User:Boleszek/Notebook/Physics 307l, Junior Lab, Boleszek/2008/09/22

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The Balmer Series

The Balmer series is a set of electromagnetic waves emitted by a given atom when its electron returns back to the n=2 energy level after being initial excited away from it to a higher level. On the way "back down" a single electron may occupy a few energy levels (from 7 to 4 back to 2, for example) so the excitation of a single electron may result in multiple emissions of different energies. Due to the individual electrical natures of each atom, the Balmer series is noticeably different for each one. Though it is uncommon, it is possible for an electron to skip certain energy levels during excitations if it is more energetically favorable for it to occupy a different energy level (these possibilities are determined by what are referred to as "quantization rules").SJK 02:40, 4 October 2008 (EDT)
02:40, 4 October 2008 (EDT)
Are you talking about "selection rules" here? If so, I don't think it comes into play in this lab, but I had fun reading about it again.

First developed through trial and error attempts to match data by Balmer in 1885, the Balmer Formula (shown much further below) can also be arrived to by applying the Bohr formula for atomic energy states to a difference (n1-n2) in energy states. The constant of proportionality (Rydberg constant) that relates the change in energy state to the wavelength of emitted light tends to slowly increase as the atomic mass of each atom increases.

The Goal

SJK 02:55, 4 October 2008 (EDT)
02:55, 4 October 2008 (EDT)
From here down to "data analysis" section, see Darrell's notebook for comments

It is our aim to successfully calibrate a constant deviation spectrometer with known values of the mercury spectrum and proceed to measure the spectra of hydrogen and deuterium. We will use these values to determine the Rydberg constant. Then we will use the same apparatus aimed at a sodium lamp in order to attempt to resolve the two characteristic closely spaced lines of its spectrum.

Procedure

The procedure can be found in Dr. Gould's manual

Calibration

Before calibration begins we notice that the prism looks a little dusty and has white impurity on its backside. We are afraid that cleaning it might result in a change in its surface optical properties so we try to calibrate with the prism as is (If we can not isolate a spectral line adequately we will try to clean it).

1. Our first attempts at calibration of the constant deviation spectrometer are met with frustration at not being able to direct light through the prism to the eyepiece. Finally, after multiple attempts we "see the light".
2. The first spectral light we see is green. The manual tells us that this wavelength of green is 546.1nm.
3. We proceed with calibration by turning the screw drive which rotates the prism to the 546nm position while simultaneously making sure that the prism is still properly oriented such that the green light is viewed in the center. Once the green line is aligned with the cross hairs we think our calibration is complete.
4. During our attempt to view the yellow lines at 577nm and 579nm we are dismayed to find that our scale does not read anywhere near the expected values. So something must have been wrong with our initial attempts at calibration.
5. As I, Boleszek, type this Darrel realizes that we read the scale wrong. So we never turned it to the 546nm mark in the first place!
6. We set the vernier scale to 546nm and then adjusted our prism so that the green line was visible at the intersection of the cross hairs.
7. We turned the scale to the 577/579 regime of yellow spectra and adjusted for minor misalignments.
8. In order to see how good our calibration is we turn the scale all the way to the violet range (404.7nm-435.8nm). We are surprised to see a multitude (up to 6) of violet lines of various intensities. We try to change the focus and slit width but the multitude of lines remains. We believe this could be due to diffraction effects.
9. We employ the help of Professor Koch and he's not so sure that single slit diffraction could be the culprit. He suggests two things: 1)maybe these are extra reflections or, even worse, 2)maybe we're not even using the right light source!
10. Turning to the red spectrum we find that what we see does not at all match the 690.75nm line our procedure indicates . Furthermore, the procedure indicates we should see two well-defined yellow lines, but we only see one well defined yellow line, and actually, to be fully accurate, it occurs at 575nm according to our scale instead of 577nm or 579. We are acquiring reason to believe that we in fact are not using the correct lamp.
11. The "mercury" lamp is turned off, and after we wait for it to cool down we remove it. As I look for bulbs in the lab cabinet I hear from across the room a chuckle from Darrel, followed by "it's Krypton!" (the label was only visible after the bulb was removed). So we were measuring the wrong light all along!
12. We acquire an array of new bulbs through the agency of Prof. Koch and decide to restart with hydrogen.
13. Using the internet we find accepted values of the H spectrum and recalibrate the spectrometer to the lowest visible wavelength (violet1) of H at 410.8nm.
14. We check if the other spectral lines are in the ball park in reference to the vernier scale and we find that we are at most 10nm off.
15. We decide to start making measurements of the spectra.

Measurements

For the first 3 sets of Hydrogen spectrum measurements (for each element) we recalibrate the spectrometer to a different color so as to minimize the systematic error of our vernier scale accuracy. The rest of the measurements are performed one after another with the same calibration. For all of these measurements we determine that, due to the physical limitations of the vernier scale, we can not be sure of their certainty to within +/-1nm.

To minimize variations, all readings are taken after moving the dial in the clockwise direction. Clockwise was chosen as it is pushing against the spring. Darrell had previous experience with this type of instrument and knew that moving into the spring would provide more consistent measurements.

• HYDROGEN

-Accepted Values of Hydrogen Spectra

1. 410.8 Violet1
2. 434.1 Violet2
3. 486.0 Green/Blue
4. 656.0 Red

-Measured Values of Hydrogen Spectra

1st Run (calibrated to violet1)

1. 410.8
2. 435.5
3. 488.8
4. 666.0

2nd Run (calibrated to violet2)

1. 410.0
2. 434.0
3. 486.0
4. 657.0

3rd Run (calibrated to green/blue): For this run Darrell narrowed the slit to obtain higher resolution lines.

1. 412.0
2. 436.0
3. 489.0
4. 666.0

4th Run (calibrated to Red)

1. 410
2. 434
3. 486
4. 656

Now we'll get some data with the same calibration to compare our repeatability in reading 5th Run (Repeat of calibrated to Red)

1. 410
2. 434
3. 486
4. 658

6th Run (Repeat of calibrated to Red)

1. 410
2. 434
3. 486
4. 657
• DEUTERIUM DATA

Keeping the final calibration used for hydrogen as it produced decent results compared to known spectra and decent repeatability

1st Run

1. 410.0 Violet1
2. 434.0 Violet2
3. 485.5 Green/Blue
4. 657.5 Red

2nd Run

1. 410.0
2. 434.0
3. 485.5
4. 657.0

3rd Run

1. 410.0
2. 433.25
3. 485.5
4. 657.0

4th Run

1. 410.5
2. 434
3. 485.5
4. 655.5

5th Run

1. 410.6
2. 434.9
3. 485.3
4. 655

• RESOLUTION OF SPECTROMETER

We are instructed by the manual to see if we can successfully measure the two closely spaced yellow lines (586.0 & 586.6) characteristic of the sodium spectrum. Unfortunately we were unable to find a sodium lamp so we decided to find another element whose spectrum contains two closely spaced lines. After searching for spectra on the internet we saw that Krypton has fairly closely spaced lines so we decided to use this gas for our measurement of the resolution.

Within the purple region of the spectrum we found quite a few closely spaced lines. So we looked for 2 that were as close together as we could resolve. These lines were at 445.4nm and 445nm, indicating that we have a resolution well within 1nm. Repeating this procedure in the orange part of the spectrum we found two lines that were equally closely spaced together as the two purple lines, however they were at 605nm and 607nm, indicating to us that the resolution of our spectrometer at longer wavelengths was lower.

We thought to look for this difference in resolution because the change in the scale of the vernier dial indicated that it would have a sensitivity that changed across the range of measurements. This also fits well with our understanding of the dispersion of light that the instrument depends on. The angle of diffraction, as a function of wavelength, does not follow a straight line but is curved, so we expected that the visual distance between two lines would not be directly proportional to the wavelength difference. (see dispersion of light)

Data Analysis

The wavelength (λ) data taken above will now fulfill its purpose by being applied to the Balmer Formula:

λ^-1=R(1/4-1/n)

Where n is an energy level >2 and R is the constant to be determined.

• Determination of Rydberg ConstantSJK 02:58, 4 October 2008 (EDT)
02:58, 4 October 2008 (EDT)
It took me quite a while to figure out what you were doing here...even with the explanation...but now I see, and I think it was a clever way to check for "skipping", though I don't think we expect skipping based on QM

Using MATLAB, along with Darrell's expertise in coding, we calculated possible values of R for a variety of quantum numbers n for each frequency. Comparing plots of these values graphed on a single axis, we located the points at which each graph yielded a value of R that was consistent with the others. We found that each higher energy (lower wavelength) line corresponded to the next higher quantum number (n=3,4,5..) such that no energy level was "skipped" during the excitation process of the gas. The plot of the data along with the code that produced it are below.

matlab code
graph of Rydberg constant for various values of n for each spectrum line
SJK 03:02, 4 October 2008 (EDT)
03:02, 4 October 2008 (EDT)
What do you mean "by hand," and why would you expect hand calculations to be any different (besides in speed) than with the computer?
I chose to try some old school data analysis along with the computerized modern version to see how they compare. For the Hydrogen spectrum data I used Matlab, but for the Deuterium data I chose to do calculations by hand. Here are the results:

R_H=1.093*10^7

R_D=1.097088*10^7

Since these two atoms are so similar in structure they really should have very similar Rydberg constants. My results show that the Deuterium calculation yields a value closer to the accepted value. This is probably not because I am better at doing accurate calculations than Matlab, but most likely because all of our Deuterium data was taken using one calibration, whereas the Hydrogen data used for the Matlab calculation was recalibrated for each run. In the name of consistency I decided to calculate the Rydberg constant of Hydrogen using the data (runs4-5) that was taken for one single calibration which, importantly, was the same calibration used for the Deuterium data acquisition. After summing the values for each type of line, dividing each sum by the number of runs (3), calculating the Rydberg constant for each averaged wavelength using consecutively increasing values of n for lines of higher and higher energy (n=3-6), and finally averaging the 4 Rydberg constants, I obtained:

R_H=1.097014*10^7

This result is actually intriguing to me because it expresses a behavior that is expected. In my quantum physics textbook the author explains that atoms of greater mass should have slightly greater Rydberg constants and according to my calculations R_D is indeed about one ten-thousandth larger than R_H. But the excitement is short lived, for I have yet to calculate the experimental error, and if it does not fall within the accuracy of one ten-thousandth, then I cannot fully trust my results to be more than "good luck" error.SJK 03:00, 4 October 2008 (EDT)
03:00, 4 October 2008 (EDT)
Excellent point. I was worried for a second that you'd jump to a conclusion and very glad to see you reason this way.

Error Analysis

We have three sets of data here to error analyze:

1. H data with recalibration for every run
2. H data with constant calibration
3. D data with constant calibration

The first H data will be done with Matlab (using the std function) and the rest will be done by hand.

SJK 03:07, 4 October 2008 (EDT)
03:07, 4 October 2008 (EDT)
I think I now know what you mean "by hand" above. And this is indeed a good standard deviation formula. However, I will point out the "N" versus "N-1" difference in the denominator, which you probably noticed during your trek through wikipedia. In our case, we're more interested in the N-1 formula, because we are trying to estimate the standard deviation of the parent distribution from our actual data. There are a variety of explanations for the N-1. E.g., you have one less degree of freedom, since you're computing the mean from the same data set. Also, if you only took one data point, it's somehow more sensible that you get an undefined standard deviation estimate, rather than zero. N-1 is accepted as correct for the best estimator, and who knows, maybe you can prove it. Another point, though, is that if you take a bunch of data points, there isn't too much difference between 1/sqrt(N) and 1/sqrt(N-1)

The other thing I will point out is that for what you're doing you actually want the standard error of the mean which involves another 1/sqrt(N)...we learned about this last week after you turned in the lab.

Before presenting the results it is fitting that I prove I know, at least to some degree, what I intend to be doing. I admit that I was not to clear on how to calculate the standard deviation of a discrete set of data points at the beginning of this lab, but then I went to Wikipedia. At first I was taken aback by the large variation in definitions of std, but after careful sifting, I believe I found the correct std for my purposes in the form of a nice list. Here are the Wiki-instructions:

1. Find the mean, $\scriptstyle\overline{x}$, of the values.
2. For each value xi calculate its deviation ($\scriptstyle x_i - \overline{x}$) from the mean.
3. Calculate the squares of these deviations.
4. Find the mean of the squared deviations. This quantity is the variance σ2.
5. Take the square root of the variance.

This calculation is described by the following formula:

$\sigma = \sqrt{\frac{1}{N} \sum_{i=1}^N (x_i - \overline{x})^2}\,,$

I will follow these instructions for the hand-made calculations and compare the results with Matlab's std function.

• H data (matlab)

Violet1--.9452nm

Violet2--1.0308nm

Green/Blue--1.6763nm

Red--sqrt(2)nm≈5.51nm

• H data (hand-calculated)

Violet1--0nm (this excellent result is rely due to the fact that we calibrated to this gas)

Violet2--0nm

Green/Blue--0nm

Red--sqrt(2)nm≈1.41421nm

• D data (hand-calculated)

Violet1--Here's proof that I'm actually doing this:

mean-- 410+410+410+410.5+410.6=2051.1 2051.1/5=410.22

deviation-- .22,.22,.22,.28,.38

squares of deviations-- .0484,.0484,.0484,.0784,.1444

variance-- 3*.0484 + .0784 + .1444 = .368 .368/5=.0736

standard deviation-- sqrt(.0736)≈.27129nm

Violet2--.402nm

Green/Blue--.08nm

Red--.94

SJK 03:10, 4 October 2008 (EDT)
03:10, 4 October 2008 (EDT)
I wonder about the validity of calibrating with hydrogen anyway, but I really wonder about your recalibration method. But since I didn't work on it with you, I guess it's water under the bridge now...I guess when I think about it, calibrating using Mercury isn't any better.

The standard deviations for the runs with constant calibration are noticeably lower than the runs for which we kept on recalibrating the spectrometer. One may wonder why we even bothered to recalibrate the apparatus for each of the first 3 Hydrogen runs. Our reasoning was that we might get values that were overall closer to the accepted ones if we kept recalibrating. In our minds it seemed that we were going to get high accuracy but low repeatability with this technique. But it turned out that there was much too much room for human error in this experiment and so both our accuracy and precision increased after we let the apparatus measure with a calibration we felt we close enough.

I did mention that I was going to compare my calculations with those of Matlab, but since I used Matlab for a set of data with altogether different error characteristics than the other two sets of data this comparison would be superfluous.

SJK 03:14, 4 October 2008 (EDT)
03:14, 4 October 2008 (EDT)
the reasoning is important and good here. However, you forgot units on R above, and in fact it's not nm but inverse nm. You'd want to consider how to convert the precision in nm for individual wavelengths to precision in inverse nm in the overal R constant. Fractional error is a start. We'll learn about error propagation soon, so the most important point here is that you weren't careful with the units for R!

The range (about .01-5) of our standard deviations indicate to us that we can only be sure of the accuracy of the Rydberg constants determined above to within one-hundredth of a nm. The seemingly correct difference of one ten-thousandth of a nm in Rydberg constants calculated for H and D above cannot be confirmed with utmost certainty, but neither can it be proven to have no physical significance. An apparatus with higher precision would have to be used to resolve this problem.