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fix ttm doc formatting fixes

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Axel Kohlmeyer
2024-03-29 02:50:30 -04:00
parent a13d53dd65
commit 996f5b9d52
1 changed files with 38 additions and 37 deletions
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75
doc/src/fix_ttm.rst
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@ -136,23 +136,23 @@ transfer between the subsystems:
\bigtriangledown (\kappa_e \bigtriangledown T_e) -
g_p (T_e - T_a) + g_s T_a'
where C_e is the specific heat, rho_e is the density, kappa_e is the
thermal conductivity, T is temperature, the "e" and "a" subscripts
represent electronic and atomic subsystems respectively, g_p is the
coupling constant for the electron-ion interaction, and g_s is the
electron stopping coupling parameter. C_e, rho_e, and kappa_e are
specified as parameters to the fix. The other quantities are derived.
The form of the heat diffusion equation used here is almost the same
as that in equation 6 of :ref:`(Duffy) <Duffy>`, with the exception that the
electronic density is explicitly represented, rather than being part
of the specific heat parameter.
where :math:`C_e` is the specific heat, :math:`\rho_e` is the density,
:math:`\kappa_e` is the thermal conductivity, *T* is temperature, the
"e" and "a" subscripts represent electronic and atomic subsystems
respectively, :math:`g_p` is the coupling constant for the electron-ion
interaction, and :math:`g_s` is the electron stopping coupling
parameter. :math:`C_e`, :math:`\rho_e`, and :math:`\kappa_e` are
specified as parameters to the fix *ttm* or *ttm/grid*. The other
quantities are derived. The form of the heat diffusion equation used
here is almost the same as that in equation 6 of :ref:`(Duffy) <Duffy>`,
with the exception that the electronic density is explicitly
represented, rather than being part of the specific heat parameter.
Currently, the TTM fixes assume that none of the user-supplied
parameters will vary with temperature. Note that :ref:`(Duffy)
<Duffy>` used a tanh() functional form for the temperature dependence
of the electronic specific heat, but ignored temperature dependencies
of any of the other parameters. See more discussion below for fix
ttm/mod.
parameters will vary with temperature. Note that :ref:`(Duffy) <Duffy>`
used a tanh() functional form for the temperature dependence of the
electronic specific heat, but ignored temperature dependencies of any of
the other parameters. See more discussion below for fix *ttm/mod*.
.. note::
@ -265,27 +265,27 @@ heat sources (e.g. laser heating in ablation simulations):
\bigtriangledown (\kappa_e \bigtriangledown T_e) -
g_p (T_e - T_a) + g_s T_a' + \theta (x-x_{surface})I_0 \exp(-x/l_{skin})
where theta is the Heaviside step function, I_0 is the (absorbed)
laser pulse intensity for ablation simulations, l_skin is the depth
of skin-layer, and all other designations have the same meaning as in
the former equation. The duration of the pulse is set by the parameter
*tau* in the *init_file*.
where :math:`\theta` is the Heaviside step function, :math:`I_0` is the
(absorbed) laser pulse intensity for ablation simulations,
:math:`l_{skin}` is the depth of the skin-layer, and all other
designations have the same meaning as in the former equation. The
duration of the pulse is set by the parameter *tau* in the *init_file*.
Fix ttm/mod also allows users to specify the dependencies of C_e and
kappa_e on the electronic temperature. The specific heat is expressed
as
Fix *ttm/mod* also allows users to specify the dependencies of
:math:`C_e` and :math:`\kappa_e` on the electronic temperature. The
specific heat is expressed as
.. math::
C_e = C_0 + (a_0 + a_1 X + a_2 X^2 + a_3 X^3 + a_4 X^4) \exp (-(AX)^2)
where *X* = T_e/1000, and the thermal conductivity is defined as
kappa_e = D_e\*rho_e\*C_e, where D_e is the thermal diffusion
coefficient.
where :math:`X = \frac{T_e}{1000}`, and the thermal conductivity is
defined as :math:`\kappa_e = D_e \cdot rho_e \cdot C_e`, where
:math:`D_e` is the thermal diffusion coefficient.
Electronic pressure effects are included in the TTM model to account
for the blast force acting on ions because of electronic pressure
gradient (see :ref:`(Chen) <Chen>`, :ref:`(Norman) <Norman>`). The total force
Electronic pressure effects are included in the TTM model to account for
the blast force acting on ions because of electronic pressure gradient
(see :ref:`(Chen) <Chen>`, :ref:`(Norman) <Norman>`). The total force
acting on an ion is:
.. math::
@ -293,13 +293,14 @@ acting on an ion is:
{\vec F}_i = - \partial U / \partial {\vec r}_i + {\vec
F}_{langevin} - \nabla P_e/n_{ion}
where F_langevin is a force from Langevin thermostat simulating
electron-phonon coupling, and nabla P_e/n_ion is the electron blast
force.
where :math:`F_{langevin}` is a force from Langevin thermostat
simulating electron-phonon coupling, and :math:`\nabla P_e/n_{ion}` is
the electron blast force.
The electronic pressure is taken to be P_e = B\*rho_e\*C_e\*T_e
The electronic pressure is taken to be :math:`P_e = B \cdot rho_e \cdot
C_e \cdot T_e`
The current fix ttm/mod implementation allows TTM simulations with a
The current fix *ttm/mod* implementation allows TTM simulations with a
vacuum. The vacuum region is defined as the grid cells with zero
electronic temperature. The numerical scheme does not allow energy
exchange with such cells. Since the material can expand to previously
@ -319,10 +320,10 @@ electronic pressure gradient is calculated as
\frac{x}{x+\lambda}\frac{(C_e{}T_e)_{x+\Delta
x}-(C_e{}T_e)_{x}}{\Delta x} \right]
where lambda is the electron mean free path (see :ref:`(Norman) <Norman>`,
:ref:`(Pisarev) <Pisarev>`)
where :math:`\lambda` is the electron mean free path (see :ref:`(Norman)
<Norman>`, :ref:`(Pisarev) <Pisarev>`)
The fix ttm/mod parameter file *init_file* has the following syntax.
The fix *ttm/mod* parameter file *init_file* has the following syntax.
Every line with an odd number is considered as a comment and
ignored. The lines with the even numbers are treated as follows: