Diffusion¶

Fick couple and single-sided exponential depth laws. The figures use Mix volume fraction (phi); the same Diffusion field can instead drive free optical channels (delta_o, …) on a single material. Does not cover spline / polynomial grades (Functional forms) or second-order transitions (Phase transitions).

Builder: case_diffusion

Shared preamble:

from refloxide.model import MaterialSLD, FreeUniTensor, Mix, ReflectModel
from refloxide.profiles import (
    Diffusion,
    SecondOrderTransition,
    DepthProfile,
    Spline,
    Polynomial,
    CallableField,
    CallableDepthProfile,
)

vac = MaterialSLD("", 0, name="vacuum")(0, 0)
si = MaterialSLD("Si", 2.33, name="si")(0, 3.0)
mat = FreeUniTensor(energies=[284.4], name="film")
mat_a = FreeUniTensor(energies=[284.4], name="a")
mat_b = FreeUniTensor(energies=[284.4], name="b")
mat_c = FreeUniTensor(energies=[284.4], name="c")

Couple vs exponential¶

Annotated: left/edge, right/base, couple \(\pm\)length, exp length. Depth figure overlays kind="couple" vs kind="exponential" with shared \(n_o\)/\(n_e\) colors and linestyle per kind (sharp | broadened columns; rows: optical constants, birefringence, \(\phi\)). Reflectivity uses the same linestyles for \(R_{ss}\)/\(R_{pp}\) plus \(\Delta\log_{10} R\) vs couple.

phi = Diffusion(thickness=160.0, left=1.0, right=0.0, length=20.0, kind="couple")
# or: Diffusion(..., kind="exponential", edge=1.0, base=0.0, length=20.0)
layer = DepthProfile(
    material=Mix([mat_a, mat_b], rule="linear"),
    thickness=160.0,
    roughness=1.0,
    phi=phi,
)
stack = vac | layer | si

couple is the erf Fick couple; exponential is single-sided base+(edge-base)exp(-z/length) (finite reserved):

diffusion depth

diffusion reflectivity

Stacking¶

Builder: case_diffusion_short_stack

Three Mix grades under |: surface oxidation (exponential from the vacuum edge), an interdiffusion couple between materials, then another exponential from that new surface into the bulk. \(\phi\) is continuous at the internal interfaces.

oxide = DepthProfile(
    material=Mix([mat_a, mat_b], rule="linear"),
    thickness=80.0,
    phi=Diffusion(
        thickness=80.0, kind="exponential", edge=1.0, base=0.05, length=18.0
    ),
)
couple = DepthProfile(
    material=Mix([mat_a, mat_b], rule="linear"),
    thickness=160.0,
    phi=Diffusion(
        thickness=160.0, kind="couple", left=0.05, right=0.95, length=22.0
    ),
)
into_bulk = DepthProfile(
    material=Mix([mat_a, mat_b], rule="linear"),
    thickness=120.0,
    phi=Diffusion(
        thickness=120.0, kind="exponential", edge=0.95, base=0.10, length=35.0
    ),
)
stack = vac | oxide | couple | into_bulk | si
model = ReflectModel(stack, energies=[284.4], parallel=False)

diffusion stack depth

diffusion stack reflectivity