Science

Saha Thermal Ionization Equation (Stellar & Laboratory Plasma)

Estimate the ionized fraction of pure atomic hydrogen in a ground-state LTE approximation. Use a log-domain Saha ratio and a stable quadratic solution to avoid cancellation at high ionization.

Define the hydrogen model before using the equation

Assume pure atomic hydrogen, local thermodynamic equilibrium (LTE), charge neutrality n_e=n_p, and ground-state partition functions. Let n=n_H0+n_p be total hydrogen-nucleus density and x=n_p/n. Then x^2/(1−x)=K, where K=2.41468E21×T^(3/2)×EXP(−157809/T)/n with T in kelvin and n in m^−3.

The coefficient uses the electron translational factor with the ground-state statistical-weight ratio: 2U_p/U_H0≈2×1/2=1. Do not add another factor of two without changing that convention. The temperature constant is a rounded ionization-energy-to-Boltzmann-constant ratio; NIST gives hydrogen’s ionization energy near 13.5984 eV.

This model excludes molecules, excited-state corrections, other elements, non-LTE radiation effects and pressure ionization. Temperature and density both matter. The examples are illustrative gas states, not measurements or complete models of the Sun, Vega or a stellar spectral class. Balmer-line strengths also require excitation populations and radiative transfer; x alone does not predict their peak.

Enter temperature and hydrogen-nucleus density

B2 initially contains 9500 K and B3 contains 1E20 m^−3. B3 is not electron density; electron density follows from x×B3 under these assumptions. Convert a value in cm^−3 to m^−3 by multiplying by 1E6.

C2 validates the two inputs with the formula below. C3 holds ln(K), C4 holds the unrounded ionized fraction, and C5 displays it rounded to four decimals. Blank, text, zero and negative inputs produce a message.

=IF(COUNT(B2:B3)<>2,"Enter numeric temperature and density",IF(MIN(B2:B3)<=0,"Temperature and density must be positive","OK"))

Compute the logarithm of the Saha ratio

C3 computes L=ln(K) rather than first multiplying a large T^(3/2) by an exponential. This avoids intermediate overflow and avoids discarding very small K before solving for x.

K is dimensionless; the coefficient is expressed for SI number density. The direct expression K=2.41468E21*T^1.5*EXP(-157809/T)/n is useful for algebra, but the worksheet uses its logarithm for numerical stability.

=IF(C2<>"OK",C2,LN(2.41468E21)+1.5*LN(B2)-157809/B2-LN(B3))

Solve the quadratic without subtracting nearly equal numbers

The positive algebraic root is (SQRT(K^2+4K)−K)/2. For large K, its subtraction loses precision and can return zero instead of a value near one. C4 uses the equivalent stable formula below.

When L>=0, x=2/(1+SQRT(1+4*EXP(-L))). When L<0, set q=EXP(L/2) and use x=2q/(SQRT(q^2+4)+q). The implementation defines smallRoot=EXP(-ABS(L)/2), so every exponential argument is non-positive even if both branches are evaluated.

N(C3) supplies a numeric intermediate while the outer audit reports invalid inputs. It does not make invalid data acceptable. Very extreme numerical cases can still round to exactly zero or one in floating-point arithmetic; that is not proof of an exactly neutral or completely ionized physical gas.

=IF(C2<>"OK",C2,LET(logRatio,N(C3),smallRoot,EXP(-ABS(logRatio)/2),IF(logRatio>=0,2/(1+SQRT(1+4*smallRoot^2)),2*smallRoot/(SQRT(smallRoot^2+4)+smallRoot))))

Read and experiment with the results

For T=9500 K and n=1E20 m^−3, C5 displays 0.6706, meaning approximately 67.06%. Raising T to 15000 K at the same density displays 0.9992. At 5778 K and 1E23 m^−3, it displays 0.0001; use the unrounded C4 to inspect this small fraction.

Use =IF(ISNUMBER(C4),ROUND(C4,4),C4) for C5. Do not feed the four-decimal display back into later calculations. To display a percentage, format C4 as a percentage rather than multiplying it by 100 and also applying percentage formatting.

Try increasing density at a fixed temperature: the ionized fraction decreases in this model. The tests also verify x^2/(1−x)=K for moderate values, plus numerical cases where the original subtraction loses precision. Tests outside plausible plasma conditions check arithmetic only, not model validity.

Try the example

Use the original values or change them to test your understanding.