4 Reciprocal lattice. 4.1 The lattice function
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1 4 Reciprocl lttice Reciprocl vectors nd the bsis of the reciprocl vectors were first used by J. W. Gibbs. Round 880 he mde used of them in his lectures bout the vector nlysis [], pp. 0, 83. In structure nlysis the concept of the reciprocl lttice hs been estblished by P. Ewld nd M. v. Lue in 93, t the very begining of the discipline []. The reson ws to fcilitte clcultions in nlytic geometry of liner forms in coordinte systems with non orthonorml bsis, which must inevitbly be used when crystls of lower symmetry non cubic re investigted. Since tht time the reciprocl lttice belongs to the bsic concepts of crystllogrphy, solid stte physics nd other disciplines. The bove mentioned clssicists nd lso uthors of textboos nd reference boos define the bsis vectors s of the reciprocl lttice in n lgebric wy see reltions 4. below or Appendix C. Then, however, the textboos nd reference boos declre the reciprocl lttice to be the Fourier trnsform of lttice. The proof of tht sttement is however lmost lwys missing. Probbly the only exception is the boo by Guinier [3], pp The reson for tht my lie with the fct tht the proof is fr from being simple. The fct tht the reciprocl lttice is the Fourier trnsform of lttice is, however, importnt not only for the study of cystlline solids but lso for presenttion of the Fourier series of periodic functions of two nd more vribles tht re not periodic in orthogonl directions, for the formultion of the smpling theorem in dimensions N etc. In this chpter we first give tht lgebric definition of the reciprocl lttice section 4. nd then we submit the proof tht reciprocl lttice is the Fourier trnsform of lttice function section 4.3. At tht, we obtin the expression for the Fourier series of generl lttice functions section 4.4, i.e. even of such ones which chrcterize lttices tht re periodic in non ortogonl directions only. 4. The lttice function The N dimensionl lttice with bsis vectors r, r,,..., N, constituted by points is chrcterized by the so clled lttice function f x x n n n N N x x n, where x n n n n N N denotes lttice vector nd the symbol implies tht ll the components n, n,..., n N of the multiindex n cquire ll integer vlues. Hence, the lttice function is N multiple series of the Dirc distributions of N vribles. Let us form the mtrix N N A rs....., 3. N N NN the rows of which consist of the coordintes rs of bsis vectors r of the lttice in system of coordintes with orthogonl bsis e, e,..., e N. Let us consider the vrible x nd the multiindex n to be column mtrices formed by coordintes x r in tht orthogonl bsis nd by integers n r, respectively. Then, we my express the lttice function in the form f x x A T n. 4 The determinnt of the mtrix A is clled the outer product of the vectors r, r,,..., N, see e.g. [4], p. 95. [,,..., N ] det A. 5 Its bsolute vlue does not depend on the choice of the orthonorml bsis nd defines cf. e.g. [9], p. 6 the volume of the N dimensionl prllepiped unit cell, the edges of which re the bsis vectors r of the lttice:
2 4 RECIPROCAL LATTICE det A. 6 The independence of the bsolute vlue of the outer product on the choice of the orthonorml bsis e, e,..., e N is obvious from the squre of tht product: [ ],,..., N det A det A det A T det AA T det r s det G. 7 The symbol A T denotes the trnsposed mtrix of the mtrix A nd N N det G N N N N 8 is the Grm determinnt of the bsis vectors of the lttice whose independence on the coordinte system is evident. If the dimension N, nd 3, the bsolute vlue of the outer product hs the mening of length, re, nd volume, respectively. For N 4 it is lso resonble to consider the bsolute vlue of the outer product [,,..., N ] s volume becuse it hs properties ttched to the volume [4], p. 95, [9], pp. 6 7: i [,,..., N ] 0 if nd only if the vectors forming the outer product re linerly dependent. ii If one of the vectors involved in the outer product is multiplied by number α then the outer prouduct is multiplied by the number α. iii [,,..., N ] N, where r mens the length of the vector r. 4. Algebric definition of reciprocl lttice In the middle of the lst century the Interntionl Union of Crystllogrphy recommended to define the bsis vectors s of the lttice reciprocl to the lttice with the bsis vectors r by sclr products r s K rs, r, s,,..., N, where K is the so clled reciprocl constnt see [6], p., the vlue of which cn be chosen s the cse my be. The recommendtion of the Union hs, however, not been ccepted nd in crystllogrphy, mteril science nd relted technologicl brnches K see e.g. [3], p. 386, [5], p. 63 is consistently used, wheres in solid stte physics, surfce physics, etc. K is chosen see e.g. [7], p. 6, [8], p. 87. Emotionl discussions hve been crried on which of the two eventulities is the right one cf. [0], p. 6. We will follow here the customs of crystllogrphy, though it slightly complictes the formul expressing the Fourier trnsform of the lttice function. Thus, we define the bsis vectors s of the reciprocl lttice by sclr product r s rs, r, s,,..., N. It is evident from the definition tht the reciprocl lttice to the reciprocl lttice is the originl lttice nd tht the bsis vector s of the reciprocl lttice is orthogonl to ll the bsis vectors r of the originl lttice with exception of the vector s. Definition is n implicit definition of the bsis vectors of the reciprocl lttice; it is not formul expressing the vectors s in terms of the bsis vectors r of the originl lttice. To get such n explicit expression we rewrite N equtions in the mtrix wy where A st A A T I, N N N N NN 3
3 4. Algebric definition of reciprocl lttice 3 is the mtrix the rows of which re coordintes of bsis vectors s of the reciprocl lttice in the orthonorml system of coordintes nd I is the unit mtrix. It is evident from tht A A T, i.e. st ts. 4 Expressed by words: If the rows of the mtrix A re coordintes of the bsis vectors of the originl lttice, then the columns of the inverse mtrix A st re the coordintes of the bsis vectors of the originl lttice: N N s st e t st det A st det A e t, s,,... N, 5 t t where det A st is minor formed from det A by striing out the s th row nd t th column. Expression 5 cn be written in more simple wy. Let us form the mtrix A s by replcing the s th row of the mtrix A by vectors e, e,..., e N of the orthonorml bsis. Then the sum N st det A st e t det A s t is the Lplce development of the det A s long the elements of the s th row nd the expression 5 for the bsis vectors s of the reciprocl lttice tes prticulrly simple form s det A s, s,,..., N. 6 det A The disdvntge of these expressions is tht they re relted to n orthonorml system of coordintes. In fct, the expression 6 provides the coordintes of the vector s in terms of the coordintes rt of vectors r in n orthonorml system of coordintes. It is true tht this orthonorml system my be rbitrry. In spite of tht we will im t n expression which does not require the use of ny orthonorml system of coordintes. Fortuntely in E 3 the outer product is nd det A 3 det A 3, det A 3, det A 3. Hence, the reltions 6 re the well nown expressions of the bsis vectors of the reciprocl lttice stright by the bsis vectors of the lttice: 3 3, 3 3, However, this hppy disposition seems to be limited to N 3 only. To be free from the necessity to use n orthonorml system of coordintes lso in the cse of generl dimension N, we multiply the numertor nd denomintor of 6 by the determinnt det A T. In the denomintor we get det A det A T det G nd in the numertor det A s det A T det G s, where det G s is the determinnt mde up by replcing the elements of the s th row in the Grm determinnt by the bsis vectors,,..., N of the lttice. In this wy we get the bsis vectors s of the reciprocl lttice expressed stright by the bsis vectors r of the lttice rther then by their coordintes: s det G s, s,,..., N. 8 det G Note: The expressions 8 cn be derived without ny use of n orthonorml system of coordintes nd the mtrix A: We resolve the vectors s in the bsis formed by the bsis vectors of the lttice s α s α sn N, s,,..., N, 9 nd multiply ech of these decompositions successively by vectors r, r,,..., N. ccount the definition we obtin Ting into
4 4 4 RECIPROCAL LATTICE α s r α sn N r rs, r, s,,..., N. 0 For certin s, the system 0 represents N liner lgebric equtions for N coefficients α s, α s,..., α sn. The coefficient determinnt of this system of liner lgebric equtions is the Grm determinnt 4.8. The right hnd side of the equtions 0 is zero, if r s, nd one, if r s. Crmer s rule then provides very simple solution of the sytem of equtions 0: α st st det G st, t,,..., N, det G where det G st is minor of the element s t in the Grm determinnt. On inserting solutions in 9, we get s det G N t st t det G st det G s det G, in cordnce with 8. Let us use 8 to get expressions for bsis vectors in E nd E. In E we obtin In E it is i.e.,.,, On the contrry,. 3,. 4 In E 3 the trditionl expressions 7 re much more simple thn 8 nd this simplicity invites us to me use of them for two dimensionl lttices, too. However, in E the vector product is not defined. This difficulty is usully bypssed by dding the unit vector n to the two bsis vectors, of the two dimensionl lttice in such wy tht vectors,, n form the three dimensionl right hnded bsis. Then the bsis vectors of the reciprocl lttice re clculted ccording to 7 see e.g. [8], p. 87: n, n. 5 Equivlence of expression 5 nd 3 is esy to see, if we write the unit vector n s the frction n, 6 insert it into 5 nd use the identities b c c b b c, b c d c b d d b c. We get,,
5 4.3 Reciprocl lttice nd the Fourier trnsform of the lttice function 5 in ccordnce with 3. As to the unimportnt third bsis vector of the reciprocl three dimensionl bsis is concerned, it is evident from 6 tht 3 n. 4.. Exmple: Reciprocl lttice to the primitive rectngulr lttice in E Let us clculte the bsis vectors of the reciprocl lttice to primitive rectngulr lttice in E with the rtio of the lengths of the bsis vectors : nd with the longer side of the unit cell in the verticl direction. Let thelength of the bsis vectors, be, see Figure. Then, 4, 0. nd expressions 3 give the bsis vectors of the reciprocl lttice in the form,. The bsis vectors of the reciprocl lttice thus hve the sme direction s the bsis vectors of the originl lttice nd their mgnitudes re,. The reciprocl lttice is gin primitive rectngulr lttice, but the rtio of the primitive cell sides is :, i.e. the longer side hs horizontl direction see Figure b. If, for some reson, we do not choose orthogonl bsis vectors of our rectngulr lttice but soume other ones we get, of course, the sme reciprocl lttice. However, the bsis vectors specifying it re different. For exmple, let us choose the bsis vectors, shown in Figure c. Obviously, 8, 5, 6, nd expressions 3 give the following bsis /, 3 /. Their mgnitudes re / cf. Figure d. vectors of the reciprocl lttice /, 4.. Exmple: Reciprocl lttice to the centred rectngulr lttice in E Now we find the reciprocl lttice to the centred rectngulr lttice in E. Let the rtio of the sides of the rectngulr unit cell be : nd let the longer side hve the verticl direction. We choose the non orthogonl bsis vectors, shown in Figure. Then, 5 /4,. /. According to 3 the bsis vectors of the reciprocl lttice re 5 4 /, /. Their mgnitudes re 5/, / cf.figure b. Hence, the reciprocl lttice is gin centered rectngulr lttice, but with the side rtio :, i.e. with the longer side in the horizontl direction. 4.3 Reciprocl lttice nd the Fourier trnsform of the lttice function We will clculte the Fourier trnsform of the lttice function 4. three times: First in E Section In this cse the clcultion of the Fourier integrls gives the Fourier trnsform hving the shpe of the Fourier series. This Fourier series is simultneously geometric series whose rtio hs the bsolute vlue equl to one. The sum of the geometric series is the series of the Dirc distributions which is proportionl to the lttice functions of the reciprocl lttice. The clcultion in E Section 4.3. tes dvntge of the quoted properties of the one dimensionl lttice function. Nevertheless, it requires lso severl rtifices typicl for deriving of the Fourier trnsform of more dimensionl lttice function. The rtifices re, however, geometriclly illustrtive. The clcultion in E N Section is formlly identicl with the clcultion in E. In ll the cses we will find tht the Fourier trnsform of the lttice function is proportionl to the lttice function of the reciprocl lttice with the reciprocl constnt K / The Fourier trnsform of the lttice function in E In one dimmensionl cse the lttice function hs the form fx n x n nd it represents periodic distribution of the Dirc distributions of single vrible with period see Figure 3. Thus, we my formlly expnd it into the Fourier series fx h c h exp i h x. It is noteworthy tht in the cse of lttice function ll the coefficients c h hve the sme vlue:
6 6 4 RECIPROCAL LATTICE Figure : Two dimensionl primitive rectngulr lttice, c whose bsis vectors re chosen in different wys nd its reciprocl lttice b, d. c h / / / fx exp i h x dx n / / / x n exp i h x dx x exp i h x dx. Therefore, the Fourier series of the lttice function hs the form n x n h exp i h x h exp i h x,
7 4.3 Reciprocl lttice nd the Fourier trnsform of the lttice function 7 Figure : Two dimensionl centered rectngulr lttice with bsis vectors, specifying primitive lttice nd its reciprocl lttice b. which is geometric series of the rtio equl to complex unity exp i x. This is n importnt result which will be used when clculting the Fourier trnsform of more dimensionl lttice function. Therefore, we prepre needful formul nd, ccording to, we give the formul for the sum of geometric series of the rtio q expibx: h exp ibhx b n x n b. 3 Now, we cn clculte the Fourier trnsform of the one dimensionl lttice function either stright from its definition or from its Fourier series. We choose the first possibility. By interchnging the order of integrtion nd ddition nd using the sifting property of the Dirc distribution we get the Fourier trnform of the lttice function in the form of the Fourier series: [ F X A x n exp ixx ] dx n A x n exp ixx dx A n n exp ixn A n exp ixn. 4 The Fourier series 4 of the Fourier trnsform of the lttice function is geometric series of the rtio expix. According to 3 it is the Fourier series of the function F X A h X h A h X h A h X Thus, we conclude tht the Fourier trnsform of the one dimensionl lttice function is proportionl to the one dimensionl lttice function with the period : { FT x n } X h X B B h. 5 n h h The lttice function nd its Fourier trnsform 5 re shown in Figure 3. h.
8 8 4 RECIPROCAL LATTICE fx FX B x b X Figure 3: Liner lttice 4.3. with the period, nd its Fourier trnsform 4.3.5, b The Fourier trnsform of the lttice function in E In two dimensionl spce the lttice function hs the form f x fx, x x n n n n n n x n n, x n n. 6 By direct clcultion of the Fourier trnform of this function [ F X A A A n n n n n n x n n ] exp i X x d x exp [ i X n n ] exp [ i X n n ] 7 we obtin the Fourier trnform expressed gin by the Fourier series, this time, of course, by the double series. It is, however, evident tht it my be rewritten s the product of two simple series: F X A n exp [ i X n ] n exp [ i X n ]. 8 Ech of them is geometric series with the rtio expi X nd expi X, respectively. According to 3 they re the Fourier series of functions h X h The Fourier trnsform 8 thus tes the form F X A h nd h X h h X h. 9 X h. 0 Before going on with processing of 0, we will loo up geometricl mening of simple series in this expression see Figure 4. Evidently, the equtions X i h i, i.e. X i i i h i, h i 0, ±, ±,...
9 4.3 Reciprocl lttice nd the Fourier trnsform of the lttice function 9 Figure 4: To the derivtion of the Fourier trnsform of the lttice function in E. In the upper prt of the figure there is lttice with bsis vectors,. The lower prt of the figure shows the corresponding reciprocl lttice with bsis vectors,. represent in E system of stright lines perpendiculr to the vector i nd hving the spcing i. The product of the series in 0 then represents the system of the Dirc distribution of two vribles with non zero vlues t points X specified by conditions i.e. t points X h, X h, h, h 0, ±, ±,..., X h h constituting the two dimensionl reciprocl lttice with the reciprocl constnt. Of course, we cn get this result just by lgebric tretment of 0, without using the geometricl interprettion. We will do it now nd we will even complete the result. For this purpose we rewrite the product of two simple series in 0 s the double series of the Dirc distributions of two vribles. The Fourier trnsform 0 tes then the form F X B h h X h, X h. If we consider the rgument of the Dirc distributions in to be the row mtrix we my process it further. The vrible X is ten s the row mtrix the elements of which re coordintes X, X in n orthonorml bsis nd the multiindex h is the row mtrix with integers h, h s elements.
10 0 4 RECIPROCAL LATTICE X h, X h X, X h, h X, X h, h X A T h X h A T A T X h A A T. The Fourier trnsform with the rgument of the Dirc distributions processed in this wy cn be further djusted nd brought bc from the mtrix representtion to the vectoril one which does not depend on the choice of the orthonorml bsis: F X B h h B det A B B X h A A T h h X X X h A h h, X h h h h. 3 Here det A is the re of the unit cell of the originl lttice cf Hence, the Fourier trnsform 3 of the lttice function 6 is proportionl with the proportionlity fctor /B to the lttice function of the reciprocl lttice with the reciprocl constnt K / The Fourier trnsform of the lttice function in E N The sme result s in E is obtined lso in generl cse of the lttice function in E N. Algebric nd nlytic processing is the sme s in E, only the geometric notion is missing. We clculte the Fourier trnsform of the lttice function 4.: { } F X FT x n n n N N A N A N A N x n n n N N exp i X x d N x exp [ i X n n n N N ] exp [ i X n n n N N ]. 4 Expression 4 represents the Fourier series of the N dimensionl lttice function in the form of N multiple Fourier series. The series cn be fctorized nd by the interchnge of the order of summtion nd multipliction rewritten s the product of N simple geometric series:
11 4.4 The Fourier series of the lttice function in E N F X A N N j exp [i ] X j nj A N N j n j exp [i ] X j nj. 5 Ech of the simple geometric series in 5 cn be expressed ccording to 3 s simple series of Dirc distributions of single vrible F X A N N N j h j X j h j. On the nlogy of the product of simple series of the Dirc distributions of single vrible cn be expressed s the N multiple series of the Dirc distributions of N vribles: F X X B N h,..., X N h N. The rgument of these Dirc distributions cn be considered to be the row mtrix nd the tretment similr to cn be performed: F X B N B N B N det A X A T h X h A T A T X h A. 6 The rgument of the Dirc distributions cn be expressed similrly to 3 in terms of bsis vectors of the reciprocl lttice nd the use of 4.6 leds to the Fourier trnsform of the lttice function in the form F X B N B N X h h h N N X X h, 7 where X h h h h N N 8 is the lttice vector of the reciprocl lttice. The Fourier trnsform 7 of the lttice function in its most generl shpe is then proportionl with proportionlity fctor /B N to the lttice function of the reciprocl lttice with the reciprocl constnt K /. 4.4 The Fourier series of the lttice function in E N While deriving the Fourier trnsform of the lttice function we hve got the trnsform in the form of the Fourier series. In one dimensionl cse it is the expression 4.34, in two dimensionl cse 4.37 nd in N dimensionl cse In ll these cses the exponents of the summtion contin the sclr
12 REFERENCES product of the vrible X nd of the lttice vector x n of the lttice, not of the reciprocl lttice, s could be expected t the Fourier trnsform. Now the question rises how does it loo lie the Fourier trnsform of the lttice function. For the one dimensionl cse we hve found it nd the expression 4.3 shows tht in the exponent of the summnds there is the product of the vrible x nd of the reciprocl lttice vector h. If we restrict ourselves in the N dimensionl cse to the lttice functions with mutully orthogonl bsis vectors it would be possible to fctorize them, to clculte the Fourier series in ech vrible nd then to put gin the individul fctors together. Unfortuntely it is not possible to proceed this wy in the generl cse when the vectors r re not orthogonl. But, fortuntely enough, the Fourier trnsform of the lttice function 4. cn be obtined in ny cse by the inverse Fourier trnsform of the Fourier trnsform of the lttice function in the from 4.37: x n n n N N FT B exp X X h h h N N h h h N N exp ix x d N X [ i h h h N ] N x. The Fourier series of the lttice function is frequently used for expressing the electron density in crystlline solids [5], p. 69, [6], pp. 353 nd following, for deriving the smpling theorem of functions of severl vribles when non orthogonl smpling net is used [], nd elsewhere. References [] Gibbs J. W.: Elements of Vector Anlysis. Tuttle, Morehouse nd Tylor, New Hven [] Ewld P.: Historisches und Systemtisches zum Gebruch des Reziproen Gitters in der Kristllstruturlehre. Zeitschrift für Kristllogrphie , [3] Guinier A.: X Ry Diffrction In Crystls, Imperfect Crystls, nd Amorphous Bodies. W. H. Freemn nd Co., Sn Frncisco 963. [4] Čech E.: Záldy nlyticé geometrie I. Přírodovědecé vydvtelství, Prh 95. [5] Gicovzzo C. et l.: Fundmentls of Crystllogrphy. Interntionl Union of Crystllogrphy, Oxford University Press 99. [6] Henry N. F. M., Lonsdle K. eds.: Interntionl Tbles for X-Ry Crystllogrphy. Vol.. The Kynoch Press, Birminghm 95. [7] Kittel Ch.: Úvod do fyziy pevných láte. Acdemi, Prh 985. [8] Lüth H.: Surfces nd Interfces of Solid Mterils. 3rd ed. Springer Verlg, Berlin 995. [9] Gntmcher F. R.: Těorij mtric. 4. izd. Izdtěl stvo Nu, Mosv 988. [0] Kittel Ch.: Introduction to Solid Stte Physics. 4th ed., John Wiley, Inc., New Yor 97. [] Petersen D. P., Middleton D.: Smpling nd Reconstruction of Wve Number Limited Functions in N Dimensionl Eucliden Spces. Informtion nd Control 5 96,
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