Structural properties of Amorphous Materials
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S(k)
r θ
r
2Z R C
i =
√
−1
j z = x + iy x, y ∈ R
[z] = x [z] = y z
∗
= x −iy
z z = re
iθ
r, θ ∈ R r = |z| z
θ = arg[z] z
r r ∈ R
3
x = max {n ∈ Z; n ≤ x} x = min {n ∈ Z; n ≥ x}
[1 . . . n) ≡ {1 . . . n −1}
f ⊂ [0 . . . ∞)
f(x) = 0 ∀x < 0
Φ
(x, y) ≡
∂
2
∂x
2
Φ(x, y)
Φ
(1, y) ≡
∂Φ
∂x
x=1
f|g =
∞
−∞
f
∗
(x)g(x)
x
g, f ≡ f|g
ˆ
f = (2π)
−
1
2
e
iωt
|f
f = (2π)
−
1
2
e
−iωt
|
ˆ
f
sin (ωt) |f
f(t) g(t) f ∗ g
[f ∗ g](τ) =
∞
−∞
f(t) g(τ − t)
t
2πiνt iωt
f(t) g(t) f ⊗g
[f ⊗ g](τ) =
∞
−∞
f(t) g(t + τ)
t
L
p
f
L
p
f
L
p
=
∞
−∞
|f(x)|
p
dx
1
p
L
2
||
ˆ
f||
2
= ||f||
2
∀f ∈
L
2
(R) L
2
(R) f : R → C f
2
∈ R
f v
1
. . . v
n
f
v
1
, ,v
n
f A B f : A → B
f : R × R → C
f : (R × R) → C
ψ
σ
(t; a, b)
t a b σ ψ t a
b σ
σ
2
f
f f
2
= 1
σ
2
f
=
∞
−∞
t
2
f(t)
t −
∞
−∞
tf(t)
t
2
σ
f
[z] : C → C
[z] =
2
√
π
z
0
e
−t
2
t
Γ Γ[z] : C → C
Γ[z] =
∞
0
t
z−1
e
−t
t
x→∞
|f(x)/g(x)| ≤ C C ∈ R
f(−t) = f(t)
e
iωt−
1
2
t
2
C
p
α
f(−t) = −f(t)
1, x, x
2
, x
3
, . . . f(x)
f n
x
i
|f
= 0 ∀i ∈ {0 . . . n −1}
≡ 10
−10
A
f
(t
0
) f(t) t = t
0
f(t)
ˆ
f(ω) f(t)
f
Ψ
(a, b)
f(t) Ψ
g(r)
g(r)
g
n
(r) n
g
αβ
(r) α β
lim
r→∞
g
α,β
(r) = 1
F (k)
S(k) S(k)
S(k; r) S(k) r
S(k; r, θ) S(k) r θ k r
ω
f
(t
0
) f(t) t = t
0
φ
Ψ(t)
ψ(t; a, b) a b
ρ(r) ρ(r)
σ
σ
Ψ
|Ψ|
2
Σ(k; r) S(k; r) r = 0
θ
ϑ
−2
−2r
−1
r = 2 ρ = 16
ρ = 4 r
−12
− 2r
−6
r = 2 ρ = 4
6
ρ(r) r
i
r
j
r δ
ρ(r) =
i
j=i
δ(r − r
i
)δ(r − r
j
)
. . .
g(r) ρ(r)
g(r) =
1
4πr
2
ρ
0
ρ(r)
ρ
0
N
ρ(r)
S(k) =
2π
N
ˆρ(k)ˆρ(−k)
=
1
N
N
i=1
N
j=1
e
−ik·r
i
e
ik·r
j
g(r)
S(k)
S(k) =
1
N
N
i=1
N
j=1
e
−ik·r
i
e
ik·r
j
= 1 +
1
N
i
j=i
e
−ik·(r
i
−r
j
)
= 1 +
1
N
i
j=i
R
3
R
3
e
−ik·(r−r
)
δ(r − r
i
)δ(r
− r
j
)
r
r
= 1 +
1
N
R
3
R
3
e
−ik·(r−r
)
ρ(r
− r)
r
r
= 1 + ρ
0
R
3
e
−ik·r
g(r)
r
= 1 + (2π)
3
ρ
0
δ(k) + ρ
0
R
3
e
−ik·r
(g(r) − 1)
r
δ
S(k) = 1 + ρ
0
R
3
e
−ik·r
(g(r) − 1) r
(S(k) −1) ρ
0
(g(r) −1) g(r) S(k)
g(r) =
1
(2π)
3
ρ
0
R
3
e
ik·r
(S(k) −1)
r
g(r) S(k)
r ≡ |r| k ≡ |k| g(r) S(k)
S(k) g(r)
S(k) g(r)
S(k) = 1 + ρ
0
2π
0
π
0
∞
0
(g(r) −1)e
ikr cos θ
r
2
sin θ
r θ φ
= 1 + ρ
0
∞
0
4πr
2
(g(r) −1)
sin(kr)
kr
r
= 1 +
4πρ
0
k
∞
0
r(g(r) − 1) sin(kr)
r
d(r)
d(r) = 4πρ
0
r(g(r) − 1)
F (k)
F (k) = k(S(k) − 1)
F (k) =
∞
0
sin(kr)d(r)
r
d(r) =
2
π
∞
0
sin(kr)F (k)
k
r → ∞
≡ 10
−10
−1
g(r) S(k) r k
2π
2π
2π/h
i(k) i =
√
−1
