Solution of problem 5.2 of Simon, The Oxford Solid State Basics

plot.gp

#!/usr/bin/env gnuplot

#  MIT License
#
#  Copyright (c) 2024 I.F.F. dos SANTOS
#
#  Permission is hereby granted, free of charge, to any person obtaining a copy
#  of this software and associated documentation files (the “Software”), to
#  deal in the Software without restriction, including without limitation the
#  rights to use, copy, modify, merge, publish, distribute, sublicense, and/or
#  sell copies of the Software, and to permit persons to whom the Software is
#  furnished to do so, subject to the following conditions:
#
#  The above copyright notice and this permission notice shall be included in
#  all copies or substantial portions of the Software.
#
#  THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
#  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
#  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
#  AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
#  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
#  FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS
#  IN THE SOFTWARE.

set term png size 720, 720 enhanced font "serif,12"
set output "simon-2_5.png"

set key box
set grid

set xrange [1:21]

set xlabel "Atomic number (Z)"
set ylabel "First Ionization energy [eV]"

plot \
   "simon-2_5.dat" u 1:2 w p lt rgb "blue" pt 5 ps 2 title "Approximation a", \
   "simon-2_5.dat" u 1:3 w p lt rgb "red"  pt 7 ps 2 title "Approximation b"

exit

simon-2_5.c

/* *****************************************************************************
   MIT License

   Copyright (c) 2024 I.F.F. dos SANTOS

   Permission is hereby granted, free of charge, to any person obtaining a copy
   of this software and associated documentation files (the “Software”), to
   deal in the Software without restriction, including without limitation the
   rights to use, copy, modify, merge, publish, distribute, sublicense, and/or
   sell copies of the Software, and to permit persons to whom the Software is
   furnished to do so, subject to the following conditions:

   The above copyright notice and this permission notice shall be included in
   all copies or substantial portions of the Software.

   THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
   IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
   FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
   AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
   LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
   FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS
   IN THE SOFTWARE.
***************************************************************************** */
#include <stdio.h>
/* ***
   Program to plot the graph requested by the problem 2.5 of Simon,
   The Oxford Solid State Basics.
COMPILE:
   cc simon-2_5.c -o simon-2_5
*** */

#define Rydberg_energy 13.605693122990 // eV

int main(void){
   /* Note that:
      n is the Principal quantum number */
   int Z_nuc, N_inside, N_same, n;
   double Z, E, tmp;
   FILE *data;

   data = fopen("simon-2_5.dat", "w");
   if(data == NULL){
      fputs("Error: can't write data in a file.", stderr);
      return 1;
   }

   Z_nuc = 1;
   while(Z_nuc < 22){

      if(Z_nuc < 3){
         /* Shells: 1s1 -- 1s2*/
         n = 1;
         N_inside = 0;
         N_same = ((Z_nuc == 1) ? 1 : (N_same + 1.0));
      }else if(Z_nuc < 11){
         /* Shells: 2s1 -- 2p6*/
         n = 2;
         N_inside = 2;
         N_same = ((Z_nuc == 3) ? 1 : (N_same + 1.0));
      }else if((Z_nuc < 19) || (Z_nuc == 21)){
         /* Shells: 3s1 -- 3d1*/
         n = 3;
         N_inside = 10;
         N_same = ((Z_nuc == 11) ? 1 : ((Z_nuc == 21) ? 19 : (N_same + 1.0)));
      }else{
         /* Shells: 4s1 -- 4s2*/
         n = 4;
         N_inside = 18;
         N_same = ((Z_nuc == 19) ? 1 : (N_same + 1.0));
      }

      /* Approximation a */
      Z = (double)(Z_nuc - N_inside);
      tmp = (double)(n*n);
      E = Rydberg_energy * Z*Z / tmp;
      fprintf(data, "%d %g ", Z_nuc, E);

      /* Approximation b */
      Z -= 0.5 * (double)(N_same - 1);
      E = Rydberg_energy * Z*Z / tmp;
      fprintf(data, "%g\n", E);

      ++Z_nuc;
   }

   fclose(data);

   return 0;
}