(d4769d99) · Commits · 郑智淋 / lammps

doc/Manual.html

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		4.18 <A HREF = "Section_howto.html#4_18">Elastic constants</A>
		<BR>
		4.19 <A HREF = "Section_howto.html#4_19">Library interface to LAMMPS</A>
		<BR>
		4.20 <A HREF = "Section_howto.html#4_20">Calculating thermal conductivity</A>
		<BR>
		4.21 <A HREF = "Section_howto.html#4_21">Calculating viscosity</A>
		<BR></UL>
		<LI><A HREF = "Section_example.html">Example problems</A>

		@@ -295,6 +299,10 @@ listed above.

doc/Manual.txt

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		@@ -113,7 +113,9 @@ listed above.
		4.16 "Thermostatting, barostatting, and compute temperature"_4_16 :b
		4.17 "Walls"_4_17 :b
		4.18 "Elastic constants"_4_18 :b
		4.19 "Library interface to LAMMPS"_4_19 :ule,b
		4.19 "Library interface to LAMMPS"_4_19 :b
		4.20 "Calculating thermal conductivity"_4_20 :b
		4.21 "Calculating viscosity"_4_21 :ule,b
		"Example problems"_Section_example.html :l
		"Performance & scalability"_Section_perf.html :l
		"Additional tools"_Section_tools.html :l
		@@ -176,6 +178,8 @@ listed above.
		:link(4_17,Section_howto.html#4_17)
		:link(4_18,Section_howto.html#4_18)
		:link(4_19,Section_howto.html#4_19)
		:link(4_20,Section_howto.html#4_20)
		:link(4_21,Section_howto.html#4_21)

		:link(9_1,Section_python.html#9_1)
		:link(9_2,Section_python.html#9_2)

doc/Section_howto.html

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		@@ -33,6 +33,8 @@ LAMMPS.
		4.17 <A HREF = "#4_17">Walls</A><BR>
		4.18 <A HREF = "#4_18">Elastic constants</A><BR>
		4.19 <A HREF = "#4_19">Library interface to LAMMPS</A><BR>
		4.20 <A HREF = "#4_20">Calculating thermal conductivity</A><BR>
		4.21 <A HREF = "#4_21">Calculating viscosity</A> <BR>

		<P>The example input scripts included in the LAMMPS distribution and
		highlighted in <A HREF = "Section_example.html">this section</A> also show how to
		@@ -1707,6 +1709,184 @@ grab data from LAMMPS, change it, and put it back into LAMMPS.
		</P>
		<HR>

		<A NAME = "4_20"></A><H4>4.20 Calculating thermal conductivity
		</H4>
		<P>The thermal conductivity kappa of a material can be measured in at
		least 3 ways using various options in LAMMPS. (See <A HREF = "Section_howto.html#4_21">this
		section</A> of the manual for an analogous
		discussion for viscosity). The thermal conducitivity tensor kappa is
		a measure of the propensity of a material to transmit heat energy in a
		diffusive manner as given by Fourier's law
		</P>
		<P>J = -kappa grad(T)
		</P>
		<P>where J is the heat flux in units of energy per area per time and
		grad(T) is the spatial gradient of temperature. The thermal
		conductivity thus has units of energy per distance per time per degree
		K and is often approximated as an isotropic quantity, i.e. as a
		scalar.
		</P>
		<P>The first method is to setup two thermostatted regions at opposite
		ends of a simulation box, or one in the middle and one at the end of a
		periodic box. By holding the two regions at different temperatures
		with a <A HREF = "Section_howto.html#4_13">thermostatting fix</A>, the energy added
		to the hot region should equal the energy subtracted from the cold
		region and be proportional to the heat flux moving between the
		regions. See the paper by <A HREF = "#Ikeshoji">Ikeshoji and Hafskjold</A> for
		details of this idea. Note that thermostatting fixes such as <A HREF = "fix_nh.html">fix
		nvt</A>, <A HREF = "fix_langevin.html">fix langevin</A>, and <A HREF = "fix_temp_rescale.html">fix
		temp/rescale</A> store the cumulative energy they
		add/subtract. Alternatively, the <A HREF = "fix_heat.html">fix heat</A> command can
		used in place of thermostats on each of two regions, and the resulting
		temperatures of the two regions monitored with the "compute
		temp/region" command or the temperature profile of the intermediate
		region monitored with the <A HREF = "fix_ave_spatial.html">fix ave/spatial</A> and
		<A HREF = "compute_ke_atom.html">compute ke/atom</A> commands.
		</P>
		<P>The second method is to perform a reverse non-equilibrium MD
		simulation using the <A HREF = "fix_thermal_conductivity.html">fix
		thermal/conductivity</A> command which
		implements the rNEMD algorithm of Muller-Plathe. Kinetic energy is
		swapped between atoms in two different layers of the simulation box.
		This induces a temperature gradient between the two layers which can
		be monitored with the <A HREF = "fix_ave_spatial.html">fix ave/spatial</A> and
		<A HREF = "compute_ke_atom.html">compute ke/atom</A> commands. The fix tallies the
		cumulative energy transfer that it performs. See the <A HREF = "fix_thermal_conductivity.html">fix
		thermal/conductivity</A> command for
		details.
		</P>
		<P>The third method is based on the Green-Kubo (GK) formula which relates
		the ensemble average of the auto-correlation of the heat flux to
		kappa. The heat flux can be calculated from the fluctuations of
		per-atom potential and kinetic energies and per-atom stress tensor in
		a steady-state equilibrated simulation. This is in contrast to the
		two preceding non-equilibrium methods, where energy flows continuously
		between hot and cold regions of the simulation box.
		</P>
		<P>The <A HREF = "compute_heat_flux.html">compute heat/flux</A> command can calculate
		the needed heat flux and describes how to implement the Green_Kubo
		formalism using additional LAMMPS commands, such as the <A HREF = "fix_ave_correlate.html">fix
		ave/correlate</A> command to calculate the needed
		auto-correlation. See the doc page for the <A HREF = "compute_heat_flux.html">compute
		heat/flux</A> command for an example input script
		that calculates the thermal conductivity of solid Ar via the GK
		formalism.
		</P>
		<HR>

		<A NAME = "4_21"></A><H4>4.21 Calculating viscosity
		</H4>
		<P>The shear viscosity eta of a fluid can be measured in at least 3 ways
		using various options in LAMMPS. (See <A HREF = "Section_howto.html#4_">this
		section</A>??? of the manual for an analogous
		discussion for thermal conductivity). Eta is a measure of the
		propensity of a fluid to transmit momentum in a direction
		perpendicular to the direction of velocity or momentum flow.
		Alternatively it is the resistance the fluid has to being sheared. It
		is given by
		</P>
		<P>J = -eta grad(Vstream)
		</P>
		<P>where J is the momentum flux in units of momentum per area per time.
		and grad(Vstream) is the spatial gradient of the velocity of the fluid
		moving in another direction, normal to the area through which the
		momentum flows. Viscosity thus has units of pressure-time.
		</P>
		<P>The first method is to perform a non-equlibrium MD (NEMD) simulation
		by shearing the simulation box via the <A HREF = "fix_deform.html">fix deform</A>
		command, and using the <A HREF = "fix_nvt_sllod.html">fix nvt/sllod</A> command to
		thermostat the fluid via the SLLOD equations of motion. The velocity
		profile setup in the fluid by this procedure can be monitored by the
		<A HREF = "fix_ave_spatial.html">fix ave/spatial</A> command, which determines
		grad(Vstream) in the equation above. E.g. the derivative in the
		y-direction of the Vx component of fluid motion or grad(Vstream) =
		dVx/dy. In this case, the Pxy off-diagonal component of the pressure
		or stress tensor, as calculated by the <A HREF = "compute_pressure.html">compute
		pressure</A> command, can also be monitored, which
		is the J term in the equation above. See <A HREF = "Section_howto.html#4_13">this
		section</A> of the manual for details on NEMD
		simulations.
		</P>
		<P>The second method is to perform a reverse non-equilibrium MD
		simulation using the <A HREF = "fix_viscosity.html">fix viscosity</A> command which
		implements the rNEMD algorithm of Muller-Plathe. Momentum in one
		dimension is swapped between atoms in two different layers of the
		simulation box in a different dimension. This induces a velocity
		gradient which can be monitored with the <A HREF = "fix_ave_spatial.html">fix
		ave/spatial</A> command. The fix tallies the
		cummulative momentum transfer that it performs. See the <A HREF = "fix_viscosity.html">fix
		viscosity</A> command for details.
		</P>
		<P>The third method is based on the Green-Kubo (GK) formula which relates
		the ensemble average of the auto-correlation of the stress/pressure
		tensor to eta. This can be done in a steady-state equilibrated
		simulation which is in contrast to the two preceding non-equilibrium
		methods, where momentum flows continuously through the simulation box.
		</P>
		<P>Here is an example input script that calculates the viscosity of
		liquid Ar via the GK formalism:
		</P>
		<PRE># Sample LAMMPS input script for viscosity of liquid Ar
		</PRE>
		<PRE>units real
		variable T equal 86.4956
		variable V equal vol
		variable dt equal 4.0
		variable p equal 400 # correlation length
		variable s equal 5 # sample interval
		variable d equal $p*$s # dump interval
		</PRE>
		<PRE># convert from LAMMPS real units to SI
		</PRE>
		<PRE>variable kB equal 1.3806504e-23 # [J/K/</B> Boltzmann
		variable atm2Pa equal 101325.0
		variable A2m equal 1.0e-10
		variable fs2s equal 1.0e-15
		variable convert equal ${atm2Pa}${atm2Pa}${fs2s}${A2m}${A2m}*${A2m}
		</PRE>
		<PRE># setup problem
		</PRE>
		<PRE>dimension 3
		boundary p p p
		lattice fcc 5.376 orient x 1 0 0 orient y 0 1 0 orient z 0 0 1
		region box block 0 4 0 4 0 4
		create_box 1 box
		create_atoms 1 box
		mass 1 39.948
		pair_style lj/cut 13.0
		pair_coeff * * 0.2381 3.405
		timestep ${dt}
		thermo $d
		</PRE>
		<PRE># equilibration and thermalization
		</PRE>
		<PRE>velocity all create $T 102486 mom yes rot yes dist gaussian
		fix NVT all nvt temp $T $T 10 drag 0.2
		run 8000
		</PRE>
		<PRE># viscosity calculation, switch to NVE if desired
		</PRE>
		<PRE>#unfix NVT
		#fix NVE all nve
		</PRE>
		<PRE>reset_timestep 0
		variable pxy equal pxy
		variable pxz equal pxz
		variable pyz equal pyz
		fix SS all ave/correlate $s $p $d &
		v_pxy v_pxz v_pyz type auto file S0St.dat ave running
		variable scale equal ${convert}/(${kB}$T)$V$s${dt}
		variable v11 equal trap(f_SS[3/</B>)*${scale}
		variable v22 equal trap(f_SS[4/</B>)*${scale}
		variable v33 equal trap(f_SS[5/</B>)*${scale}
		thermo_style custom step temp press v_pxy v_pxz v_pyz v_v11 v_v22 v_v33
		run 100000
		variable v equal (v_v11+v_v22+v_v33)/3.0
		variable ndens equal count(all)/vol
		print "average viscosity: $v [Pa.s/</B> @ $T K, ${ndens} /A^3"
		</PRE>
		<HR>

		<HR>

		<A NAME = "Berendsen"></A>
		@@ -1724,6 +1904,11 @@ Spellmeyer, Fox, Caldwell, Kollman, JACS 117, 5179-5197 (1995).
		<P><B>(Horn)</B> Horn, Swope, Pitera, Madura, Dick, Hura, and Head-Gordon,
		J Chem Phys, 120, 9665 (2004).
		</P>
		<A NAME = "Ikeshoji"></A>

		<P><B>(Ikeshoji)</B> Ikeshoji and Hafskjold, Molecular Physics, 81, 251-261
		(1994).
		</P>
		<A NAME = "MacKerell"></A>

		<P><B>(MacKerell)</B> MacKerell, Bashford, Bellott, Dunbrack, Evanseck, Field,

doc/Section_howto.txt

+185 −1

Original line number	Diff line number	Diff line
		@@ -29,7 +29,9 @@ LAMMPS.
		4.16 "Thermostatting, barostatting and computing temperature"_#4_16
		4.17 "Walls"_#4_17
		4.18 "Elastic constants"_#4_18
		4.19 "Library interface to LAMMPS"_#4_19 :all(b)
		4.19 "Library interface to LAMMPS"_#4_19
		4.20 "Calculating thermal conductivity"_#4_20
		4.21 "Calculating viscosity"_#4_21 :all(b)

		The example input scripts included in the LAMMPS distribution and
		highlighted in "this section"_Section_example.html also show how to
		@@ -1693,6 +1695,184 @@ have example C++ and C and Python codes which show how a driver code
		can link to LAMMPS as a library, run LAMMPS on a subset of processors,
		grab data from LAMMPS, change it, and put it back into LAMMPS.

		:line

		4.20 Calculating thermal conductivity :link(4_20),h4

		The thermal conductivity kappa of a material can be measured in at
		least 3 ways using various options in LAMMPS. (See "this
		section"_Section_howto.html#4_21 of the manual for an analogous
		discussion for viscosity). The thermal conducitivity tensor kappa is
		a measure of the propensity of a material to transmit heat energy in a
		diffusive manner as given by Fourier's law

		J = -kappa grad(T)

		where J is the heat flux in units of energy per area per time and
		grad(T) is the spatial gradient of temperature. The thermal
		conductivity thus has units of energy per distance per time per degree
		K and is often approximated as an isotropic quantity, i.e. as a
		scalar.

		The first method is to setup two thermostatted regions at opposite
		ends of a simulation box, or one in the middle and one at the end of a
		periodic box. By holding the two regions at different temperatures
		with a "thermostatting fix"_Section_howto.html#4_13, the energy added
		to the hot region should equal the energy subtracted from the cold
		region and be proportional to the heat flux moving between the
		regions. See the paper by "Ikeshoji and Hafskjold"_#Ikeshoji for
		details of this idea. Note that thermostatting fixes such as "fix
		nvt"_fix_nh.html, "fix langevin"_fix_langevin.html, and "fix
		temp/rescale"_fix_temp_rescale.html store the cumulative energy they
		add/subtract. Alternatively, the "fix heat"_fix_heat.html command can
		used in place of thermostats on each of two regions, and the resulting
		temperatures of the two regions monitored with the "compute
		temp/region" command or the temperature profile of the intermediate
		region monitored with the "fix ave/spatial"_fix_ave_spatial.html and
		"compute ke/atom"_compute_ke_atom.html commands.

		The second method is to perform a reverse non-equilibrium MD
		simulation using the "fix
		thermal/conductivity"_fix_thermal_conductivity.html command which
		implements the rNEMD algorithm of Muller-Plathe. Kinetic energy is
		swapped between atoms in two different layers of the simulation box.
		This induces a temperature gradient between the two layers which can
		be monitored with the "fix ave/spatial"_fix_ave_spatial.html and
		"compute ke/atom"_compute_ke_atom.html commands. The fix tallies the
		cumulative energy transfer that it performs. See the "fix
		thermal/conductivity"_fix_thermal_conductivity.html command for
		details.

		The third method is based on the Green-Kubo (GK) formula which relates
		the ensemble average of the auto-correlation of the heat flux to
		kappa. The heat flux can be calculated from the fluctuations of
		per-atom potential and kinetic energies and per-atom stress tensor in
		a steady-state equilibrated simulation. This is in contrast to the
		two preceding non-equilibrium methods, where energy flows continuously
		between hot and cold regions of the simulation box.

		The "compute heat/flux"_compute_heat_flux.html command can calculate
		the needed heat flux and describes how to implement the Green_Kubo
		formalism using additional LAMMPS commands, such as the "fix
		ave/correlate"_fix_ave_correlate.html command to calculate the needed
		auto-correlation. See the doc page for the "compute
		heat/flux"_compute_heat_flux.html command for an example input script
		that calculates the thermal conductivity of solid Ar via the GK
		formalism.

		:line

		4.21 Calculating viscosity :link(4_21),h4

		The shear viscosity eta of a fluid can be measured in at least 3 ways
		using various options in LAMMPS. (See "this
		section"_Section_howto.html#4_??? of the manual for an analogous
		discussion for thermal conductivity). Eta is a measure of the
		propensity of a fluid to transmit momentum in a direction
		perpendicular to the direction of velocity or momentum flow.
		Alternatively it is the resistance the fluid has to being sheared. It
		is given by

		J = -eta grad(Vstream)

		where J is the momentum flux in units of momentum per area per time.
		and grad(Vstream) is the spatial gradient of the velocity of the fluid
		moving in another direction, normal to the area through which the
		momentum flows. Viscosity thus has units of pressure-time.

		The first method is to perform a non-equlibrium MD (NEMD) simulation
		by shearing the simulation box via the "fix deform"_fix_deform.html
		command, and using the "fix nvt/sllod"_fix_nvt_sllod.html command to
		thermostat the fluid via the SLLOD equations of motion. The velocity
		profile setup in the fluid by this procedure can be monitored by the
		"fix ave/spatial"_fix_ave_spatial.html command, which determines
		grad(Vstream) in the equation above. E.g. the derivative in the
		y-direction of the Vx component of fluid motion or grad(Vstream) =
		dVx/dy. In this case, the Pxy off-diagonal component of the pressure
		or stress tensor, as calculated by the "compute
		pressure"_compute_pressure.html command, can also be monitored, which
		is the J term in the equation above. See "this
		section"_Section_howto.html#4_13 of the manual for details on NEMD
		simulations.

		The second method is to perform a reverse non-equilibrium MD
		simulation using the "fix viscosity"_fix_viscosity.html command which
		implements the rNEMD algorithm of Muller-Plathe. Momentum in one
		dimension is swapped between atoms in two different layers of the
		simulation box in a different dimension. This induces a velocity
		gradient which can be monitored with the "fix
		ave/spatial"_fix_ave_spatial.html command. The fix tallies the
		cummulative momentum transfer that it performs. See the "fix
		viscosity"_fix_viscosity.html command for details.

		The third method is based on the Green-Kubo (GK) formula which relates
		the ensemble average of the auto-correlation of the stress/pressure
		tensor to eta. This can be done in a steady-state equilibrated
		simulation which is in contrast to the two preceding non-equilibrium
		methods, where momentum flows continuously through the simulation box.

		Here is an example input script that calculates the viscosity of
		liquid Ar via the GK formalism:

		# Sample LAMMPS input script for viscosity of liquid Ar :pre

		units real
		variable T equal 86.4956
		variable V equal vol
		variable dt equal 4.0
		variable p equal 400 # correlation length
		variable s equal 5 # sample interval
		variable d equal $p*$s # dump interval :pre

		# convert from LAMMPS real units to SI :pre

		variable kB equal 1.3806504e-23 # \[J/K/] Boltzmann
		variable atm2Pa equal 101325.0
		variable A2m equal 1.0e-10
		variable fs2s equal 1.0e-15
		variable convert equal $\{atm2Pa\}$\{atm2Pa\}$\{fs2s\}$\{A2m\}$\{A2m\}*$\{A2m\} :pre

		# setup problem :pre

		dimension 3
		boundary p p p
		lattice fcc 5.376 orient x 1 0 0 orient y 0 1 0 orient z 0 0 1
		region box block 0 4 0 4 0 4
		create_box 1 box
		create_atoms 1 box
		mass 1 39.948
		pair_style lj/cut 13.0
		pair_coeff * * 0.2381 3.405
		timestep $\{dt\}
		thermo $d :pre

		# equilibration and thermalization :pre

		velocity all create $T 102486 mom yes rot yes dist gaussian
		fix NVT all nvt temp $T $T 10 drag 0.2
		run 8000 :pre

		# viscosity calculation, switch to NVE if desired :pre

		#unfix NVT
		#fix NVE all nve :pre

		reset_timestep 0
		variable pxy equal pxy
		variable pxz equal pxz
		variable pyz equal pyz
		fix SS all ave/correlate $s $p $d &
		v_pxy v_pxz v_pyz type auto file S0St.dat ave running
		variable scale equal $\{convert\}/($\{kB\}$T)$V$s$\{dt\}
		variable v11 equal trap(f_SS\[3/])*$\{scale\}
		variable v22 equal trap(f_SS\[4/])*$\{scale\}
		variable v33 equal trap(f_SS\[5/])*$\{scale\}
		thermo_style custom step temp press v_pxy v_pxz v_pyz v_v11 v_v22 v_v33
		run 100000
		variable v equal (v_v11+v_v22+v_v33)/3.0
		variable ndens equal count(all)/vol
		print "average viscosity: $v \[Pa.s/] @ $T K, $\{ndens\} /A^3" :pre

		:line
		:line

		@@ -1708,6 +1888,10 @@ Spellmeyer, Fox, Caldwell, Kollman, JACS 117, 5179-5197 (1995).
		[(Horn)] Horn, Swope, Pitera, Madura, Dick, Hura, and Head-Gordon,
		J Chem Phys, 120, 9665 (2004).

		:link(Ikeshoji)
		[(Ikeshoji)] Ikeshoji and Hafskjold, Molecular Physics, 81, 251-261
		(1994).

		:link(MacKerell)
		[(MacKerell)] MacKerell, Bashford, Bellott, Dunbrack, Evanseck, Field,
		Fischer, Gao, Guo, Ha, et al, J Phys Chem, 102, 3586 (1998).

doc/compute_heat_flux.html

+46 −40

Original line number	Diff line number	Diff line
		@@ -89,8 +89,8 @@ integral estimated, and the Green-Kubo formula above evaluated.
		the autocorrelation. The trap() function in the
		<A HREF = "variable.html">variable</A> command can calculate the integral.
		</P>
		<P>An an example LAMMPS input script for solid Ar is appended below.
		The result should be: average conductivity ~0.29 in W/mK.
		<P>An example LAMMPS input script for solid Ar is appended below. The
		result should be: average conductivity ~0.29 in W/mK.
		</P>
		<HR>

		@@ -128,12 +128,9 @@ energy/area/time <A HREF = "units.html">units</A>
		</P>
		<HR>

		<H4>Sample LAMMPS input script
		</H4>
		<PRE>atom_style atomic
		units real
		variable kB equal 1.3806504e-23 # [J/K] Boltzmann
		variable kCal2J equal 4186.0/6.02214e23
		<PRE># Sample LAMMPS input script for thermal conductivity of solid Ar
		</PRE>
		<PRE>units real
		variable T equal 70
		variable V equal vol
		variable dt equal 4.0
		@@ -141,8 +138,16 @@ variable p equal 200 # correlation length
		variable s equal 10 # sample interval
		variable d equal $p*$s # dump interval
		</PRE>
		<P># ---------------------------------------------------------
		</P>
		<PRE># convert from LAMMPS real units to SI
		</PRE>
		<PRE>variable kB equal 1.3806504e-23 # [J/K] Boltzmann
		variable kCal2J equal 4186.0/6.02214e23
		variable A2m equal 1.0e-10
		variable fs2s equal 1.0e-15
		variable convert equal ${kCal2J}*${kCal2J}/${fs2s}/${A2m}
		</PRE>
		<PRE># setup problem
		</PRE>
		<PRE>dimension 3
		boundary p p p
		lattice fcc 5.376 orient x 1 0 0 orient y 0 1 0 orient z 0 0 1
		@@ -155,14 +160,14 @@ pair_coeff * * 0.2381 3.405
		timestep ${dt}
		thermo $d
		</PRE>
		<P># ------------- equilibration and thermalization ----------------
		</P>
		<PRE># equilibration and thermalization
		</PRE>
		<PRE>velocity all create $T 102486 mom yes rot yes dist gaussian
		fix NVT all nvt temp $T $T 10 drag 0.2
		run 8000
		</PRE>
		<P># -------------- flux calculation, switch to NVE if desired --------
		</P>
		<PRE># thermal conductivity calculation, switch to NVE if desired
		</PRE>
		<PRE>#unfix NVT
		#fix NVE all nve
		</PRE>
		@@ -176,13 +181,14 @@ variable Jy equal c_flux[2]/vol
		variable Jz equal c_flux[3]/vol
		fix JJ all ave/correlate $s $p $d &
		c_flux[1] c_flux[2] c_flux[3] type auto file J0Jt.dat ave running
		variable scale equal ${kCal2J}${kCal2J}/${kB}/$T/$T/$V$s${dt}1.0e25
		variable scale equal ${convert}/${kB}/$T/$T/$V$s${dt}
		variable k11 equal trap(f_JJ[3])*${scale}
		variable k22 equal trap(f_JJ[4])*${scale}
		variable k33 equal trap(f_JJ[5])*${scale}
		thermo_style custom step temp v_Jx v_Jy v_Jz v_k11 v_k22 v_k33
		run 100000
		variable k equal (v_k11+v_k22+v_k33)/3.0
		print "average conductivity: $k [W/mK]"
		variable ndens equal count(all)/vol
		print "average conductivity: $k[W/mK] @ $T K, ${ndens} /A^3"
		</PRE>
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