Why is there a need for UHV?

There are many different classification ranges of vacuum conditions, and to my knowledge there is no one unified standard that everyone goes by. The terminology will vary slightly depending on who you may talk to, however a general guideline for the vacuum classifications goes like this:

1 – 10^{-3 Torr}
------------------------------- Rough (Low) Vacuum

10^{-3}
– 10^{-5} Torr ----------------------------
Medium Vacuum (MV)

10^{-6}
– 10^{-8} Torr ----------------------------
High Vacuum (HV)

10^{-9}
– 10^{-12} Torr ---------------------------
Ultra High Vacuum (UHV)

< 10^{-13}
Torr --------------------------------- Extremely High
Vacuum (XHV)

0 Torr --------------------------------------- Perfect Vacuum (unattainable)

It is worth mentioning that all of
these pressure measurements are measured in Torr
and not the SI units of Pascals (Pa), typically Torr is used in most labs. Occasionally labs will report
pressures in mbar, which is 1/100^{th} of a Pascal. Conversions for pressure are:

1 Torr = 1.333224 mbar

1 Torr = 133.3224 Pascals

1 Torr = 0.0013157 atm

1 Torr = 0.0193367 PSI

While vacuum conditions are important in everyday life for items such as the incandescent light bulb, CRT’s, and vacuum cleaners….., the need for vacuum conditions for life as a surface scientist is inescapable. There are two main reasons why vacuum conditions are necessary: 1) To reduce the number of molecules in the chamber to a point where the gas density is small enough to permit experiments involving electrons and ions modest travel distances without interference from gas phase scattering. And 2) most importantly, provide an environment that a sample surface may be cleaned and maintained in a cleaned state for durations of time needed for experiments.

To start with, the density of gas molecules in the vacuum chamber may be estimated by the ideal gas law:

_{}

Where :

_{}

Rearranging the ideal gas law gives:

_{}

with N being the total number of molecules of the gas: _{}

The density of gas in the chamber is the number of molecules per volume:

_{}

As the density of gas molecules in the vacuum chamber decreases, the mean free path of particles increase. The calculations for the mean free paths of particles (electrons, ions, atoms, or molecules) can be obtained from a simple bimolecular collision model (hard-sphere collisions).

The effective collisional area of two molecules the same size can be modeled by:

_{}

Over time, the molecule of interest will travel through the chamber sweeping through a volume of space that is dependant on the cross sectional area A and the velocity of the molecule.

_{}

Then by taking the length the molecules travels νt, and dividing that by the volume times the density of molecules (ρ). An estimate of the mean free path may be found.

_{}

Two adjustments to the above
equation need to be made. First the average velocity of the molecule has to be
adjusted. In the current equation the velocity of only one molecule is
considered and the other is required to be at rest, However this is not correct
and a releative velocity needs to be added which
consists of multiplying the average velocity by the square root of 2. _{}. Secondly the density
of the molecules should be converted into something more familiar. By using the
derived density of molecules from the ideal gas law (seen previously) a mean
free path equation can be obtained.

_{}

Pressure: mean free path λ:

760 Torr 3.35
x 10^{-8 }m

1 Torr 2.55
x 10^{-5 m}

0.01 Torr 2.55
x 10^{-3 m}

1 x 10^{-6} Torr 2.55
x 10^{1 }m

1 x 10^{-11} Torr \ 2.55
x 10^{6 }m
(or 1530 miles)

These estimates are for molecules at T=273 K and a molecular diameter of 5 angstroms. Therefore it is seen that at atmospheric pressures, the mean free path of molecules are very short. Also, that most molecule/ion/electron experiments have a sufficient mean free path once the high vacuum pressure range is achieved. However, some experiments and instruments such as a particle accelerators, and free electron lasers will require pressures in the ultra high vacuum range to provide very long mean free paths for molecules to travel.

The main reason
that surface scientist need UHV conditions is to maintain the sample surfaces
in a known condition without contamination during the time scale of an
experiment. Sometimes a typical experimental run can be a matter of minutes,
such as running a thermal programmed desorption (TPD)
experiment. Other experiments such as one using a scanning tunneling
microscope (STM) can run for several hours. A general guidline
that is used for UHV work is that at a pressure of 1 x 10^{-6} Torr the crystal surface will be contaminated with a
monolayer of gas in 1 second. Then every order of magnitude drop in pressure
from there will result in an order of magnitude increase in time it take the crystal to be contaminated with a monolayer of material.

However, to do a more rigorous calculation of the time required to form a monolayer of contaminate, the incident flux of gas molecules needs to be calculated. The incident flux is related to the gas density at the surface by:

_{}

Where :

_{}

Now when equations are substituted into the flux equation above we get:

_{}

By pulling out what we can, the equation ends up as:

_{}

In the equation above it is important to note that:

1) the pressure is in pascals

2) _{A}m

3) Temperature is in Kelvins (K)

Pressure Flux of molecules number of times each surface

per square meter seconds atom is struck per second

760 Torr 2.90 x 10^{27} 2.908
x 10^{8}

1 Torr 3.83 x 10^{24} 3.826
x 10^{5}

1 x 10^{-3} 3.83 x 10^{21
} 383

1 x 10^{-6} 3.83 x 10^{18} 0.383

1 x 10^{-9} 3.83
x 10^{15} 3.83
x 10^{-4}

1 x 10^{-11} 3.83
x 10^{13} 3.83
x 10^{-6}

Now that the flux of gas molecules impacting with the surface has been calculated the sticking coefficient will be discussed. The sticking coefficient, θ is a ratio of the number of molecules that impact with the surface verses the number of molecules that adsorb to the surface. Typically the sticking coefficient falls in the range of 1 to 0, where a coefficient of one indicates complete adsorption, and a coefficient of zero is no adsorption. The sticking coefficient can depend upon a multitude of factors such as temperature, surface coverage, crystal face.

By assuming
the sticking coefficient is unity, a lower bound of time per monolayer
contamination coverage can be calculated. (Note: the number of atom on the
surface is typically on the order of 10^{15 } per cm^{2} or 10^{19}
per m^{2}.)

Pressure Time / ML

(Torr) (seconds)

760 3.44
x 10^{-9}

1 2.61
x 10^{-6}

1 x 10^{-3} 2.61
x 10^{-3}

1 x 10^{-6} 2.61

1 x 10^{-9} 2.61
x 10^{3}

1 x 10^{-11} 2.61
x 10^{5}

To briefly recap what has been show
here, the requirements for typical experiments in surface science require a
pressure less than 10^{-4} Torr for useable
collision free conditions, and pressures less than 10^{-9} Torr to maintain a clean sample surface.