Visualization of Wulff s crystals

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1 Visualization of Wulff s crystals Vesna Veličković and Eberhard Malkowsky, Faculty of Science and Mathematics, University of Niš, Serbia Department of Mathematics, Fatih University, Istanbul, Turkey Državni Univerzitet u Novom Pazaru, Vuka Karadžića bb, Novi Pazar, Serbia Abstract. In this extended abstract, we deal with Wulff s construction and the graphical representation of Wulff s crystals and their surface energy functions as potential surfaces. Keywords: Visualization, Wulff s crystals PACS: Aj, Lt INTRODUCTION According to Wulff s principle [1], the shape of a crystal is uniquely determined by its surface energy function. A surface energy function is a real valued function depending on a direction in space. Let B n denote the unit sphere in euclidean R n+1,thatis, B n = x =(x 1,x 2,...,x n+1 ) R n+1 : x 2 = ( n+1 k=1 1/2 xk) 2 = 1, and let F : B n R be a surface energy function. Then, we may consider the set PM = { x = F( e) e R n+1 : e B n } as a natural representation of F. If n = 1, then e = e(u)=(cosu,sinu) for u (0,2π) and we obtain a potential curve with a parametric representation PC = { x = f (u)(cosu,sinu) : u (0,2π)}, where f (u)=f( e(u)). If n = 2, then e = e(u 1,u 2 )=(cosu 1 cosu 2,cosu 1 sinu 2,sinu 1 ), for (u 1,u 2 ) R =( π/2,π/2) (0,2π), and we obtain a potential surface with a parametric representation PS = { x = f (u 1,u 2 )(cosu 1 cosu 2,cosu 1 sin u 2,sinu 1 ) : (u 1,u 2 ) R}, where f (u 1,u 2 )=F( e(u 1,u 2 )). WULFF S PRINCIPLE AND PARAMETRIC REPRESENTATIONS FOR WULFF S CRYSTALS Wulff gave a geometric principle of construction for crystals [1]. Theorem 1 (Wulff s principle) For every e B n,lete e denote the hyperplane orthogonal to e and through the point P with position vector p = F( e) e, and H e be the half space which contains the origin 0 and has the boundary E e = H e. Then, the crystal C F which has F as its surface energy function is uniquely determined and given by C F = e B n H e = e B n{ x : x e F( e)}. Since Wulff s construction in Theorem 1 is far from applicable for the graphical representation of crystals, we give two results which are more useful. Theorem 2 ([2, Theorem 5.3]) Let F : B n R + be a continuous function. Then, a point X is on the boundary C F of Wulff s crystal C F corresponding to F if and only if F( e) x e for all e B n and F( e 0 )= x e 0 for some e 0 B n. Advancements in Mathematical Sciences AIP Conf. Proc. 1676, ; doi: / AIP Publishing LLC /$

2 FIGURE 1. Wulff s crystals constructed by Theorems 2 and 3 Theorem 3 ([2, Theorem 5.4]) Let F : B n R + be a continuous function and CF : B n R + be defined by CF( e)=inf { F( u)( e u) 1 : u B n and e u > 0 }. Then, the boundary C F of Wulff s crystal corresponding to F is given by C F = { x = CF( e) e R n+1 : e B n }; (1) in particular, if n = 2, then a parametric representation for the boundary C F of Wulff s crystal corresponding to F is for (u 1,u 2 ) R =( π/2,π/2) (0,2π). x(u 1,u 2 )=CF( e(u 1,u 2 )) e(u 1,u 2 ), (2) Although we have used both Theorems 2 and 3 to develop algorithms and programmes for the graphic representation of Wulff s crystals, in some cases a parametric representation can explicitly be given for the boundary of a Wulff s crystal, that is, for the function CF. One such case is when the function F is equal to a norm in three dimensional space. If F =, then the boundary of Wulff s crystal corresonding to F is given by the dual norm of. Corollary 4 ([2, Corollary 5.5]) Let be a norm on R n+1 and, for each w B n,letφ w : R n+1 R be defined by φ w (x)= w x = n+1 k=1 w kx k ( x R n+1 ). Then, the boundary C of Wulff s crystal corresponding to is given by { C = x = 1 } φ e e Rn+1 : e B n, (3) where φ e is the norm of the functional φ e, that is, the dual norm of. Remark 5 For n = 2, we obtain from (2) and (3) the following parametric representation for Wulff s crystal corresponding to a norm in R 3 x(u 1,u 2 )=C ( e(u 1,u 2 ) ) e(u 1,u 2 )= 1 φ e e(u1,u 2 ), for (u 1,u 2 ) R; the potential surface has a parametric representation y = F( e(u 1,u 2 )) e(u 1,u 2 )= e(u 1,u 2 ) e(u 1,u 2 ), for (u 1,u 2 ) R

3 FIGURE 2. Wulff s crystals corresponding to the l 1 and l norms FIGURE 3. Potential surface of the w (Λ) norm and potential surface with corresponding Wulff s crystal Example 6 As a first example, we represent the potential surfaces and corresponding Wulff s crystals for the l 1 and l norms 1 and given by x 1 = x 1 + x 2 + x 3 and x = max 1 k 3 x k for x = {x 1,x 2,x 3 }.Thetwo pictures in Figure 2 are dual to one another. Example 7 Finally, we represent some potential surfaces and the corresponding Wulff s crystals for the norm w (Λ) and for the dual norm W (Λ) ([3, Proposition 5.3, Theorem 5.5]). We identify the sequences x =(x k ) k=1 with the three dimensional vectors x given by their three sections x [3], that is x =(x 1,x 2,x 3 ). We also choose Λ =(1,2,3,4) in each picture (Figures 3 and 4). Further studies on Wulff s crystals and their graphical representations can be found in [4]

4 FIGURE 4. Potential surface of the W (Λ) norm and potential surface with corresponding Wulff s crystal ACKNOWLEDGMENTS Research of both authors supported by the research project #114F104 of Tübitak, and of the second author also by # of the Serbian Ministry of Science, Technology and Environment. REFERENCES 1. G. Wulff, Zeitschrift für Krystallographie 53 (1901). 2. E. Malkowsky, and V. Veličković, MATCH Commun. Math. Comput. Chem. 67, (2012). 3. E. Malkowsky, and V. Veličković, Topology and its Applications 158, (2011). 4. E. Malkowsky, F. Özger, and V. Veličković, MATCH Commun. Math. Comput. Chem. 70, (2013)

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