Misplaced Pages

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63-682: In particular, a family of lattice planes of a given (direct) Bravais lattice is determined by three integers h , k , and  ℓ , the Miller indices . They are written ( hkℓ ), and denote the family of (parallel) lattice planes (of the given Bravais lattice) orthogonal to g h k ℓ = h b 1 + k b 2 + ℓ b 3 {\displaystyle \mathbf {g} _{hk\ell }=h\mathbf {b} _{1}+k\mathbf {b} _{2}+\ell \mathbf {b} _{3}} , where b i {\displaystyle \mathbf {b} _{i}} are

126-460: A ) + q ( Δ k ⋅ b ) + r ( Δ k ⋅ c ) = 2 π n {\displaystyle p(\mathbf {\Delta k} \cdot \mathbf {a} )+q(\mathbf {\Delta k} \cdot \mathbf {b} )+r(\mathbf {\Delta k} \cdot \mathbf {c} )=2\pi n} does not hold for any arbitrary integers p , q , r {\displaystyle p,q,r} . This ensures that if

189-433: A 2 and a 3 as Hence zone indices of the direction perpendicular to plane ( hkℓ ) are, in suitably normalized triplet form, simply [ 2 h + k , h + 2 k , ℓ ( 3 / 2 ) ( a / c ) 2 ] {\displaystyle [2h+k,h+2k,\ell (3/2)(a/c)^{2}]} . When four indices are used for the zone normal to plane ( hkℓ ), however,

252-403: A rhombus if the equation is satisfied. If the scattering satisfies this equation, all the crystal lattice points scatter the incoming wave toward the scattering direction (the direction along k o u t {\displaystyle \mathbf {k} _{\mathrm {out} }} ). If the equation is not satisfied, then for any scattering direction, only some lattice points scatter

315-742: A rhombus . Each G {\displaystyle \mathbf {G} } is by definition the wavevector of a plane wave in the Fourier series of a spatial function which periodicity follows the crystal lattice (e.g., the function representing the electronic density of the crystal), wavefronts of each plane wave in the Fourier series is perpendicular to the plane wave's wavevector G {\displaystyle \mathbf {G} } , and these wavefronts are coincident with parallel crystal lattice planes. This means that X-rays are seemingly "reflected" off parallel crystal lattice planes perpendicular G {\displaystyle \mathbf {G} } at

378-421: A , b and c appropriately: divide by the largest of the three numbers, and then multiply by the least common denominator . Thus, integer Miller indices implicitly include indices with all rational ratios. The reason why planes where the components (in the reciprocal-lattice basis) have rational ratios are of special interest is that these are the lattice planes : they are the only planes whose intersections with

441-699: A crystal in diffraction, and this is the meaning of the Laue equations. This fact is sometimes called the Laue condition . In this sense, diffraction patterns are a way to experimentally measure the reciprocal lattice for a crystal lattice. The Laue condition can be rewritten as the following. Applying the elastic scattering condition | k o u t | 2 = | k i n | 2 {\displaystyle |\mathbf {k} _{\mathrm {out} }|^{2}=|\mathbf {k} _{\mathrm {in} }|^{2}} (In other words,

504-555: A crystal lattice L {\displaystyle L} , where atoms are located at lattice points described by x = p a + q b + r c {\displaystyle \mathbf {x} =p\,\mathbf {a} +q\,\mathbf {b} +r\,\mathbf {c} } with p {\displaystyle p} , q {\displaystyle q} , and r {\displaystyle r} as any integers . (So x {\displaystyle \mathbf {x} } indicating each lattice point

567-447: A lattice L ∗ {\displaystyle L^{*}} , called the reciprocal lattice of the crystal lattice L {\displaystyle L} , as each Δ k {\displaystyle \mathbf {\Delta k} } indicates a point of L ∗ {\displaystyle L^{*}} . (This is the meaning of the Laue equations as shown below.) This condition allows

630-478: A reciprocal lattice vector. The requirement of lowest terms means that it is the shortest reciprocal lattice vector in the given direction. Equivalently, ( hkℓ ) denotes a plane that intercepts the three points a 1 / h , a 2 / k , and a 3 / ℓ , or some multiple thereof. That is, the Miller indices are proportional to the inverses of the intercepts of the plane, in the basis of the lattice vectors. If one of

693-435: A scattering vector Δ k {\displaystyle \mathbf {\Delta k} } . Hence there are infinitely many scattering vectors that satisfy the Laue equations as there are infinitely many choices of Miller indices ( h , k , l ) {\displaystyle (h,k,l)} . Allowed scattering vectors Δ k {\displaystyle \mathbf {\Delta k} } form

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756-399: A single incident beam to be diffracted in infinitely many directions. However, the beams corresponding to high Miller indices are very weak and can't be observed. These equations are enough to find a basis of the reciprocal lattice (since each observed Δ k {\displaystyle \mathbf {\Delta k} } indicates a point of the reciprocal lattice of the crystal under

819-677: Is a plane equation in geometry. Another equivalent equation, that may be easier to understand, is k out ⋅ G ^ = 1 2 | G | {\displaystyle {{\mathbf {k} }_{\text{out}}}\cdot {\widehat {\mathbf {G} }}={\frac {1}{2}}\left|\mathbf {G} \right|} (also ( − k in ) ⋅ G ^ = 1 2 | G | {\displaystyle (-{{\mathbf {k} }_{\text{in}}})\cdot {\widehat {\mathbf {G} }}={\frac {1}{2}}\left|\mathbf {G} \right|} ). This indicates

882-693: Is a stub . You can help Misplaced Pages by expanding it . Laue equations In crystallography and solid state physics , the Laue equations relate incoming waves to outgoing waves in the process of elastic scattering , where the photon energy or light temporal frequency does not change upon scattering by a crystal lattice . They are named after physicist Max von Laue (1879–1960). The Laue equations can be written as Δ k = k o u t − k i n = G {\displaystyle \mathbf {\Delta k} =\mathbf {k} _{\mathrm {out} }-\mathbf {k} _{\mathrm {in} }=\mathbf {G} } as

945-418: Is a collection of equally spaced parallel lattice planes that, taken together, intersect all lattice points. Every family of lattice planes can be described by a set of integer Miller indices that have no common divisors (i.e. are relative prime ). Conversely, every set of Miller indices without common divisors defines a family of lattice planes. If, on the other hand, the Miller indices are not relative prime,

1008-418: Is an integer linear combination of the primitive vectors.) Let k i n {\displaystyle \mathbf {k} _{\mathrm {in} }} be the wave vector of an incoming (incident) beam or wave toward the crystal lattice L {\displaystyle L} , and let k o u t {\displaystyle \mathbf {k} _{\mathrm {out} }} be

1071-423: Is an equation for a plane (as the set of all points indicated by k o u t {\displaystyle \mathbf {k} _{\mathrm {out} }} satisfying this equation) as its equivalent equation G ⋅ ( 2 k out − G ) = 0 {\displaystyle \mathbf {G} \cdot (2{{\mathbf {k} }_{\text{out}}}-\mathbf {G} )=0}

1134-473: Is important to determine the planes and thus to have a notation system. Ordinarily, Miller indices are always integers by definition, and this constraint is physically significant. To understand this, suppose that we allow a plane ( abc ) where the Miller "indices" a , b and c (defined as above) are not necessarily integers. If a , b and c have rational ratios, then the same family of planes can be written in terms of integer indices ( hkℓ ) by scaling

1197-475: Is marked by Miller indices . By convention, negative integers are written with a bar, as in 3 for −3. The integers are usually written in lowest terms, i.e. their greatest common divisor should be 1. Miller indices are also used to designate reflections in X-ray crystallography . In this case the integers are not necessarily in lowest terms, and can be thought of as corresponding to planes spaced such that

1260-639: Is the Laue condition, equivalent to the Laue equations.) And, the elastic scattering | k o u t | 2 = | k i n | 2 {\displaystyle |\mathbf {k} _{\mathrm {out} }|^{2}=|\mathbf {k} _{\mathrm {in} }|^{2}} has been assumed so G {\displaystyle \mathbf {G} } , k o u t {\displaystyle \mathbf {k} _{\mathrm {out} }} , and − k i n {\displaystyle -\mathbf {k} _{\mathrm {in} }} form

1323-471: Is the integer p h + q k + r l {\displaystyle ph+qk+rl} . The claim that each parenthesis, e.g. ( Δ k ⋅ a ) {\displaystyle (\mathbf {\Delta k} \cdot \mathbf {a} )} , is to be a multiple of 2 π {\displaystyle 2\pi } (that is each Laue equation) is justified since otherwise p ( Δ k ⋅

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1386-537: Is the wave vector for a plane wave associated with parallel crystal lattice planes. (Wavefronts of the plane wave are coincident with these lattice planes.) The equations are equivalent to Bragg's law ; the Laue equations are vector equations while Bragg's law is in a form that is easier to solve, but these tell the same content. Let a , b , c {\displaystyle \mathbf {a} \,,\mathbf {b} \,,\mathbf {c} } be primitive translation vectors (shortly called primitive vectors) of

1449-592: The Penrose tiling , are formed by "cuts" of periodic lattices in more than three dimensions, involving the intersection of more than one such hyperplane .) Lattice plane In crystallography , a lattice plane of a given Bravais lattice is any plane containing at least three noncollinear Bravais lattice points. Equivalently, a lattice plane is a plane whose intersections with the lattice (or any crystalline structure of that lattice) are periodic (i.e. are described by 2d Bravais lattices). A family of lattice planes

1512-419: The basis or primitive translation vectors of the reciprocal lattice for the given Bravais lattice. (Note that the plane is not always orthogonal to the linear combination of direct or original lattice vectors h a 1 + k a 2 + ℓ a 3 {\displaystyle h\mathbf {a} _{1}+k\mathbf {a} _{2}+\ell \mathbf {a} _{3}} because

1575-400: The direction : That is, it uses the direct lattice basis instead of the reciprocal lattice. Note that [hkℓ] is not generally normal to the ( hkℓ ) planes, except in a cubic lattice as described below. For the special case of simple cubic crystals, the lattice vectors are orthogonal and of equal length (usually denoted a ), as are those of the reciprocal lattice. Thus, in this common case,

1638-750: The reciprocal lattice for a crystal lattice L {\displaystyle L} (defined by x = p a + q b + r c {\displaystyle \mathbf {x} =p\,\mathbf {a} +q\,\mathbf {b} +r\,\mathbf {c} } ) in real space, we know that G ⋅ x = G ⋅ ( p a + q b + r c ) = 2 π ( h p + k q + l r ) = 2 π n {\displaystyle \mathbf {G} \cdot \mathbf {x} =\mathbf {G} \cdot (p\mathbf {a} +q\mathbf {b} +r\mathbf {c} )=2\pi (hp+kq+lr)=2\pi n} with an integer n {\displaystyle n} due to

1701-638: The Laue equations are satisfied, then the incoming and outgoing (diffracted) wave have the same phase at each point of the crystal lattice, so the oscillations of atoms of the crystal, that follows the incoming wave, can at the same time generate the outgoing wave at the same phase of the incoming wave. If G = h A + k B + l C {\displaystyle \mathbf {G} =h\mathbf {A} +k\mathbf {B} +l\mathbf {C} } with h {\displaystyle h} , k {\displaystyle k} , l {\displaystyle l} as integers represents

1764-489: The Miller indices ( hkℓ ) and [ hkℓ ] both simply denote normals/directions in Cartesian coordinates . For cubic crystals with lattice constant a , the spacing d between adjacent ( hkℓ ) lattice planes is (from above) Because of the symmetry of cubic crystals, it is possible to change the place and sign of the integers and have equivalent directions and planes: For face-centered cubic and body-centered cubic lattices,

1827-431: The [ 1 1 0] directions are really similar. If S is the intercept of the plane with the [ 1 1 0] axis, then There are also ad hoc schemes (e.g. in the transmission electron microscopy literature) for indexing hexagonal lattice vectors (rather than reciprocal lattice vectors or planes) with four indices. However they do not operate by similarly adding a redundant index to the regular three-index set. For example,

1890-429: The condition of elastic wave scattering by a crystal lattice, where Δ k {\displaystyle \mathbf {\Delta k} } is the scattering vector , k i n {\displaystyle \mathbf {k} _{\mathrm {in} }} , k o u t {\displaystyle \mathbf {k} _{\mathrm {out} }} are incoming and outgoing wave vectors (to

1953-445: The corresponding Miller indices, and i is a redundant index. This four-index scheme for labeling planes in a hexagonal lattice makes permutation symmetries apparent. For example, the similarity between (110) ≡ (11 2 0) and (1 2 0) ≡ (1 2 10) is more obvious when the redundant index is shown. In the figure at right, the (001) plane has a 3-fold symmetry: it remains unchanged by a rotation of 1/3 (2 π /3 rad, 120°). The [100], [010] and

Miller index - Misplaced Pages Continue

2016-736: The crystal and from the crystal, by scattering), and G {\displaystyle \mathbf {G} } is a crystal reciprocal lattice vector . Due to elastic scattering | k o u t | 2 = | k i n | 2 {\displaystyle |\mathbf {k} _{\mathrm {out} }|^{2}=|\mathbf {k} _{\mathrm {in} }|^{2}} , three vectors. G {\displaystyle \mathbf {G} } , k o u t {\displaystyle \mathbf {k} _{\mathrm {out} }} , and − k i n {\displaystyle -\mathbf {k} _{\mathrm {in} }} , form

2079-438: The crystal are 2d-periodic. For a plane (abc) where a , b and c have irrational ratios, on the other hand, the intersection of the plane with the crystal is not periodic. It forms an aperiodic pattern known as a quasicrystal . This construction corresponds precisely to the standard "cut-and-project" method of defining a quasicrystal, using a plane with irrational-ratio Miller indices. (Although many quasicrystals, such as

2142-525: The crystal can be thought as the sum of outgoing plane waves from the crystal. (In fact, any wave can be represented as the sum of plane waves, see Fourier Optics .) The incident wave and one of plane waves of the diffracted wave are represented as where k i n {\displaystyle \displaystyle \mathbf {k} _{\mathrm {in} }} and k o u t {\displaystyle \displaystyle \mathbf {k} _{\mathrm {out} }} are wave vectors for

2205-404: The crystal) and outgoing (from the crystal by scattering) wavevectors forms a rhombus. Since the angle between k o u t {\displaystyle \mathbf {k} _{\mathrm {out} }} and G {\displaystyle \mathbf {G} } is π / 2 − θ {\displaystyle \pi /2-\theta } , (Due to

2268-470: The crystal, where they resonate with the oscillators, so the phases of these waves must coincide. At each point x = p a + q b + r c {\displaystyle \mathbf {x} =p\,\mathbf {a} +q\,\mathbf {b} +r\,\mathbf {c} } of the lattice L {\displaystyle L} , we have or equivalently, we must have for some integer n {\displaystyle n} , that depends on

2331-507: The difference between the incoming and outgoing wave vectors. The three conditions that the scattering vector Δ k {\displaystyle \mathbf {\Delta k} } must satisfy, called the Laue equations , are the following: where numbers h , k , l {\displaystyle h,k,l} are integer numbers . Each choice of integers ( h , k , l ) {\displaystyle (h,k,l)} , called Miller indices , determines

2394-406: The direct lattice vectors need not be mutually orthogonal.) This is based on the fact that a reciprocal lattice vector g {\displaystyle \mathbf {g} } (the vector indicating a reciprocal lattice point from the reciprocal lattice origin) is the wavevector of a plane wave in the Fourier series of a spatial function (e.g., electronic density function) which periodicity follows

2457-433: The family of planes defined by them is not a family of lattice planes, because not every plane of the family then intersects lattice points. Conversely, planes that are not lattice planes have aperiodic intersections with the lattice called quasicrystals ; this is known as a "cut-and-project" construction of a quasicrystal (and is typically also generalized to higher dimensions). This geometry-related article

2520-479: The incident and outgoing plane waves, x {\displaystyle \displaystyle \mathbf {x} } is the position vector , and t {\displaystyle \displaystyle t} is a scalar representing time, and φ i n {\displaystyle \varphi _{\mathrm {in} }} and φ o u t {\displaystyle \varphi _{\mathrm {out} }} are initial phases for

2583-969: The incoming and diffracted waves are at the same (temporal) frequency. We can also say that the energy per photon does not change.) To the above equation, we obtain The second equation is obtained from the first equation by using k o u t − k i n = G {\displaystyle \mathbf {k} _{\mathrm {out} }-\mathbf {k} _{\mathrm {in} }=\mathbf {G} } . The result 2 k o u t ⋅ G = | G | 2 {\displaystyle 2\mathbf {k} _{\mathrm {out} }\cdot \mathbf {G} =|\mathbf {G} |^{2}} (also 2 k in ⋅ ( − G ) = | G | 2 {\displaystyle 2{{\mathbf {k} }_{\text{in}}}\cdot (-\mathbf {G} )=|\mathbf {G} {{|}^{2}}} )

Miller index - Misplaced Pages Continue

2646-595: The incoming wave. (This physical interpretation of the equation is based on the assumption that scattering at a lattice point is made in a way that the scattering wave and the incoming wave have the same phase at the point.) It also can be seen as the conservation of momentum as ℏ k o u t = ℏ k i n + ℏ G {\displaystyle \hbar \mathbf {k} _{\mathrm {out} }=\hbar \mathbf {k} _{\mathrm {in} }+\hbar \mathbf {G} } since G {\displaystyle \mathbf {G} }

2709-568: The indices is zero, it means that the planes do not intersect that axis (the intercept is "at infinity"). Considering only ( hkℓ ) planes intersecting one or more lattice points (the lattice planes ), the perpendicular distance d between adjacent lattice planes is related to the (shortest) reciprocal lattice vector orthogonal to the planes by the formula: d = 2 π / | g h k ℓ | {\displaystyle d=2\pi /|\mathbf {g} _{hk\ell }|} . The related notation [hkℓ] denotes

2772-991: The known orthogonality between primitive vectors for the reciprocal lattice and those for the crystal lattice. (We use the physical, not crystallographer's, definition for reciprocal lattice vectors which gives the factor of 2 π {\displaystyle 2\pi } .) But notice that this is nothing but the Laue equations. Hence we identify Δ k = k o u t − k i n = G {\displaystyle \mathbf {\Delta k} =\mathbf {k} _{\mathrm {out} }-\mathbf {k} _{\mathrm {in} }=\mathbf {G} } , means that allowed scattering vectors Δ k = k o u t − k i n {\displaystyle \mathbf {\Delta k} =\mathbf {k} _{\mathrm {out} }-\mathbf {k} _{\mathrm {in} }} are those equal to reciprocal lattice vectors G {\displaystyle \mathbf {G} } for

2835-501: The lattice vectors. Both definitions are given below. In either case, one needs to choose the three lattice vectors a 1 , a 2 , and a 3 that define the unit cell (note that the conventional unit cell may be larger than the primitive cell of the Bravais lattice , as the examples below illustrate). Given these, the three primitive reciprocal lattice vectors are also determined (denoted b 1 , b 2 , and b 3 ). Then, given

2898-406: The left. And, note that for hexagonal interplanar distances, they take the form Crystallographic directions are lines linking nodes ( atoms , ions or molecules ) of a crystal. Similarly, crystallographic planes are planes linking nodes. Some directions and planes have a higher density of nodes; these dense planes have an influence on the behavior of the crystal: For all these reasons, it

2961-415: The light (typically X-ray) wavelength, and | G | = 2 π d n {\displaystyle |\mathbf {G} |={\frac {2\pi }{d}}n} with d {\displaystyle d} as the distance between adjacent parallel crystal lattice planes and n {\displaystyle n} as an integer. With these, we now derive Bragg's law that

3024-462: The literature often uses [ h , k , − h − k , ℓ ( 3 / 2 ) ( a / c ) 2 ] {\displaystyle [h,k,-h-k,\ell (3/2)(a/c)^{2}]} instead. Thus as you can see, four-index zone indices in square or angle brackets sometimes mix a single direct-lattice index on the right with reciprocal-lattice indices (normally in round or curly brackets) on

3087-415: The measurement), from which the crystal lattice can be determined. This is the principle of x-ray crystallography . For an incident plane wave at a single frequency f {\displaystyle \displaystyle f} (and the angular frequency ω = 2 π f {\displaystyle \displaystyle \omega =2\pi f} ) on a crystal, the diffracted waves from

3150-911: The mirror-like scattering, the angle between k i n {\displaystyle \mathbf {k} _{\mathrm {in} }} and G {\displaystyle \mathbf {G} } is also π / 2 − θ {\displaystyle \pi /2-\theta } .) k o u t ⋅ G = | k o u t | | G | sin ⁡ θ {\displaystyle \mathbf {k} _{\mathrm {out} }\cdot \mathbf {G} =|\mathbf {k} _{\mathrm {out} }||\mathbf {G} |\sin \theta } . Recall, | k o u t | = 2 π / λ {\displaystyle |\mathbf {k} _{\mathrm {out} }|=2\pi /\lambda } with λ {\displaystyle \lambda } as

3213-586: The notations essentially indicate some integer.) By rearranging terms, we get Now, it is enough to check that this condition is satisfied at the primitive vectors a , b , c {\displaystyle \mathbf {a} ,\mathbf {b} ,\mathbf {c} } (which is exactly what the Laue equations say), because, at any lattice point x = p a + q b + r c {\displaystyle \mathbf {x} =p\,\mathbf {a} +q\,\mathbf {b} +r\,\mathbf {c} } , we have where n {\displaystyle n}

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3276-531: The original Bravais lattice, so wavefronts of the plane wave are coincident with parallel lattice planes of the original lattice. Since a measured scattering vector in X-ray crystallography , Δ k = k o u t − k i n {\displaystyle \Delta \mathbf {k} =\mathbf {k} _{\mathrm {out} }-\mathbf {k} _{\mathrm {in} }} with k o u t {\displaystyle \mathbf {k} _{\mathrm {out} }} as

3339-508: The outgoing (scattered from a crystal lattice) X-ray wavevector and k i n {\displaystyle \mathbf {k} _{\mathrm {in} }} as the incoming (toward the crystal lattice) X-ray wavevector, is equal to a reciprocal lattice vector g {\displaystyle \mathbf {g} } as stated by the Laue equations , the measured scattered X-ray peak at each measured scattering vector Δ k {\displaystyle \Delta \mathbf {k} }

3402-573: The plane that is perpendicular to the straight line between the reciprocal lattice origin G = 0 {\displaystyle \mathbf {G} =0} and G {\displaystyle \mathbf {G} } and located at the middle of the line. Such a plane is called Bragg plane. This plane can be understood since G = k o u t − k i n {\displaystyle \mathbf {G} =\mathbf {k} _{\mathrm {out} }-\mathbf {k} _{\mathrm {in} }} for scattering to occur. (It

3465-658: The point x {\displaystyle \mathbf {x} } . Since this equation holds at x = 0 {\displaystyle \mathbf {x} =0} , φ i n = φ o u t + 2 π n ′ {\displaystyle \varphi _{\mathrm {in} }=\varphi _{\mathrm {out} }+2\pi n'} at some integer n ′ {\displaystyle n'} . So (We still use n {\displaystyle n} instead of ( n − n ′ ) {\displaystyle (n-n')} since both

3528-553: The primitive lattice vectors are not orthogonal. However, in these cases the Miller indices are conventionally defined relative to the lattice vectors of the cubic supercell and hence are again simply the Cartesian directions. With hexagonal and rhombohedral lattice systems , it is possible to use the Bravais–Miller system, which uses four indices ( h k i ℓ ) that obey the constraint Here h , k and ℓ are identical to

3591-406: The reciprocal lattice vector ( hkℓ ) as suggested above can be written in terms of reciprocal lattice vectors as h b 1 + k b 2 + ℓ b 3 {\displaystyle h\mathbf {b} _{1}+k\mathbf {b} _{2}+\ell \mathbf {b} _{3}} . For hexagonal crystals this may be expressed in terms of direct-lattice basis-vectors a 1 ,

3654-631: The reflections from adjacent planes would have a phase difference of exactly one wavelength (2 π ), regardless of whether there are atoms on all these planes or not. There are also several related notations: In the context of crystal directions (not planes), the corresponding notations are: Note, for Laue–Bragg interferences Miller indices were introduced in 1839 by the British mineralogist William Hallowes Miller , although an almost identical system ( Weiss parameters ) had already been used by German mineralogist Christian Samuel Weiss since 1817. The method

3717-509: The same angle as their angle of approach to the crystal θ {\displaystyle \theta } with respect to the lattice planes; in the elastic light ( typically X-ray ) -crystal scattering, parallel crystal lattice planes perpendicular to a reciprocal lattice vector G {\displaystyle \mathbf {G} } for the crystal lattice play as parallel mirrors for light which, together with G {\displaystyle \mathbf {G} } , incoming (to

3780-400: The three Miller indices h , k , ℓ , ( h k ℓ ) {\displaystyle h,k,\ell ,(hk\ell )} denotes planes orthogonal to the reciprocal lattice vector: That is, ( hkℓ ) simply indicates a normal to the planes in the basis of the primitive reciprocal lattice vectors. Because the coordinates are integers, this normal is itself always

3843-420: The wave vector of an outgoing (diffracted) beam or wave from L {\displaystyle L} . Then the vector k o u t − k i n = Δ k {\displaystyle \mathbf {k} _{\mathrm {out} }-\mathbf {k} _{\mathrm {in} }=\mathbf {\Delta k} } , called the scattering vector or transferred wave vector , measures

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3906-447: The waves. For simplicity we take waves as scalars here, even though the main case of interest is an electromagnetic field, which is a vector . We can think these scalar waves as components of vector waves along a certain axis ( x , y , or z axis) of the Cartesian coordinate system . The incident and diffracted waves propagate through space independently, except at points of the lattice L {\displaystyle L} of

3969-465: Was also historically known as the Millerian system, and the indices as Millerian, although this is now rare. The Miller indices are defined with respect to any choice of unit cell and not only with respect to primitive basis vectors, as is sometimes stated. There are two equivalent ways to define the meaning of the Miller indices: via a point in the reciprocal lattice , or as the inverse intercepts along

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