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positive and the zeros of F(C e) are given by the zeros of G(C e). Thus, the electron fraction, C e, is given by the solution of


                   G(Ce) = 0                     (4.19)


Simplifying equation (4.17), the polynomial equation (4.19) is


                     13            2-l1
        G(C) = Ce +  1    1 + Z-i)Ce    Z. C e
                    i=2 1                   e


                             x K' i-1 = 0        (4.20)


Equation (4.20) is a 13th degree polynomial in Ce with one real positive root. From physical intuition there should be only one real root, since for a given temperature and pressure, the number of electrons in a plasma is uniquely determined. This can be shown rigorously by application of Descartes' rule of signs (37) to equation (4.20). Solutions to equation (4.20) are obtained using the iterative Newton-Rhapson method (38). The NewtonRhapson recurrence formula for the solution to (4.20) is

                             G (Ce(n))
          Ce (n + 1) = C (n) - (Ce(n))           (4.21)