Geometric Properties of Square Lattice

Transcription

1 Applied Mathematical Sciences, Vol. 8, 014, no. 91, HIKARI Ltd, Geometric Properties of Square Lattice Ronalyn T. Langam College of Engineering and Information Technology Saint Michael s College Iligan City, Philippines Rosalio G. Artes, Jr. Department of Mathematics and Statistics Mindanao State University - Iligan Institute of Technology Iligan City, Philippines Copyright c 014 Ronalyn T. Langam and Rosalio G. Artes, Jr. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We investigated some of the geometric properties of the lattice points of the square lattice Z and the coincidence lattice generated by rotating about the origin the square lattice Z at some specified angle of rotations. We have shown that for every nonnegative rational number r Q, there exists θ 0, π ] such that the point (q, p) belongs to the coincidence lattice if and only if r = p. In addition, we have shown that { ( ) q } π n the set R = tan 1 : n, k N is dense in 0, π ] using the density property of Q in R. Mathematics Subject Classification: 74E15, 11P1 Keywords: square lattice, density, coincidence lattice 1 Introduction Lattices play an important role in the study of crystallography. The discovery of X-rays in the late 19th century opened a new venture in the research world.

2 454 Ronalyn T. Langam and Rosalio G. Artes, Jr. The determination of the atomic structures of various crytals is slowly conquered through X-ray crystallography 5]. Since then, hundreds of thousands of molecular structures have been determined. Meanwhile, studies involving coincidence site lattices (CSL) evolved after Friedel s recognition of its usefulness in the study of grain boundary 4]. The latter is a particular crystal defect which is actually the interface betweeen two interpenetrating misoriented grains of the same phase. Due to this, bonds between individual atoms in the grain boundary are different from the bulk crystal, eventually limiting the application of the material. To possibly overcome this, a method called grain boundary engineering is formulated. In here, the knowledge of structural independence of various grain boundaries in a crystal is used to optimize its properties 6]. With this, thorough understanding of the properties of the crystal being considered is necessary. However, problems in the manipulation of grain boundaries arise since a polycrystalline material contains a huge number of grain boundaries. Polycrystalline materials are solids that are composed of many crystallites of varying size and orientation. With the quest in finding a probable solution to this complications, studies involving grain boundaries using the CSL approach boomed. One of the most common lattice being considered is the square lattice, a lattice generated by vectors of the form (u, 0) and (0,u), where u R. Ifu = 1, then it is the lattice of all integer points, denoted by Z. A CSL is generated by superimposing a rotated copy of a lattice to the original lattice, that is, a rotated copy of a lattice is being drawn on top of the original lattice. It is actually the points of coincidence derived after superimposition. Since this CSL is generated through rotation, it is often termed as rotational coincidence lattice. In general, studies of CSLs are extended involving multilattices which is a superimposition of finite rotated copies of a lattice to the original lattice. In chapter four, the pulled lattice denoted by Z (α) is being defined and considered. This lattice is generated by altering one of the spanning vectors of Z (0, 1) by (cos α, sin α). Bolman in 3], derived all possible patterns of lattice points of both lattices (the rotated and the original lattice), that varies with θ. In relation with this, the authors were interested to investigate the specific angles of rotation that generate a nontrivial coincidence lattice, a lattice that contains a point other than the origin. Moreover, some properties of these angles are studied in detail. Preliminary Notes Given n linearly independent vectors b 1,b,...,b n R m, the lattice generated by these vectors is defined as } L(b 1,b,...,b n )={ xi b i x i Z.

3 Geometric properties of square lattice 4543 We refer to b 1,b,...,b n as a basis of the lattice. Equivalently, if we define B as the m n matrix whose columns are b 1,b,...,b n, then the lattice generated by B is L(B) =L(b 1,b,...,b n )={Bx x Z n }. A square lattice is a lattice generated by {(α, 0), (0,α)} for some α R. The span of a lattice L(B) is the linear space spanned by its vectors, span(l(b))=span(b) ={By y R n }. A coincidence lattice is the intersection lattice of Z and the lattice rotated about the origin at an angle θ which is denoted by Z θ. In many literatures, coincidence lattice is also called coincidence site lattice (CSL). 3 Results Consider the angles formed by the points symmetric with respect to the line y = x. To narrow down, consider the points (p, q) on the first quadrant, that is, p, q > 0. Now, for each n, let θ n = π ( ) n n tan 1. Note that n +1 n +1 converges ( ) n to 1 as n goes to infinity. Hence, tan 1 goes to π as n goes to infinity. n +1 4 Moreover, tan 1 is increasing on 0, π ]. Thus, θ n n=1 is strictly decreasing and converges to 0. In general, consider the angles of the form θ n,k = π ( ) n tan 1, where n, k N. Theorem 3.1 Let n, k N and consider the sequence θ n,k n,k=1. Then θ n,k converges to zero and is strictly decreasing. Proof: For θ n,k to be strictly increasing it follows from the fact that tan 1 function is increasing on 0, π ) n. Further, as n, 1. Thus, ( ) n tan 1 π ( ) 4. Hence, π n tan 1 0. Consider now the angle with vertex at the origin and the rays pass through any two distinct points in the same quadrant. A coincidence angle θ 0, π ], is an angle with vertex at the origin and the rays pass through any two distinct points in the same quadrant.

4 4544 Ronalyn T. Langam and Rosalio G. Artes, Jr. The coincidence span of θ 0, π ] denoted by span(θ) is defined to be the length of the line segment joining two distinct points P (x 1,y 1 ) and Q(x,y ) in the same quadrant provided that gcd(x 1,y 1 ) = 1 and gcd(x,y ) = 1. If gcd(x 1,y 1 ) 1 or gcd(x,y ) 1, then span(θ) is the distance between the points ( ) x 1 P gcd(x 1,y 1 ), y 1 gcd(x 1,y 1 ) ( and Q x gcd(x,y ), y gcd(x,y ) Theorem 3. Let θ R. Then span(θ) =k if and only if θ = π ( ) 1 tan 1. k +1 Proof: Consider the point P (1, 1+k) and its reflection along the line y = x, the point Q(1 + k, 1). Since k N, θ = π ( ) 1 tan 1 if and only if k +1 span(θ) = (1 + k) 1] +1 (1 + k)] = k + k = k. Meanwhile, consider the equivalence classes of rationals. Consider the set of positive integers Z and define a relation onz by (p, q) (s, t) if(s, t) =r(p, q) =(rp, rq) for some r Q. Theorem 3.3 The relation is an equivalence relation. Proof: We must show that is reflexive, symmetric, and transitive. i. Let p, q Z. Since (p, q) =(p, q), (p, q) (p, q). Hence, is reflexive. ii. Let (p, q), (s, t) Z Z such that (p, q) (s, t). Then there exists 0 w Q such that (p, q) =w(s, t). Hence, (s, t) = 1 (p, q). Take w r = 1. Then (s, t) =r(p, q) and (s, t) (p, q). Thus, is symmetric. w iii. Let (p, q), (s, t), (a, b) Z Z such that (p, q) (s, t) and (s, t) (a, b). Then (p, q) =r 1 (s, t) and (s, t) =r (a, b) for some r 1,r Q. Hence, (p, q) =r 1 (s, t) =r 1 r (a, b)] = (r 1 r )(a, b). Set r = r 1 r. Then (p, q) =r(a, b). Hence (p, q) (a, b). Thus, is transitive. ).

5 Geometric properties of square lattice 4545 Accordingly, is an equivalence relation. Consequently, the relation partitions Z Z into distinct equivalence classes. Moreover, (x, y)] lies entirely on the line passing through the origin with slope y x. Now, for every (p, q) Z, there exists an equivalence class of Q containing (p, q). Define ϕ :Z Z ] Q] by ] q ϕ((p, q)]) =. p Let x, y Z Z ] such that ϕ(x) =ϕ(y). Then there exist x 1,x,y 1,y Z such ] that x ] =(x 1,y 1 )] ] and y =(x,y )]. Hence ϕ(x) =ϕ(y) implies that y1 y y1 = = r for some r Q. Consequently, x = y. Hence, ϕ is x 1 x x 1 injective. s ] Let q Q ]. Then q =, for some s, t Z. Moreover, (t, s)] Z Z ] t s ] and ϕ((t, s)]) =. Hence, ϕ is surjective. t Accordingly, ϕ is an isomorphism. With ϕ((0, 0)]) = ϕ ({(0,q):q Z} {(p, 0) : p Z}) = 0] and with the above result, there exists an embedding of Q on Z. Formally, the following is immediate. Theorem 3.4 Let φ :Z Z ] Q ] be defined by φ((p, q)]) = Then φ is an isomorphism. ] q. p The above theorem asserts that the set of rationals can be partitioned into distinct equivalence classes and that Q can be embeded on Z. Lemma 3.5 If α, β 0, π ] with α<β, then there exists θ R such that α<θ<β. Proof: Let α, β 0, π ] with α<β. Then there exists r 1,r R such that tan α = r 1 and tan β = r. Thus, ( tan 1 (r 1 ) < tan 1 (r ). Since tan 1 function is bijective and increasing on 0, π ), it follows that r 1 <r. By the density of rationals, there exists r Q such that r 1 <r<r. Hence, tan 1 (r 1 ) < tan 1 (r) < tan 1 (r ). ( ) q Set θ = tan 1 (r) = tan 1 = π p tan 1 (r) R.

6 4546 Ronalyn T. Langam and Rosalio G. Artes, Jr. in Theorem 3.6 Let R = 0, π ]. Proof: Note that R { ( ) } π i tan 1 : i, k N. Then R is dense i + k 0, π ] (. Let θ 0, π ] 3.5, there exists θ 1 R such that 0 <θ 1 <θ. Similarly, since θ 1. Then, θ>0. By Lemma 0, π ], there exists θ R such that 0 <θ 1 <θ <θ. Continuing in this manner, an increasing sequence θ n is obtained which converge to θ by the density of rationals. The above theorem asserts that for every θ 0, π ], there exists a sequence θ j j=1 such that θ j θ as j. References 1] M. Baake and P. Zeiner, Multiple Coincidences for Dimensions d 3, Phil. Mag., 87 (007), ] H.K.D.H Bhadeshia, Worked Examples in the Geometry of Crystals, Institute of Materials, ] W. Bollman, Crystal Defects and Crystalline Interfaces, Switzerland: Springer-Verlag Berlin Heidelberg, ] G. Friedel, Lecons de Cristallographie, Herman, Paris, ] C. Hammond, The Basics of Crystallography and Diffraction, New York: Oxford University Press Inc., ] P. Lejček, Grain Boundary Segregation in Metals, New York: Springer- Verlag Berlin Heidelberg, ] D. Schwarzenback, Crystallography, England: Jhon Wiley and Sons Ltd., ] M. Shamsuzzoha and R. Rahman, A New Geometrical Method for constructing Coincidence Site Lattices for Cubic Crystals, International Journal of Engineering Research and Development, 3 (01), ] N.G. Szwacki N.G. and T. Szwacka, Basic Elements of Crystallography, Singapore: Pan Stanford Publishing Pte. Ltd., ] P. Zeiner, Symmetries of Coincidence Site Lattices of Cubic Lattices, Z. Krist. 0 (005), Received: June 1, 014