Parameter definitions for "born"
It's probably useful to look first at the special features page. You may also want to look at one of the general help files, Sliders, or
Graphs.
Inputs
- SLIDERS
- amplitude: defines the amplitude, in arbitrary units, of the displacements in the initial conditions.
- wave vector: defines the wave vector, in units of pi/a where a is the lattice constant, of the initial displacement pattern. For wavelike initial conditions, allowed values are restricted by the choice of boundary conditions. (Restricted to zero or positive values)
- packet width: defines the root mean square width, in units of the lattice constant, of the Gaussian envelope of the wave packet, for either the traveling packet or the standing packet choice of initial conditions. (Restricted to positive values.)
- peak position: the position, in units of the lattice constant, of the center of the packet (i.e., the number of the central atom of the packet), or the position of the impurity atom.
- imp. mass: if impure is chosen as the crystal type, this gives the mass of the impurity atom, in units of the mass of the atoms of the host chain. (Restricted to values greater than 0.005.)
- quartic cnst: defines the magnitude of the quartic spring constant K' for the anharmonic choice of crystal type. Units of K' are such that the harmonic and quartic forces are equal for a spring compressed by unit amplitude, A=1. (Restricted to the range 0 to 30.)
- speed: gives the number of calculational time steps between successive updates of the animation display. Changing the speed does NOT change the numerical accuracy of the simulation or the meaning of the time readout. It DOES influence the time resolution of the graph which stores data sets only at each update of the display. (These graphs are memory hogs: increase the speed for long runs.) (Restricted to positive integers.)
- MENUS
- boundary: sets the boundary conditions on the end of the chain. Choices are:
- periodic: periodic boundary conditions: in the equations of motion for u(0) and u(N-1), u(-1) and u(N) are replaced by u(N-1) and u(1) respectively.
- fixed: fixed boundary conditions: u(n) is forced to be zero for n=0 and n=N-1 where N is the number of atoms in the chain.
-
initial: gives the choice of a variety of initial conditions for the simulation:
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traveling_wave: initial displacements and velocities chosen to give a sinusoidal traveling wave moving to the right.
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standing_wave: sinusoidal initial displacements and initial velocities are zero.
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traveling_packet: initial displacements are a cosinusoidal wave multiplied by a Gaussian envelope of amplitude A and mean deviation given by the packet width; velocities are chosen to give a packet traveling to the right.
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standing_packet: same initial displacements as for the traveling_packet but initial velocities are zero.
-
random: random initial displacements distributed uniformly over the range -A to A where A is the amplitude.
- crystal_type:
- pure: all atoms of the chain have the same mass, and the springs are harmonic.
- impure: the atom at the peak position in the chain has a mass relative to that of each of the other atoms on the chain given by imp. mass.
- anharmonic: a quartic anharmonic term with strength given by quartic cnst is added to the potential of each of the springs coupling the masses.
- display: enables the graph to display:
- single: the displacement of the central atom in the chain versus time.
- triple: the displacement of the central atom and its two right hand neighbors.
- quintuple: the displacement vs. time of the central atom and its next four neighbors.
- BUTTONS/TOGGLES
- run/stop: starts and stops the simulation: keyboard equivalent is the spacebar.
- initialize: reinitializes the program and resets the clock: keyboard equivalent is the i key.
- copy graph: see general help--graphs.
- presets: Takes you to a tear-off menu with which you may select the presets. For more details, see the preset item in the miscellaneous help file.
- configure:
- number of atoms: the total number of atoms in the chain may be changed. (Restricted to be more than 2 and less than 256.)
- graph update increments: this is the number of calculational cycles between successive updates of the graph. Smaller values give a smoothly changing graph, but slow down the execution. The resolution of the final graph is NOT influenced by this choice. The resolution IS reduced by increasing the speed. (Restricted to positive integer.)
- show grid: toggles on and off the grid in the animation display.
- show continuous function: toggles on and off the continuous function A*cos(qna), included for comparison with the initial displacements of the atoms.
- {revert: updates the listed configure parameters to the current values (which may have been changed by a preset command).}
- {close: closes the configure window.}
- help: see help.
- quit: click here to close the "born" simulation.
Outputs
- DISPLAY:
The vertical positions in the animation display give the displacements of the masses (atoms) from their equilibrium positions. The display may be thought of either as a graphical representation of longitudinal motion on the chain or as a pictorial representation of transverse motion.
The vertical lines in the display appear every ten atoms. The horizontal lines are at -1, -0.5, 0, 0.5, and 1 in the same arbitrary units as the amplitude parameter.
- GRAPH
The graph gives the time dependence of the displacement of selected atoms, according to the choice from the display menu. The displacement of the central atom is shown in blue for all choices. The displacement of the two (four) nearest atoms to the the right are also shown for the triple (quintuple) choices.
- READOUT
- time: The time readout in the upper left corner of the main graph window displays the correct "simulation time" independent of the speed slider. (The speed slider changes the display rate and the graph resolution, but that's all.) The units of the time counter are 1/50 of the period corresponding to the "cutoff frequency" of the chain, 2*sqrt(K/M).