Local Defects in the Thomas-Fermi-von
Weiszäcker Theory of Crystals.
Faizan Nazar
working with Christoph Ortner
Many Body Quantum Systems
21st March 2014
Motivation - Defects in Lattices
Image by Kai Nordlund
Perfect and Defective Crystals
3
∞
Let
R Λ = Z and let η ∈ Cc (B1/4 (0)) be non-negative and satisfy
η = 1.
Perfect and Defective Crystals
3
∞
Let
R Λ = Z and let η ∈ Cc (B1/4 (0)) be non-negative and satisfy
η = 1.
I
Periodic nuclear density
m0 (x) =
X
η(x − l).
l∈Λ
I
Defective nuclear density
X
m1 (x) =
η(x − l − UΛ (l)) + ρnuc
def (x).
l∈Λ
Perfect and Defective Crystals
3
∞
Let
R Λ = Z and let η ∈ Cc (B1/4 (0)) be non-negative and satisfy
η = 1.
I
Periodic nuclear density
m0 (x) =
X
η(x − l).
l∈Λ
I
Defective nuclear density
X
m1 (x) =
η(x − l − UΛ (l)) + ρnuc
def (x).
l∈Λ
∞
1. Nuclear defect - ρnuc
def ∈ Cc (BR (0)),
2. Nuclear displacement - UΛ : Λ → R3 , satisfying UΛ ∈ W 1,2 (Λ):
k∇UΛ k`2 (Λ) =
3
XX
l∈Λ i=1
!1/2
2
|UΛ (l + ei ) − UΛ (l)|
< ∞.
The TFW Energy
The Thomas-Fermi-von Weiszäcker energy functional is defined as
Z
Z
Z
1
TFW
2
10/3
φ(m − u 2 ),
E
(u, m) =
|∇u| +
u
+
2
3
3
R
R
where u ≥ 0 and
−∆φ = 4π(m − u 2 ).
The TFW Energy
The Thomas-Fermi-von Weiszäcker energy functional is defined as
Z
Z
Z
1
TFW
2
10/3
φ(m − u 2 ),
E
(u, m) =
|∇u| +
u
+
2
3
3
R
R
where u ≥ 0 and
−∆φ = 4π(m − u 2 ).
The minimiser (u, φ) is unique and satisfies
5
− ∆u + u 7/3 − φu = 0,
3
− ∆φ = 4π(m − u 2 ).
Existence and uniqueness is shown in [C/LB/L].
The minimisers of the energies are given by:
Periodic
(u0 , φ0 )
7/3
−∆u0 + 53 u0 − φ0 u0 = 0,
−∆φ0 = 4π(m0 − u02 ),
Defective
(u1 , φ1 )
7/3
−∆u1 + 35 u1 − φ1 u1 = 0,
−∆φ1 = 4π(m1 − u12 ).
The minimisers of the energies are given by:
Periodic
(u0 , φ0 )
7/3
−∆u0 + 53 u0 − φ0 u0 = 0,
−∆φ0 = 4π(m0 − u02 ),
Defective
(u1 , φ1 )
7/3
−∆u1 + 35 u1 − φ1 u1 = 0,
−∆φ1 = 4π(m1 − u12 ).
Additionally, for i = 0, 1 mi is smooth, hence ui , φi are also
smooth.
The minimisers of the energies are given by:
Periodic
(u0 , φ0 )
7/3
−∆u0 + 53 u0 − φ0 u0 = 0,
−∆φ0 = 4π(m0 − u02 ),
Defective
(u1 , φ1 )
7/3
−∆u1 + 35 u1 − φ1 u1 = 0,
−∆φ1 = 4π(m1 − u12 ).
Additionally, for i = 0, 1 mi is smooth, hence ui , φi are also
smooth.
We apply a change of variables argument to compare (u0 , φ0 ) with
(u1 , φ1 ).
Residual Estimates
First, we interpolate Id + UΛ from Λ to R3 . This gives a bijective
deformation field Y ∈ C ∞ (R3 , R3 ) and J = det(∇Y ).
Now define the predictors
ue = u0 ◦ Y −1 J −1/2 ,
φe = φ0 ◦ Y −1 .
Residual Estimates
First, we interpolate Id + UΛ from Λ to R3 . This gives a bijective
deformation field Y ∈ C ∞ (R3 , R3 ) and J = det(∇Y ).
Now define the predictors
ue = u0 ◦ Y −1 J −1/2 ,
φe = φ0 ◦ Y −1 .
The residual terms u1 − ue, φ1 − φe satisfy
5 7/3
eu = g1 ,
−∆(u1 − ue) + (u1 − ue7/3 ) − φ1 u1 + φe
3
e = 4π(e
−∆(φ1 − φ)
u 2 − u12 ) + g2 + 4πρnuc
def ,
where g1 , g2 satisfy for each k ∈ N
kg1 kH k (R3 ) + kg2 kH k (R3 ) ≤ Ck k∇UΛ k`2 (Λ) .
Main Estimate
By adapting the proof of uniqueness from [C/LB/L], we obtain
Theorem
For each k ∈ N, for all ξ ∈ Cc∞ (R3 )
X Z e 2 ξ2
|∂ α (u1 − ue)|2 + |∂ α (φ1 − φ)|
|α|≤k
≤ Ck
X Z
2
2
|∂ α g1 |2 + |∂ α g2 |2 + |∂ α ρnuc
def | ξ
|α|≤k
+ Ck
Z e 2 |∇ξ|2 .
(u1 − ue)2 + (φ1 − φ)
Additional Estimates
I
Sending ξ → 1 gives
e H k (R3 ) ≤ Ck k∇UΛ k`2 (Λ) + kρnuc kH k (R3 ) .
ku1 − uekH k (R3 ) + kφ − φk
def
Additional Estimates
I
Sending ξ → 1 gives
e H k (R3 ) ≤ Ck k∇UΛ k`2 (Λ) + kρnuc kH k (R3 ) .
ku1 − uekH k (R3 ) + kφ − φk
def
I
Suppose that |∇UΛ (l)| ≤ C (1 + |l|)−d , then
−d
e
|(u1 − ue)(x)| + |(φ1 − φ)(x)|
≤ C (1 + kρnuc
.
def kL2 (R3 ) )(1 + |x|)
Additional Estimates
I
Sending ξ → 1 gives
e H k (R3 ) ≤ Ck k∇UΛ k`2 (Λ) + kρnuc kH k (R3 ) .
ku1 − uekH k (R3 ) + kφ − φk
def
I
Suppose that |∇UΛ (l)| ≤ C (1 + |l|)−d , then
−d
e
|(u1 − ue)(x)| + |(φ1 − φ)(x)|
≤ C (1 + kρnuc
.
def kL2 (R3 ) )(1 + |x|)
I
Alternatively, suppose UΛ (l) = 0 for |l| > R, then there exists γ > 0
such that
−γ|x|
e
|(u1 − ue)(x)| + |(φ1 − φ)(x)|
≤ CR k∇UΛ k`2 (Λ) + kρnuc
.
def kL2 (R3 ) e
Consider the two energy differences
E 0 (VΛ ) = E TFW (uV , mV ) − E TFW (u0 , m0 ),
TFW
E def (VΛ ) = E TFW (uV , mV + ρnuc
(u0 , m0 ).
def ) − E
They are well-defined for any VΛ ∈ W 1,2 (Λ)
|E 0 (VΛ )| ≤ C k∇VΛ k`2 (Λ) ,
|E def (VΛ )| ≤ C (k∇VΛ k`2 (Λ) + kρnuc
def kL2 (R3 ) ).
Similarly, we can also estimate the forcing and Hessian for both E 0 ,
and E def .
Outlook
We wish to show the existence of a local minimiser UΛ of E def and
prove decay estimates for UΛ .
Outlook
We wish to show the existence of a local minimiser UΛ of E def and
prove decay estimates for UΛ .
Thanks for your attention!