Bragg's Law

12 Key excerpts on "Bragg's Law"

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  Kittel's Introduction to Solid State Physics

  • Charles Kittel(Author)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  But each plane reflects 10 3 to 10 5 of the incident radiation, so that 10 3 to 10 5 planes may contribute to the formation of the Bragg-reflected beam in a perfect crystal. Reflection by a single plane of atoms is treated in Chapter 17 on surface physics. The Bragg law is a consequence of the periodicity of the lattice. Notice that the law does not refer to the composition of the basis of atoms associated with every lattice point. We shall see, however, that the composition of the basis determines the relative intensity of the various orders of diffraction (denoted by n above) from a given set of parallel planes. Bragg reflection from a single crystal is shown in Fig. 3 and from a powder in Fig. 4. SCATTERED WAVE AMPLITUDE The Bragg derivation of the diffraction condition (1) gives a neat state-ment of the condition for the constructive interference of waves scattered from the lattice points. We need a deeper analysis to determine the scattering Figure 3 Sketch of a monochromator which by Bragg reflection selects a narrow spectrum of x-ray or neutron wavelengths from a broad spectrum incident beam. The upper part of the figure shows the analysis (obtained by reflection from a second crystal) of the purity of a 1.16 Å beam of neutrons from a calcium fluoride crystal monochromator. (After G. Bacon.) Figure 4 X-ray diffractometer recording of powdered silicon, showing a counter recording of the diffracted beams. (Courtesy of W. Parrish.) 28 2 Wave Diffraction and the Reciprocal Lattice intensity from the basis of atoms, which means from the spatial distribution of electrons within each cell. Fourier Analysis We have seen that a crystal is invariant under any translation of the form T u 1 a 1 u 2 a 2 u 3 a 3 , where u 1 , u 2 , u 3 are integers and a 1 , a 2 , a 3 are the crystal axes. Any local physical property of the crystal, such as the charge concentra-tion, electron number density, or magnetic moment density is invariant under T .

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  Optics for Materials Scientists

  • Myeongkyu Lee(Author)
  • 2019(Publication Date)
  • Apple Academic Press

    (Publisher)

  This new form of Bragg law is more dominantly used in X-ray diffraction. The Bragg law can be experimentally applied in two different ways. By using a monochromatic X-ray beam of known wavelength λ and measuring θ, we can determine the spacing d of various planes. Alternatively, a crystal with planes of known spacing d can be used to determine the radiation wavelength λ. Most applications of X-ray diffraction are associated with the measurement of the diffraction angle (2 θ rather than θ is experimentally measured). The essential components of typical X-ray diffractometer are shown in Figure 5.29. A collimated beam from the X-ray source is incident onto a sample stationed on the sample holder, which may be set at any desired angle to the incident beam. A detector measures the intensity of the diffracted beam; it can be rotated around the sample and set at any desired angular position. The sample holder can also be rotated around its center independently of or in conjunction with the detector. The diffractometer measures the angle 2 θ between the incident and detected beams. In the symmetric scan (often called θ −2 θ scan or 2 θ − θ scan), the angle θ between the incident beam and the sample holder is maintained at half the measured diffraction angle 2 θ, as depicted in Figure 5.29. FIGURE 5.29 Typical X-ray diffractometer. The Bragg law of eq 5.39 is given in a simple scalar form. The Bragg equation for (hkl) planes can be reformatted as (2 sin θ B) (2 π / λ) = 2 π / d h k l. (5.40) Figure 5.30 shows a graphical configuration for the Bragg law in which the incident and diffracted beams are represented by their wave vectors K i and K d, respectively. The magnitude of (K d – K i) is equal to the left-hand side of eq 5.40

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  Solid-State Physics, Fluidics, and Analytical Techniques in Micro- and Nanotechnology

  • Marc J. Madou(Author)
  • 2011(Publication Date)
  • CRC Press

    (Publisher)

  Bragg’s Law In 1913 W.H. and W.L. Bragg (a father and son team)proposedthattheconditionforconstructive FIGURE2.16 Electron density map of adenosine triphos-phate (ATP). 50 Solid-State Physics, Fluidics, and Analytical Techniques in Micro- and Nanotechnology specular reflection of x-rays from a set of crystal planes separated by a distance d hkl could be repre-sentedas: 2d hkl sin θ = n λ (2.20) This expression basically tells us that construc-tive interference of waves reflected by successive crystalplanesoccurswheneverthepathdifference (2 d hkl sin θ )isanintegralmultiple( n )ofthewave-length λ .Also,foreach( hkl )familyofplanes,x-rays will only diffract at one angle θ . The integer n is knownastheorderofthecorrespondingreflection. BecauseBraggreflectioncanonlyoccurfor λ ≤ 2 d , oneneedsx-rayswithwavelengthsintheÅngstrom rangetoresolvecrystalplanes.TheBraggequation iseasilyderivedfromaninspectionof Figure2.18 . Bragg’slawisaresultoftheperiodicityofthelattice withtheatomsinthecrystalbasiscontrollingthe relativeintensityofthevariousorders( n )ofdiffrac-tion from a set of parallel ( hkl ) planes. This basic equation is the starting point for understanding crystaldiffractionofx-rays,electrons,neutrons,and anyotherparticlesthathaveadeBroglie wavelength ( Chapter 3 ) less than the interatomic spacing. Althoughthereflectionfromeachplaneisspecular, onlyforcertainvaluesof θ willthereflectionsfrom allplanesaddupinphasetogiveastrongreflected beam. Each plane reflects only 10 − 3 to 10 − 5 of the incident radiation, i.e., it is not a perfect reflector. Hence,10 3 to10 5 planescontributetotheformation oftheBragg-reflectedbeaminaperfectcrystal. Thecompositionofthebasisdeterminestherela-tiveintensityofthevariousordersofdiffraction. Laue Equations In1912vonLauepredictedthatdiffractionpatterns of x-rays on crystals would be entirely analogous tothediffractionoflightbyanopticalgrating.

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  Modern Metallography

  The Commonwealth and International Library: Metallurgy Division

  • R. E. Smallman, K. H. G. Ashbee, W. S. Owen, D. W. Hopkins, H. M. Finniston(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  The Bragg law is only obeyed for critical values of Θ and d. In practice, to ensure that Bragg's Law is satisfied and that reflections from various crystal planes will occur, it is necessary to provide a range of either Θ or values. The various ways in which this can be done leads to the standard methods of X-ray diffraction. X-RAY METALLOGRAPHY 67 THE STRUCTURE FACTOR In calculating the intensity scattered by a given atomic structure we have first to consider the intensity scattered by one atom, and then go on to consider the contribution from all the other atoms in the particular arrangement which make up that structure. The efficiency of an atom in scattering X-rays is usually denoted by/, the atomic scattering factor, which is the ratio of A a , the amplitude scattered by an atom to that scattered by a single electron, A e . If atoms were merely points, their scattering factors would be equal to the number of electrons they contain, i.e. to their atomic numbers. However, because the size of the atom is comparable with the wavelength of X-rays, scattering from different parts of the atom is not in phase, and the result is that the intensity scattered by an atom I a < ZH e , where I e is the intensity scattered by one electron and Z is the atomic number. The atomic scattering factor depends on both the Bragg angle Θ and the X-ray wavelength, because the path difference for the individual waves scattered from the various electrons in the atom is zero when 0 = 0 and increases with increasing Θ; f is therefore expressed as a function of sin θ/λ as shown in Fig. 41. 30 20 I io 0 0-3 0-6 0-9 sinfl/ FIG. 41. Variation of atomic scat-tering factor with sin θ/λ. 68 MODERN METALLOGRAPHY To find the intensity scattered by a material of a given structure it is necessary to add waves of different phase and amplitude which arise from atoms of different scattering power situated at different positions in the unit cell.

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  Structure of Materials

  An Introduction to Crystallography, Diffraction and Symmetry

  • Marc De Graef, Michael E. McHenry(Authors)
  • 2012(Publication Date)
  • Cambridge University Press

    (Publisher)

  11.2.2 Bragg’s law in reciprocal space Bragg’s law predicts the geometrical conditions that need to be satisfied in order to observe diffraction from sets of lattice planes in a crystal. For a given wavelength, the equation defines the Bragg angle, θ , for all sets of planes in the crystal. In the first section of this chapter, we have seen that the incident X-ray photon can be represented by a wave vector k , parallel to the propagation direction of the wave and with length equal to the inverse of the wavelength. This vector has the dimension of a reciprocal length and is thus a vector in reciprocal space. We have also seen that each set of parallel planes ( hkl ) can be repre-sented by a reciprocal lattice vector g hkl . Let us now consider the diffraction process from a reciprocal space point of view. Consider the X-ray beam characterized by the wave vector k , as shown in Fig. 11.13 (a). This vector makes an angle θ with the plane ( hkl ), which is represented by the normal vector g hkl . In the figure, we have chosen the origin of reciprocal space to be at the endpoint of the wave vector k . The diffracted wave k also encloses an angle θ with the diffracting plane, and is drawn with its initial point in the origin. Since vectors can be translated parallel to themselves, we can translate the vector k so that its initial point coincides with the initial point of k . The angle between k and k is thus equal to 2 θ . Since the wavelength does not change during the diffraction process, the length of the vector k must also be equal to 1 /λ . Therefore, the endpoints of both wave vectors are located on a circle, with center at the initial point of k and radius 1 /λ , as shown in Fig. 11.13 (b). This circle is known as the Ewald circle (or the Ewald sphere in 3-D), after the Austrian crystallographer Paul Peter Ewald (1888–1985), who first introduced this circle (sphere) into crystallography ( Ewald , 1913 ; 1962 ).

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  Electron Diffraction in the Transmission Electron Microscope

  Electron Diffraction in the Transmission Electron Microscope

  • P.E. Champness(Author)
  • 2020(Publication Date)
  • Garland Science

    (Publisher)

  Bragg angle.

  Although Bragg’s approach considers the radiation to be reflected from crystal planes, there is a very important difference between these ‘reflections’ and the reflections of light from a mirror: light is reflected whatever the angle of incidence, whereas radiation incident on a set of crystal planes is only reflected or diffracted at the specific angles given by Bragg’s Law. This difference is a consequence of the fact that the crystal is three dimensional and specifically that, for rays reflected from the top plane (as with light), the path difference is always zero and the waves always interfere constructively.

  In light optics, when n = 1 in Bragg’s Law we refer to the first-order diffracted beam, when n = 2 to the second-order diffracted beam, etc. For diffraction from crystal planes, however, we conventionally write:

  For example, rather than referring to the second-order reflection from the (100) planes, we consider that the radiation is diffracted by planes of half the spacing, i.e. the (200) planes (Figure 2.4 ). (The Miller indices are reciprocals of the fractional intercepts of the planes on the unit-cell axes, Appendix A .) In terms of Bragg’s equation we can write:

  Figure 2.4. (a) The Bragg condition for the second-order reflection from the (1 00) planes. If the path difference, GY + YH, between rays reflected by successive planes is 2λ, the Bragg equation can be written: 2d100 Sin θ = 2λ; (b) The same condition, but considered as the first-order reflection from the (200) planes. If GY + YH = 2λ, then QY + YR = λ. This distance is clearly equal to Q’R’ + Y’R’, the path difference for the (200) set of planes.

  It is clearly more convenient to refer to the ‘200 reflection’ than to ‘the second-order reflection from the (100) planes’. Notice that, when we refer to reflections from planes, we omit the brackets that we use when we refer to the planes themselves; i.e. the (hkl) planes give rise to the hkl reflection. (Note that multiple indices such as 200, 300, etc. are sometimes called Laue indices

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  High-Resolution XAS/XES

  Analyzing Electronic Structures of Catalysts

  • Jacinto Sa(Author)
  • 2014(Publication Date)
  • CRC Press

    (Publisher)

  Z M , (2.5) with ρ as the crystal density (in g/cm 3 ), Z as the number of electrons in the crystal molecule (or crystal atom) of molar mass M (in g), and λ as the wavelength of the radiation (in Å). The Bragg law is a necessary but not sufficient condition to observe constructive interferences. Actually, the amplitudes and relative phases of the X–rays scattered by the individual atoms in the unit cell of the crystal are given by the structure factor F hkl defined by: F f e hk j i h x k y z j N j j j lscript lscript = ⋅ ⋅ + ⋅ + ⋅ = ∑ 2 1 π ( ) , (2.6) 31 Crystal Spectrometers where f j stands for the atomic scattering factor of the j th atom whose position in the unit cell is given by the coordinates x j , y j , z j , and N is the number of atoms in the unit cell. The atomic scattering factor f j corresponds to the ratio between the amplitude scattered by the atom j and the one scattered by an isolated electron, under identi-cal conditions. Clearly, the maximum value that f j can take is Z j (atomic number of the atom j ). When f j = Z j , all the electrons scatter in phase with each other. As the intensity of a wave is given by the square of its amplitude, the intensity of the photon beam diffracted by the planes ( hkℓ ) is proportional to the square of the modulus of the structure factor F hk lscript : I F F F hk hk hk hk lscript lscript lscript lscript ∝ = ⋅ * 2 , (2.7) where F hk lscript * is the complex conjugate of F hk lscript . For face-centered cubic crystals like silicon and germanium, the structure factor F hkℓ only when all Miller’s indices are odd or all are even. In other words, for face-centered cubic crystals there is no dif-fracted intensity if the Miller’s indices are mixed. For instance, no diffraction is observed for the (110) planes of Ge but a strong diffraction is observed for Ge (220), which is equivalent to Ge (110) in the second order of diffraction.

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  Introduction to the Physics of Highly Charged Ions

  • Heinrich F. Beyer, Viateheslav P. Shevelko(Authors)
  • 2016(Publication Date)
  • CRC Press

    (Publisher)

  2.4.7 Bragg reflection, diffraction grating As we have seen, diffraction effects become important when the wavelength of the light is of the same order of magnitude as the diffracting objects such as a narrow gap for visible light. Such structures can also be represented by the regular lattice of a crystal. For these the wavelengths of x-rays are comparable to the spacing d of lattice planes. As illustrated in figure 2.13, x-rays of a certain wavelength λ may constructively interfere when partially reflected between surfaces that produce a path difference equal to an integral number m of wavelengths. This condition is defined by Bragg’s law, m λ = 2 d sin θ. (2.29) where the angle θ is the grazing or Bragg angle. Equation (2.29) forms the basis for x-ray spectrometry developed by William Henry Bragg (1862–1942) and William Lawrence Bragg (1890–1971). The underlying physical process of the 40 Radiation d Θ Figure 2.13. Bragg reflection of x-rays on the regular structure of a crystal lattice. Bragg diffraction is the scattering of the incident radiation on the individual atoms of the crystal. Coherent radiation, however, does not originate from each atom independently. Instead, the waves radiated from each atom are correlated through their common origin from one x-ray source and the phase-sensitive scattering process. This way scattered radiation is produced that is weak or zero in a random direction but very strong in a particular direction. Whereas the scheme in figure 2.13 works fine in getting a grasp of what is happening, more complicated structures require Fourier analysis. The latter was introduced in its basic form when light diffraction at a slit or double slit was discussed. Mathematically, the diffraction pattern is always given as the Fourier transform of the scattering power of the sample. A perfectly ordered crystal, analyzed in three dimensions, will produce a diffraction pattern of isolated spots of constructive interference.

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  X-Ray Diffraction for Materials Research

  From Fundamentals to Applications

  • Myeongkyu Lee(Author)
  • 2017(Publication Date)
  • Apple Academic Press

    (Publisher)

  d -spacings) than at low angles.

  8.3 INDEXING OF POWDER PATTERN

  All substances produce a characteristic diffraction pattern.

  X

  ‐ray diffraction discloses the presence of a substance as its existing form, not in terms of the constituent elements. This is the reason why graphite can be differentiated from diamond by the diffraction method, though both consist of carbon. By the same token,

  X

  ‐ray diffraction reveals the presence of sodium chloride

  ( NaCl )

  , whereas typical chemical analysis detects only the presence of elements Na and Cl. Diffraction methods have the advantage that the substance does not have to be dissociated or dissolved. As stated in Chapter 5, the diffraction intensity is proportional to the squared magnitude of the structure factor. In powder diffraction, there is another factor strongly affecting the diffraction intensity, which is known as the multiplicity factor . All lattice planes of equal d ‐spacing give reflections at the same position, i.e., the same Bragg angle. Since reflections from such planes are independent of one another, the intensity of a powder peak will be simply the sum of the intensities of all reflections involved. A crystal may possess some symmetry elements and all lattice planes related by symmetry have the same d ‐spacing. Thus, the number of planes contributing to a powder reflection peak hkl will be the number of planes belonging to the form {hkl }. For example, all six planes of (100), (010), (001), (100), (010) and (001) equally contribute to the 100 reflection of a cubic crystal. Thus, its multiplicity factor is 6. All these planes have different orientations in a single crystal but will be correctly oriented for powder diffraction with equal probability. Likewise, there are eight planes in the form {111}: (111),

  (

  1

  ¯

  11 )

  ,

  ( 1

  1

  ¯

  1 )

  , (111) and their opposites. The corresponding intensity in a powder pattern will thus be eight times that of a single 111 reflection. (111) and its opposite (111) planes may be crystallographically identical or different, depending on the point group of the crystal. Regardless of their surface properties, both planes are equally involved in diffraction and are counted separately in the multiplicity factor. The multiplicity factor also depends on the crystal system. In a tetragonal crystal, the (100) and (001) planes may have different d

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  Understanding Solid State Physics

  • Sharon Ann Holgate(Author)
  • 2009(Publication Date)
  • CRC Press

    (Publisher)

  A “diffraction spot” will be produced whenever there is constructive interference between the X-rays E Reflected beam Incident X-ray beam θ θ θ B C d A α FIGURE 5.1 When a beam of X-rays hits a crystal, it behaves as if it has been reflected by some of the atomic planes. This diagram shows what happens when two rays from a beam hit a pair of atomic planes separated by a distance, d . (The atoms marked A, B, C, and E are used in conjunction with the main text to help derive the Bragg law.) 128 Understanding Solid State Physics reflected from neighbouring planes. Figure 5.2 shows an experimentally measured diffraction pattern consisting of an array of diffraction lines. The conditions under which constructive interference of reflected X-rays occurs are summarized in the Bragg law, which can be derived by looking closely at Figure 5.1. If we just consider the top two atomic planes in the diagram, which are separated by the distance d (usually referred to as the inter-plane spacing), it is clear that the lower of the two X-rays has to travel further than the upper ray before leaving the crystal. In fact its path length is BAC longer. If the angle of incidence of the X-rays with respect to the crystal planes is taken to be θ , the angle marked α will also be θ , as it is a complementary angle. If perpendiculars are drawn from the atom at point E to the lower ray (hitting the ray at points B and C), and another line is drawn connecting the atoms at point E and point A, two little right-angled triangles are formed. The angles at the top of these triangles will be θ because there is already a right angle—since α = θ —plus the angle β = (90 − θ )° at the bottom of each triangle. To make this easier to visualise, Figure 5.3 shows one of these little triangles separately. From Figure 5.3 we can now find the distance, x , from B to A, as sin θ = x / d , and therefore x = d sin θ .

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  Modern Electron Microscopy in Physical and Life Sciences

  • Milos Janecek, Robert Kral, Milos Janecek, Robert Kral(Authors)
  • 2016(Publication Date)
  • IntechOpen

    (Publisher)

  Figure 3. Incident, transmitted and diffracted electron beams in a thin specimen for the Bragg law. In the Bragg law, electrons are collided with the crystal planes. Some of them are diffracted and the rest are transmitted through the specimen, which does not participate in the formation of the diffraction pattern [1–3]. Modern Electron Microscopy in Physical and Life Sciences 6 1.4. The Laue conditions Diffraction can be considered as a total scattering of individual atoms. So, diffraction is mathematically expressed in terms of total scattering of atoms. The distance from atom A to atom B is described by vector r in three-dimensional space. Also, incident and diffracted beams are described by unit vectors p 0 and p , respectively. With the use of these vector notations, we can write paths difference in vector notation as AC = r . p 0 and AD = r . p . According to the vector relationship in Figure 4, the path difference of wave should be an integer multiple of wave‐ length and we may write as . CAD n l = = r P (4) The vector r , defined in spherical coordinates, is a converted form of vector components x , y and z in Cartesian coordinates, and a , b and c are unit vectors defined in the direction of coordinates axes which can be considered as distances between atoms. So, we may write Equation (4) as . . . h k l l l l = = = P a P b P c (5) These algebraic relations are known as Laue conditions. These relations must be satisfied when strong diffraction occurs [2, 3]. Figure 4. Incident and diffracted waves of atoms in the thin specimen are p 0 and p , respectively, the angle between them is 2 θ and the distance from atom A to atom B is described by vector r . Electron Diffraction http://dx.doi.org/10.5772/61781 7 1.5. The structure factor Diffraction intensity is different for each of the crystalline planes because the distribution of atoms per unit area is not the same for individual planes.

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  Quantum Nanochemistry, Volume Four

  Quantum Solids and Orderability

  • Mihai V. Putz(Author)
  • 2016(Publication Date)
  • Apple Academic Press

    (Publisher)

  This phenom-enon corresponds to the experimental observations, as recorded through the contractions effect of the diffraction fringes for the diffraction on the slight bending crystals (Janáček & Kuběna, 1978; Janáček et al., 1978; Authier & Simon, 1968). From practical reasons, the relation (5.259) is also conveniently writ-ten through the resuming of all the corrections due to the slight elastically deformations into a so-called deformation function F ( ε ), with which the deformation solution of the amplitude of the structure factor in the equiva-lent solutions can be written: (5.260) (5.261) The last expression, by its shape, will additionally allow the evaluation of the model for the phase associated to the structure factor, relating with the deformation function, so solving in an original way the “phase matter” mentioned at the beginning of this Chapter. 5.4.4.4 Application to the Electronic Maps of Contour As a first implementation of the previous results, the Bragg’s law for the central diffraction fringe can be considered, i.e., adapting the relation (5.5) to the actual needs: (5.262) which can be further included in the expression of the direction cosine of the scattering Bragg’s angle specific to the diffracting g -plane: (5.263) 590 Quantum Nanochemistry—Volume IV: Quantum Solids and Orderability In any case, beyond the direct involvement of the Bragg’s law in the above results, the problem of the phase determination appears, for the structure factor amplitude of Eqs. (5.259)–(5.261). Such a determination of the phase may use the real and imaginary com-ponents of the structure factor expression, since identified at exponential level with a generalized form of the relation (5.261) (Putz, 2003): (5.264) Moreover, by Eq.

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