Theoretical Modeling of Laser-Stimulated Nanostructures
283
Let us introduce the following notation: x 0 − x ≡ X ; y 0 − y ≡ Y ; z 0 − z ≡ Z .
To obtain the relationship between the coordinates of the two systems, we use both
of the above definitions of the vector ρ. Substituting into the expression (11), the
values e x , e y , e z from (7) and comparing that is obtained with (10), one can get
⎧
⎪ ⎨
⎪ ⎩
ξ 0 =
1
T
X +
S x
T
Z ,
η 0 = −
S x S y
N T
X +
T
N
Y +
S y
N T
Z ,
ζ 0 = −
S x
N
X −
S y
N
Y +
1
N T
Z .
(12)
Now we return to the relations (8). In it, the components v
(ξ )
, v
(η)
, v
(ζ ) depend on
ξ 0 , η 0 , ζ 0 . We can replace in these components the variables ξ 0 , η 0 , ζ 0 in accordance
with (12). Then the components of the laboratory system v = v
(i) e i will depend both
on the coordinates of the observation point r 0 and on the coordinates of some local
system r. Since there are many such local systems on the surface, then in order to
find the resulting value of the velocity vector at the observation point r 0 at a certain
point in time in the laboratory reference frame, it is necessary to carry out continuous
summation over all surface points (i.e., over the variable r), or rather integration over
the entire surface with a weight factor equal to the area of this surface at the same
moment of time.
The obtained relations allow us to go to the laboratory reference system, since in it
all the processes associated with destructive processing can be observed. Theoretical
calculations are more convenient to do in the local reference system.
We also note that assuming r = r 0 , and accordingly ρ = 0, we can obtain all
the above relations for the velocities at the condensate-gas interface. Relation (12)
in this case turns into identities of the form 0 = 0.
4 Crater Dynamics Equations
To formulate the equation for the crater form function in the laboratory coordinate
system, we use the mass flow balance condition (1). This condition in the local
reference system can be written as
ρ s v
(ζ )
s = ρ 0s v
(ζ )
0s .
(13)
The surface values of the velocity v
(ζ )
s and the density of the gas phase are determined [13] by the relations: v
(ζ )
s = −
P
σ
s
σ
√
κa
and ρ s = a P
η , based on the conditions
of the balance of the flow of energy (3) and the conditions of the balance of the flow
of pulses (2). In these definitions, P s is the surface pressure at the phase boundary, κ
is the polytropic exponent, which varies from 1 to 5/3, η ≡ 1/κ, σ ≡ (κ − 1)/2κ.
In addition, P s depends on time t and x, y coordinates, if we consider them in a
laboratory frame of reference. That is, the left side of (13) is already formulated in
the variables of the laboratory system. As for the right-hand side of (13), it should
283
Let us introduce the following notation: x 0 − x ≡ X ; y 0 − y ≡ Y ; z 0 − z ≡ Z .
To obtain the relationship between the coordinates of the two systems, we use both
of the above definitions of the vector ρ. Substituting into the expression (11), the
values e x , e y , e z from (7) and comparing that is obtained with (10), one can get
⎧
⎪ ⎨
⎪ ⎩
ξ 0 =
1
T
X +
S x
T
Z ,
η 0 = −
S x S y
N T
X +
T
N
Y +
S y
N T
Z ,
ζ 0 = −
S x
N
X −
S y
N
Y +
1
N T
Z .
(12)
Now we return to the relations (8). In it, the components v
(ξ )
, v
(η)
, v
(ζ ) depend on
ξ 0 , η 0 , ζ 0 . We can replace in these components the variables ξ 0 , η 0 , ζ 0 in accordance
with (12). Then the components of the laboratory system v = v
(i) e i will depend both
on the coordinates of the observation point r 0 and on the coordinates of some local
system r. Since there are many such local systems on the surface, then in order to
find the resulting value of the velocity vector at the observation point r 0 at a certain
point in time in the laboratory reference frame, it is necessary to carry out continuous
summation over all surface points (i.e., over the variable r), or rather integration over
the entire surface with a weight factor equal to the area of this surface at the same
moment of time.
The obtained relations allow us to go to the laboratory reference system, since in it
all the processes associated with destructive processing can be observed. Theoretical
calculations are more convenient to do in the local reference system.
We also note that assuming r = r 0 , and accordingly ρ = 0, we can obtain all
the above relations for the velocities at the condensate-gas interface. Relation (12)
in this case turns into identities of the form 0 = 0.
4 Crater Dynamics Equations
To formulate the equation for the crater form function in the laboratory coordinate
system, we use the mass flow balance condition (1). This condition in the local
reference system can be written as
ρ s v
(ζ )
s = ρ 0s v
(ζ )
0s .
(13)
The surface values of the velocity v
(ζ )
s and the density of the gas phase are determined [13] by the relations: v
(ζ )
s = −
P
σ
s
σ
√
κa
and ρ s = a P
η , based on the conditions
of the balance of the flow of energy (3) and the conditions of the balance of the flow
of pulses (2). In these definitions, P s is the surface pressure at the phase boundary, κ
is the polytropic exponent, which varies from 1 to 5/3, η ≡ 1/κ, σ ≡ (κ − 1)/2κ.
In addition, P s depends on time t and x, y coordinates, if we consider them in a
laboratory frame of reference. That is, the left side of (13) is already formulated in
the variables of the laboratory system. As for the right-hand side of (13), it should