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CALCULATION
OF MAXIMUM
ALLOWABLE PRESSURES ON
CIRCULAR WINDOW MATERIALS
Calculation
of maximum pressure will depend upon a number of
user selectable parameters. For instance, the window
material, window size, flange size and a safety
factor all may be varied depending upon application.
To
calculate maximum allowable pressure, we must assume
that the maximum stress in a uniform circular plate
is given by the equation:
where
k is a constant, the value for which depends upon
whether or not the window is clamped — use
0.75 for clamped windows and 1.125 for unclamped
(See Fig. 2). Smax is the maximum stress, D is the
window diameter under pressure (ie, the portion
of window not supported by the flange as shown in
the schematic in Figure 2), P is the load (expressed in psi), and t
is the thickness of the window material. In the formula solving for t, the window diameter D can be expressed in any unit of measure such as mm or inches. To avoid
plastic deformation a safety factor must be introduced
where SF is a safety factor and Fa is apparent elastic
limit for the material itself. Where apparent elastic
limit is not available, use yield stress. Allowing
for a safety factor (SF), the equations for calculation
of maximum allowable pressure and for minimum window
thickness (t) where the operating pressure (P) is
known are:
The
apparent elastic limits of some IR optical materials
are listed below:
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ATR
SPECTROSCOPY
The
formulae set forth below are useful for ATR (MIR)
spectroscopy.
Penetration
Depth
The
depth of penetration is defined by the formula:
where
l
is
the wavelength of the infrared light, n1
is the refractive index of the ATR crystal, 
is the angle of incidence of the infrared beam at
the boundary and n21
is the ratio of the refractive indices of the sample,
ns and ATR crystal,
Since the evanescent wave decreases in intensity
exponentially from the surface of the crystal, the
penetration depth, dp, is defined as
the distance at which the amplitude of this wave
has decreased to (1/e) or 37% of its original value.
Effective
Angle of Incidence
This
is the angle of incidence of the infrared beam internally
in the ATR crystal when a variable angle HATR such
as the Varimax™ is used for analysis. When
the scale angle, scale,
is not equal to the crystal face angle, face,
the effective angle,
is different than the scale angle due to refraction.
Number
of Reflections
The
number of reflections in the crystal gives a measure
of the intensity of the resulting spectrum. This
number is a function of the effective angle of incidence
, and
the length, I, and thickness, t, of the crystal.
Effective
Pathlength
The
effective pathlength, Peff, is defined
as the product of the penetration depth dp,
and the number of bounces, N, the IR beam makes
within the crystal:
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