grown in the Institute of Crystals Growth (Berlin) by the Czochralski technique with the variation of the germanium concentration in the process of growth [1,2].
2. Structural characteristics of SiGe solid solutions 2.1. Measurement of lattice parameters Precision measurements of the lattice parameters of SiGe crystals at small concentrations of the second component were carried out on a di!ractometer Siemens-D500 using the copper anode
0168-9002/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 0 ) 0 0 2 1 3 - 8
E.V. Shulakov et al. / Nuclear Instruments and Methods in Physics Research A 448 (2000) 152}157
radiation. To eliminate possible systematic errors the procedure of related measurements of di!raction angles was used for the sample under study and standard Si crystal. The results of measurements of the lattice parameters of mixed composition crystals are shown in Fig. 1. The absolute error in the determination of the Ge content was better than 0.2 at% [3]. The composition dependence of the lattice parameter in the considered concentration range is well approximated by the linear dependence: a(C)"5.43119#0.1985, where C is the germanium concentration. Attention is drawn to the fact that the experimental dependence a(C) differs considerably from the Vegard's law [4] for which we have a(C)"5.43066#0.2269, according to the values of Si and Ge lattice parameters. The obtained slope of the dependence a(C) is in good agreement with the data of the work [5].
Fig. 1. The dependence of the lattice parameter of the SiGe crystal on the composition. The dashed line corresponds to the Vegard's law.
153
The data obtained when measuring the lattice parameter in di!erent points of the SiGe sample with a Ge content of 1.3}1.4 at% for the (1 1 1) cut are listed in Table 1. The values of the composition obtained by the recalculation of the values of the lattice parameter to the concentration according to the dependence a(C) in Fig. 1 are listed in the last column of the table. The table data demonstrate a high sensitivity of the method of the lattice parameter measurement of SiGe crystals by the 4 4 4 re#ection using the Cu K lines to determine the ? Ge concentration in the sample. Note that the lattice parameter gradient over the sample depth is surely detected by measurements from face and back crystal sides. 2.2. Analysis of structural perfection The investigation of the SiGe crystals perfection was conducted by the method of double-crystal spectrometer in the non-dispersion scheme (n, !n). Fig. 2 shows the rocking curves for the SiGe crystal with the Ge concentration of 1.4 at%, the [1 1 1] growth axis, the (111) cut, and for the standard Si crystal. The 111 re#ection, Cu K radi? ation was used. The width of rocking curves was 13.3 angular sec, for the SiGe solid solution and 10.3 angular sec for the standard Si crystal, the theoretical width of the rocking curve was 9.4 angular sec for ideal silicon and nonpolarized radiation. So, the width of the rocking curve of the gradient crystal is 29% larger than for the standard silicon sample of a high perfection. These measurements indicate the high quality of SiGe crystals. The real structure of SiGe was studied by the section topography technique. The X-ray generator `Rigakua D-4C and the Lang camera A-4 were
Table 1 The lattice parameter of the SiGe sample at the side perpendicular to the growth axis of the crystal Place on the sample
2H (rpap)
Center 8 mm from the center 13 mm from the center Back side of the sample, center Standard Si
158.259 158.270 158.282 158.262 158.628
( ( ( ( (
2 2 2 2 2
) ) ) ) )
Lattice parameter (As )
Ge content (at%)
5.43399 5.43389 5.43378 5.43396 5.43066
1.411 1.360 1.305 1.396 0.000
( ( ( ( (
2 2 2 2 2
) ) ) ) )
( ( ( (
8 8 8 8
) ) ) )
SECTION V.
154
E.V. Shulakov et al. / Nuclear Instruments and Methods in Physics Research A 448 (2000) 152}157
Fig. 2. Rocking curves in the scheme (n, !n), radiation Cu K , ? the Bragg case 111 re#ection: (1) SiGe with C "1.4 at% and b"0; (2) standard Si; (3) the theoretical curve for Si.
used. The di!raction geometry was: the point source with a focus size of 10 lm, nonpolarized radiation Mo K , the 8 lm inlet slit, the registra? tion plane is perpendicular to the di!racted beam. The following were studied: (a) the SiGe sample with a germanium concentration of 1.4 at%, the [1 1 1] growth axis, the (111) cut, a crystal thickness of 610 lm, a diameter of 41 mm, (b) the gradient SiGe crystal with C "2.4 at% Ge and b"0.34 at%/cm, the [1 1 1] growth axis, the (11-2) cut, a crystal thickness of 520 lm. The analysis of section X-ray topographs of the sample end (a) shows that the interference fringes are observed only from central part 12 mm in diameter, and at the sample periphery the fringes disappear. The SiGe section X-ray topograph fragment (22-4) of the central part of the disk is shown in Fig. 3a. The fragment size of 390;770 lm corresponds to the area of the outlet crystal surface of 412;770 lm. Observation of dynamic interference oscillations testi"es to a high crystal perfection. At the same time a change in the fringes structure in the upper X-ray topograph part points to weak distortions of the crystal structure even in the central part of the sample. Surveying of section X-ray topographs of the gradient crystal (b) was conducted so that the germanium concentration increase was directed up-
Fig. 3. Transmission section X-ray topographs: (a) SiGe(22-4), C "1.4 at%, b"0: (b) SiGe(4-40), C "2.4 at%, b"0.34 at%/cm.
ward along vertical. The X-ray topograph fragment size in Fig. 3b is 383;770 lm (412;770 lm at the crystal), in the lower X-ray topograph part the Ge concentration is smaller, the re#ection 22-4. The contrast of fringes at the X-ray topograph corresponds to the picture being registered for perfect crystals. The growth strips are observable in the perpendicular direction. A change in the contrast along the central part of the X-ray topograph is connected with the Ge concentration gradient. Crystals in a whole can be characterized as rather perfect.
3. Principles of radiation monochromatization by a gradient crystal Let us consider the characteristics of the di!raction scheme providing monochromatization of the divergent beam. Fig. 4 shows the geometry of divergent X-ray beam di!raction by the gradient crystal. The source}crystal distance S0 is denoted by R. Assume that in the center of the sample in the point 0 with the germanium concentration C and lattice spacing d , the Bragg condition is exactly ful"lled
E.V. Shulakov et al. / Nuclear Instruments and Methods in Physics Research A 448 (2000) 152}157
155
Fig. 4. The scheme of X-ray divergent beam di!raction by the gradient crystal.
for the wavelength j . Then the wavelength change *k in the point x is described by the expression: *j/j "*d/d #*h ctg h . (1) A change in the Bragg angle *h is related to the coordinate x by the relation *h "!x sin h /R . It is seen from Eq. (1) that the quantity *j/j can be minimized for the crystal with the constant deformation gradient: d"d (1#kx), satisfying the con dition: k"cos h /R. (2) The lattice parameter in crystals of SiGe solid solutions with the constant germanium concentration gradient C"C #bx can be represented in the form d"d (1#bbx), (3) where b"d\(Rd/RC) . The coe$cient b was mea sured in the region of small Ge concentrations and is equal to 0.1985/d . Fig. 5 represents the plots of the optimal gradient of the concentration b"k/b at C "0.024 for 111 and 220 re#ections depend ing on the radiation wavelength. All calculations were made for BESSY II dipole radiation of the 1.7 GeV electron beam. Vertical beam divergence c"3;10\ rad and R"20 m were taken as calculated parameters. It is seen that the magnitude b changes weakly in the range of short wavelengths and is approximately equal to 1.37 at%/cm. The last property allows us to reconstruct the di!racted radiation wavelength simply by the crystal turn. We will consider that the crystal size L overlaps completely the vertical divergence c of the primary beam: ¸5Rc/sin h. Then the spectral resolution
Fig. 5. Optimal gradient of the germanium concentration in silicon in at%/cm.
for the new value of the Bragg angle h"h #*h is determined in the point 0 by the expression: j/*j"[c" cos h !cos h"/sin h#(*j/j) ]\. (4) Here (*j/j) is the proper spectral width of the Bragg re#ection for the perfect crystal which is equal to 4.24 d "s "/j, where s is the h-coe$cient F of the Fourier expansion of the crystal polarizability. Fig. 6 represents the dependences of the spectral resolution on the radiation wavelength for two di!erent gradient SiGe crystals optimized for wavelengths 0.0709 nm (b"1.35 at%/cm) and 0.154 nm (b"1.25 at%/cm) and for the perfect SiGe crystal (b"0). Maximal values of j/*j for dependences 1 and 2 are approximately equal to 1.6;10. It is seen that gradient crystals have essentially a higher spectral resolution and allow one to change the radiation wavelength in a rather wide range without essential loss in resolution. Besides, gradient crystals increase considerably the spectral re#ection for a given radiation wavelength. An increase in intensity as compared to the perfect crystal is estimated by the formula: c/*h , where *h "2.12 "s "/sin 2h is the angular Q width of re#ection of the perfect crystal. Gain in intensity on the given wavelength for SiGe crystals with the optimal concentration gradient is shown in Fig. 7. Under consideration of scattering the analysis represented above describes only approximately
SECTION V.
156
E.V. Shulakov et al. / Nuclear Instruments and Methods in Physics Research A 448 (2000) 152}157
modeling of scattering processes based on the numerical solution of Takagi equations was used. Under this modeling, the crystal structure of the sample was described as follows:
Fig. 6. Spectral resolution j/*j for the 220 re#ection of SiGe crystals. C "2.4 at%: (1) gradient crystal, b"1.25 at%/cm; (2) gradient crystal, b"1.35 at%/cm; (3) perfect crystal, b"0.
Fig. 7. Integral intensity at the given wavelength for SiGe crystals with optimal gradient relative to the case of the perfect crystal (b"0), C "2.4 at%.
the intensity of the di!racted beam and its space structure. With this purpose the problem of dynamic scattering of the divergent spherical wave by crystals of the SiGe solid solution with the constant gradient of the lattice parameter in the re#ection plane is considered in the framework of the present investigation. Such a problem in the dynamic theory of X-ray di!raction by crystals was "rst set up. This problem has no analytical solution, therefore for the study of peculiarities on such crystals the
; "kx(x!a ), V ; "kxz. (5) X Here the functions ; and ; correspond to the V X atom displacement in the gradient lattice relative to their positions in the perfect crystal. The axis X is directed along the crystal surface, and the axis Z is directed inside the sample. This description was used when the Cauchy problem with boundary conditions for wave "elds on the outlet crystal surface was numerically solved. The preliminary results on modeling of the spectral composition of the di!racted radiation, space distribution of the scattered wave front, rocking curves of the sample in non-dispersion scheme, intensity distribution of wave "elds in the sample bulk were obtained. The analysis of the modeling results in the sample bulk showed that at the scattering of the monochromatic spherical wave on the gradient crystal, the primary radiation penetrates deeply into the sample and forms in it a complicated pattern of the intensity distribution. This result di!ers essentially from the traditional representation that in the Bragg geometry only the thin nearsurface layer of the sample participates in scattering due to the e!ects of the primary extinction. The di!racted radiation has a wider angular spectrum and larger integral intensity. 4. Discussion of results Analyses of the literature data and the presented experiments show that the application of optics of gradient crystals to the synchrotron radiation is in a rather perceptible direction. Such crystals are expected to provide a high-energy resolution and increasing luminosity at a given wavelength. Gradient crystals can be used in the divergent synchrotron radiation beam and provide the same parameters of the spectral resolution as perfect silicon crystals at the collimated beam. It indicates a possibility of obtaining radiation in the necessary wavelength range without expensive usually used
E.V. Shulakov et al. / Nuclear Instruments and Methods in Physics Research A 448 (2000) 152}157
collimating mirrors. In principle, the same e!ect can be achieved by bent monochromators. However, these systems require precision mechanical equipment and do not allow one to realize the reconstructed monochromatization regime.
Acknowledgements This work was supported by Russian Foundation for Basic Researches, Grants No. 96-02-00166 and 98-02-16652.
157
References [1] N.V. Abrosimov, S.N. Rossolenko, V. Alex, A. Gerhardt, W. SchroK der, J. Crystal Growth 166 (1996) 657. [2] N.V. Abrosimov, S.N. Rossolenko, W. Thieme, A. Gerhardt, W. SchroK der, J. Crystal Growth 174 (1997) 182. [3] J. Donecker, A. Gerhardt, J. Wollweber, Mater. Sci. Eng. B 28 (1994) 18. [4] J.P. Dismukes, L. Ekstorm, R.J. Pa!, J. Phys. Chem. 68 (1964) 3021. [5] R. Fabbri, F. Cembali, M. Servidori, A. Zani, J. Appl. Phys. 74 (1993) 2359.
SECTION V.
Monochromators for Synchrotron radiation based on SiGe gradient crystals E.V. Shulakov *, V.Sh. Shekhtman, S.S. Khasanov, I.A. Smirnova Institute of Microelectronics Technology, RAS, 142432 Chernogolovka, Moscow Region, Russia Institute of Solid State Physics, RAS, 142432 Chernogolovka, Moscow Region, Russia
Abstract This work is devoted to the investigation of monochromatization of the divergent synchrotron beam at the radiation di!raction by single crystals of silicon}germanium solid solutions with the given gradient of the lattice parameter along the sample surface. It has been shown that such crystals provide a high spectral resolution and increasing luminosity at a given radiation wavelength. The concentration dependence of the lattice parameter of SiGe crystals for compositions from 1 to 12 at % of Ge has been re"ned. The real structure of crystals has been analyzed by methods of two-crystal spectrometer and section topography. The description of the dynamic problem on the Bragg symmetric di!raction of the divergent beam of the X-ray radiation is represented for the case when the scattering vector is perpendicular to the direction of the lattice parameter gradient of the crystal. 2000 Elsevier Science B.V. All rights reserved. PACS: 07.85.-m; 07.85-Qe Keywords: Si-Ge gradient crystals; X-ray di!raction; Synchrotron radiation; X-ray monochromator
1. Introduction Crystals with the composition concentration gradient along the sample surface are of great interest for a new generation of synchrotrons with the character small beam cross-section (several tens of mkm) and divergence of an order of angular minute. The basic e!ect of monochromatization of the divergent beam is providing the variation of the lattice parameter in the crystal which allows one to select the di!racted beam with the given
* Corresponding author. Tel.: #7-095-962-8074; fax: #7095-962-8047. E-mail address: [email protected] (E.V. Shulakov).
wavelength from the polychromatic radiation. The present work is devoted to the investigation of SiGe crystals grown in the Institute of Crystals Growth (Berlin) by the Czochralski technique with the variation of the germanium concentration in the process of growth [1,2].
2. Structural characteristics of SiGe solid solutions 2.1. Measurement of lattice parameters Precision measurements of the lattice parameters of SiGe crystals at small concentrations of the second component were carried out on a di!ractometer Siemens-D500 using the copper anode
0168-9002/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 0 ) 0 0 2 1 3 - 8
E.V. Shulakov et al. / Nuclear Instruments and Methods in Physics Research A 448 (2000) 152}157
radiation. To eliminate possible systematic errors the procedure of related measurements of di!raction angles was used for the sample under study and standard Si crystal. The results of measurements of the lattice parameters of mixed composition crystals are shown in Fig. 1. The absolute error in the determination of the Ge content was better than 0.2 at% [3]. The composition dependence of the lattice parameter in the considered concentration range is well approximated by the linear dependence: a(C)"5.43119#0.1985, where C is the germanium concentration. Attention is drawn to the fact that the experimental dependence a(C) differs considerably from the Vegard's law [4] for which we have a(C)"5.43066#0.2269, according to the values of Si and Ge lattice parameters. The obtained slope of the dependence a(C) is in good agreement with the data of the work [5].
Fig. 1. The dependence of the lattice parameter of the SiGe crystal on the composition. The dashed line corresponds to the Vegard's law.
153
The data obtained when measuring the lattice parameter in di!erent points of the SiGe sample with a Ge content of 1.3}1.4 at% for the (1 1 1) cut are listed in Table 1. The values of the composition obtained by the recalculation of the values of the lattice parameter to the concentration according to the dependence a(C) in Fig. 1 are listed in the last column of the table. The table data demonstrate a high sensitivity of the method of the lattice parameter measurement of SiGe crystals by the 4 4 4 re#ection using the Cu K lines to determine the ? Ge concentration in the sample. Note that the lattice parameter gradient over the sample depth is surely detected by measurements from face and back crystal sides. 2.2. Analysis of structural perfection The investigation of the SiGe crystals perfection was conducted by the method of double-crystal spectrometer in the non-dispersion scheme (n, !n). Fig. 2 shows the rocking curves for the SiGe crystal with the Ge concentration of 1.4 at%, the [1 1 1] growth axis, the (111) cut, and for the standard Si crystal. The 111 re#ection, Cu K radi? ation was used. The width of rocking curves was 13.3 angular sec, for the SiGe solid solution and 10.3 angular sec for the standard Si crystal, the theoretical width of the rocking curve was 9.4 angular sec for ideal silicon and nonpolarized radiation. So, the width of the rocking curve of the gradient crystal is 29% larger than for the standard silicon sample of a high perfection. These measurements indicate the high quality of SiGe crystals. The real structure of SiGe was studied by the section topography technique. The X-ray generator `Rigakua D-4C and the Lang camera A-4 were
Table 1 The lattice parameter of the SiGe sample at the side perpendicular to the growth axis of the crystal Place on the sample
2H (rpap)
Center 8 mm from the center 13 mm from the center Back side of the sample, center Standard Si
158.259 158.270 158.282 158.262 158.628
( ( ( ( (
2 2 2 2 2
) ) ) ) )
Lattice parameter (As )
Ge content (at%)
5.43399 5.43389 5.43378 5.43396 5.43066
1.411 1.360 1.305 1.396 0.000
( ( ( ( (
2 2 2 2 2
) ) ) ) )
( ( ( (
8 8 8 8
) ) ) )
SECTION V.
154
E.V. Shulakov et al. / Nuclear Instruments and Methods in Physics Research A 448 (2000) 152}157
Fig. 2. Rocking curves in the scheme (n, !n), radiation Cu K , ? the Bragg case 111 re#ection: (1) SiGe with C "1.4 at% and b"0; (2) standard Si; (3) the theoretical curve for Si.
used. The di!raction geometry was: the point source with a focus size of 10 lm, nonpolarized radiation Mo K , the 8 lm inlet slit, the registra? tion plane is perpendicular to the di!racted beam. The following were studied: (a) the SiGe sample with a germanium concentration of 1.4 at%, the [1 1 1] growth axis, the (111) cut, a crystal thickness of 610 lm, a diameter of 41 mm, (b) the gradient SiGe crystal with C "2.4 at% Ge and b"0.34 at%/cm, the [1 1 1] growth axis, the (11-2) cut, a crystal thickness of 520 lm. The analysis of section X-ray topographs of the sample end (a) shows that the interference fringes are observed only from central part 12 mm in diameter, and at the sample periphery the fringes disappear. The SiGe section X-ray topograph fragment (22-4) of the central part of the disk is shown in Fig. 3a. The fragment size of 390;770 lm corresponds to the area of the outlet crystal surface of 412;770 lm. Observation of dynamic interference oscillations testi"es to a high crystal perfection. At the same time a change in the fringes structure in the upper X-ray topograph part points to weak distortions of the crystal structure even in the central part of the sample. Surveying of section X-ray topographs of the gradient crystal (b) was conducted so that the germanium concentration increase was directed up-
Fig. 3. Transmission section X-ray topographs: (a) SiGe(22-4), C "1.4 at%, b"0: (b) SiGe(4-40), C "2.4 at%, b"0.34 at%/cm.
ward along vertical. The X-ray topograph fragment size in Fig. 3b is 383;770 lm (412;770 lm at the crystal), in the lower X-ray topograph part the Ge concentration is smaller, the re#ection 22-4. The contrast of fringes at the X-ray topograph corresponds to the picture being registered for perfect crystals. The growth strips are observable in the perpendicular direction. A change in the contrast along the central part of the X-ray topograph is connected with the Ge concentration gradient. Crystals in a whole can be characterized as rather perfect.
3. Principles of radiation monochromatization by a gradient crystal Let us consider the characteristics of the di!raction scheme providing monochromatization of the divergent beam. Fig. 4 shows the geometry of divergent X-ray beam di!raction by the gradient crystal. The source}crystal distance S0 is denoted by R. Assume that in the center of the sample in the point 0 with the germanium concentration C and lattice spacing d , the Bragg condition is exactly ful"lled
E.V. Shulakov et al. / Nuclear Instruments and Methods in Physics Research A 448 (2000) 152}157
155
Fig. 4. The scheme of X-ray divergent beam di!raction by the gradient crystal.
for the wavelength j . Then the wavelength change *k in the point x is described by the expression: *j/j "*d/d #*h ctg h . (1) A change in the Bragg angle *h is related to the coordinate x by the relation *h "!x sin h /R . It is seen from Eq. (1) that the quantity *j/j can be minimized for the crystal with the constant deformation gradient: d"d (1#kx), satisfying the con dition: k"cos h /R. (2) The lattice parameter in crystals of SiGe solid solutions with the constant germanium concentration gradient C"C #bx can be represented in the form d"d (1#bbx), (3) where b"d\(Rd/RC) . The coe$cient b was mea sured in the region of small Ge concentrations and is equal to 0.1985/d . Fig. 5 represents the plots of the optimal gradient of the concentration b"k/b at C "0.024 for 111 and 220 re#ections depend ing on the radiation wavelength. All calculations were made for BESSY II dipole radiation of the 1.7 GeV electron beam. Vertical beam divergence c"3;10\ rad and R"20 m were taken as calculated parameters. It is seen that the magnitude b changes weakly in the range of short wavelengths and is approximately equal to 1.37 at%/cm. The last property allows us to reconstruct the di!racted radiation wavelength simply by the crystal turn. We will consider that the crystal size L overlaps completely the vertical divergence c of the primary beam: ¸5Rc/sin h. Then the spectral resolution
Fig. 5. Optimal gradient of the germanium concentration in silicon in at%/cm.
for the new value of the Bragg angle h"h #*h is determined in the point 0 by the expression: j/*j"[c" cos h !cos h"/sin h#(*j/j) ]\. (4) Here (*j/j) is the proper spectral width of the Bragg re#ection for the perfect crystal which is equal to 4.24 d "s "/j, where s is the h-coe$cient F of the Fourier expansion of the crystal polarizability. Fig. 6 represents the dependences of the spectral resolution on the radiation wavelength for two di!erent gradient SiGe crystals optimized for wavelengths 0.0709 nm (b"1.35 at%/cm) and 0.154 nm (b"1.25 at%/cm) and for the perfect SiGe crystal (b"0). Maximal values of j/*j for dependences 1 and 2 are approximately equal to 1.6;10. It is seen that gradient crystals have essentially a higher spectral resolution and allow one to change the radiation wavelength in a rather wide range without essential loss in resolution. Besides, gradient crystals increase considerably the spectral re#ection for a given radiation wavelength. An increase in intensity as compared to the perfect crystal is estimated by the formula: c/*h , where *h "2.12 "s "/sin 2h is the angular Q width of re#ection of the perfect crystal. Gain in intensity on the given wavelength for SiGe crystals with the optimal concentration gradient is shown in Fig. 7. Under consideration of scattering the analysis represented above describes only approximately
SECTION V.
156
E.V. Shulakov et al. / Nuclear Instruments and Methods in Physics Research A 448 (2000) 152}157
modeling of scattering processes based on the numerical solution of Takagi equations was used. Under this modeling, the crystal structure of the sample was described as follows:
Fig. 6. Spectral resolution j/*j for the 220 re#ection of SiGe crystals. C "2.4 at%: (1) gradient crystal, b"1.25 at%/cm; (2) gradient crystal, b"1.35 at%/cm; (3) perfect crystal, b"0.
Fig. 7. Integral intensity at the given wavelength for SiGe crystals with optimal gradient relative to the case of the perfect crystal (b"0), C "2.4 at%.
the intensity of the di!racted beam and its space structure. With this purpose the problem of dynamic scattering of the divergent spherical wave by crystals of the SiGe solid solution with the constant gradient of the lattice parameter in the re#ection plane is considered in the framework of the present investigation. Such a problem in the dynamic theory of X-ray di!raction by crystals was "rst set up. This problem has no analytical solution, therefore for the study of peculiarities on such crystals the
; "kx(x!a ), V ; "kxz. (5) X Here the functions ; and ; correspond to the V X atom displacement in the gradient lattice relative to their positions in the perfect crystal. The axis X is directed along the crystal surface, and the axis Z is directed inside the sample. This description was used when the Cauchy problem with boundary conditions for wave "elds on the outlet crystal surface was numerically solved. The preliminary results on modeling of the spectral composition of the di!racted radiation, space distribution of the scattered wave front, rocking curves of the sample in non-dispersion scheme, intensity distribution of wave "elds in the sample bulk were obtained. The analysis of the modeling results in the sample bulk showed that at the scattering of the monochromatic spherical wave on the gradient crystal, the primary radiation penetrates deeply into the sample and forms in it a complicated pattern of the intensity distribution. This result di!ers essentially from the traditional representation that in the Bragg geometry only the thin nearsurface layer of the sample participates in scattering due to the e!ects of the primary extinction. The di!racted radiation has a wider angular spectrum and larger integral intensity. 4. Discussion of results Analyses of the literature data and the presented experiments show that the application of optics of gradient crystals to the synchrotron radiation is in a rather perceptible direction. Such crystals are expected to provide a high-energy resolution and increasing luminosity at a given wavelength. Gradient crystals can be used in the divergent synchrotron radiation beam and provide the same parameters of the spectral resolution as perfect silicon crystals at the collimated beam. It indicates a possibility of obtaining radiation in the necessary wavelength range without expensive usually used
E.V. Shulakov et al. / Nuclear Instruments and Methods in Physics Research A 448 (2000) 152}157
collimating mirrors. In principle, the same e!ect can be achieved by bent monochromators. However, these systems require precision mechanical equipment and do not allow one to realize the reconstructed monochromatization regime.
Acknowledgements This work was supported by Russian Foundation for Basic Researches, Grants No. 96-02-00166 and 98-02-16652.
157
References [1] N.V. Abrosimov, S.N. Rossolenko, V. Alex, A. Gerhardt, W. SchroK der, J. Crystal Growth 166 (1996) 657. [2] N.V. Abrosimov, S.N. Rossolenko, W. Thieme, A. Gerhardt, W. SchroK der, J. Crystal Growth 174 (1997) 182. [3] J. Donecker, A. Gerhardt, J. Wollweber, Mater. Sci. Eng. B 28 (1994) 18. [4] J.P. Dismukes, L. Ekstorm, R.J. Pa!, J. Phys. Chem. 68 (1964) 3021. [5] R. Fabbri, F. Cembali, M. Servidori, A. Zani, J. Appl. Phys. 74 (1993) 2359.
SECTION V.