Shay Nevo and Shreya Chaudhary: 2007
Introduction
When light passes from one medium to another, such as from
air into water, the velocity and the direction of propagation
both change. The phenomenon is called refraction.
When light passes through a prism it is deviated by refraction
at both surfaces. Minimum deviation occurs when the light ray
passes symmetrically through the prism, (figure 1). In this symmetrical position
the angle of deviation D, and the prism angle A
are related to the refractive index of the prism material n,
by the well known formula....
The refractive index of the prism material can be found by measuring the angle of minimum deviation. If a hollow glass prism is filled with layers of salt water and fresh water, the vertical refractive index gradient, which changes over time, can be found.
Research question
How does the refractive index gradient at the interface between layers of brine and tap water change over time?
Hypothesis
It is expected that the refractive index gradient will be some nonlinear function of time.
Explanation
Diffusion will slowly mix the two layers at the interface. If the mixing is due solely to diffusion there will be rapid change over a narrow 2-3 cm band in the first day, followed by reduced percentage changes over wider bands on day 2 and subsequent days. The diffusion rate is temperature dependent. Room temperature without air conditioning varies in a narrow band between 26 and 30°C. The temperature dependence of the diffusion rate is not expected to be significant because the tank was set up over an undisturbed holiday period.
Procedure
A fish tank with a 60° prism extension was half filled with tap water. Four kilograms of salt were added to the fresh water to make a near saturated solution.
 Fig 3 - filling the tank. |
Fresh water was then added very slowly so that the two layers
would not mix. The interface between the two solutions is very
clearly shown in figure
4.
A helium/neon laser was placed on a table
(figure 5) that could be moved up and down in a vertical direction.
The table was moved until the laser spot on the wall two meters
beyond the tank was at minimum deviation. Since the refractive
index changes in the brine/water layers are small, and the deviation
is very close to minimum over a ±5° range of angles
of incidence, the angle was taken to be minimum at all heights.
Every day, for seven days, a large rectangular piece of paper
was taped to the wall. The position of the laser light on the
paper as the laser table was moved to different heights was marked
with a felt pen. The height of the laser table was recorded for
each point. At the end of the seven days, the fish tank was moved
away and the position of the undeviated laser light point was
marked on the wall. The angle of deviation was then determined
for each point by measuring two angles and placing a scale marked
in degrees on each of the overlaid sheets of paper. Angle of deviation
for each point and the corresponding and laser table heights were
then entered by hand into Logger Pro.
Control of variables
Because the beam was at times passing horizontally through a refractive index gradient it was bent downwards and the laser spots on the wall were displaced vertically. The vertical displacements were greatest on day zero and day one because the refractive index gradients close to the interface were then very large. The data was corrected for vertical displacement by using the recorded height of the laser table as the depth measurement.
The laser was clamped in position between two large wooden blocks fixed to the table with double sided tape. it was important to ensure that the laser beam remained in the same direction for all measurements. This was checked by marking the place where the laser light met the wall before the experiment began, and after it ended, to make sure the point stayed at the same place.
The papers on the wall were carefully overlaid so that the same deviation angle scale applied to all sheets.
Table 1 below lists deviation angles and laser heights above the floor for day 1.
The refractive index values were calculated using the minimum deviation formula with errors in deviation angles of ±0.5° the angle of the prism (59.0 ± 0.1°). The calculation was done in Logger Pro by entering calculating the maximum possible value of n and subtracting the value from column 3.
Sample calculation
Maximum value
Value without error
Seven tables were entered into Logger Pro and seven graphs were plotted showing refractive index plotted against the laser height.
Graph 1 shows the data for day zero. The index of refraction of the two solutions of salt water and fresh water are close to the expected values (1.33 for water and 1.80 for saturated brine). Note the very rapid change at a height of 77 cm. the change is too rapid to accurately determine the refractive index gradient at the point of inflection at 75.5 cm.
Graph 2 shows the refractive index versus height on day one. Diffusion is mixing the two layers.
Graph 3 shows the same variables plotted on day four. Further mixing has taken place.
Straight lines were fitted to each set of data points between 75 and 80 cm on the vertical axis. In this region the gradient of the refractive index with respect to height is approximately constant, and the slope of the lines increases as time passes.
Error calculations
Graph 4 shows extreme lines fitted by hand for day one to determine the likely error in the gradient.
The error is ...
The refractive index gradient (dn/dy) is the rate of change of refractive index with height (elevation). The gradients are the reciprocals of the slopes of the lines on the depth-refractive index graphs. The gradient is negative since the refractive index reduces with increasing elevation. The slopes of the lines and the refractive index gradients are listed in table 2. Note the change of units from cm to m in the final column.
Table 2
Graph 5 is a plot of the refractive index gradients (dn/dy) centered on 72.5 cm (the height of the initial well defined interface) over six days. As expected the rate of change was most rapid on the first day and reduced as time passed. The data is a good fit to an exponential function. A plot of the refractive index gradient against the natural log of the time in days is a straight line within errors but the fit is not exact. The refractive index gradient varies approximately as an exponential function to within ±15%.
Graph 5 shows that the refractive index gradient varies as a non linear function of time as expected. An exponential function is a good fit to the data points. Diffusion is seen to proceed in a regular predictable way. The region of very nearly linear gradient expands as time passes. The mixing by diffusion is remarkably slow. The layered solutions (more dense below) is effectively stable for several days, even at the high room temperatures experienced in Bangkok. In cooler conditions the diffusion would be significantly slower.
Temperature inversion in the atmosphere
The refractive index gradients observed here are similar to the gradients in the atmosphere that occur in temperature inversions, when warm air overlays cooler more dense air. Temperature inversions in calm conditions stable over many hours and are responsible for superior mirages often seen in deserts in the early morning and sometimes visible at sea, when a ship appears to be sailing in the clouds.
Placing the laser on an elevating table and using a wall to mark the positions of the beam was satisfactory. Data was collected in five-ten minutes on each of seven consecutive days. Placing the deviation angle scale directly on the wall charts allowed the angular displacements to be read off efficiently. Calculation in Logger Pro converted the raw data to refractive indexes, avoiding the need to make dozens of tedious calculations by hand.
Further work could be done with two substances that have a much larger density difference. The difference in density of water (1.00 g/cc) and brine (1.20 g/cc) is relatively small. It would be interesting to plot the refractive index gradient against time for layers of water and saturated zinc chloride solution, which has a density of 1.82 g/cc).
This experiment was done indoors with an average room temperature of 28° C. Room temperature could be set to a much lower value with air conditioning (~15°C) and the diffusion rate of brine and water compared with the rate at the higher temperature. A study over a range of temperatures would require the processing of a large amount of data.