Before we get started here's a compilation of typical values of atmospheric electrical parameters at different altitudes.

We'll spend the first portion of this lecture finishing up the discussion of small ions and small ion balance equations.  Last class we left with a small ion balance equation that contained ionization rate (production) and recomination rate (loss) terms.



Today we will add two additional small ion loss terms.  A small ion can attach to an uncharged particle, creating a charged particle or a so-called "large ion". 



Or a small ion of one polarity can attach to a charged particle of the opposite polarity creating an unchared particle (provided the small ion and the particle have equal quantities of charge).  These two new terms are included in the small ion balance equation below.  The β+o and β+- terms are referred to as "attachment coefficients."



We often assume that the concentrations of positive and negative small ions and the concentrations of positively and negatively charged particles are equal.  Let's also make the following assumptions concerning the attachment coefficients.

The balance equation then becomes

The bottom equation is just a simplification of the top equation.  A total particle concentration term, Z, is used rather than keeping track of the concentrations of charged and uncharged particles.

The figure below (from The Earth's Electrical Environment reference) illustrates how ion-particle attachment begins to significantly reduce small ion concentrations beginning at particles concentrations of about 1000 cm-3


In some past editions of this course we have spent close to a full class period looking at how you might derive expressions for the ion-particle attachment coefficients.  I'm not sure that's really necessary here.  Though I have added supplementary notes that you can look at if you are interested.  You can find them here.  The first part of these supplementary notes deals with the attachment to uncharged particles, the second part considers attachment to charged particles.



We will spend some time considering what fraction of particles are uncharged and charged.  We'll start with a large ion (charged particle) balance equation.



N in this equation can represent the concentration of either positively or negatively charged particles.  Large ions are created when a small ion attaches to an uncharged particle.  They are destroyed when a small ion attaches to a charged particle of the opposite polarity .

Under steady state conditions

Now we'll look at the fraction of large and small particles that are uncharged.  In the supplementary notes we show that β0 = β1 for large particles.  In that case



For large particles you would expect to find equal numbers of positively charged, negatively charged, and non-charged particles.

For small particles




The agreement between predictions and measurements of the uncharged fraction (No/Z) is not very good for small particles.  If the table had been extended to larger particles, the predicted No/Z would approach 0.33.

Better agreement is obtained using Boltzmann statistics. 


A charged particle has a certain amount of "stored" energy associated with it.  Thus we can use the Boltzmann distribution above to predict the distribution of charged particles (the particles can carry only integral multiples of an electronic charge, i.e. Q = me, where m is an integer and e is the charge on an electron).

The energy stored on a charged sphere is (click here to see the details of the derivation)

We can insert this expression into the Boltzmann distribution equation above.


A temperature of 300 K was assumed in the calculation above.  The exponential starts to become pretty small for particles with radii less that 2.8 x 10-6 cm (especially if m > 1).  So we can see that most small particles will be uncharged.  Those that are charged will mostly just carry 1 electronic charge.  For example



Now we can compare predictions of the uncharged fraction of particles with measurements.  Here are the details of the predicted value for a particle with radius = 10-6 cm.

This agrees much better with the measured value.  The table shown earlier is reproduced below.  This time Boltzmann statistics are used to predict the uncharged fraction.


The agreement between measured and predicted values is much better.  Again, if the table had been extended to larger particles, we would expect the predicted No/Z to approach 0.33.