__Types of Spectra__

The physics of discharges is severely complicated. Without going into too much detail, the spectra emitted fall under three categories:

*Linear spectra*, from excited element gases or vapors.*Banded spectra*, from excited molecules.*Continuous spectra*, from highly heated or ionized matter.

__The Self-Absorption Phenomenon__

Unofficially, there is a fourth category, the *mixed* or
*degenerate* kind, which are spectra which may contain linearity and
continuity simultaneously (Na). In reality,
whenever continuum is to be found around or close to certain lines in
the emission spectrum, the phenomenon of *self-reversal* takes
place. This phenomenon, otherwise known as *spontaneous reversal* or the *corona phenomenon*^{[1]}, occurs whenever more energy than the energy
required to ionize a certain electron is being provided for a certain
transition in the atom involved. Look at the figure below, depicting a
simplistic schematic for a sodium discharge tube:

High Pressure sodium lamp discharge tube

Now look at the self-absorption phenomenon as seen through the Phasmatron spectroscope, displaying precisely this behavior optically:

You can see the continuum emitted by the arc around the absorption band, and then the dark absorption band with the two Sodium lines in the middle. So the phenomenon actually occurs because Kirchoff absorption occurs inside the tube, along with continuous emission. A more detailed spectral distribution to the right shows clearly what's happening. The absorption that occurs on the D1/D2 doublet is clearly visible.

Self-absorption usually happens with resonance lines, i.e. with lines whose transitions terminate at the ground state. For example, the sodium D1/D2 lines above are resonance lines. However, when the density of the discharge plasma is high, enough absorption on the resonance lines causes self-absorption on other lines as well. Here is a similar absorption which happens on the lines of Mercury under high pressure, although less pronounced. This is the blue Mercury line, followed by the corresponding spectral distribution.

In general, most conventional discharge sources used for lighting
exhibit this phenomenon, even on non-resonance lines, whenever the plasma pressure is high. Most of the times though, the spectroscopes
are not powerful enough to show self-reversal because the width of the
absorption within the line is of an order of a couple of Angstroms,
sometimes even less than an Angstrom^{[2]}. The second photo before the end
in the Hg document (same as the one above), which shows the blue Mercury line,
needs at least an R=50,000 to show the effect. The effect is less
obvious as the ionization levels for a particular element go up in eV and as one moves to non-resonance lines.
Sodium displays this behavior because it has a relatively low
ionization energy level and because the D doublet is a resonance line. Mercury on the other hand needs much higher
energies to ionize. In any case, alkali are prone to showing the effect
on relatively low power spectroscopes, as are some easily ionizable
metals. The effect can be seen on Li, Ca
and sodium sometimes using an R as low as 700-800. For color photographs which show this effect, consult The Double Amici Prism Hand-Held Spectroscope^{[3]}.

__Approximating the Profile of Lines Suffering Self-absorption__

The light distribution around the emission wavelength of specific lines can be approximated using the Lorentzian distribution. Let's use Maple to try to approximate the complete profile of the phenomenon as a function of time for the High Pressure Sodium Lamp. We start with the basics:

> restart;

> with(plots):

> K:=C->C+273.15;#Celsius -> Kelvin

> C:=K->K-273.15;#Kelvin -> Celsius

[3] gives the temperature of the hottest spot of the discharge tube of this lamp, as 1200°C. The usable temperature range therefore, is 18°C-1200°C.

> t0:=0;#start time

> t1:=1;#end time

> Tr:=K(18);#T radiating in K

> Ti:=K(1200);#T ionization in K

> eq:=(T-Tr)/(t-t0)=(Ti-Tr)/(t1-t0);#equation of temperature as function of time

> T:=unapply(solve(eq,T),t);#solve as a function of our t

Check:

> C(T(t0)),C(T(t1));

18.0000000, 1200.000000

> plot(C(T(t)),t=t0..t1);

High Pressure Sodium lamp discharge tube temperature as a function of time

Now we need the Thermal Conductivity for sodium *vapor*. We consult "Thermal Transport" document in [4] and we get a small table for the Thermal Conductivity of sodium vapor, which we re-measure *carefully*^{[4]}.

Sodium vapor Thermal Conductivity as a function of temperature

The data is measured and put in a list which we can manipulate with Maple:

> with(CurveFitting):

> LT:=[seq(700+n*100,n=0..8)];#temperatures

LT := [700, 800, 900, 1000, 1100, 1200, 1300, 1400, 1500]

> s:=(0.0350-0.030)/10;#temp step size

> Ld:=[4.9,15,24.5,30.5,34.8,37.2,38.5,39.2,40];#chart measure in steps

Ld := [4.9, 15, 24.5, 30.5, 34.8, 37.2, 38.5, 39.2, 40]

> Lk:=[seq(0.030+Ld[n]*s,n=1..nops(Ld))]:#store conductivity as a list

> k:=unapply(PolynomialInterpolation(LT, Lk,T),T):#find interpolating polynomial for data

The sodium vapor Thermal Conductivity data is available only for the range 700°K-1500°K. This corresponds to 426.85°C-1226.85°C. Ti=T(t1) falls inside this range, so we are ok on this, but T(t0)=Tr does not. There's nothing we can do about that, so we will just use the resulting interpolating polynomial as we get it from the data^{[5]}. The resulting interpolating polynomial is:

> plot(k(T(t)),t=t0..t1);

Sodium vapor Thermal Conductivity k(T(t)) as a function of time from the interpolating polynomial

Next we need to model the High Pressure Sodium lamp inner discharge tube. How do we do that? In the most obvious way. As an energy pulse of thickness equal to the tube thickness. The inner discharge tube has an average thickness of 5/1000, so,

> H:=Heaviside;

> g:=unapply(piecewise(x<=2.5/1000,T(t)),x,t);#half thickness 2.5/1000

> plot(g(x,0),x=0..10/1000);#tube temp @t0

High pressure sodium vapor discharge tube initial energy profile

The problem is now a simple Heat Equation problem. Specifically, we are dealing with the solution for the *homogeneous* case (sodium vapor throughout), so the general solution is given in terms of the Thermal Conductivity of sodium vapor k(T(t)) (which we have, above) as:

Homogeneous Heat Equation Solution

> c:='c';#normalization constant

> Phi:=(x,t)->c/sqrt(4*k(T(t))*t)*exp(-x^2/(4*k(T(t))*t));#solution for homogeneous Heat Equation

> u:=unapply(int(Phi(x-y,t)*g(y,t0),y=0..1),x,t):#convolute

> c:=solve(int(Phi(x,t0+1e-10),x=0..1)=1/2,c);#normalize

> p1:=plot(g(x,0),x=0..10/1000,color=red):

> p2:=plot(u(x,1/(10^9)),x=0..10/1000,color=green):

> p3:=plot(u(x,1/(10^8)),x=0..10/1000,color=blue):

> p4:=plot(u(x,1/(10^7)),x=0..10/1000,color=magenta):

> p5:=plot(u(x,1/(10^6)),x=0..10/1000,color=cyan):

> p6:=plot(u(x,1/(10^5)),x=0..10/1000,color=brown):

> display({p1,p2,p3,p4,p5,p6});

Homogeneous Heat Equation Solution graphs for T(t0) for the sodium vapor discharge tube as a function of time

We are done with the preliminaries. We now approximate the Lorentzian profile of an emission line^{[6]}:

> Lp:=(w,w0,gamma)->1/(1+((w-w0)/gamma)^2);#Lorentzian distribution

> L:=proc(T1,T2,wr)

> local s;

> if T2<=Tb then #if T2 of outer vapor less than Tb, show just emission

> s:=plot(Lp(w,0,(T1-Tr)/(Ti-Tr)),w=-wr..wr);

> else #if T2 of outer vapor greater than Tb, then we have absorption!

> s:=plot(Lp(w,0,(T1-Tr)/(Ti-Tr))-Lp(w,0,(T2-Tb)/(Ti-Tb)),w=-wr..wr);

> fi;

> s;

> end:

Now we approximate the behavior as a function of time for all temperatures between t0 and t1:

> N:=30;

> dt:=(t1-t0)/N;

> u:='u';u:=array(0..N):

> for n from 0 to N do

> c:='c';Phi:='Phi';Phi:=(x,t)->c/sqrt(4*k(T(t))*t)*exp(-x^2/(4*k(T(t))*t));

> u[n]:=unapply(int(Phi(x-y,t)*g(y,t0+n*dt),y=0..1),x,t):

> c:=solve(int(Phi(x,t0+n*dt+1e-10),x=0..1)=1/2,c);

> od:

> LP:=[seq(L(T(t0+n*dt),u[n](1.25/1000,t0+8e-8),15),n=1..N)]:#animation list

> for n from N+1 to N+15 do

> LP:=[op(LP),L(T(t0+N*dt),u[N](1.25/1000,t0+8e-8),15)];#add some extra frames

> od:

> display(LP,insequence=true);

Profile of the sodium D line suffering self-absorption in a High Pressure Sodium Lamp

as a function of the lamp discharge tube warm-up time t, with T(t0)=18°C and T(t1)=1200°C

- There is another phenomenon called
*corona discharge*, which refers to electric discharges of a certain type. This is not the phenomenon the author is discussing here. - The actual width of the lines in arc generated spectra is never infinitely thin. It is always of finite width äë, which depends on the energy given to the arc. The only known äë~0 sources were constructed around 1926 by Bogros and Paul using atomic beams and electron guns, but had the disadvantage that the lines were weak.
- The best spectroscopes to view self-absorption in detail are spectroscopes with Fabry-Perot etalons which can reach resolving powers of the order of 1,000,000.
- Note that we are using the chart for sodium
*vapor*conductivity, not the charts for liquid sodium. Any radiating phenomena are caused by sodium*vapor*, not by liquid sodium. - We need the sodium Thermal Conductivity for our
*entire range*of temperatures, because in order to model the line profile, we need to know how heat is transported in the discharge tube for the entire temperature range. The interpolating polynomial gives much bigger values for the temperatures which are out of our range (below 700°K), which is in fairly good agreement with the rest of the charts in document [4] for the Thermal Conductivity of*liquid*sodium. - The Lorentzian profile of the self-reversal phenomenon is simply
*two*Lorentzian profiles superimposed on top of one another. The first profile (inner arc) radiates, the second profile (outer vapor) absorbs. With an important difference: In order for the second profile to start absorbing, it has to have reached a T=Tb. Otherwise there's no*vapor*to do the absorbing. - For example, witness the shape of the 4226.728A Ca line on a Carbon arc below, which has suffered self-absorption:

- G. Bruhat, Cours De Physique Generale: Optique, Sixieme Edition, Paris 1965 (in French).
- C.J. Smith, A Degree Physics Part III: Optics, Indian Edition 1985, Radha Publishing House.
- This author, "The Science of Color, the Emission Spectra of the Elements and Some Lamp Engineering Applications". (online)
- INSC, "Material Properties of Sodium". (online)