Mathematical Model for Diamond-Type Crystals with Impurities or Defects

D. Opriş and I.D. Albu

Abstract

This work develops the geometry and dynamics of diamond-type crystals with impurities or defects and symmetry from the perspective of Lagrangian mechanics. We begin by formulating continuous-discret network for diamondtype crystals, then we formulate the continuous-discret Lagrange-d’Alembert principle, Noether’s theorem and momentum equation for diamond-type crystals with impurities are given. Several detailed examples are given to illustrate the theory.

Mathematics Subject Classification: 70H35, 70D10, 70F25 Key Words: diamond-type crystal, Lagrange-d’Alembert principle, Noether theorem.

1 Introduction

The diamond-type crystals are among the most widely studied crystals in literature and the usual theories supply results in good agreement with the experiments [5]. These theories are formulated in terms of the invariants of some representations of the space group O_{h}^{7} [2]. In [6] the case of crystals with impurities is studied and consideration of a distribution of lattice group, described by the nonintegrable distribution of lattice bases. A broad overview of the paper is as follows. We begin by describing continuous-discret network diamond-type crystal, using some representations of the space group O_{h}^{7} [2]. The methodology from [1], is adapted for continuous-discret mechanics [3],[4], and continuous-discret Lagrange-d’Alembert principle for diamondtype crystals is given. The description of the distorted crystal structure can be realized by considering a constraint S. The equations of motions can be written in terms of the usual Euler-Lagrange operator. Following this, we add the hypothesis of symmetry and we develop evolution equation for the momentum that generalizes the usual conservation laws associated to a symmetry group. Several detailed examples are given to ilustrate the theory.

Balkan Journal of Geometry and Its Applications, Vol.3, No.1, 1998, pp. 73-88 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

2 Continuous-discret network for diamond-type crystals

Let Z be the ring of integers and let N be the set of all natural numbers. The metric space (D_{∞}, δ), where

(2.1)  D_{∞} = {n = (n_{0}, n_{1}, n_{2}, n_{3}) ∈ Z^{4}|n_{0} + n_{1} + n_{2} + n_{3} ∈ {0, 1}}

and

(2.2)  δ : D_{∞} × D_{∞} ⟶ N , δ(n, n′) = ∑_{i=0}^{3} |n_{i} − n′_{i}|

is a discret parametric space for the ,,infinite” crystal having the structure of diamond [2]. The group of all isometries of the space (D_{∞}, δ) is isomorphic to the space group O_{h}^{7} [1]. For each n ∈ D_{∞} we consider the neighbours order k of n, that is the elements of the set

(2.3)  V^{(k)}(n) = {n′ ∈ D_{∞}|δ(n, n′) = k}.

In particular

(2.4)  V^{(1)}(n) = {n^{α} | α = 0, 1, 2, 3},

where

(2.5)  n^{α} = n + ε(n)e^{α}  ε(n) = (−1)^{n_{0}+n_{1}+n_{2}+n_{3}}

{e^{α}} is the canonical basis of R^{4}. By considering

(n^{α})^{β} = n^{αβ} = n + ε(n)e^{α} + ε(n^{α})e^{β} , α, β ∈ {0, 1, 2, 3} , α ≠ β
(2.6)  n^{αα} = n , n^{αβ} ≠ n^{βα} , α ≠ β ,

we obtain the second neighbours of n

(2.7)  V^{(2)}(n) = {n^{αβ}| α ≠ β α, β ∈ {0, 1, 2, 3}}

and the third neighbours of n

(2.8)  V^{(3)}(n) = {n^{αβγ}|α ≠ β ≠ γ α, β, γ ∈ {0, 1, 2, 3}}

Let N ∈N , N > 3, be a fixed natural number and let Z_{N} be the quotient space Z/(NZ). We will obtain a parametric space for the ,,finite” crystal having the structure of diamond by using the set

(2.9)  D = {n = [n_{0}, n_{1}, n_{2}, n_{3}] ∈ (Z_{N})^{4}|n_{0} + n_{1} + n_{2} + n_{3} ∈ {0, 1}}

If [a, b] ⊂R is an interval, then the set R = [a, b]×D will be called the continuoustype network for diamond-type crystals. Let q : R ⟶ R^{m} , (m ≥ 1), be a C^{1}-function (with respect to t ∈ [a, b]) and

q(t, n) = (q^{i}(t, n)) , i = \overline{1, m} , q^{.} (t, n) = \frac{dq(t, n)}{dt} , (t, n) ∈ R

q^{α}(t, n) = q(t, n^{α}) − q(t, n) , n^{α} ∈ V_{(n)}^{(1)} , q^{αβ}(t, n) = q^{αβ}(t, n^{αβ}) − q(t, n) ,

(2.10)  n^{αβ} ∈ V_{(n)}^{(2)}.

The space of functions

(2.11)  L^{2}(R) = {q : R ⟶ R^{m}| ∫_{a}^{b} ∑_{n∈D} δ_{ij}q^{i}(t, n)q^{j}(t, n) < ∞}

with the canonical scalar product is a Hilbert space. Let the space

(2.12)  Ω = {q ∈ L^{2}(R)|q(a, n) = q_{1}(n) , q(b, n) = q_{2}(n) , ∀n ∈ D)},

with q_{1}, q_{2} fixed. The tangent space to Ω in q ∈ Ω is given by

(2.13)  T_{q}(Ω) = {η : R ⟶ R^{m}|η(a, n) = 0 , η(b, n) = 0 , ∀ ∈ D},

where

(2.14)  η(t, n) = \frac{dq(ε, t, n)}{dε}|_{ε=0}

and q(ε, t, n) ∈ Ω , q(0, t, n) = q(t, n) , ε ∈ I ⊂ R , 0 ∈ I. For a C^{1}-function F : Ω ⟶ R, the variation δF is

(2.15)  δF : T_{q}Ω ⟶ R , δF (η) = \frac{dF (q(ε))}{dε}|_{ε=0} .

The element q ∈ Ω is called a critical point for F if δF (η) = 0 , ∀η ∈ T_{q}Ω.

3 Continuous-discret Lagrange-d’Alembert principle for diamond-type crystals

Consider the sets

(3.1)  \begin{matrix}Ω^{1} = \{q^{α}(t, n) , q ∈ Ω , α ∈ \{0, 1, 2, 3\} , (t, n) ∈ R\} \\ Ω^{2} = \{q^{αβ}(t, n) , q ∈ Ω , α ≠ β α, β ∈ \{0, 1, 2, 3\} , (t, n) ∈ R\} \\ Ω^{•}= \{q^{.} (t, n) , q ∈ Ω , (t, n) ∈ R\} \end{matrix}

and the C^{1}-function L : R × Ω × Ω^{1} × Ω^{2}× Ω^{•}⟶R, given by

(3.2)  L(t, n) = L(t, n, q(t, n), q^{α}(t, n), q^{αβ}(t, n), q^{.} (t, n)) , (t, n) ∈ R.

The functional

(3.3)  A(q) = ∫_{a}^{b} ∑_{n∈D} L(t, n)dt

is called the action of L with respect to q ∈ Ω. Theorem 3.1 [4] (First variation formula). The variation δA(q) of the action A(q) is

(3.4)  δA(q)(η) = ∫_{a}^{b} ∑_{n∈D} E_{i}(\overline{L})η^{i}(t, n)dt , η ∈ T_{q}Ω,

where

(3.5)  E_{i}(\overline{L}) = \frac{d\overline{L}(t, n)}{dq^{i}(t, n)} − \frac{d}{dt}(\frac{∂\overline{L}(t, n)}{∂q^{.}^{i}(t, n)})

(3.6)  \overline{L}(t, n) = L(t, n) + ∑_{α=0}^{3} L(t, n^{α}) + ∑_{\begin{matrix}α, β = 0 \\ α ≠ β \end{matrix}}^{3} L(t, n^{αβ}).

From theorem 3.1 we deduce Theorem 3.2 (Discret continuous variation principle). An element q ∈ Ω is a critical point for A(q) if and only if

(3.7)  \frac{d\overline{L}(t, n)}{dq^{i}(t, n)} − \frac{d}{dt} (\frac{∂\overline{L}(t, n)}{∂q^{.}^{i}(t, n)}) = 0 , ∀(t, n) ∈ R , i = \overline{1, m}.

Example. The Lagrange function of the atoms of the crystal with respect to their equilibrum positions is given by [2],[5]

(3.8) \begin{aligned}L(t, n) & = & \frac{1}{2}mδ_{ij}q^{.}^{i}(t, n)q^{.}^{j}(t, n) − \frac{1}{2} ∑_{α=0}^{3} ϕ_{ijα}q^{iα}(t, n)q^{jα}(t, n)− \\ & − & \frac{1}{2} ∑_{\begin{matrix}α, β = 0 \\ α ≠ β \end{matrix}}^{3} ϕ_{ijαβ}q^{iαβ}(t, n)q^{jαβ}(t, n) , (t, n) ∈ R , i, j = 1, 2, 3, \end{aligned}

where

ϕ_{ijα} = ϕ_{jiα} = const , ϕ_{ijαβ} = ϕ_{jiαβ} = const.

From (3.7) we obtain

(3.9) mδ_{ij} \frac{d^{2}q^{i}(t, n)}{dt^{2}} = ∑_{α=0}^{3} ϕ_{ijα}q^{jα}(t, n) + ∑_{\begin{matrix}α, β = 0 \\ α ≠ β \end{matrix}}^{3} ϕ_{ijαβ}q^{jαβ}(t, n) , ∀(t, n) ∈ R.

The system of equations (3.9) corresponds to the system of equations used in lattice dynamics of diamond-type crystals (the model of Born-von Karman).

For the study of the dynamics in crystals it is useful to introduce some so-called motions of order α and αβ. Let f^{a} : R × Ω × Ω^{1} × Ω^{2}× Ω^{.}⟶R, a = \overline{1, p}, be a C^{1}-function with respect to t ∈ [a, b] , q ∈ Ω , q^{α} ∈ Ω^{1} , q^{αβ} ∈ Ω^{2} , q^{.}∈Ω^{•}. We put

(3.10)  f^{a}(t, n) = f^{a}(t, n, q(t, n), q^{α}(t, n), q^{αβ}(t, n), q^{.}^{i} (t, n)) , (t, n) ∈ R

and suppose that

(3.11)  rang ∥\frac{∂f^{a}(t, n)}{∂q^{αi}(t, n)}∥ = p < m , α = 0, 1, 2, 3;

(3.12)  rang ∥\frac{∂f^{a}(t, n)}{∂q^{αβi}(t, n)}∥ = p < m , α, β = 0, 1, 2, 3 , α ≠ β;

(3.13)  rang ∥\frac{∂f^{a}(t, n)}{∂q^{.}^{i}(t, n)}∥ = p < m.

Let us consider the set (3.14)

S = \{(q(t, n), q^{α}(t, n), q^{αβ}(t, n), q^{.} (t, n)) ∈ Ω × Ω^{1} × Ω^{2}× Ω^{.} |f^{a}(t, n) = 0 , a = \overline{1, p}\}

For a generic element q ∈ Ω, for S, let η ∈ T_{q}Ω be the tangent vector to Ω satisfying the conditions

(3.15)  \frac{∂f^{a}(t, n)}{∂q^{αi}(t, n)}η^{i}(t, n) = 0 , a = \overline{1, p} , α ∈ \{0, 1, 2, 3\}, fixed.

η is called the virtual variation of the order α for the system (Ω, L, S), where L is given by (3.2) and S is given by (3.14).

The Lagrange-d’Alembert principle of order α is the following: an admissible element q ∈ Ω is called a motion of the order α for the system (Ω, L, S) if [E]_{i}(\overline{L})η^{i}(t, n) = 0 , ∀(t, n) ∈ R, for all virtual variations of the order α. Proposition 3.3. The motion of the order α is given by

(3.16)  E_{i}(\overline{L})(t, n) = μ_{a}^{α} \frac{∂f^{a}(t, n)}{∂q^{iα}(t, n)} i = \overline{1, m},

f^{a}(t, n) = 0 , a = \overline{1, p} , α fixed , (t, n) ∈ R.

The elements η ∈ T_{q}Ω satisfying the conditions

(3.17)  \frac{∂f^{a}(t, n)}{∂q^{αβi}(t, n)}η^{i}(t, n) = 0 , a = \overline{1, p} , α ≠ β , fixed

are called the virtual variations of the order αβ for the system (Ω, L, S).

The Lagrange-d’Alembert principle of order αβ is the following: an admissible element q ∈ Ω is called a motion of the order αβ for the system (Ω, L, S) if [E]_{i}(\overline{L})η^{i}(t, n) = 0 , ∀(t, n) ∈ R, for all virtual variations of the order αβ.

Proposition 3.4. The motion of the order αβ is given by

(3.18)  E_{i}(\overline{L})(t, n) = μ_{a}^{αβ} \frac{∂f^{a}(t, n)}{∂q^{iαβ}(t, n)} , i = \overline{1, m},
f^{a}(t, n) = 0 , a = \overline{1, p} , α, β fixed , (t, n) ∈ R.

The elements η ∈ T_{q}Ω satisfying the conditions

(3.19)  \frac{∂f^{a}(t, n)}{∂q^{.}^{i}(t, n)}η^{i}(t, n) = 0 , a = \overline{1, p},

are called the virtual variations for (Ω, L, S).

The Lagrange-d’Alembert principle is: an admissible element q ∈ Ω is called a motion for the system (Ω, L, S) if [E]_{i}(\overline{L})η^{i}(t, n) = 0 , ∀(t, n) ∈ R, for all virtual variations. Proposition 3.5 [3]. The motion is given by

(3.20)  E_{i}(\overline{L}(t, n) = μ_{a} \frac{∂f^{a}(t, n)}{∂q^{.}^{i}(t, n)}) , i = \overline{1, m},

f^{a}(t, n) = 0 , a = \overline{1, p} , (t, n) ∈ R.

4 Constraint distribution on the space

Ω × Ω^{1} × Ω^{2}× Ω^{•}

The description of the distorted crystal structure can be realised by considering a constraint S given by (3.14). If we choose the affine constraints of the form

(4.1)  f^{a}(t, n) = q^{aα}(t, n) + A_{r}^{aα}q^{rα}(t, n) − γ^{aα}(t, n),

where

(4.2)  A_{r}^{aα}(t, n) = A_{r}^{aα}(q(t, n)) , γ^{aα}(t, n) = γ^{aα}(q(t, n)),

a = \overline{1, p} , r = \overline{p + 1, m} , α ∈ \{0, 1, 2, 3\}, fixed, then from the Lagrange-d’Alembert principle of order α we get. Proposition 4.1. The motion of the order α is given by

(4.3)  E_{r}(\overline{L}) = A_{r}^{aα}(t, n)E_{a}(\overline{L})  a = \overline{1, p} , r = \overline{p + 1, m},
q^{aα}(t, n) + A_{r}^{aα}(t, n)q^{rα}(t, n) − γ^{aα}(t, n) = 0 , (t, n) ∈ R.

Now we define the constrained Lagrangian of order α, L_{c}, by substituting the constraints (4.2) into the Lagrangian (3.2).

L_{C}(t, n) = L(t, n, q(t, n), −A_{r}^{aα}(t, n)q^{rα}(t, n) + γ^{aα}(t, n), q^{rα}(t, n),

(4.4)  q^{β}(t, n), q^{γβ}(t, n), q^{.} (t, n))

Theorem 4.2. The equations of the motion of order α are

(4.5)  \begin{aligned}E_{r}(\overline{L_{C}}) & − & A_{r}^{aα}(t, n)E_{a}^{α}(\overline{L_{C}}) = [A_{r}^{aα}(t, n^{α}) − A_{r}^{aα}(t, n)] \frac{∂L(t, n^{α})}{∂q^{aα}(t, n)}+ \\ & + & B_{rs}^{aα}(t, n)q^{sα}(t, n)\frac{∂L(t, n)}{∂q^{aα}(t, n)} + γ_{r}^{aα}(t, n)\frac{∂L(t, n)}{∂q^{aα}(t, n)} \end{aligned}

q^{aα}(t, n) + A_{r}^{aα}(t, n)q^{rα}(t, n) − γ^{aα}(t, n) = 0 , a = \overline{1, p} , r, s = \overline{p + 1, m},

where

(4.6)  B_{rs}^{aα}(t, n) = A_{r}^{bα}(t, n)\frac{∂A_{s}^{aα}(t, n)}{∂q^{b}(t, n)} − \frac{∂A_{s}^{aα}(t, n)}{∂q^{r}(t, n)},

(4.7)  γ_{r}^{aα}(t, n) = \frac{∂γ^{aα}(t, n)}{∂q^{r}(t, n)} − A_{r}^{bα}(t, n)\frac{∂γ^{aα}(t, n)}{∂q^{b}(t, n)},

(4.8)  \begin{aligned}E_{a}^{α}(\overline{L_{C}}) & = & \frac{∂L_{C}(t, n)}{∂q^{a}(t, n)} − ∑_{\begin{matrix}β = 0 \\ β ≠ α \end{matrix}}^{3} \frac{∂(L_{C}(t, n) + L_{C}(t, n^{β}))}{∂q^{aβ}(t, n)}− \\ & − & ∑_{\begin{matrix}β, γ = 0 \\ β ≠ γ \end{matrix}}^{3} \frac{∂(L_{C}(t, n) + L_{C}(t, n^{βα}))}{∂q^{aβγ}(t, n)} − \frac{d}{dt} (\frac{∂L_{C}(t, n)}{∂q^{.}^{a}(t, n)}) \end{aligned} .

Let the affine constraints of the form

(4.9)  f^{a}(t, n) = q^{aαβ}(t, n) + A_{r}^{aαβ}(t, n)q^{rαβ}(t, n) − γ^{aαβ}(t, n),

where

(4.10)  A_{r}^{aαβ}(t, n) = A_{r}^{aαβ}(q(t, u)) , γ^{aαβ}(t, n) = γ^{aαβ}(q(t, n))
a = \overline{1, p} , r = \overline{p + 1, m} , α, β ∈ \{0, 1, 2, 3\} , α ≠ β fixed

From the Lagrange-d’Alembert principle of order αβ we obtain Proposition 4.3. The motion of the order αβ is given by

(4.11)  E_{r}(\overline{L}) = A_{r}^{aαβ}(t, n)E_{a}(\overline{L}) a = \overline{1, p} , r = \overline{p + 1, m},

(4.12)  q^{aαβ}(t, n) + A_{r}^{aαβ}(t, n)q^{rαβ}(t, n) − γ^{aαβ}(t, n) = 0 , (t, n) ∈ R.

Define the constrained Lagrangian of order αβ , L_{C}, by substituting the constraints (4.12) into the Lagrangian (3.2):

(4.13)  \begin{aligned}L_{C}(t, n) & = & L(t, n, q(t, n), q^{γ}(t, n), −A_{r}^{aαβ}(t, n)q^{rαβ}(t, n)+ \\ & + & γ^{aαβ}(t, n), q^{rαβ}(t, n), q^{γδ}(t, n), q^{.} (t, n)). \end{aligned}

Theorem 4.4. The equations of the motion of order αβ are

E_{r}(\overline{L_{C}}) − A_{r}^{aαβ}(t, n)E_{a}^{αβ}(\overline{L_{C}}) = [A_{r}^{aαβ}(t, n^{αβ}) − A_{r}^{aαβ}(t, n)] \frac{∂L(t, n^{αβ})}{∂q^{aαβ}(t, n)}+

(4.13)′  +B_{rs}^{aαβ}(t, n)q^{sαβ}(t, n)\frac{∂L(t, n)}{∂q^{aαβ}(t, n)} + γ_{r}^{aαβ}(t, n)\frac{∂L(t, n)}{∂q^{aαβ}(t, n)},

q^{aαβ}(t, n) + A_{r}^{aαβ}(t, n)q^{rαβ}(t, n) − γ_{r}^{aαβ}(t, n) = 0 , a = \overline{1, p} , r, s = \overline{p + 1, m},

where

(4.14)  B_{rs}^{aαβ}(t, n) = A_{r}^{aαβ}(t, n)\frac{∂A_{s}^{aαβ}(t, n)}{∂q^{b}(t, n)} − \frac{∂A_{s}^{aαβ}(t, n)}{∂q^{r}(t, n)},

(4.15)  γ_{r}^{aαβ}(t, n) = \frac{∂γ^{aαβ}(t, n)}{∂q^{r}(t, n)} − A_{r}^{bαβ}(t, n)\frac{∂∂^{aαβ}(t, n)}{∂q^{b}(t, n)},

(4.16) \begin{aligned}E_{a}^{αβ}(\overline{L_{C}}) & = & \frac{∂L_{C}(t, n)}{∂q^{a}(t, n)} − ∑_{γ=0}^{3} \frac{∂(L_{C}(t, n) + L_{C}(t, n^{γ}))}{∂q^{aγ}(t, n)}− \\ & − & ∑_{\begin{matrix}γ, δ = 0 \\ γ ≠ δ, γ ≠ α, β \end{matrix}}^{3} \frac{∂(L_{C}(t, n) + L_{C}(t, n^{γδ}))}{∂q^{aγδ}(t, n)} − \frac{d}{dt} (\frac{∂L_{C}(t, n)}{∂q^{.}^{a}(t, n)}) . \end{aligned}

Finally let us consider affine constraints of the form

(4.17) f^{a}(t, n) = q^{.}^{a}(t, n) + A_{r}^{a}(t, n)q^{.}^{r}(t, n) − γ^{a}(t, n) a = \overline{1, p} , r = \overline{p + 1, m},

where

(4.18)  A_{r}^{a}(t, n) = A_{r}^{a}(q(t, n)) , γ^{a}(t, n) = γ^{a}(q(t, n)).

From the Lagrange-d’Alembert principle we obtain Propositin 4.5. The motion is given by

(4.19)  E_{r}(\overline{L}) = A_{r}^{a}(t, n)E_{a}(\overline{L}),

(4.20)  q^{.}^{a}(t, n) + A_{r}^{a}(t, n)q^{.}^{r}(t, n) − γ^{a}(t, n) = 0 , a = \overline{1, p} , r = \overline{r + 1, m}.

Let now the constrained Lagrangian L_{C} obtained by substituting the constraints (4.20) in the Lagrangian

(4.21)  L(t, n) = L(n, q(t, n), q^{α}(t, n), q^{αβ}(t, n), q^{.} (t, n)),

that is (4.22)

L_{C}(t, n) = L(n, q(t, n), q^{α}(t, n), q^{αβ}(t, n), −A_{r}^{a}(t, n)q^{.}^{r}(t, n) + γ^{a}(t, n), q^{.}^{r}(t, n)).

Mathematical Model for Diamond-Type Crystal 81 Theorem 4.6. The equations of the motion are

(4.23) E_{r}(\overline{L_{C}}) − A_{r}^{a}(t, n)\frac{d\overline{L_{C}}(t, n)}{dq^{s}(t, n)} = B_{rs}^{a}(t, n)\frac{∂L(t, n)}{∂q^{.}^{a}(t, n)}q^{.}^{s}(t, n) + \frac{∂L(t, n)}{∂q^{.}^{a}(t, n)}γ_{r}^{a}(t, n),

q^{.}^{a}(t, n) + A_{r}^{a}(t, n)q^{.}^{r}(t, n) − γ^{a}(t, n) = 0 , a = \overline{1, p} , r, s = \overline{p + 1, m},

where

(4.24) B_{rs}^{a}(t, n) = \frac{∂A_{r}^{a}(t, n)}{∂q^{r}(t, n)} − \frac{∂A_{s}^{a}(t, n)}{∂q^{r}(t, n)} + A_{r}^{b}(t, n)\frac{∂A_{s}^{a}(t, n)}{∂q^{b}(t, n)} − A_{s}^{b} \frac{∂A_{r}^{a}(t, n)}{∂q^{b}(t, n)},
(4.25)  γ_{r}^{a}(t, n) = \frac{∂γ^{a}(t, n)}{∂q^{r}(t, n)} − A_{r}^{b}(t, n)\frac{∂γ^{a}(t, n)}{∂q^{b}(t, n)} + γ^{b}(t, n)\frac{∂A_{r}^{a}(t, n)}{∂q^{b}(t, n)}.

Examples. 1) Let the Lagrangian

L(t, n) = \frac{1}{2}δ_{ij}q^{.}^{i}(t, n)q^{.}^{j}(t, n) − \frac{1}{4} ∑_{α=0}^{3} δ_{ij}q^{iα}(t, n)q^{jα}(t, n)−

(4.26)  −\frac{1}{4} ∑_{\begin{matrix}α, β = 0 \\ α ≠ β \end{matrix}}^{3}δ_{ij}q^{iαβ}(t, n)q^{jαβ}(t, n)

and the constraints q^{30}(t, n) − q^{2}(t, n)q^{10}(t, n) = 0.

We have

L_{C}(t, n) = \frac{1}{2}δ_{ij}q^{.}^{i}(t, n)q^{.}^{j}(t, n) − \frac{1}{4}(1 + q^{2}(t, n)^{2})q^{10}(t, n)^{2} − \frac{1}{4}q^{20}(t, n)^{2}−

−\frac{1}{4} ∑_{α=1}^{3} δ_{ij}q^{iα}(t, n)q^{jα}(t, n) − \frac{1}{4} ∑_{\begin{matrix}α, β = 0 \\ α ≠ β \end{matrix}}^{3} δ_{ij}q^{iαβ}(t, n)q^{jαβ}(t, n)

and the equations of the motion of order ,,0” are given by

q̈^{1}(t, n) + q^{2}(t, n)q̈^{3}(t, n) = (1 + \frac{1}{2}q^{2}(t, n)^{2} + \frac{1}{2}q^{2}(t, n^{0})^{2}) q^{10}(t, n)+

+ ∑_{α=1}^{3}(q^{1α}(t, n) + q^{2}(t, n)q^{3α}(t, n)) + ∑_{\begin{matrix}α, β = 0 \\ α ≠ β \end{matrix}}^{3} (q^{1αβ}(t, n) + q^{2}(t, n)q^{3αβ}(t, n)),

q̈^{2}(t, n) = ∑_{α=0}^{3} q^{2α}(t, n) + ∑_{\begin{matrix}α, β = 0 \\ α ≠ β \end{matrix}}^{3} q^{2αβ}(t, n) − \frac{1}{2}q^{2}(t, n)q^{10}(t, n)^{2},

q^{30}(t, n) = q^{2}(t, n)q^{10}(t, n) .

2) Let the Lagrangian given by (4.26) and the constraint

q^{312}(t, n) − q^{2}(t, n)q^{112}(t, n) = 0.

We have

L_{C}(t, n) = \frac{1}{2}δ_{ij}q^{.}^{i}(t, n)q^{.}^{j}(t, n) − \frac{1}{4} ∑_{α=0}^{3} δ_{ij}q^{iα}(t, n)q^{jα}(t, n)−

−\frac{1}{4}(1 + q^{2}(t, n)^{2})q^{112}(t, n)^{2} − \frac{1}{4}q^{212}(t, n)^{2} − ∑_{\begin{matrix}α, β = 0 \\ α ≠ β, α ≠ 1, β ≠ 2 \end{matrix}}^{3} δ_{ij}q^{iαβ}(t, n)q^{jαβ}(t, n)

and the equations of the motions of order ,,12” are the followings

q̈^{1}(t, n) + q^{2}(t, n)q̈^{3}(t, n) = (1 + \frac{1}{2}q^{2}(t, n)^{2} + \frac{1}{2}q^{2}(t, n^{12})^{2})q^{112}(t, n)+

+ ∑_{α=0}^{3}(q^{1α}(t, n) + q^{2}(t, n)q^{3α}(t, n)) + ∑_{\begin{matrix}α, β = 0 \\ α ≠ β, α ≠ 1, β ≠ 2 \end{matrix}}^{3} (q^{1αβ}(t, n) + q^{2}(t, n)q^{3αβ}(t, n)),

q̈^{2}(t, n) = ∑_{α=0}^{3} q^{2α}(t, n) + ∑_{\begin{matrix}α, β = 0 \\ α ≠ β, α ≠ 1, β ≠ 2 \end{matrix}}^{3} q^{2αβ}(t, n) − \frac{1}{2}q^{2}(t, n)q^{112}(t, n)^{2},

q^{312}(t, n) = q^{2}(t, n)q^{112}(t, n).

Consider the Lagrangian (4.26) and the constraint

q^{.}^{3}(t, n) − q^{2}(t, n)q^{.}^{1}(t, n) = 0.

We have

L_{C}(t, n) = \frac{1}{2}(1 + q^{2}(t, n)^{2})q^{.}^{1}(t, n)^{2} + \frac{1}{2}q^{.}^{2}(t, n)^{2} − \frac{1}{4} ∑_{α=0}^{3} δ_{ij}q^{iα}(t, n)q^{jα}(t, n)−

−\frac{1}{4} ∑_{\begin{matrix}α , β = 0 \\ α ≠ β \end{matrix}}^{3} δ_{ij}q^{iαβ}(t, n)q^{jαβ}(t, n)

and the equations of the motiom are given by

(1 + q^{2}(t, n))q̈^{1}(t, n) + q^{.}^{2}(t, n)q̈^{3}(t, n) = −2q^{2}(t, n)q^{.}^{1}(t, n)q^{.}^{2}(t, n)−

− ∑_{α=0}^{3}(q^{1α}(t, n) + q^{2}(t, n)q^{3α}(t, n)) − ∑_{\begin{matrix}α , β = 0 \\ α ≠ β \end{matrix}}^{3} (q^{1αβ}(t, n) + q^{2}(t, n)q^{3αβ}(t, n))

q̈^{2}(t, n) = q^{2}(t, n)q^{.}^{1}(t, n)^{2} + ∑_{α=0}^{3} q^{2α}(t, n) + ∑_{\begin{matrix}α , β = 0 \\ α ≠ β \end{matrix}}^{3} q^{2αβ}(t, n) .

5 Noether’s Theorem for diamond-type crystals

Let G be a Lie group acting (at the left) on Ω by (g, q) ∈ G × Ω ⟶ gq = \overline{q} , (gq)(t, n) = \overline{q}(t, n, g). Let G be the Lie algebra of G and G^{∗} the linear dual af G. To each vector ξ ∈ G corresponds an one-parameter subgroup of G, exp(εξ) , ε ∈ I ⊂ R, whose action on Ω determines

(5.1)  ξ_{Ω}(t, n) = \frac{d}{dε}[exp(εξ)q(t, n)]_{ε=0} , ∀(t, n) ∈ R.

From (5.1), we obtain

(5.2)  ξ_{Ω}^{i}(t, n) = K_{a}^{i}(t, n)ξ^{a} , i = \overline{1, m} , a = \overline{1, r} , r = dimG,

where

(5.3)  ξ = ξ^{a}e_{a} ∈ G , K_{a}^{i} = \frac{d\overline{q}^{i}(t, n, exp(εe_{a}))}{dε}|_{ε=0} .

Let \overline{q}^{p} be the canonical prolongation of the action of G on Ω × Ω^{1} × Ω^{2}× Ω^{•}. The Lie group G is called a symmetry group of the system (Ω, L), where L is autonomous in t, if

(5.4)  L ∘ \overline{q}^{p}(t, n, g) = L(t, n) , ∀(t, n) ∈ R , ∀g ∈ G.

The corresponding α-momentum map is the function J_{α} : Ω × T_{q}Ω^{1} ⟶ G^{∗}, given by

(5.5)  J_{α}(t, n) = \frac{∂L(t, n)}{∂q^{iα}(t, n)}K_{a}^{i}(t, n^{α})e^{a} , α ∈ \{0, 1, 2, 3\} , fixed.

The corresponding αβ-momentum map is the function J_{αβ} : Ω × T_{q}Ω^{2} ⟶ G^{∗},

(5.6)  J_{αβ}(t, n) = \frac{∂L(t, n)}{∂q^{iαβ}(t, n)}K_{a}^{i}(t, n^{αβ})e^{a} , α, β ∈ \{0, 1, 2, 3\} , α ≠ β , fixed.

The corresponding continuous momentum map is J : Ω × T_{q} Ω^{•}⟶ G^{∗},

(5.7)  J (t, n) = \frac{∂L(t, n)}{∂q^{.}^{i}(t, n)}K_{a}^{i}(t, n)e^{a}.

Theorem 5.1. (Noether’s Theorem for diamond-type crystals). For each solution of the Euler-Lagrange equations (3.7),

(5.8)  ∑_{α=0}^{3} J_{α}^{α}(t, n) + ∑_{\begin{matrix}α , β = 0 \\ α ≠ β \end{matrix}}^{3} J_{αβ}^{αβ}(t, n) + \frac{dJ (t, n)}{dt} = 0,

J_{α}^{α}(t, n) = J_{α}(t, n^{α}) − J_{α}(t, n) , J_{αβ}^{αβ}(t, n) = J_{αβ}(t, n^{αβ}) − J_{αβ}(t, n).

Suppose that the Lagrangian L does not depends on q^{j} , j fixed. Locally the system (Ω,L) admits a symmetry group G_{j}. The action on Ω is given by

(5.9)  \overline{q}^{i}(t, n) = q^{i}(t, n) , \overline{q}^{j}(t, n) = q^{j}(t, n) + α^{j} , i ≠ j, α^{j} ∈ R.

We have K_{i}^{j}(t, n) = δ_{j}^{i} and from (5.6) we obtain

(5.10) ∑_{α=0}^{3} \frac{∂(L(t, n) + L(t, n^{α}))}{∂q^{jα}(t, n)} + ∑_{\begin{matrix}α , β = 0 \\ α ≠ β \end{matrix}}^{3} \frac{∂(L(t, n) + L(t, n^{αβ}))}{∂q^{jαβ}(t, n)} − \frac{d}{dt} (\frac{∂L(t, n)}{∂q^{.}^{j}(t, n)}) = 0.

The coordinate q^{i} is called a cyclic coordinate.

6 Momentum equation for diamond-type crystals with impurities

In this sections we shall use the Lagrange-d’Alembert principle to derive an equation for a generalized momentum as a consequence of the symmetries. We assume that the action of G on Ω is free and proper. The orbit through a point q ∈ Ω is denoted by Orb(q) = {gq|g ∈ G}. Let S ⊂ Ω × Ω^{1} × Ω^{2}× Ω^{•} and T_{q}S the virtual variation. If S_{q} = T_{q}S ∩ T_{q}(orb(\overline{q}^{p})) ≠ \{0\}, where \overline{q}^{p} is the canonical prolongation of the action of

G on Ω × Ω^{1} × Ω^{2}× Ω^{•} then let G(q) = {ξ ∈ G|ξ_{Ω}(q) ∈ S_{q}}.

The α-nonholomic momentum map J_{α} is defined by

(6.1)  J_{α}(t, n) = \frac{∂L(t, n)}{∂q^{iα}(t, n)}ξ^{i}(t, n^{α}), (t, n) ∈ R,

where

ξ^{i}(t, n^{α}) = K_{a}^{i}(t, n^{α})ξ^{a}(t, n^{α}) , ξ(t, n^{α}) = ξ(q(t, n^{α})), α ∈ {0, 1, 2, 3} , α fixed.

The αβ-nonholonomic momentum map J_{αβ} is defined by

(6.2)  J_{αβ}(t, n) = \frac{∂L(t, n)}{∂q^{iαβ}(t, n)}ξ^{i}(t, n^{αβ}) , (t, n) ∈ R,

where

ξ^{i}(t, n^{αβ}) = K_{a}^{i}(t, n^{αβ})ξ^{a}(t, n^{αβ}), α , β ∈ {0, 1, 2, 3} , α ≠ β , fixed.

The continuous nonholonomic momentum map J is defined by

(6.3)  J (t, n) = \frac{∂L(t, n)}{∂q^{.}^{i}(t, n)}ξ^{i}(t, n)  , (t, n) ∈ R,

ξ^{i}(t, n) = K_{a}^{i}(t, n)ξ^{a}(t, n).

Theorem 6.1. Assume that the Lagrangian L is invariant under the group action and ξ(q) ∈ G(q). Then any solution of the Lagrange-d’Alembert equation for a S satisfies the following momentum equation

∑_{α=0}^{3} J_{α}^{α}(t, n) + ∑_{α,β=0}^{3} J_{αβ}^{αβ}(t, n) + \frac{dJ (t, n)}{dt} = ∑_{α=0}^{3} \frac{∂L(t, n)}{∂q^{iα}(t, n)}K_{a}^{i}(t, n)ξ^{aα}(t, n)+

(6.4)  + ∑_{\begin{matrix}α, β = 0 \\ α ≠ β \end{matrix}}^{3} \frac{∂L(t, n)}{∂q^{iαβ}(t, n)}K_{a}^{i}(t, n)ξ^{aαβ}(t, n) + \frac{∂L(t, n)}{∂q^{.}(t, n)}K_{a}^{i}(t, n)\frac{dξ^{a}(t, n)}{dt}.

At a fixed point q_{0} ∈ Ω, we consider a basis {e_{1}, … e_{p}, e_{p+1}, … e_{r}} of G such that the first p elements form a basis of G(q_{0}). Thus r = dimG , p = dimG(q_{0}), which, by assumption, is locally constant. We can introduce a similar basis {e_{1}(q), … e_{p}(q), e_{p+1}(q), …, e_{r}(q)} for q ∈ Ω. Let a change of basis matrix

(6.5) e_{u}(q(t, n)) = e_{u}(t, n) = Ψ_{u}^{v}(q(t, n))e_{v}(q_{0}(t, n)) = Ψ_{u}^{v}(t, n)e_{v} , u, v = \overline{1, r}.

Here the matrix (Ψ_{a}^{b}) is an r × r invertible matrix. By the definitions (5.5), (5.6), (5.7) we can write

J_{αa}(t, n) = \frac{∂L(t, n)}{∂q^{iα}(t, n)}[e_{a}(t, n^{α})]_{Ω}^{i},

(6.6)  J_{αβa}(t, n) = \frac{∂L(t, n)}{∂q^{iαβ}(t, n)}[e_{a}(t, n^{αβ})]_{Ω}^{i},

J_{a}(t, n) = \frac{∂L(t, n)}{∂q^{.}^{i}(t, n)}[e_{a}(t, n)]_{Ω}^{i} , a = \overline{1, p} , (t, n) ∈ R,

where

[e_{a}(t, n^{α})]_{Ω}^{i} = K_{a}^{i}(t, n^{α}), [e_{a}(t, n^{αβ})]_{Ω}^{i} = K_{a}^{i}(t, n^{αβ}) , [e_{a}(t, n)]_{Ω}^{i} = K_{a}^{i}(t, n).

Theorem 6.2. The momentum equation in a moving basis \{e_{n}(t, n)\}_{u=\overline{1,r}} is given by

∑_{α=0}^{3} J_{αa}^{α}(t, n) + ∑_{\begin{matrix}α, β = 0 \\ α ≠ β \end{matrix}}^{3}J_{αβa}^{αβ} + \frac{dJ_{a}(t, n)}{dt} = ∑_{α=0}^{3} Λ_{a}^{b}(t, n, n^{α})J_{αb}(t, n)+

+ ∑_{\begin{matrix}α, β = 0 \\ α ≠ β \end{matrix}}^{3}Θ_{a}^{b}(t, n, n^{α})J_{αβb}(t, n)+Γ_{a}^{b}(t, n)J_{b}(t, n)+∑_{α=0}^{3} \frac{∂L(t, n)}{∂q^{iα}(t, n)}Λ_{b}^{s}(t, n, n^{α})[e_{s}(t, n^{α})]_{Ω}^{i}+

+ ∑_{\begin{matrix}α, β = 0 \\ α ≠ β \end{matrix}}^{3} \frac{∂L(t, n)}{∂q^{iαβ}(t, n)}Θ_{b}^{s}(t, n, n^{αβ})[e_{s}(t, n^{αβ})]_{Ω}^{i}+

86 D.Opriş and I.D.Albu

(6.7)  +\frac{∂L(t, n)}{∂q^{.}^{i}(t, n)}Γ_{bj}^{s}(t, n)q^{.}^{j}(t, n)[e_{s}(t, n)]_{Ω}^{i} a = \overline{1, p} s = \overline{p + 1, r},

where (6.8)

Λ_{a}^{v}(t, n, n^{α}) = Ψ_{a}^{αu}(t, n)Ψ^{∼}_{u}^{v}(t, n^{α}),

Θ_{a}^{v}(t, n, n^{α}) = Ψ_{a}^{αβu}(t, n)Ψ^{∼}_{u}^{v}(t, n^{αβ}),

Γ_{aj}^{v}(t, n) = \frac{∂Ψ_{a}^{u}(t, n)}{∂q^{j}(t, n)}Ψ^{∼}_{u}^{v}(t, n) , a = \overline{1, p} , v = \overline{1, r}.

Examples. 1) Let the Lagrangian

(6.9)  \begin{aligned}L(t, n) & = & \frac{1}{2}δ_{ij}q^{.}^{i}(t, n)q^{.}^{j}(t, n) − \frac{1}{4} ∑_{α=0}^{3} δ_{ij}q^{iα}(t, n)q^{jα}(t, n)− \\ & − & \frac{1}{4} ∑_{\begin{matrix}α, β = 0 \\ α ≠ β \end{matrix}}^{3} δ_{ij}q^{iαβ}(t, n)q^{jαβ}(t, n) \end{aligned}

and the constraint

(6.10)  q^{30}(t, n) − q^{2}(t, n)q^{10}(t, n) = 0.

The constraint and the Lagrangian are invariants under the R^{2}-action on R^{3} given by

(q^{1}, q^{2}, q^{3}) ⟶ (q^{1} + λ, q^{2}, q^{3} + μ).

The tangent spaces to the orbits of this action are given by

T_{q(t,n)}(Orb(q(t, n))) = span{(1, 0, 0), (0, 0, 1)}

and the virtual vectors of the constraints are given by

S_{q(t,n)} = span{(1, 0, q^{2}(t, n)), (0, 1, 0)}.

It follows

T_{q(t,n)}(Orb(q(t, n))) ∩ S_{q(t,n)} = span{1, 0, q^{2}(t, n)}

and

ξ_{Ω}^{q(t,n)} = (1, 0, q^{2}(t, n)) , ξ^{q(t,n)} = (1, q^{2}(t, n)).

The nonholonomic momentum maps in this case are J_{α}(t, n) = −\frac{1}{2}q^{1α}(t, n) − \frac{1}{2}q^{3α}(t, n)q^{2}(t, n^{α}), J_{αβ}(t, n) = −\frac{1}{2}q^{1αβ}(t, n) − \frac{1}{2}q^{3αβ}(t, n)q^{2}(t, n^{αβ}),

J (t, n) = q^{.}^{1}(t, n) + q^{.}^{3}(t, n)q^{2}(t, n).

The momentum equation is given by Mathematical Model for Diamond-Type Crystal 87

∑_{α=0}^{3} q^{1α}(t, n) + ∑_{\begin{matrix}α, β = 0 \\ α ≠ β \end{matrix}}^{3} (q^{1αβ}(t, n) + q^{3αβ}(t, n)q^{2}(t, n^{αβ})) + ∑_{α=1}^{3} q^{3α}(t, n)q^{2}(t, n^{α})+

+q^{2}(t, n)q^{2}(t, n^{0})q^{10}(t, n) + q̈^{1}(t, n) + q̈^{3}(t, n)q^{2}(t, n) = 0.

2) Let the Lagrangian (6.9) and the constraint

q^{312}(t, n) − q^{2}(t, n)q^{112}(t, n) = 0.

The momentum equation is given by

∑_{α=0}^{3} q^{1α}(t, n) + ∑_{\begin{matrix}α, β = 0 \\ α ≠ β, α ≠ 1, β ≠ 2 \end{matrix}}^{3} q^{1αβ}(t, n) + ∑_{α=0}^{3} q^{3α}(t, n)q^{2}(t, n^{α}) + ∑_{\begin{matrix}α, β = 0 \\ α ≠ β, α ≠ 1, β ≠ 2 \end{matrix}}^{3} q^{3αβ}(t, n)q^{2}(t, n^{αβ})+

+(1 + q^{2}(t, n)q^{2}(t, n^{12}))q^{112}(t, n) + q̈^{1}(t, n) + q̈^{3}(t, n)q^{2}(t, n) = 0.

3) Let the Lagrangian (6.9) and the constraint

q^{.}^{3}(t, n) − q^{2}(t, n)q^{.}^{1}(t, n) = 0.

The momentum equation is given by

∑_{α=0}^{3}(q^{1α}(t, n) + q^{3α}(t, n)q^{2}(t, n^{α})) + ∑_{\begin{matrix}α, β = 0 \\ α ≠ β \end{matrix}}^{3} (q^{1αβ}(t, n) + q^{3αβ}(t, n)q^{2}(t, n^{αβ}))+

+(1 + q^{2}(t, n)^{2})q̈^{1}(t, n) + q^{2}(t, n)q^{.}^{2}(t, n)q^{.}^{1}(t, n) = 0.

4) For the same Lagrangian and the constraints

q^{.}^{3}(t, n) − q^{2}(t, n)q^{.}^{1}(t, n) = 0,

q^{30}(t, n) − q^{2}(t, n)q^{10}(t, n) = 0, q^{312}(t, n) − q^{2}(t, n)q^{312}(t, n) = 0,

the momentum equation is

(1 + q^{2}(t, n)^{2})q̈^{1}(t, n) + q^{2}(t, n)q^{.}^{1}(t, n)q^{.}^{2}(t, n) + (1 + q^{2}(t, n^{0})^{2})q^{2}(t, n)q^{10}(t, n)+

+(1+q^{2}(t, n^{12}))q^{112}(t, n)+∑_{α=0}^{3}(q^{1α}(t, n)+q^{3α}q^{2}(t, n^{α}))+ ∑_{\begin{matrix}α, β = 0 \\ α ≠ β, α ≠ 1, β ≠ 2 \end{matrix}}^{3} (q^{1αβ}+q^{3αβ}q^{2}(t, n^{αβ}))=0.

Acknowledgements. A version of this paper was presented at the First Conference of Balkan Society of Geometers, Politehnica University of Bucharest, September 23-27, 1996.

References

[1] A.M.Bloch, P.S.Krishnaprasad, J.E.Marsden, R.M.Murray, Nonholonomic mechanical system with symmetry, Archive for Rational Mechanics and Analysis, C.D.S. Tehnical Report 90-013 California Institute of Technology, (1995).

[2] N.Cotfas, A mathematical model for diamond-type crystals, J. Phys. A. Math. Gen 28(1995), 1371-1379.

[3] D.Crăciun and D.Opriş, Discrete-continuous dynamical systems with symmetry, Balkan Journal of Geometry and Its Applications, 2,2 (1997), 35–49.

[4] D.Opriş and I. Albu, On the discrete-continuous dynamical systems, Analalele Universităţii ,,Al.I.Cuza”, Iaşi (1995).

[5] C.Patel, W.F.Sherman and G.R.Wilkinsen, J. Phis. C. Solid State Phis. 17(1984), 6063-69.

[6] A.Trzesowski, Gauge theory of dislocations, International Journal of Theoretical Physics vol. 26, 11(1987).

West University of Timişoara Department of Mathematics B-dul V.Pârvan 4 1900 Timişoara, Romania