Crystal problems for binary systems Laurent B etermin Villum - PowerPoint PPT Presentation

Crystal problems for binary systems Laurent B´ etermin Villum Centre for the Mathematics of Quantum Theory, University of Copenhagen Talk based on joint works with H. Kn¨ upfer , F. Nolte and M. Petrache Workshop on Optimal and Random Point Configurations February 28, 2018, ICERM, Providence Laurent B´ etermin (KU) Binary systems 02/28/2017 1 / 30

Introduction: ionic solids Salt NaCl Combination of bonding, size of ions, orbitals... ⇒ Structure Laurent B´ etermin (KU) Binary systems 02/28/2017 2 / 30

Introduction: ionic solids Mathematical justification of these structures: very difficult problem. Identical particles / optimization of structure: several open problems. Two approaches (in this talk) via energy optimization : fixing the lattice structure and optimizing the charge distribution; 1 fixing the charge distribution and optimizing the structure ( d = 1). 2 In both cases: the interaction is electrostatic (and more general). Laurent B´ etermin (KU) Binary systems 02/28/2017 3 / 30

Born’s problem for the electrostatic energy (1921) Max Born (1882-1970) “How to arrange positive and negative charges on a simple cubic lattice of finite extent so that the electrostatic energy is minimal?” ¨ Uber elektrostatische Gitterpotentiale, ur Physik , 7:124-140, 1921 Zeitschrift f¨ Laurent B´ etermin (KU) Binary systems 02/28/2017 4 / 30

Born’s Conjecture (1921) Conjecture [Born ’21] The alternate configuration of charges is the unique solution among all periodic distributions of charges on Z 3 . The total amount of charge is fixed and the neutrality have to be assumed. Laurent B´ etermin (KU) Binary systems 02/28/2017 5 / 30

Born’s result in dimension 1 (1921) Born proved the conjecture in dimension 1 (Ewald summation method). N N � ϕ 2 � Assuming that ϕ 0 > 0, i = N and ϕ i = 0, he proved the i =1 i =1 optimality of the alternate configuration ϕ i = ( − 1) i , achieved for N ∈ 2 N . Laurent B´ etermin (KU) Binary systems 02/28/2017 6 / 30

Crystallization in dimension 1 Identical particles and symmetric potential: 1 Ventevogel ’78: convex functions, Lennard-Jones-type potentials. 2 Ventevogel-Nijboer ’79: Gaussian, more general repulsive-attractive potentials, positivity of the Fourier transform. 3 Gardner-Radin ’79: classical (12 , 6) Lennard-Jones by alternatively adding points from both sides of the configuration. Laurent B´ etermin (KU) Binary systems 02/28/2017 7 / 30

Crystallization for one-dimensional alternate systems We consider periodic configurations of alternate kind of particles. ρ : length. N : number of point per period ( N = 8 in the example). They interact by three kind of potentials: f 12 , f 11 and f 22 . Theorem [B.-Kn¨ upfer-Nolte ’18] (soon on arXiv) If f 12 ( x ) = − f 11 ( x ) = − f 22 ( x ) = − x − p , p ≥ 0 . 66, then, for any ρ > 0 and any N ≥ 1, the equidistant configuration is the unique maximizer of the total energy per point among all the periodic configurations of points. Proof based on Jensen’s inequality. The same occurs for many systems (at high density). Laurent B´ etermin (KU) Binary systems 02/28/2017 8 / 30

Back to Born’s conjecture: charge and potential d � Bravais lattice X = Z u i of covolume 1 , i.e. | Q | = 1 (unit cell). i =1 distribution of charge ϕ : X → R , s.t. x ∈ X has charge ϕ x = ϕ ( x ). N -periodicity : ϕ ∈ Λ N ( X ), i.e. ∀ x ∈ X , ∀ i , ϕ ( x + Nu i ) = ϕ ( x ). � ϕ 2 y = N d . We assume ϕ 0 > 0. The total charge is fixed : y ∈ K N � d � � Periodicity cube K N := x = m i u i ∈ X ; 0 ≤ m i ≤ N − 1 . i =1 We note K ∗ N the same cube for the dual lattice X ∗ . � ∞ e −| x | 2 t d µ f ( t ), Borel measure µ f ≥ 0, Potential f ( x ) = 0 i.e. f ( x ) = F ( | x | 2 ) where F is completely monotone. Laurent B´ etermin (KU) Binary systems 02/28/2017 9 / 30

Energy minimization problem Definition of the energy    1 � � ϕ y ϕ x + y f ( x ) e − η | x | 2  , E X , f [ ϕ ] := lim 2 N d η → 0 y ∈ K N x ∈ X \{ 0 } � ϕ y = 0 if f �∈ ℓ 1 ( X \{ 0 } ) (charge neutrality). where we assume y ∈ K N Problem: Minimizing E X , f among all N and all ϕ ∈ Λ N ( X ) satisfying � ϕ 2 y = N d and ϕ 0 > 0 . y ∈ K N [B.-Kn¨ upfer ’17] 1 General strategy connecting E X , f with lattice theta function. 2 Explicit solution for X orthorhombic or triangular (uniqueness). Laurent B´ etermin (KU) Binary systems 02/28/2017 10 / 30

The space Λ N ( X ) of N -periodic charges Λ N ( X ) is equipped with inner product and norm: � � ( ϕ, ψ ) K N = ϕ ( y ) ψ ( y ) , � ϕ � = ( ϕ, ϕ ) K N . y ∈ K N ϕ ∈ Λ N ( X ∗ ) s.t. ∀ k ∈ X ∗ , Discrete Fourier transform: ϕ ∈ Λ N ( X ) ⇒ ˆ 1 ϕ y e − 2 π i � N y · k . ϕ ( k ) = ˆ d N 2 y ∈ K N Discrete inverse Fourier transform of ψ ∈ Λ N ( X ∗ ): for any x ∈ X , 1 2 π i ˇ � N y · x . ψ ( x ) = ψ y e d N 2 y ∈ K ∗ N Laurent B´ etermin (KU) Binary systems 02/28/2017 11 / 30

The autocorrelation function s = ϕ ∗ ϕ � ϕ 2 y = N d , then we define Let ϕ : X → R be N -periodic such that y ∈ K N � s x = ϕ y ϕ y + x . y ∈ K N Properties of s s ∈ Λ N ( X ), s − x = s x , 2   �  � s x = ϕ x ,  x ∈ K N x ∈ K N s 0 = N d . Laurent B´ etermin (KU) Binary systems 02/28/2017 12 / 30

The inverse Fourier transform ξ s , i.e. for any k ∈ X ∗ and any x ∈ X , We define ξ := N − d 2 ˇ ξ k := 1 2 π i ξ k e − 2 π i � N y · k , � N k · x . s x = s y e N d y ∈ K N k ∈ K ∗ N Properties of ξ ξ k ∈ R , ξ − k = ξ k , ϕ k | 2 ≥ 0, ξ k = | ˇ 2   ξ 0 = 1  � ϕ x ,  N d x ∈ K N � ξ k = N d . k ∈ K ∗ N Laurent B´ etermin (KU) Binary systems 02/28/2017 13 / 30

Absolutely summable case We assume that f ∈ ℓ 1 ( X \{ 0 } ), then    1 ϕ y ϕ x + y f ( x ) e − η | x | 2 � � E X , f [ ϕ ] = lim  2 N d η → 0 y ∈ K N x ∈ X \{ 0 } 1 � � = ϕ y ϕ x + y f ( x ) 2 N d y ∈ K N x ∈ X \{ 0 } 1 � = s x f ( x ) 2 N d x ∈ X \{ 0 } 1 2 π i � � N x · k f ( x ) = ξ k e 2 N d k ∈ K ∗ x ∈ X \{ 0 } N 1 � = ξ k E [ k ] . 2 N d k ∈ K ∗ N Laurent B´ etermin (KU) Binary systems 02/28/2017 14 / 30

Rewriting E in terms of translated theta function � ∞ e −| x | 2 t d µ f ( t ), we obtain, ∀ k ∈ K ∗ Since f ( x ) = N , 0 2 π i � N x · k f ( x ) E [ k ] = e x ∈ X \{ 0 } � ∞ �� � e −| x | 2 t e 2 π ix · k N − 1 = d µ f ( t ) 0 x ∈ X   � ∞ e − π 2 2 d 2 t − d t | p + k N | �  d µ f ( t ) =  π − 1 2 0 p ∈ X ∗ � ∞ � π � d 2 t − d � � 2 θ X ∗ + k = π − 1 d µ f ( t ) . t N 0 z 0 minimizer of z �→ θ X ∗ + z ( α ) for all α > 0 ⇒ k 0 = Nz 0 minimizer of E . Laurent B´ etermin (KU) Binary systems 02/28/2017 15 / 30

From ξ to ϕ : existence of a minimizer for E X , f Lemma � If ξ ∈ Λ N ( X ∗ ), ξ ≥ 0, ξ − k = ξ k and ξ k = N d , then ϕ defined by k ∈ K ∗ N 1 � 2 π � � � ϕ x = ξ k cos N x · k d N 2 k ∈ K ∗ N ϕ y ϕ y + x , ξ = N − d � ϕ 2 � y = N d , s x = 2 ˇ satisfies s . y ∈ K N y ∈ K N If k 0 = Nz 0 minimizes E , we have that ξ , defined by ξ k 0 = N d and ξ k = 0 otherwise, is a minimizer of ξ �→ E X , f [ ϕ ], which corresponds to ϕ x = c cos(2 π x · z 0 ) , x ∈ X , c constant. Laurent B´ etermin (KU) Binary systems 02/28/2017 16 / 30

From ξ to ϕ : uniqueness of the minimizer for E X , f We assume that k �→ θ X ∗ + k N ( α ) has at most two minimizers k 0 and k 1 in X ∗ for some N ∈ N , then: N = � d k 0 and k 1 are symmetry related : k 1 i =1 u ∗ i − k 0 N , by periodicity and parity of ξ , the minimizer of ξ �→ E X , f [ ϕ ] is given by ξ k 0 = ξ k 1 = N d ∀ k ∈ K ∗ 2 , N \{ k 0 , k 1 } , ξ k = 0 . � 2 π Then, we obtain s x = N d cos � N k 0 · x . ϕ | 2 , we reconstruct the unique solution ϕ such Using the fact that ξ k = | ˇ that ϕ 0 > 0: ϕ x = c cos(2 π x · z 0 ) , x ∈ X , c constant. � ⇒ Any minimizer is neutral : ϕ y = 0 (general fact). y ∈ K N Laurent B´ etermin (KU) Binary systems 02/28/2017 17 / 30

Translated lattice theta function For a Bravais lattice X ⊂ R d and a point z ∈ R d and α > 0, we define e − πα | x + z | 2 . � θ X + z ( α ) := x ∈ X 1 1 � � We have θ X + z ( α ) = 2 P X z , where P X solves the heat equation d 4 πα α for ( z , t ) ∈ R d × (0 , ∞ )  ∂ t P X ( z , t ) = ∆ z P X    � for z ∈ R d . P X ( z , 0) = δ p    p ∈ X Laurent B´ etermin (KU) Binary systems 02/28/2017 18 / 30