Challenge Team Interim Report
Team Number: 061
School Name: Portales High School
Area of Science: Physics
Project Title: Particle Implantation into Ionic and Covalent Solids
Please click on one of the following to go to that section of our Interim Report
I. Executive Summary
II. Lattice and Collisional Dynamics
III. Ionic Dynamics Simulation
IV. Super Computer Implementation
V. Computer Graphical Animation
VI. Progress and Result to Date
This year's project for Portales team 61 is based upon the principles of particle implantation into ionic and covalent solids. In this experiment we will be shooting different particles (electrons, ions, protons, whole atoms, etc...) into lattices of various particles (sodium chloride crystals, silicon wafers, ice crystals, etc.). By doing this, we hope to study the results of particle implantation into lattices pertaining to real life projects such as semi-conductor technology and various other fields of study.
The supercomputer is used in this project to keep track of all inter-particle forces of all particles in the model experiment, which includes the lattice itself and the particles being shot into the lattice.
In our programming, a user will specify initial conditions for each atom or molecule in the lattice, including the mass, starting positions (xi, yi, zi), and initial velocities (vxi, vyi, vzi) in a given coordinate system. The program will calculate these data in the context of attractive and repulsive forces between the atoms and molecules to find the inter-particle forces at work and then calculate new coordinates and new velocities of each particle for a given time interval (Dt). The new time interval will represent a new movie frame in our lattice dynamics "motion picture". The motion picture of the impact of the projectile particle into the lattice target will be produced via computer graphics animation by hooking together many frames representing many successive time increments Dt. This way we will find out how the objects within a given lattice will interact before, during, and after impact.
If it is assumed that the ions in a crystal lattice move at slow speeds and modest accelerations, then these motions can be approximated by Newton's kinematic equations listed below in vector and vector component form:
These equations give the position and the velocity of particle i in terms of initial position, initial velocity, and acceleration. These equations will be adapted to form the basis for a lattice and collision dynamics simulation.
In an ionic crystal lattice such as sodium chloride, the interaction potential energy between any two ions can be described as a sum of a Coulombic attraction or repulsion term and an electron cloud repulsion term. This second term will account for the fact that the electron clouds of two ions or neutral atoms will repel one another if the two ions are brought into close proximity.
This equation gives the position and the velocity of particle i in terms of initial position, initial velocity, and acceleration. In equation (2), Zi and Zj are the charges on the ions, e is the fundamental unit of electrical charge,Î 0 is the vacuum permitivity, Bij is an electron cloud repulsion term for ions i and j, n is a scaling factor for the strength of the electron cloud repulsion term, and rij is the separation between the two ions. The inter-ionic distance rij is expressed as:
For a multi-ion crystal lattice, the interaction potential energy affecting ion i is:
The force that ion i experiences as a result of interactions with the all other ions is given as the negative gradient of the interaction potential energy:
The individual components of the force vectors are:
The lattice and collision dynamics simulation is carried out by calculating Newton's kinematic equations of motion iteratively through n time-steps, each of duration Dt. At each time-step the acceleration of ion i is defined as the force on ion i divided by its mass. The ion positions at each time step are noted with a time-step superscript. These ion positions are stored in an array for subsequent plotting and animation programs. The forces and positions at each time-step are computed as:
The program we have constructed is in the C++ language and consists of 3 main loops. User inputs into the program include: all ion masses, charges, initial positions, and initial velocities. Initial positions are taken to be standard crystal lattice positions. Initial velocities vector directions are assigned by a random number generator and magnitudes are scaled proportional to the square root of the Kelvin temperature. In its present form, the program can handle particles that are either ions or neutral atoms in a lattice. In the future, our program will have the provision to handle a high velocity incoming projectile ion impinged upon the lattice. The program can also simulate absorption of a high energy photon by an ion in the lattice. This simulation is achieved by imparting a high velocity to that ion such that the kinetic energy of the ejected ion is related directly to the photon energy absorbed by the ion. The first loop is the time-step loop which goes from 1 to n time steps of duration Dt, n being the total number of time-steps in the entire program: for (int t=1;t <= n;t++) (n is a user input). Within the time-step loop are the ion i and j loops. In the ion loop i we initialize the forces Fi(x, y, z). The third and final loop is within loop i and is the ion loop j. In this loop are the main equations. Here we compute all the forces, velocities, and positions of each ion but only if i does not equal j. All the equations shown in section II (Lattice and Collisional Dynamics) are used in this loop to figure out what will happen to each ion. Overall in this code there are N+1 equations to compute for each time one ion interacts with another ion. Apply that to N ions within the lattice plus 1 projectile ion; then multiplying that number by n time-steps, there are literally billions of calculations which would crash a normal PC.
The visual presentation of our project will be shown to the judges by a program called Tecplot. We will run our program, which will output all the information into a file-format which Tecplot could recognize. Animation effects will be achieved by having the graphics package step through the sequence of ionic positions as calculated in each particle dynamics time-step.
As of this date (1/3/99) some ion dynamics equations appropriate for this project have been coded in C++. The code is not complete has not been tested. A random number generator routine has been developed to simulate the random thermal motions of the ionic lattices. The SuperComputer Challenge team has also developed a way to model and visualize these affects at the macroscopic level by assembling a lattice of empty soft-drink cans.