**
Alexandrite Effect Method for
Temperature Measurement **

The “alexandrite
effect” is used in gemology to describe a distinct change of color appearance
change when alexandrite is switched from being illuminated by daylight (6500K)
to incandescent light (2856K)[1]. The alexandrite shows the same color if
blackbody and a gray body, have the same temperature, because the relative
spectral power distributions of a blackbody and a gray body are the same at the
same temperature. Thermal radiators emit the approximately same relative
spectral power distributions at the same temperature.

The typical
spectral transmittance of an alexandrite crystal has two bands in the visible
spectrum [1]. Figure 1 illustrates the typical two-band spectrum of an
alexandrite crystal. When the radiation of a radiating body passes through the
crystal, the crystal appears different colors at different temperatures of the
radiating body.

The continuous color sequence of a thermal radiator is displayed as the
Planck locus in the CIE color diagram, from red, orange, yellow, white, to blue
as temperature increases. In fact, the temperature of the thermal radiator only
relates to the hue red to blue, regardless of it lightness and saturation,
because the thermal radiator always has a lightness of 100 and a saturation of 0
as a light source regardless of its temperature.

Figure 1. The typical spectral transmittance of alexandrite
crystal in the visible wavelength range.

The color of alexandrite at different temperature is represented by the
hue-angle in the CIELAB color space, and can be calculated by [2]:

(1)

where X, Y, and Z
are the tristimulus values, , , and are CIE color-matching functions,
is the spectral power distribution of a radiating body,
is the spectral transmittance of the alexandrite, and
l is the wavelength in the visible range. Then, the three
coordinates of the CIELAB color space can be calculated as follows:

(2)

where
,and are the tristimulus values of the measured radiating
body. Finally, the hue-angle in CIELAB space is given by:

(3)

Fig. 2 illustrates the relationship between temperature of the thermal radiator
and the hue-angle of the alexandrite crystal in the CIELAB color space.
Temperature is a function of the hue-angle that can be determined by
mathematical methods [8]. A polynomial function to the sixth power of hue-angle
is usually adequate for most temperature measurement applications:

(4)

In a small
temperature range, a polynomial function to the third power of hue-angle is
sufficient.

##

##
Fig.2. Relationship between the temperature
and hue-angle in CIELAB color space

The alexandrite
effect spectropyrometer [3] consists of an optical probe, a spectrometer and a
computer with a digital alexandrite effect filter, which tabulates the spectral
transmittance along the a-crystallographic axis of the alexandrite with a
maximum hue change of about 180 degrees in CIELAB color space between 6500 K and
2856K. The spectropyrometer measures the spectral power distribution of a
thermal plasma torch through the alexandrite filter and calculates the hue-angle
to determine its thermal temperature.

##
References

[1] Y. Liu, J. Shigley, and
S. Hemphill, “The alexandrite effect in gemstones," Color Res. & Appl. **19**
186-194, 1994.

[2] CIE,
Publication No. 15.2, Colorimetry, (CIE Central Bureau, Vienna), 1996.

[3] Y.
Liu , “Alexandrite effect spectropyrometer,*” Photonic Devices and Algorithms for Computing VIII,** *
SPIE Vol. 6310,
63100E, 1-8, 2006.