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Top page ( correct Bohr model including the two electron atom )
If you copy and paste the program source code below into a text editor, you can easily compile and run this.
(This class file name is MathMethod, so save this text editor as "MathMethod.java", and compile it.)
In this program, we first input the atomic number Z (Oxygen= 8, Neon=10), the initial x-coordinate r1 (in MM) of electron 1 and the absolute value of the total energy E ( in eV ) of the two-electron atoms (ions).
Here we use the new units, ( 1 MM = 10-14 meter, 1 SS = 10-22 second, 1 MM/SS = 108 m/s ).
From the inputted values, this program outputs the last velocity ( VX,VY in MM/SS ) and number of the de Broglie's waves ( midWN ) contained in one quarter of the orbit.
import java.util.Scanner;
class MathMethod {
public static void main(String[] args) {
// input Z, r1 and |E|
Scanner stdIn=new Scanner(System.in);
System.out.println("Atomic number Z ? (O6+=8,,Ne8+=10) ");
double Z=stdIn.nextDouble();
System.out.println("r1 between nucleus and electron 1 (MM)？ ");
double r=stdIn.nextDouble();
System.out.println("total energy |E| of the two-electron atom (eV) ? ");
double E=stdIn.nextDouble();
double me=9.1093826e-31;
double pai=3.141592653589793; double epsi=8.85418781787346e-12;
double h=6.62606896e-34; double ele=1.60217653e-19;
for (int i=1;i < 100;i++) { // repeat until r1=initial r1+100
// calculation of initial VY from E and r1
double poten=-(2*Z*ele*ele)/(4*pai*epsi*r)+(ele*ele)/(4*pai*epsi*2*r);
//vya= total E-potential energy
double vya=-(E*1.60217646e-19)-poten*1.0e14;
if (vya > 0) {
// vyb=velocity from kinetic energy
double vyb=Math.sqrt(vya/me);
double VY=vyb*1.0e-8; // change m/sec to MM/SS
double prexx=r; double VX=0.0; double WN=0.0; double preyy=0.0;
double yy,vk,preVY,preWN,midWN; double xx=0.0;
do {
xx=prexx+VX; yy=preyy+VY; //electron 1 position after 1SS
preVY=VY;preWN=WN ;
vk=VX*VX+VY*VY; //calculation of WN from VX,VY
WN=WN+(me*vk*1.0e-6)/h;
//calculation of VX,VY from Coulomb force
double ra=Math.sqrt(prexx*prexx+preyy*preyy);
double rb=Math.sqrt(4.0*prexx*prexx+2.0*preyy*preyy);
// change MM to meter
ra=ra*1.0e-14; rb=rb*1.0e-14;
prexx=prexx*1.0e-14; preyy=preyy*1.0e-14;
double ac=(ele*ele)/(4*pai*epsi*me);
VX=VX+1.0e-30*ac*prexx*(-Z/(ra*ra*ra)+2.0/(rb*rb*rb));
VY=VY+1.0e-30*ac*preyy*(-Z/(ra*ra*ra)+1.0/(rb*rb*rb));
prexx=xx;preyy=yy;
} while (xx >=0); // electron has moved one quater of an orbit?
if (VY > -0.001 && VY < 0.001) { // last VY condition
System.out.print("r1: "+r+" ");
System.out.printf("VX:%.5f", VX);
System.out.printf("VY:%.5f", VY);
System.out.printf("preVY:%.5f", preVY);
midWN=(preWN+WN)/2; System.out.printf("midWN:%.5f\n", midWN);
}
} r=r+1;
}}}