A toolbox for air ion and aerosol calculations

*Version 20120205*

The first version of the toolbox was published
in web with aim to make easily available for colleagues the approximation
algorithms, which were developed when compiling the numerical simulator of
atmospheric aerosol nucleation bursts (Tammet and Kulmala,
2005). Afterwards, the toolbox was expanded with additional algorithms. Last
(but not least) update is the function Z_approx_air,
which motivation and explanation is described in the document
Size-mobility-approx.pdf and handouts of
a presentation. The present
version of the toolbox contains 23 algorithms written in standard Pascal and
meeting the requirements of Delphi, Free Pascal and Turbo Pascal. All
algorithms are simultaneously presented as individual text files and as components
of the source code of the test program. The test program is available as
hypertext *A_tools.htm* , as a text file *A_tools.txt*, and as a compiled program file *A_tools.exe*.
The last one can be obtained when unpacking the file *A_tools.exe.zip*. The source code can be made
immediately acceptable for a compiler when the name of downloaded file *A_tools.txt*
is changed to* A_tools.dpr* for Delphi or to *A_tools.pas* for Free
Pascal or Turbo Pascal. Free Pascal and Turbo Pascal require deleting of some lines
as explained in the text of the file.

The algorithms can be freely used, modified and translated into different programming languages. They are published without any warranty and some algorithms are not thoroughly checked. The author of the toolbox will be thankful for information about possible problems and solutions as well as for confirmation of successful applications.

The 23 algorithms are implemented as 22 functions and one procedure:

- electrical_mobility_air
- mass_diameter_air
- electrical_mobility_1_air
- mass_diameter_1_air
- Z_approx_air
- d_approx_air
- plain_Millikan_mobility
- plain_Millikan_diameter
- attachment_coefficient
- coagulation_coefficient
- Boltzmann_probability
- Wiedensohler_probability
- charge_probability
- ion_equilibrium
- Knudsen_growthrate
- growthfactor0
- growthfactor1
- growthfactor2
- ion_needlesink
- particle_needlesink
- ion_aerosolsink
- mechanical_mobility
- mechanical_diameter

1. **Function
**electrical_mobility_air

*Arguments*:

- air pressure (mb)
- air temperature (C)
- density of the particle matter ρ (g
cm
^{–3}) - particle number charge (dimensionless)
- mass diameter of the particle
*d =*(6*m*/(πρ))^{1/3}(nm).

*Result*:
Electrical mobility *Z* of the particle (cm^{2}V^{–1}s^{–1}).

*Comments*:
Algorithm is based on the model explained in the paper (Tammet,
1995). The published text consists of two errata that are indicated and
corrected in the programs of the toolbox. Fortunately, these errata did not
have any notable effect in typical applications. The output value is changed to
the electrical mobility *Z*. Mechanical mobility and diffusion coefficient
are

*B* (m s^{–1}N^{–1}) = 6.24×10^{14} *Z*
(cm^{2}V^{–1}s^{–1})

*D* (m^{2} s^{–1}) = 8.61×10^{–9} × *T*
(K) × *Z* (cm^{2}V^{–1}s^{–1}).

Additional changes are:

- gas parameters are included rigidly as
air parameters (use function
*mechanical mobility*in case of different gases), - the original approximations of the air viscosity and free path are replaced by the approximations recommended in standard ISO15900
- the slip factor constants are replaced by
the results of recent research:

*A*= 1.165,*B*= 0.480,*C*= 1.001.

2. **Function**
mass_diameter_air

*Arguments*:

- air pressure (mb)
- air temperature (C)
- density of the particle matter (g cm
^{–3}) - arithmetic number charge of the particle (dimensionless)
- electrical mobility of the particle (cm
^{2}V^{–1}s^{–1}).

*Result*: Mass
diameter of the particle (nm).

*Uses*:
function *electrical_mobility_air*.

*Comment*: The
algorithm inverts the function *electrical_mobility_air*.

3. **Function**
electrical_mobility_1_air

*Arguments*:

- air pressure (mb)
- air temperature (C)
- mass diameter of the particle
*d =*(6*m*/(πρ))^{1/3}(nm).

*Result*:
Electrical mobility *Z* of the particle (cm^{2}V^{–1}s^{–1}).

*Uses*:
function *electrical_mobility_air*.

*Comment*: .
The result is the exactly same as *electrical_mobility_air* for a singly
charged particle of density 2 g cm^{–3}.The extra function is included
for convenience of the user.

4. **Function**
mass_diameter_1_air

*Arguments*:

- air pressure (mb)
- air temperature (C)
- electrical mobility of the particle (cm
^{2}V^{–1}s^{–1}).

*Result*: Mass
diameter of the particle (nm).

*Uses*:
function *mass_diameter_air*.

*Comment*: .
The result is the exactly same as *mass_diameter_air* for a singly charged
particle of density 2 g cm^{–3}.The extra function is included for
convenience of the user.

5. **Function **Z_approx_air

*Arguments*:

- air pressure (mb)
- air temperature (C)
- mass diameter of the particle (nm).

*Result*:
Electrical mobility of the particle (cm^{2}V^{–1}s^{–1}).

*Restrictions*:
Particle should be singly charged and have density of about 2 g cm^{–3.}

If the restrictions are not allowed then
use *electrical_mobility_air *or *mechanical_mobility* instead.

*Comment*:
This function is a mathematical approximation of *electrical_mobility_air*
and its

advantages are simplicity of calculations (no iteration) and comprehensive interpretation in terms of Millikan equation.

6. **Function **d_approx_air

*Arguments*:

- air pressure (mb)
- air temperature (C)

- electrical mobility of the particle (cm
^{2}V^{–1}s^{–1}).

*Result*: Mass
diameter of the particle (nm).

*Uses*: function
*Z_approx_air*.

*Restrictions*:
Particle should be singly charged and have density of about 2 g cm^{–3.}

*Comment*: The
algorithm inverts the function *Z_approx_air*. The iterative procedure may
not converge in some unrealistic extreme situations. In this case the result
will be 0.

7. **Function
**plain_Millikan_mobility

*Arguments*:

- air pressure (mb)
- air temperature (C)
- mass diameter of the particle (nm).

*Result*:
Electrical mobility of the particle (cm^{2}V^{–1}s^{–1}).

*Restrictions*:
Particle should be singly charged.^{.}

*Comment*:
This function follows ISO15900 and uses corresponding approximations for air
viscosity and free path. Mobility is calculated according to Millikan equation
without diameter complement that is not recommended when particle diameter is
less than 10 nm. For smaller particles a diameter complement is required. Fernandez de la Mora et al. (2003) recommended a
complement of 0.53 nm. Ku et al. (2009) studied particles of
diameter down to 1 nm and used the complement of 0.3 nm. A flexible
size-dependent complement is implemented in function Z_approx_air.

8. **Function
**plain_Millikan_diameter

*Arguments*:

- air pressure (mb)
- air temperature (C)

- electrical mobility of the particle (cm
^{2}V^{–1}s^{–1}).

*Result*: Mass
diameter of the particle (nm).

*Restrictions*:
Particle should be singly charged.

*Comment*:
This function is an inverse of *plain_Millikan_mobility* and user should
consider the comments to this function. The iterative procedure of inversion
may not converge in some unrealistic extreme situations. In this case the
result will be 0.

9. **Function**
attachment_coefficient (ion-to-particle)

*Arguments*:

- number of elementary charges on the particle, attracting with sign –, repelling with +
- mass diameter of the ion (nm)
- density of the ionic matter (g cm
^{–3}) - mass diameter of the particle (nm)
- air temperature (K)
- air pressure (mb).

*Result*:
Attachment coefficient (synonym: combination coefficient) (cm^{3} s^{–1}).

*Uses*:
function *electrical_mobility_air*.

*Restriction*:
Ion is expected to be singly charged.

*Comments*:
The function is a minor modification of the algorithm explained in the paper (Tammet and Kulmala, 2005, Equation 7). It is designed as a
mathematical approximation of numerical results by Hoppel and
Frick (1986) and experimental results by Reischl et
al. (1996). The paper (Tammet and Kulmala, 2005)
contains an erratum in Equation 4, where a coefficient is written 8.616×10^{–8},
but should be 8.616×10^{–9}. Fortunately, this erratum had no effect in
the numerical algorithms.

10. **Function**
coagulation_coefficient
(particle-to-particle, according to Sahni approximation)

*Arguments*:

- mass diameter of the small neutral particle (nm)
- mass diameter of the large charged or neutral particle (nm)
- density of the small particle matter (g
cm
^{–3}) - number charge of the large particle (dimensionless)
- polarizability of the small particle (Å
^{3}) - air temperature (K)
- air pressure (mb).

*Result*:
Coagulation coefficient (cm^{3} s^{–1}).

*Uses*:
function *electrical_mobility*.

*Comments*:

The coagulation coefficient *K* is
calculated for mutual coagulation of two different particles d*N*_{1}/d*t*
= d*N*_{2}/d*t* = –*K N*_{1}* N*_{2}.
The algorithm is proposed and explained in the paper (Tammet
and Kulmala, 2005, see Equations 26 and 31). It is based on a proposal by
Fuchs and Sutugin (1971) to use the Sahni method for interpolation between the
exact models of continuum regime and free molecule regime.

As an addition, the effect of van der Waals
interaction is included in the first approximation by adding an extra distance *h*
to the collision distance. The value of the extra distance is estimated
0.115 nm (Tammet, 1995).

The second addition is including the effect of electric polarization of the small particle in case of charged large particle according to the (∞–4) theoretical model (see McDaniel and Mason, 1973) and using an approximation by Tammet (1995).

Polarizability of a small particle is often
roughly estimated as equal to *r*^{3} and polarizability of a
cluster is estimated as *n* Å^{3}, where *n* is the number of
atoms in the cluster.

11. **Function**
Boltzmann_probability (of charge *qe*
on a particle of diameter *d*)

*Arguments*:

- algebraic number of elementary charges (dimensionless)
- diameter of particle (nm)
- air temperature (K).

*Result*:
Probability to carry given algebraic number of charges in the steady state.

*Comment*: Boltzmann
distribution is valid only in case of relatively large particles, see textbooks
e.g. (Fuchs and Sutugin, 1971). If *d* < 100 nm
then function *charge_probability* is recommended as replacement.

12. **Function**
Wiedensohler_probability (of charge *qe*
on a particle of diameter *d*)

*Arguments*:

- algebraic number of elementary charges
*q*(dimensionless) - diameter of particle (nm)

*Result*:
Probability to carry given algebraic number of charges in the steady state.

*Uses*:
included 6×5 table of numeric coefficients.

*Restriction*:
*q* must be in range of –2 to + 2.

*Presumptions*:
see Wiedensohler (1988).

*Comment*: A simple
and quick procedure that well approximates theoretical results by Hoppel and Frick (1986).

13. **Function**
charge_probability (of charge *qe* on
a particle of diameter *d*)

*Arguments*:

- algebraic number of elementary charges (dimensionless)
- diameter of particle (nm)
- diameter of ions (nm)
- air temperature (K).

*Result*:
Probability to carry given algebraic number of charges in the steady state.

*Uses*:
function *attachment_coefficient*.

*Presumptions*:
ion density 2 g cm^{–3} and air pressure 1013 mb.

*Comment*:
Recommended when the particle diameter is less than 100 nm. In case of large
particles the Boltzmann distribution can be applied, see textbooks e.g. (Fuchs and Sutugin, 1971).

14. **Procedure**
ion_equilibrium

*Arguments*:

- mobility of positive small ions (cm
^{2}V^{–1}s^{–1}) - mobility of negative small ions (cm
^{2}V^{–1}s^{–1}) - ionization rate (cm
^{–3}s^{–1}) - air temperature (K)
- number concentration of background aerosol
particles (cm
^{–3}) - mean diameter of background aerosol particles (nm).

*Results*:

- Number concentration of positive small
ions (cm
^{–3}) - Number concentration of negative small
ions (cm
^{–3}).

*Restriction*:
Ions are expected to be singly charged.

*Comments*:

Procedure solves the system of balance equations:

*dn ^{+}/dt = I – αn^{+}n^{–}
– s_{b}^{+}n^{+}*,

*dn ^{–}/dt = I – αn^{+}n^{–}
– s_{b}^{–}n^{–}*,

where α is the recombination
coefficient (estimated as 1.5×10^{–6} cm^{3}s^{–1}) and
*s* is sink of small ions on background aerosol particles calculated according
to (Tammet and Kulmala, 2007, Equation 6). Space charge
of the background aerosol can be calculated afterwards as it equals to (*n*^{–}
– *n*^{+}) *e*.

*Application*:
calculating of air ion balance considering the deposition of ions on particles
background aerosol.

15. **Function**
Knudsen_growthrate (free molecule regime)

*Arguments*:

- mass diameter of the small neutral particle considered as the growth unit (nm)
- density of the growth unit (g cm
^{–3}) - number concentration of growth units (cm
^{–3}) - air temperature (K).

*Result*:
Plain Knudsen growth rate (nm h^{–1}).

*Comments*:
Nucleating particles are expected to grow by condensation of a substance, which
particles are called the growth units. Growth units are molecules or clusters.
Plain Knudsen growth rate assumes the free molecule regime when growth rate
does not depend on the size of the growing particle. The model is implemented
as discussed in (Tammet and Kulmala, 2005, Equation
37).

*Application*:
modeling of aerosol nucleation.

16. **Function**
growthfactor0 (non-evaporating substance,
method of effective polarizability)

*Arguments*:

- mass diameter of the small neutral particle considered as the growth unit (nm)
- mass diameter of the large charged or neutral particle considered as the growing particle (nm)
- density of the growth unit (g cm
^{–3}) - number charge of the large particle (dimensionless)
- polarizability of the growth unit (Å
^{3}) - air temperature (K)
- air pressure (mb).

*Result*:
Growth factor = (Growth rate) / (Plain Knudsen growth rate).

*Uses*:
function mechanical_mobility.

*Comments*:
The function is recommended for growth units of *d* < 0.7 nm, e.g.
molecules of sulfuric acid. The algorithm is proposed and explained in the
paper (Tammet and Kulmala, 2005) discussing the process
of aerosol nucleation. Nucleating particles are expected to grow by
condensation of a substance, which particles are called the growth units. Growth
units are molecules or clusters. Plain Knudsen growth rate assumes the free
molecule regime. The model discussed in (Tammet and
Kulmala, 2005) considers the growth as coagulation of growing particles
with the growth units and the present function is an application of the
coagulation algorithm.

The distinctive features of the function *growthfactor0*
are: 1) the condensing substance, e.g. sulfuric acid, is expected not to
evaporate back into air, 2) the electrical forces are considered according to
the method of effective polarizability, which is used in the function *coagulation_coefficient*.
Polarizability of a cluster in Å^{3} is often close to the number of
atoms in the cluster. However, there are considerable exceptions, e.g. polarizability
of a molecule of sulfuric acid is about 149 Å^{3}.

*Application*:
modeling of aerosol nucleation.

17. **Function**
growthfactor1 (non-evaporating substance,
method of Nadykto and Yu)

*Arguments*:

- mass diameter of the small neutral particle considered as the growth unit (nm)
- mass diameter of the large charged or neutral particle considered as the growing particle (nm)
- density of the growth unit (g cm
^{–3}) - number charge of the large particle (dimensionless)
- air temperature (K)
- air pressure (mb)
- first dipole enhancement coefficient by Nadykto and Yu (dimensionless)
- second dipole enhancement coefficient by Nadykto and Yu (dimensionless).

*Result*:
Growth factor = (Growth rate) / (Plain Knudsen growth rate).

*Uses*:
function mechanical_mobility.

*Comments*:
The function is recommended for growth units of *d* < 0.7 nm, e.g.
molecules of sulfuric acid. The algorithm is proposed and explained in the
paper (Tammet and Kulmala, 2005) discussing the process
of aerosol nucleation. Nucleating particles are expected to grow by
condensation of a substance, which particles are called the growth units.
Growth units are molecules or clusters. Plain Knudsen growth rate assumes the
free molecule regime. The model discussed in (Tammet and
Kulmala, 2005) considers the growth as coagulation of growing particles
with the growth units and the present function is an application of the
coagulation algorithm.

The distinctive features of the function *growthfactor1*
are: 1) the condensing substance, e.g. sulfuric acid, is expected not to
evaporate back into air, 2) the electrical forces are considered according to
the method by Nadykto and Yu (2003) as explained in the
paper (Tammet and Kulmala, 2005). The recommended
values of the first and second enhancement coefficients for sulfuric acid are
14 and 1.43.

*Application*:
modeling of aerosol nucleation.

18. **Function**
growthfactor2 (includes evaporation of growth
unit substance)

*Arguments*:

- mass diameter of the small neutral particle considered as the growth unit (nm)
- mass diameter of the large charged or neutral particle considered as the growing particle (nm)
- density of the growth unit (g cm
^{–3}) - arithmetic number charge of the large particle (dimensionless)
- polarizability of the growth unit (Å
^{3}) - air temperature (K)
- air pressure (mb).

*Result*:
Growth factor = (Growth rate) / (Plain Knudsen growth rate).

*Uses*:
function mechanical_mobility.

*Comments*:
The model considers the growth as coagulation of growing particles with the
growth units and the coagulation coefficient is calculated in like manner as in
the function *coagulation_coefficient*. The function is recommended for
growth units of *d* > 0.7 nm, e.g. molecules and clusters of organic
substances, which are able to evaporate back into the air. The algorithm is
proposed and explained in the paper (Tammet and Kulmala,
2005) discussing the process of aerosol nucleation.

The distinctive features of the function *growthfactor2*
are: 1) the growing particle are expected to have passed the threshold size of
the nano-Köhler model (Kulmala et al., 2004) and the
condensing substance can partially evaporate back into air, which is typical
for some organic compounds, 2) the electrical forces are considered according
to the method of effective polarizability in the same way as in the function *coagulation_coefficient*.

*Application*:
modeling of aerosol nucleation.

19. **Function**
ion_needlesink (–(d*n*/d*t*)/*n*)

*Arguments*:

- air ion mobility (cm
^{2}V^{–1}s^{–1}) - diameter of needles (mm)
- total length of needles in a volume unit
(m
^{–2}) - wind speed (m s
^{–1}) - air temperature (K)
- air pressure (mb).

*Result*: a
component of sink in the equation of the air ion balance (s^{–1}).

*Restriction*:
ion is expected to be singly charged.

*Comments*:
Algorithm is proposed and explained in the paper (Tammet
and Kulmala 2007). The deposition of ions is calculated according to the
Churchill-Bernstein heat transfer equation (see Incropera
and Dewitt, 2002) translated into the terms of particle diffusion.

*Application*:
calculating of air ion balance in a conifer forest.

20. **Function**
particle_needlesink (–(d*n*/d*t*)/*n*)

*Arguments*:

- diameter of particles (nm)
- diameter of needles (mm)
- total length of needles in a volume unit
(m
^{–2}) - wind speed (m s
^{–1}) - air temperature (K)
- air pressure (mb).

*Result*: a
component of sink in the equation of the air ion balance (s^{–1}).

*Uses*:
function mechanical_mobility.

*Restriction*:
particles are expected to be in the nanometer size range so that the deposition
is controlled only by diffusion.

*Comments*:
Algorithm is proposed and explained in the paper (Tammet
and Kulmala, 2007). The deposition of particles is calculated according to
the Churchill-Bernstein heat transfer equation (see Incropera
and Dewitt, 2002) translated into the terms of particle diffusion.

*Application*:
calculating of nanometer particle balance in a conifer forest.

21. **Function**
ion_aerosolsink (–(d*n*/d*t*)/*n*)

*Arguments*:

- air ion mobility (cm
^{2}V^{–1}s^{–1}) - polarity of ions (+1 or –1)
- number concentration of aerosol particles
(cm
^{–3}) - mean diameter of aerosol particles (nm)
- algebraic mean charge number of aerosol particles (dimensionless)
- air temperature (K).

*Result*: sink
in the equation of the air ion balance (s^{–1}).

*Restriction*:
Ion is expected to be singly charged.

*Comment*: Algorithm
presents an approximation proposed and explained in the papers (Tammet et al., 2006, Equation 10; Tammet
and Kulmala, 2007, Equation 6).

*Application*:
Calculating of air ion balance considering the deposition of ions on particles
background aerosol.

22. **Function**
mechanical_mobility

*Arguments*:

- molar mass of the carrier gas (amu)
- polarizability of gas molecules (nm
^{3}) - constant
*a*in the equation of the collision size (nm), see comments below - constant
*b*in the equation of the collision size (K) - constant
*c*in the equation of the collision size (dimensionless) - gas pressure (mb)
- gas temperature (K)
- density of the particle matter (g cm
^{–3}) - particle number charge (dimensionless)
- mass diameter of the particle (nm).

*Result*:
Mechanical mobility *B* of the particle (m s^{–1} fN^{–1}).

*Comments*:

Algorithm is proposed and explained in the
paper (Tammet, 1995), where it consists of two errata
that are indicated and corrected in the presented files. Fortunately, these
errata did not have any notable effect in typical applications. Collision size
of a gas molecule is approximated as *d = a* (1 + (*b/T*)*c*),
see (Tammet, 1995, Equation 3).

Typical values of the first six arguments are shown in the Pascal code.

Mass diameter is defined as *d = *(6*m*/(πρ))^{1/3
}, where ρ is density of the particle matter.

If a numerical result is *B* then the
numerical value of mechanical mobility in SI is 10^{–15}*B* and in
CGS 10^{–12}*B*. The electrical mobility of a particle is 1.602×*B*
cm^{2}V^{–1}s^{–1}.

23. **Function**
mechanical_diameter

*Arguments*:

- molar mass of the carrier gas (amu)
- polarizability of gas molecules (nm
^{3}) - constant
*a*in the equation of the collision size (nm), see comments below - constant
*b*in the equation of the collision size (K) - constant
*c*in the equation of the collision size (dimensionless) - gas pressure (mb)
- gas temperature (K)
- density of the particle matter (g cm
^{–3}) - particle number charge (dimensionless)

- mechanical mobility of the particle (m s
^{–1}fN^{–1}).

*Uses*:
function *mechanical_mobility*.

*Result*: Mass
diameter of the particle (nm).

*Comment*: The
algorithm inverts the function *mechanical_mobility*. The iterative
procedure may not converge in some extreme situations. In this case the result
will be 0.

**References**

Fernandez de la Mora, J., de Juan, L., Liedtke, K. Schmidt-Ott, A. (2003) Mass and size determination of nanometer particles by means of mobility analysis and focused impaction, J. Aerosol Sci. 34, 79–98.

Fuchs N.A. & Sutugin
A.G. (1971) High-dispersed aerosols. In: Hidy G.M & Brock* *J.R.
(eds.), *Topics in current aerosol research*, Pergamon Press, pp. 1–60.

Hoppel, W.A. & Frick,
G.M. (1986) Ion-aerosol attachment coefficients and steady-state charge
distribution on aerosols in a bipolar ion environment. *Aerosol Sci. Technol*.
**5**, 1–21.

Kim, J.H., Mulholland, G.W, Kukuck, S.R., Pui, D.Y.H. (2005) Slip Correction Measurements of Certified PSL Nanoparticles Using a Nanometer Differential Mobility Analyzer (Nano-DMA) for Knudsen Number From 0.5 to 83. J. Res. Natl. Inst. Stand. Technol. 110, 31-54.

Ku B.K. and Fernandez de la Mora J. (2009). Relation between Electrical Mobility, Mass, and Size for Nanodrops 1-6.5 nm in Diameter in Air. Aerosol Science and Technology 43, 241-249.

Kulmala, M.,
Kerminen, V-M., Anttila, T., Laaksonen, A. & O'Dowd, C.D. (2004) Organic
aerosol formation via sulphate cluster activation. *J. Geophys. Res*., **109**,
D04205, http://dx.doi.org/doi:10.1029/2003JD003961.

Incropera F.P.
& Dewitt, D.P. (2002) *Fundamentals of Heat and Mass Transfer*, Wiley,
New York.

McDaniel, E.W. &
Mason, E.A. (1973) *The mobility and diffusion of ions in gases*. New
York: John Wiley.

Nadykto, A.B. &
Yu, F. (2003) Uptake of neutral polar vapor molecules by charged
clusters/particles: Enhancement due to the dipole-charge interaction. *J. Geophys.
Res*., **108**(D23), 4717, http://dx.doi.org/doi:10.1029/2003JD003664.

Reischl, G.P., Mäkelä,
J.M., Karch, R., & Necid, J. (1996) Bipolar charging of ultrafine particles
in the size range below 10 nm. *J. Aerosol Sci*., **27**, 931–949.

Tammet, H (1995)
Size and mobility of nanometer particles, clusters and ions. *J. Aerosol Sci.*,
**26**, 459–475.

Tammet H. &
Kulmala M. 2005. Simulation tool for atmospheric aerosol nucleation bursts. *J.
Aerosol Sci.* 36: 173–196. http://dx.doi.org/doi:10.1016/j.jaerosci.2004.08.004.

Tammet H., Hõrrak
U., Laakso L., & Kulmala M. (2006) Factors of air ion balance in a
coniferous forest according to measurements in Hyytiälä, Finland. *Atmos.
Chem. Phys., ***6**, 3377–3390, http://www.atmos-chem-phys.net/6/3377/2006/

Tammet, H. &
Kulmala, M. (2007) Simulating aerosol nucleation bursts in a coniferous forest,
*Boreal Env. Res*., **12**, 421–430, http://www.borenv.net/BER/pdfs/ber12/ber12-421.pdf.

Wiedensohler, A. (1988) An approximation of the bipolar charge distribution for particles in the submicron size range, J. Aerosol Sci., 19, 387–389.