COHERENT INTERACTIONS OF PHOTONS AND ELECTRONS IN CRYSTALS CLOSE TO THE BRAGG ANGLE

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COHERENT INTERACTIONS OF PHOTONS AND ELECTRONS IN CRYSTALS CLOSE TO THE

BRAGG ANGLE

K. das Gupta

To cite this version:

K. das Gupta. COHERENT INTERACTIONS OF PHOTONS AND ELECTRONS IN CRYSTALS CLOSE TO THE BRAGG ANGLE. Journal de Physique Colloques, 1971, 32 (C4), pp.C4-338-C4-341.

�10.1051/jphyscol:1971462�. �jpa-00214663�

JOURNAL DE PHYSIQUE Colloque C4, supplkment au no 10, Tome 32, Octobre 1971, page C4-338

COHERENT INTERACTIONS OF PHOTONS AND ELECTRONS IN CRYSTALS CLOSE TO THE BRAGG ANGLE (")

K. Das GUPTA

Department of Physics Texas Tech University Lubbock, Texas, USA

Introduction. - A few new ideas and results in connection with the coherent interaction processes of X-rays in crystals are presented in this talk. This includes the discussion of interaction processes of X-rays and electrons associated with the interference in crystals. The interference is the simplest phenome- non which establishes a correlation between electrons or photons in crystals and thereby introduces coherence in the final output. In the case of X-rays and electrons the most fundamental interference phenomenon is the Bragg scattering in crystals. This is essentially the interaction of electromagnetic waves in case of X-rays and the de Broglie waves in the case of electrons which are to be scattered by the periodic lattice waves involved in the scattering process. In Ewald's [I] dynamical theory of X-ray interference the scattering elements of the crystal act as vibrating dipoles at the lattice sites and the problem is to find in the crystal a balanced or self-consistent state of electromagnetic field and the dipole waves. In Ewald's [I] dynamical theory explai- ning the Bragg scattering, the scattered waves have the same angular frequency coo of the incident wave.

Slater [2] described the interaction between the inci- dent electromagnetic wave exp i(oo t - k,.r) and the lattice wave exp i(o, t - k,.r) by representing in a general way the scattered wave by the real part of exp i[(o,

+

+

+

+

The appearance of a sharp line in the Bragg spectrum obtained by the crystal diffraction is guided by the Bragg law nA = 2 d sin 0, with necessary refractive index correction for different orders of reflection. The angle of ejection of the Bragg scattered photon is governed by the conservation rules for the wavevec- tors of the incident and the scattered photon

(*) This work has been supported by the Robert A. Welch Foundation.

where G is the vector in the reciprocal lattice, k is the wavevector of the incident photon and k' is the wavevector of the scattered photon. The crystal recoils as a whole with momentum, tiG. If the scattering of the photon is inelastic as indicated in a general way by Slater [2], the interaction processes must be connected with either the creation of annihilation of an elemen- tary excitation in solids. The wavevector K associated with an elementary excitation appears in the wave- vector conservation law

The physical meaning of the wavevector conserva- tion law introducing the reciprocal lattice vector G is in other words introducing the idea of Bragg type resonance of the modified radiation in the crystal as a whole. Such an effect of the internal Bragg resonance of the modified radiation in the crystal close to the unmodified Bragg scattering should cause an increase in intensity of the modified radiation subject to the internal Bragg reflection in the crystal. This is similar to the process of the origin of the Kossel lines in single crystals. The elementary excitations involved in X-ray scattering as indicated by the wavevector K in equa- tion (2) could be associated with any one of the well- known quasi-particles in solids. These are phonons, acoustic and optical ; excitons ; polarons ; plasmons and magnons in solids. The investigations of the inter- action processes of these elementary excitations in solids presently cover a very active field in solid state studies. In the case of elementary excitation of

the (< electronic ⁾⁾ type, the inelastic Raman scattering

gives useful information. The energy values of ele- mentary excitations lie between 0.1-10 eV. The resol- ving power of the conventional two-crystal spectro- meter is of the order of 0.3-1 eV for the wavelength range between 1-3 A. In recent years we have deve- loped three novel X-ray spectrometers ; (i) Three- Crystal Spectrometer, (ii) Spherically Bent Crystal Spectrometer and (iii) Two Curved Crystal Spectro- meter. Two emission peaks separated by 0.1 eV in CrK, wavelength region have been resolved with the newly developed three crystal spectrometer and the spherically bent crystal spectrometer. Since we have improved the resolution we are getting valuable information of the function o(K).

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971462

COHERENT INTERACTIONS OF PHOTONS AND ELECTRONS IN CRYSTALS C4-339 I n interactions of X-rays with atoms we are prima-

rily concerned with photoelectric effect, Compton scat- tering, Raman scattering and the Rayleigh scattering of unchanged wavelength. In interaction of X-rays with crystals the existence of the phenomenon of ther- mal diffuse scattering (TDS) is well known. The modi- fied frequencies of TDS are (o,

+

ponds to the angular frequency of the acoustic or thermal vibrations, the phonons. A spectroscopic analysis of the thermal diffuse scattering with a three- crystal spectrometer reported by Das Gupta and Welch [3] revealed that the intensity, size and shape of TDS cannot be explained unless the contribution due to the general type of inelastic Raman processes involving elementary excitations other than phonons are also taken into account. The essential features of the newly developed spectrometers will now be des- cribed.

Spectrometers. - (i) THREE-CRYSTAL SPECTROME-

TER. - Three pieces of quartz cut parallel to (10i1) planes are selected from the same crystal. The (1, - 1) rocking curve widths for crystal 1 and crystal 2, for 1 and 3 for 2 and 3 are separately determined. The observed full-width at half-maximum intensity of the rocking curve in parallel (1, - 1) position is 3.2 s as observed in the region of Mo K, radiation. The third crystal is set in parallel (1, - 1, 1) position and the observed width is 4.7 s. Any radiation from the target as observed by rotating the third crystal in antiparallel (I, 1, 1) position will have a fixed width depending only on the spectral window of the system. When crystal 1 and crystal 2 are set for the Mo K,, peak, the width as measured by the third crystal in (1, 1, 1) position is 18 s. Recently Shah and Das Gupta [4]

have analyzed the output of a two-crystal spectro- meter in dispersive position with a third crystal and reported fine structures of chromium K,,,, lines. This establishes the higher resolving power of the three- crystal spectrometer compared to that of the two- crystal spectrometer.

(ii) SPHERICALLY BENT CRYSTAL SPECTROMETER. -

Recently Das Cupta and Gott [5] have reported fine structures of chromium K, line using a newly desi- gned spherically bent crystal spectrometer. Such fine structures in transition elements cannot be resolved with the two crystal spectrometer. The shape of Ka lines of transition elements is characterized by an asymmetry, or the tailing on the long wavelength side taken with a two-crystal spectrometer. The resolu- tion in X-ray spectroscopy is essentially determined by the (1, - 1) rocking curve width of the analyzer crystal.

The rocking curve width is a function of the mosaicity, the intensity of TDS and imperfections of the crystal.

The resolution increases with the increase in the order of Bragg reflection. The essential part of the instru- ment is the spherically bent crystal which is a thin piece of mica (0.14 mm) pressed between two curved

pieces of glass of 100 cm radius. A slit of 40 micron width and 0.3 mm height has been used as the secon- dary source. The radius of the Rowland circle of the spectrometer is 50 cm, which is half the radius of the spherically bent crystal. For the spectroscopic analysis of a radiation the bent crystal is placed from the slit at a distance equal to the calculated chord length.

The effective width of the bent crystal which is exposed to X-rays is 4 mm. The axis of the cone of the inci- dent X-ray beam, of the Bragg reflected beam and the normal to the bent crystal all lie in the same hori- zontal plane. The slit, the center of the bent crystal and the focused image all lie on the focusing Rowland circle. Such an alignment of the spectrometer with a mica crystal has been tested for the 5th, 8th and 11th orders of reflection. The bent crystal is rotated with a motor and a strip chart record of the spectrum is taken in the conventional way. The horizontal width of the slit is 40 microns, which introduces at any fixed Bragg setting an error of f 4 s of arc due to the angular divergence of 8 s at the crystal caused by the finite width of the slit. This introduces an error of 0.1 eV in the measurement of the width of Cr K,, line in the 8th order. This is a controlling factor of the resolution and structures with a separation less than 0.1 eV cannot be observed for Cr Kg, in 8th order.

The astigmatism associated with the conventional cylindrically bent crystal is reduced due to the focus- ing geometry of the spherically bent crystal. The latter property contributed to a higher concentration of X-ray sand therefore spectroscopy is possible in higher orders of reflection at high resolution and at high intensity. We believe that for high resolution spectro- scopy in the angstrom order region, the spherically bent crystal spectrometer at higher orders of reflec- tion could profitably replace the conventional two- crystal spectrometer.

(iii) Two CURVED CRYSTAL SPECTROMETER. - A schematic diagram of two curved monochromators in transmission is shown in figure 1. The first crystal made of quartz for the (1011) transmitted diffraction peak is located in A, and is bent to a radius of R = 11 inches. The parts labelled S, and S, represent the slit holders. Both entry and exit slits are 1 mm in height and 4 mm in width. The assembly is aligned in order to satisfy the Bragg angle of 13" 20' for K,, copper radia- tion. Crystal I at A produces a focused spectrum on the Cauchois circle at E. The focused beam passes through a narrow slit of mm width in a lead screen L.

Crystal I1 receives the copper Ka radiation with the associated spectrum due to the continuum in the neigh- borhood of K,. The second crystal, 11, also made of quartz (10T1) from the same quartz crystal as Crystal I and located at B, is bent to a radius of R = 1

+

Crystal I1 can be rotated about a vertical axis through B and the angle 0 made by crystal I1 and the conver- gent K, beam from Crystal I is measured by a vernier scale which can be read down to one minute of arc,

K. DAS GUPTA

and a micrometer drive changes the angle by two second of arc. Crystal I1 can be moved along a fixed arm parallel to ABE so that the two Cauchois circles of Crystal I1 and Crystal I1 meet at point E where the Ka lines are focused by Crystal I. After the translational adjustment is made, Crystal I1 is rotated about point B until the peak of copper K,, spectrum is reflected by Crystal I1 at point F of the second Cauchois circle FB' E.

The double monochromator transmission scheme has been used t o obtain diffraction patterns with a camera four inches in diameter shown schematically by the circle SF, whose center is at C, in figure 1. A diffraction pattern of anthracene, obtained with 35 kV, 25 mA and 5 hours exposure, minimizesbackground blackening of the film. In fact, patterns of lithium fluoride powders taken with exposures of one hun- dred hours did not show any measurable background other than the incoherent Compton-Raman back- ground. The elimination of the background scatter- ing enables us to study both the coherent and modified scattering from low atomic number specimens.

Observation of fine structures in chromium, iron and Cobalt with three-crystal spectrometer and with spherically bent crystal spectrometer. - Shah and Das Gupta [4] have recently reported the observation

of fine structures in chromium Ka with a three-crystal spectrometer and this has been confirmed by Das Gupta and Gott [5] using the spherically bent crystal spectrometer, and by studying chromium K, in the 8th order of Bragg reflection shown in figure 2. Recently John Priest and Das Gupta [6] have observed fine

FIG. 2a. - CrK,, with two crystals; b. - CrK,, with three crystals in (1, 1, 2) position ; c. - Tracing of the strip chart record of the CrK,, line wigh spherically bent mica crystal in 8th

order of refiection.

structures in Fe K,,,, lines using a three-crystal spec- trometer, In recent literature such structures have beeen expected in transition metal due to the exchange inter- action of 2p electrons with unpaired 3d electrons.

Malkovskaya [7] has raised doubts about the inter- pretation of structures based on the idea of 2p-3d interaction.

The author presents the following qualitative inter- pretation for the existence of fine structures in these transition elements : This is a Raman process where K,, or K,, radiation excites the target atoms due to the transition of a 3d or 4s electron from the filled band to 3d or 4s vacant levels in the unfilled band.

A Raman process is characterized by the transition rule AJ = 0 or _+ 2, which would mean a transitions from either d to s or d to d, or a transition from s to s or s to d where A J = 0 or 3- 2. Transition ele- ments are typically characterized by a high level density of d and s electrons in the filled band as well as d and s wave functions in the vacant band. This fact alone would favor strongly a higher amplitude for a Raman process. Earlier calculation of the band struc- ture of copper by Slater [8] indicates fluctuations of electron density in the filled 3d-4s band. The difference in energy of any two consecutive fluctuations in level density of 3d-4s band is of the same order. It is quite likely, as observed in fine structures in K,, and K,, lines, that the incident K, photon, before it is registered in the detector, excites the atom, making a hole in the 3d band, with the electron being trans- ferred to any of the normally unoccupied 3d-4s levels of the atom. Since the fine structures always appear on the long wavelength side of the main peak, the explanation on the basis of an internal Raman type transition has been preferred.

COHERENT INTERACTIONS OF PHOTONS AND ELECTRONS IN CRYSTALS C4-341 It is believed that the internal Bragg resonance of We plan to use single crystals of chromium and iron the modified Raman of Cr K, due to the lattice to study the spatial distribution of intensity of the planes of crystalline chromium could have increased modified Raman lines with respect to the normals to the cross-section of the suspected Raman process. the lattice planes of the crystal.

References

[I] EWALD (P. P.), Rev. Mod. Phys., 1965, 37, 46. [5] DAS GUPTA (K.) and GOTT (P. F.), Phys. Letters, [2] SLATER (J. C.), Rev. Mod. Phys., 1958, 30, 197. 1970, 33A, 270.

631 DAS GUPTA (K.) and WALCH, Phys. Rev. Lefters, [6] PRIEST (J.) and DAS GUPTA (K.), (to be published).

1968, ^21, 657. [7] MALKOVSKAYA (N.), ZZV., A N SSSR, Ser. Fiz., 1960,

$41 SHAH (M.) and DAS GUPTA (K.), Phys. Letters, 24, 443.

1969, 29A, 570. ^[8] SLATER (J. C.), Phys. Rev., 1936, 49, 537.