Thursday, July 13, 2017

Refractive Index of Air as a Function of Wavelength

  In 1908 Rentschler published a study on the refractive index of a gas for different wavelengths in the Astrophysical Journal. I used his data for air in Table II to find the coefficients of a Cauchy formula to fit the data. One can also use the NIST calculator to get data for air and do another fit.

The Rentschler data appears to be for dry air. If one computes the index of refraction for the CRC Handbook data using the given wavelengths for air and computed wavelengths for the Balmer series for hydrogen the plot appears to be displaced somewhat. Note that the humidity of the air also affects the index of refraction. I used the ratio of the wavelength in air to 5000Å for the formula so it would be easier to substitute wavelengths given in nanometers (nm) using 500/λ with the same coefficients.

Monday, July 10, 2017

Balmer Series Fit Using More Accurate CRC Data

  The hydrogen wavelengths in air found in the CRC Handbook of Chemistry & Physics are lines of the Balmer series and I got better results for a fit of these lines.

One has to use the reduced mass of the electron to get RH=R/(1+me/mp) where me and mp are the electron and proton masses to compute the limit of the series in air and divide that by the index of refraction for air to get the limit of the series for air.

The index of refraction for air is a function of wavelength and that also affects the observed values and so the fit has an error that varies with the number of the series.

Sunday, July 9, 2017

Scaling Factors That Could Potentially Affect Huggins' Hydrogen Lines

  Relative to the minimum deflection wavelength for Huggins' spectroscope there appears to be a progressive shift to higher wavenumbers which is a blue shift. But relative to the limit of the series, λ, there is a redshift. There are a number of factors which might cause a scale change in the observed wavelengths and need to be taken into consideration.

Calibration of the Spectroscope
minimum deflection
calibration curve

Air Wavelengths vs Vacuum Wavelengths
Definition of standard conditions
Snell's Law (measurement in air results in a blue shift relative to a vacuum )
Rydberg Constant (R is for a vacuum)

Light nucleus
RM needed for light nucleus (R is for a heavy nucleus)
reduced mass
electron mass

Doppler shift

Gravitational redshift

All of the above contribute to a change in scale. What did Huggins miss?

Friday, July 7, 2017

A More General LS Formula for Estimating a Common Factor

  One can generalize the formula in the last post to find the best estimate of a common factor.

Using this formula gives a better estimate for the wavenumber limit of the Balmer series as the following calculation shows. The formula gives 2744.8 while the average for the individual estimates is 2744.6 and the rms errors are 1.14 and 1.13 respectively.

The improvement over an average is marginal and doesn't appear to compensate for a scaling error.

Supplemental (Jul 8): Huggins adjusted his spectroscope for minimum deviation for the H line corresponding to 4340 Å or n=5. This looks like a better fit for the wavenumbers with an rms err of 0.46 which could result from rounding to the nearest Ångström.

This rescaling brings the estimate for the limit of the wavelengths to within a quarter of an Ångström of the currently accepted value.

Supplemental (Jul 8): The deflection of light by a prism depends on the index of refraction n and the wavelength λ. Shorter wavelengths are bent more than longer ones so blue light is bent more than red. The deviation from linearity also increases as the wavelength decreases so this might explain the need to rescale the spectroscope readings. The changes are in the right direction but I haven't compare the observed errors with calculated errors.

An Error in Huggins' Hydrogen Wavelengths?

  There appears to be a systematic error in the hydrogen wavelengths in Huggins' paper when one compares them with the fit for the Balmer series. Notice that there is an progressive increase in the value of the error, Δν. If one uses the first line as the reference we can set that error equal to zero and compute the relative errors Δν'.

Trying to fit a straight line to this data is a little tricky but least squares allows us to derive a formula for the best fit for a line going through the origin.

The adjusted lines give a closer estimate for the limit of the lines of the Balmer series.

It's possible that the Iceland spar prism that Huggins used was slightly nonlinear in its dispersion of light. Dispersion is dependent on frequency. This may be why rescaling gives a better result. The least squares derivation of the formula above is fairly simple. The quantities on the right of the formula are the column vectors in the second table.

Wednesday, July 5, 2017

The Balmer Series

  It's customary on the 4th of July for Americans view local fireworks displays. What the experts can do with patterns, colors and sound is quite amazing. History tells us that fireworks originated in ancient China and the ancient alchemists experimented with colors. Knowledge of the colors that can be produced can be acquired through the use of flame tests. This tool is still used in modern chemistry.

  The scientific study of colored light was advanced in the late 17th Century by Newton's work on the prism. At the beginning of the 19th Century the introduction of the spectroscope allowed scientists to study the Sun's spectrum and discover the dark lines known as Fraunhofer lines. At the same time flame tests of various elements allowed scientists to connect these lines to chemical elements present. In 1868 Ångström published accurate values for the lines of the solar spectrum with the elements associated with them.

In 1885 Balmer published a notice giving the formula for a series of hydrogen lines. How might he have accomplished this? His data came from a notice by Huggins on the hydrogen lines present in the spectra of certain stars. The data is included in a footnote referring to a note he received from Johnstone Stoney, a fellow of the Royal Society, who states that the lines might belong to a series.

What happens if we try to do an empirical fit for the data? Notice that Stoney also includes the wave numbers, ν=1/λ, and we can try to fit these. The lines appear to converge in one direction so we might first try to fit a formula that is quadratic in 1/n (fit1). The results are quite good with an rms err of 0.4. Using n=3 for the first line gives the best fit. The value for B is relatively quite small when compared with the others and the ratio of C to A is very close to 4. Redoing the fit for just two terms makes the ratio even closer to 4 so we can just try to get a value for A by computing a value for each line and taking the average.

Although the rms error is greater, we still get a fairly good fit to the data.

Notice that Stoney includes a curve passing through the data points. Did he know the formula for the series of lines? His table suggests he used a difference formula to fit the data.

Supplemental (Jul 5): Fraunhofer's lines:

Huggins Plate 33 showing the line spectra of a number of stars. The second row appears to be α Lyræ (Vega) containing the first twelve lines of the data above:

Thursday, June 1, 2017

Newton's Temperature Scale

  The thermoscope, a bulb containing air with a long tube that was immersed in water, was developed by Galileo and others to measure temperature during the first half of the 17th Century. Boyle studied similar "weather-glasses" and introduced the hermetically sealed thermometer in England by 1665. In 1701 Newton anonymously published an article, Scala graduum caloris, which described a temperature scale ranging from the freezing point of water to that of a fire hot enough to make iron glow. An English translation of Newton's article can be found in Magie, A Source Book in Physics, p. 225.

Newton's temperature scale has a geometric series and an arithmetic series associated with it. The geometric series corresponds to the temperatures and the arithmetic series is associated with cooling times.

 "This table was constructed by the help of a thermometer and of heated iron. With the thermometer I found the measure of all the heats up to that at which lead melts and by the hot iron I found the measure of the other heats. For the heat which the hot iron communicates in a given time to cold bodies which are near it, that is, the heat which the iron loses in a given time, is proportional to the whole heat of the iron. And so, if the times of cooling are taken equal, the heats will be in a geometrical progression and consequently can easily be found with a table of logarithms."

After finding a number of temperatures with the aid of a thermometer, Newton describes how the hot iron was used.

"...I heated a large enough block of iron until it was glowing and taking it from the fire with a forceps while it was glowing I placed it at once in a cold place where the wind was constantly blowing; and placing on it little pieces of various metals and other liquefiable bodies, I noted the times of cooling until all these bodies lost their fluidity and hardened, and until the heat of the iron became equal to the heat of the human body. Then by assuming that the excess of the heat of the iron and of the hardening bodies above the heat of the atmosphere, found by the thermometer, were in geometrical progression when the times were in arithmetical progression, all heats were determined."

Newton's temperature scale can be constructed mathematically as follows where I've noted some corresponding temperatures on the Fahrenheit temperature scale for comparison.

The temperature point between the melting point of wax and the boiling point of water is an average. I used the geometric average which works best. One can put together a table as follows to compare the Fahrenheit temperatures with the index number, k, above.

A graphical comparison shows that the logs are fairly linear. Using 66°F for the temperature difference gave the best fit for human body temperature at the lower left of the plot.

The slope of the fitted line can be used to convert Farenheit temperatures to points on Newton's scale.

Newton's law of cooling can be in be expressed as the difference between the temperature of an object at some time and the ambient temperature being proportional to an exponential term involving time. This can to shown to be equivalent to the differential form of the law.

Supplemental (Jun 1): Leurechon Thermometer (1627)

Supplemental (Jun 2): 65°F gives a better fit for body temperature. Was this the ambient temperature at which the experiments were done? It's doubtful there was a standard temperature yet in Newton's time. For more on the history of early thermometers see Bolton, Evolution of the Thermometer, 1592-1743.

Supplemental (Jun 2): The average of the freezing point of water and body temperature is (32+98.6)/2= 65.3. Did this originate with Accademia del Cimento?

Supplemental (Jun 4): Corrected conversion formula for k.