Physics307L:People/Ritter/Balmer

Balmer Series
To view the exact procedure for this lab or to learn more about the apparatus used, follow the given link to last years lab manual. 

Another good site used as a reference



Purpose
The purpose of this experiment is to observe the Balmer series for Hydrogen and Deuterium. Hydrogen atoms in a discharge lamp will emit a series of lines within the visible spectra. In this lab we are looking to see the lines, and measure the observed wavelengths to determine the value of the Rydberg constant. 

Equipment and Setup
The setup for this lab was relatively simple. The apparatus called a constant-deviation spectrometer was already set when we entered the lab. The difficulty in the lab, and where it seems the most obvious error can come into play, is in the calibration. Care must be taken to place the crystal in the appropriate position. For the purposes of this lab we we given values of visible lines for the Hg spectra and asked to find these lines and orient the crystal to match up with these values. It is also important when begining to take data to play with the placement of the lamp. We initially began taking data before realizing that it was possible to achieve a clearer visual of the spectal lines.

Data Collection
Once the apparatus was properly set up, data collection was relatively simple. We were sure to only take measurements in the same direction. That is to say, we only took measurements by turning the dial in a clockwise motion. This was done in an attempt to limit the impact of "slip" between the gears on the dial. 15 measurements were taken for the Hydrogen sample and 10 for the Deuterium sample. We chose to reduce our sample size for Deuterium since our values seemed to stay so consistent.

All data and calculations can be found at the following location.

Calculations
1.) Rydberg constant

$$\frac{1}{wavelength} = R (\frac{1}{2^2} - \frac{1}{n^2}), n = 3,4,5...$$ 2.) Error calculations

mean $$\overline{x} = \frac{\sum_{i=1}^N}{N}$$

standard deviation $$s = \sqrt{\frac{1}{N-1} \sum_{i=1}^N (x_i - \overline{x})^2}$$

standard error $$SE = \frac{s}{\sqrt{N}}$$

Error Propagation $$E = \sqrt{(\frac{df}{dx}dx)^2}$$ (In our particular case the only source of error is in the measured wavelength so x = lambda).

The final equation I used for error propagation looks like this.

$$E = \sqrt{(-\frac{1}{C\sqrt{x}}*SE)^2} ; C = (\frac{1}{2^2} - \frac{1}{n^2})$$

Koch Comments
I think it is great that you are using the error propagation formula, and you are right that this is a really useful skill! I couldn't tell from your Excel file, however, exactly what formula you were using (it appears that you calculated elsewhere). Also, based on your entries, here, I'm not sure you got it right? Here is what I'm thinking:
 * $$\frac{1}{\lambda} = R_\mathrm{H}\left(\frac{1}{2^2} - \frac{1}{n^2}\right)...so,$$
 * $$R_\mathrm{H} = \left(\frac{1}{2^2} - \frac{1}{n^2}\right)^{-1} \lambda^{-1}$$...To propagate error, you are interested in
 * $$\frac{dR_H}{d\lambda}$$ which is what you were calling $$\frac{df}{dx}$$ above.
 * If you let $$\left(\frac{1}{2^2} - \frac{1}{n^2}\right)^{-1} = C$$, then
 * $$\frac{dR_H}{d\lambda} = -C*\lambda^{-2}$$

We should probably talk about this in person, in case I am confused...

Final value of R for Hydrogen
1.093E +7 +/- 24101.53

Final value of R for Deuterium
1.095E +7 +/- 18443.65

Conclusions
1.) Thoughts on calibration - The orientation of the crystal is vital to the quality of the measurements in this experiment. I do believe that more time could have been spent on this process and in turn would have reduced the amount of systematic error.

2.) Clockwise vs. Counterclockwise - After completing the experiment I noticed how all of the other groups had taken both clockwise and counterclockwise measurements. It occurs that a large source of systematic error is probably associated with the gears in the turning mechanism, but that the error in one direction would probably be different from the error in the other direction. I'm thinking this is a fairly big oversight in our experiment.

3.) Thoughts on Error calculations - This is my first attempt at calculating error propagation. I did look over the notes given in class as well as research many sites on the subject. I do feel that I did perform the calculations correctly. However, I'm not exactly sure what the achieved value means. I think this is extremely valuable skill to have, and seeing how this is the lab I plan on using for my official paper, I hope to explore this more over the coming weeks.