IONIZATION, SAHA EQUATION

1 IONIZATION, SAHA EQUATIONLet the energies of two states,AandB, beEAandEB, and their statistical weightsgAandgB,respectively. In LTE (Local Thermodynamic Equilibrium) the number of particles in the two states,NAandNB, satisfies Boltzman EQUATION :NANB=gAgBexp [ (EA EB)/kT].( )Now, we shall consider two ions, i and i+1 , of the same element. The ionization potential, the energy needed to ionize i from the ground state is , and the statistical weights of theground states of the two ions aregiandgi+1, respectively. The number densities, [ cm 3], of thetwo types of ions and free electrons areni,ni+1, andne, respectively.

2 We shall use the Boltzmanequation ( ) to estimate the number rationi+1/ni. The statistical weight of an ion in the lowerionization state to be used in the EQUATION ( ) is justgi. The statistical weight of an ion in theupper ionization state isgi+1multiplied by the number of possible states in which a free electronmay be put. As we know, in every cell of a phase space with a volumeh3there are two possiblestates for an electron, because there are two possible orientations of its 10 27erg sis the Planck constant. The energy of a free electron with a momentumpwith respect to the groundstate of an ion in a lower ionization state isE= +p2/2m.

3 The number of cells available for freeelectrons with a momentum betweenpandp+dpisVe4 p2dp/h3, whereVe= 1/neis the volumein ordinary space available per electron, andneis the free electron number density. Now, we shallintegrate over all available cells, taking the Boltzman factor into accountni+1ni=gi+1gi2 Veh3 0e ( +p2/2m)/kT4 p2dp=( )gi+1gi1ne2h3(2mkT)3/2e /kT2 0e xx1/2dx=gi+1gi1ne2h3(2 mkT)3/2e /kT,where we setx=p2/2mkT, and the value of the last integral was 1/2 last EQUATION is usually written as the Saha equationni+1neni=(2 mkT) +1gie /kT.( )It is customary to express the ionization energy in electronvolts, 1 eV = 10 of the most abundant elements, hydrogen and helium, is important for the EQUATION ofstate.

4 For these elements we have:ionization of hydrogen: = , 2gi+1/gi= 1,first ionization of helium: = , 2gi+1/gi= 4,second ionization of helium: = , 2gi+1/gi= now pure, partly ionized hydrogen. LetnH,nHI, andnHIIbe number density of allhydrogen, neutral hydrogen atoms, and hydrogen ions, respectively. We havenH=nHI+nHII,ne=nHII, andx nHII/nHis the degree of ionization. The density of gas is =HnH, whereHis the mass of a hydrogen atom. We may write the Saha EQUATION asi 1nHIInenHI= Hx21 x=(2 mkT) +1gie /kT.( )The constants in cgs units are:H= 10 24,m= 10 28,k= 10 16,h= 10 27. With these constants we havelog + logx21 x=( )= logT 5040T + log[H(2 mk) 3]= logT 68240T pressure of partially ionized hydrogen gas is simplyPg= (nH+ne)kT= (1 +x)kH T.

5 ( )The internal energy should now include not only kinetic, butalso ionization energy:Ug= (nH+ne)kT+ne = +x H .( )In these equations the degree of ionization should be treated as a function of density and tem-perature,x( , T). Differentiating EQUATION ( ) we obtain( x lnT) =x(1 x)(2 x)( + kT),( )( x ln )T= x(1 x)(2 x).( )As an example of the thermodynamic properties of partially ionized hydrogen we shall calculatespecific heat of hydrogen gas at constant volume:cV,g [ (Ug/ ) T] = ( Pg T) + H( x T) =( )=kH[ (1 +x) + ( + /kT)2x(1 x)(2 x)].When the hydrogen is 50% ionized, , thencV,g=kH[ +16( + /kT)2].( )Typically, we have /kT 1, and therefore the specific heat for partly ionized gas is much higherthan for neutral or fully ionized us consider now a mixture of partially ionized hydrogen with radiation.

6 The EQUATION ofstate isPg= (1 +x)kH T,Pr=a3T4,P=Pg+Pr, =Pg/P,( )P= (1 +x)kH T+a3T4,( )U= (1 +x)kH T+aT4+x H ,( )The equations in a differential form are:dlnP= (4 3 )dlnT+ dln + 1 +xdx( EQUATION of state),( )i 2dUP= (12 )dlnT+(32+ kT) dln +(32+ kT) 1 +xdx,( )(2 x)x(1 x)dx=(32+ kT)dlnT dln (Saha EQUATION ).( )From the Saha EQUATION we obtain equations ( ) and ( ). Those may be combined withthe equations ( ) and ( ) to get( lnP lnT) = 4 3 + (1 +x)( x lnT) ,( )( lnP ln )T= + (1 +x)( x ln )T,( )TP( U T) = 12 + (1 +x)( + kT)( x lnT) .( )Now we have all the derivatives that are needed to calculate the adiabatic relations: , ad,and ( lnT / ln )S, as well as the specific heats,cVandcP, for the mixture of partially ionizedhydrogen and 3