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Investigating the Fine Structure of H_\alpha & D_\alpha using Fabry-Perot InterferometryEdit

The Department of Physics, Cornell University, Ithaca, NY 14850, USA


Aim of the experiment Edit

  1. Measure the fine-splitting in H and D
  2. Measure the fine structure constant \alpha
  3. Measure the mass ratio m_{H}/m_{D}.


Apparatus used Edit

  • Fabry-Perot interferometer (For a high spectral resolution)
  • Prism (For primary spectral resolution)
  • Collimator
  • Hg arc-lamp
  • Wood’s tube
  • Vacuum pump
  • Pocket-spectrometer
  • CCD Camera (For data collection on computer)


Available data Edit

  • Etalon spacing - 7.000mm
  • H_{\alpha} (central) frequency =656.3nm
  • D_{\alpha} (central) frequency =656.1nm
  • (Note for 'dummies': \approx600nm-750nm is the red part of the visible spectrum.)
  • Hg green line frequency =546.1nm
  • The theory of fine structure (though historically, that was not the case)


ReadingEdit

Some TheoryEdit

A review of the theory for the experiment is in order. A brief note on the review follows.

Principally, the fine-structure comes from three corrections, of the same order, to the classical Hamiltonian of the atom. The three terms are

  • Relativistic correction to the kinetic energy
  • Spin-Orbit coupling term
  • Darwin term

The above corrections can be evaluated by a direct application of first order perturbation theory. A par-excellence exposition on the same is given in Volume 2, Section 12A-D, pp 1212 by [1]. The older (but ‘incorrect’) Sommerfield theory of the fine structure is nicely summarized in Sections 9.1-2 by [2]. The take away is that in doing this experiment, we shall not be able to differentiate between the three individual contributions. But we hope to observe the cumulative effect.

An illustration summarizing the relevant theoretical predictions is given below. (Refer for detailed derivation of the results herein Volume 2, Section 12A-D, pp 1212 by [1])

1

Figure 1: The dotted line shows the unperturbed energy level of the n = 2 state. The solid lines are the energy levels after first order perturbation theory corrections are applied. The standard spectroscopic notation is used above to label the states.

The first observations reported on having resolved the three components (as also the Lamb shift, discussed later) were by [3] . It is a very lucid paper. After all, it was published in Nature...

Historically, of course, the fine-splitting was observed before the theory was developed. The theory thereby, on its proposal, already ‘satisfied’ the observed properties of fine-splitting.


Line-width of the peaks Edit

When we get the interference intensity profile from the Fabry-Perot interferometer, note that we do not expect a Gaussian profile for each peak. We expect the same profiles for H and D, but a different one for Hg. The reasons for expecting them as such are summarized below. A note is warranted because this is in contrast to the theoretical expectation of a line for an atomic transition being Lorentzian. And the theoretical expectation is the same (Lorentzian) for all - H, D and Hg. [4]


Natural Line width Edit

Sufficient to say that the uncertainty principle disallows a monochromatic atomic transition and hence there is extant a natural line width to every transition. A simple classical model of radiative decay, as well as semi-classical derivation using Einstein’s (phenomenological) coefficients (equivalently referred to as ‘decay rates’) can be referred to in Section 21.2, pp 421 of [2] and Section 3.2, pp 126 of [5] . A complete quantum mechanical treatment of the same, which seemingly necessarily invokes vacuum fluctuations (interaction of the atomic states with the vacuum state), was done by Prof. Mukund Vengallatore in the Ultracold Atomic Physics II course, Fall 2011, Cornell University. He has promised to share the notes for the same, which thence shall be cited.

It is sufficient for our experiment to note that the natural line width has a Lorentzian profile.


Doppler Broadening Edit

There are other factors that contribute to broadening of the observed line-width. An important mechanism is Doppler broadening, which is essentially Gaussian.

Doppler broadening occurs because of the finite temperature of the gas whose atomic emissions we are observing [6]. The temperature results in a Maxwell-Boltzmann distribution of the speeds of the atoms in the gas whose motion thereby Doppler-shifts the observed frequency of emission [7]. Doppler broadening is calculated to be considerably pronounced than the inherent line-width, as far as the transition under consideration for H, D and Hg are concerned. At finite temperatures, only in a very few exceptional cases, such as Cu, can the natural line width not be neglected with respect to the Doppler broadening (Refer Section 21.4, pp 426-7 of [2]). So the inherent Lorentzian profile of the atomic transition in H, D and Hg would be masked in our observations. As an equation, we write this as,

\Delta{}_{natural}^{(H/D/Hg)}\ll\Delta{}_{Doppler}^{(H/D/Hg)} (Equation 1)

(Note: In the above and following equations, we write \Delta for the half-line widths.)

For Hg, which has an atomic mass of 200.56 amu, Doppler broadening would be smaller by a factor of

\dfrac{f_{Hg-greenline}}{f_{H_{\alpha}}}\sqrt{\dfrac{m_{H}}{m_{Hg}}}=\dfrac{546.1}{656.3}\sqrt{\dfrac{1.008}{200.56}}=0.0589(9)\pm(5) (Equation 2)

as compared to that for H (calculated for an assumed same temperature) (Refer Section 21.1, pp 420 by [2] . It is then no surprise that we can expect the Doppler broadening itself to be masked by some other broadening mechanism for the Hg green line.

An introductory but insightful derivation and analysis of Doppler broadening is given in Section 21.1, pp 420 by [2].


Apparatus-Broadening Edit

A broadening is introduced due to the apparatus itself. We refer to it as apparatus-broadening. A dominant part of the same comes from the fact that the Fabry-Perot interferometer used has finite-reflectance mirrors. The finite reflectance results in an output response function for the Fabry-Perot etalon. The response function has been lucidly derived in Section 14.7-8, pp270 by [8].

Because of Equation 1, apparatus-broadening is expected to dominate the line width for the observed Hg spectrum - i.e. it will mask even the Doppler broadening for the green line.

\Delta{}_{observed}^{(Hg)}=\Delta{}_{natural}^{(Hg)}\oplus\Delta{}_{Doppler}^{(Hg)}\oplus\Delta{}_{apparatus}\approx\Delta{}_{apparatus} (Equation 3)

Notice that we have denoted the ‘addition’ of the line-widths by \oplus and not a simple algebraic +. This is because line widths do not always add linearly. In fact, they add linearly for Lorentzian distributions superposed on one another. But as discussed, the natural line width is Lorentzian, Doppler broadening is Gaussian, and the Fabry-Perot response function is as in Section 14.7-8, pp270 by [8] - neither Gaussian nor Lorentzian. [Digression - In the notation of Section 14.7-8, pp270 by [8], the profile is of the form \frac{c}{1+\frac{1}{\sigma}\sin^{2}\frac{\delta}{2}}. It is analytically an even more complicated profile, when we express \delta in terms of ‘the position along the screen’, x, which is the independent variable in our observations/ data set. (Of course, x is converted to wavenumber, \nu, for analysis as explained later.)]

The spectral profile obtained by the cumulative broadening due to these three mechanisms is characterized by a convolution of the three individual broadening profiles. A succinct explanation is given in Section 9.5, pp 392 by [9]. And hence, to reiterate, the use of \oplus.

Also notice that we have \Delta_{apparatus} without a superscript, implying that it is independent of whether we measure it for the H, D or Hg spectrum [10]. The observed Hg spectrum will be essential to the experiment in exactly this regard. Our assumption that the apparatus-broadening gets introduced to the same extent for all H, D and Hg will allow us to get \Delta{}_{Doppler}^{(H/D)} as follows. We write it in terms of an equation, instead of text -

\Delta{}_{observed}^{(H/D)}=\Delta{}_{natural}^{(H/D)}\oplus\Delta{}_{Doppler}^{(H/D)}\oplus\Delta{}_{apparatus}

And using Equation 1 and Equation 3, we have

\Delta{}_{observed}^{(H/D)}\approx\Delta{}_{Doppler}^{(H/D)}\oplus\Delta{}_{apparatus}\approx\Delta{}_{Doppler}^{(H/D)}\oplus\Delta{}_{observed}^{(Hg)} (Equation 4)

\Rightarrow\Delta{}_{Doppler}^{(H/D)}\approx\Delta{}_{observed}^{(H/D)}\ominus\Delta{}_{observed}^{(Hg)} (Equation 5)

where the used notation (\ominus) is clear from context. Equation 5 is simply the inverse of Equation 4.

The line width of a convoluted Gaussian (the Doppler broadening profile) and a spectral/ dispersion profile (Lorentzian profile of natural broadening) can be effected by using Voigt profiles as has been elucidated by [11]. However, as we predict that the natural line width is effectively masked by the Gaussian Doppler broadening and the apparatus broadening, the described method therein is applicable only if we approximate the apparatus-broadening by a dispersion profile. There is no reason to expect this to be a better approximation than if we approximate the apparatus-broadening by a Gaussian profile. So we do, because it simplifies inverting Equation 4, i.e. solving Equation 5 , a lot easier.

As has been described by [11] , under this assumption, Equation 4 simplifies to

(\Delta{}_{observed}^{(H/D)})^{2}\approx(\Delta{}_{Doppler}^{(H/D)})^{2}+(\Delta{}_{observed}^{(Hg)})^{2} (Equation 6)

This would be the governing equation for Aim 3 of the experiment. It is also essential to be able to achieve Aim 1 and hence Aim 2.

It is apt to note here that under the Gaussian assumption made above, we can replace the line widths in Equation 6 with the Gaussian standard deviations - \sigmas. This is possible because the two are linearly related for a Gaussian profile.The \sigmas come naturally from the analysis of the images, as seen in {sec:Preliminary-Analysis}. As such, we have the modified working equation

(\sigma{}_{observed}^{(H/D)})^{2}\approx(\sigma{}_{Doppler}^{(H/D)})^{2}+(\sigma{}_{observed}^{(Hg)})^{2} (Equation 7)


The Fine Structure Constant, \alpha Edit

For using the data obtained to get the fine structure constant, we use the full quantum mechanical treatment of the atomic Hamiltonian and apply the first order perturbation theory. We do this for the case of Hydrogen. An excellently detailed and lucid analysis is given in Vol.2, Section12A-D, pp1210 by [1]. We state the operational result.

The shift in energy levels of the state (n,l,J) due to the three terms which cause the Fine structure, as mentioned above, is given by

-\dfrac{m_{e}c^{2}}{2n^{4}}(\frac{n}{J+1/2}-\dfrac{3}{4})\alpha^{4}

Note that the shift does not depend on the l quantum number. As such, for the n=2 states, which cause the H_{\alpha}and D_{\alpha}lines, the shift is given by

-\dfrac{m_{e}c^{2}}{32}(\frac{2}{J+1/2}-\dfrac{3}{4})\alpha^{4}

with J=\dfrac{1}{2}or J=\dfrac{3}{2}, the former being doubly degenerate (l=0,1) and the latter with no degeneracy (l=1). The shifts are illustrated in Figure 1.

Transitions from all three states occur to the n=0 state, which is non-degenerate. Because of the degeneracy of the ^{2}S_{1/2} and ^{2}P_{1/2} states, we observe only two transitions. The energy difference in the two transitions (fine-splitting energy) is thus given by

-\dfrac{m_{e}c^{2}\alpha^{4}}{32}(\frac{2}{3/2+1/2}-\dfrac{3}{4})-(-\dfrac{m_{e}c^{2}\alpha^{4}}{32}(\frac{2}{1/2+1/2}-\dfrac{3}{4}))

=-\dfrac{1}{128}m_{e}c^{2}\alpha^{4}+\dfrac{5}{128}m_{e}c^{2}\alpha^{4}=\dfrac{1}{32}m_{e}c^{2}\alpha^{4} (Equation 8)

Strictly speaking, the ^{2}S_{1/2} and ^{2}P_{1/2} states are not degenerate. Again, the reader is referred to Vol 2, pp1227, Vol 1, pp618 by [1] for a brief justification, which has to do with vacuum fluctuations of the electromagnetic field experienced by the electron. The phenomenon is referred to as the Lamb shift. A derivation using the second quantized approach was done by Prof. Mukund Vengallatore in the Ultracold Atomic Physics II course, Fall 2011, Cornell University. He has promised to share the notes for the same, which thence shall be cited. However, as the separation between the ^{2}S_{1/2} and ^{2}P_{1/2} states due to the Lamb shift is about 10 times smaller than the fine structure splitting given by {eq:18}, we do not expect to observe it. [The first direct observation of the Lamb shift was reported by [3]].

Equation 8 therefore gives us the expected value of the fine-splitting in units of wavenumber to be

\Delta\tilde{\nu}=\dfrac{1}{hc}(\dfrac{1}{32}m_{e}c^{2}\alpha^{4})=\dfrac{1}{32}\dfrac{m_{e}c\alpha^{4}}{h}

inverting which, we get

\alpha=\big(\dfrac{32h\Delta\tilde{\nu}}{m_{e}c}\big)^{1/4} (Equation 9)

which is the working equation for Aim 2.


Getting Comfortable with Equipment! Edit

Here, we describe setting up the experiment and aim to get you to such a point that you can easily reproduce a working apparatus ready for data collection within minutes. By the end of this section, you should be comfortable with attaining a good discharge through Wood’s tube, fine-tuning the Fabry-Perot interferometer and initializing the CCD camera for data acquisition.


Aligning the Fabry-Perot Interferometer/ Etalon, etc. Edit

Warning! Do not move the three etalon spacing screws. The two reflecting surfaces of the etalon have been aligned ‘precisely’ parallel with a constant spacing of 7.000 mm. And hence there is no need to ‘play’ with the spacing screws.

We use the Hg arc-lamp to align the etalon. The following steps are seemingly the most efficient for the same.

  • Ensure that the slit connected to the collimator (collimator slit) is sufficiently open. We do not need a slit worthy of a diffraction slit. So open it generously.
  • Remove the etalon from the optical-bench
  • Ensure that the optical path between the collimator and the black-box containing the prism is unobstructed
  • Attach the eyepiece to the holder at the non-collimator end of the optical bench. Do not loosen the tightener which keeps the holder in place wrt the optical bench.
  • Note that the eyepiece mount allows it a horizontal movement. Locate the eyepiece in the center of this horizontal range of freedom.
  • Also note that the eyepiece has a movable lens. So essentially, we can move the eyepiece along the optical-bench axis.
  • Ensure that the Hg arc-lamp has a UV filter installed. If not, wear protective goggles
  • Plug in the Hg arc-lamp and put it on
  • Place it on the wooden blocks beside the collimator such that light from the filter window falls on the collimator-slit.
  • Rotate the prism using the external (outside the black box) (large) screw till the Hg green line (546.1 nm) is seen through the eyepiece. Note that there is a ‘nearby’ unmistakeable yellow line as well. Preferably, do not move the eyepiece from its horizontally central position.
  • Ensure that the image of the Hg green line is vertical. If not, rotate the slit attached to the collimator till a vertical image is obtained
  • Also ensure that the Hg green line image is mid-way in the vertical field of view of the eyepiece. If not, adjust the height of the collimator tube to ‘focus’ the beam along the optical-bench axis.
  • Ensure that the focusing lens between the prism black box and the eyepiece (f-lens) is located midway through the range of allowed positions
  • Adjust the collimator objective lens till the sharpest possible image is got - without moving anything else
  • Then move, if required, the eyepiece along the optical-bench axis to view even a sharper image
  • Now place the etalon back in its place on the optical bench. A spring is missing from the etalon holder which would have allowed its continuous easy rotation about the vertical axis. (Vertical axis is the one normal to the optical table.) So rotate the screw responsible for this rotation (v_axis-screw) to the left - till the etalon-axis is clearly not aligned with the optical-bench axis. Now rotate the v_axis-screw back to the right so that the etalon axis moves towards the optical-bench axis. There should come a point when fringes are clearly visible, albeit very slanted. Now rotate the screw slowly, further till the fringes are almost horizontal. Now rotate the screw even more slowly, further till the fringes are ‘exactly’ horizontal. Congrats!
  • Now move your eye(s) (head) along the vertical keeping in view the fringes through the eyepiece. The fringes should be stationary and not appear to be collapsing into one another or spreading out. The latter is a sign of the etalon spacing not being uniform along the vertical. Consult your instructor if they do collapse into each other or spread out!
  • The fringes may not be very sharp. If you are observing the central fringes, adjust the h_axis-screw of the etalon holder - i.e. the screw which moves the etalon about the horizontal axis. (The horizontal axis is the third axis, orthogonal to both the vertical axis and the optical-bench axis.) The h_axis-screw protrudes below the optical bench (not optical table!) Adjusting the same will move the fringes vertically. Set it to a position where the lower orders (outer fringes) are visible. These are more closely spaced than the central fringes. [A very simple derivation of this result has been given in Section 2, pp 407 by [12], on pp 98 by [13] and in Section 14.1-7 by [8].] The outer fringes allow for ease in adjusting for a better focus. As described above, first, if required, move the collimator objective lens. Get the sharpest image possible. And then move the eyepiece!
  • (After you have the CCD camera set up, you may use it to get the best focus.)
  • A trick for focusing is to look at the edges of the slit - they should appear sharp in the obtained image. That implies a very good focus.
  • Once a great focus is attained, you may change your field of view to the central fringes again for data acquisition.
  • Beware: Replicating a ‘great’ focus is difficult. So as soon as you get one, record readings for all three of H, D and Hg.

By now, you should have attained an excellent imaging.


Attaining a ‘great’ discharge Edit

For this, we need a vacuum pump, Wood’s tube and water and heavy-water samples `connected to the Wood’s tube as shown in the diagram below. We also use an eye-spectrometer (as described below) for quick fine-tune adjustments of the discharge.

Alpha

Schematic of apparatus employed for the experiment. From the slit onwards, the apparatus is mounted on an optical bench. The Hg arc-lamp provides the Hg green-line for calibration of the apparatus. Discharge through H or D in the vacuumed Wood's tube is collimated by the slit and the first focusing lens. The etalon driver, a manual system of two screws, orients the etalon wrt the optical-bench axis. The prism and the etalon provide a spectral resolution through dispersion and interferometry respectively - the latter providing the resolution required for this experiment. We use an etalon of thickness t=7.000\pm0.001\ mm. We tune the apparatus first through an eyepiece and then employ a high resolution CCD camera for data acquisition.

  • Ensure that the exit-valve of the Wood’s tube (air-valve) (which allows air into the tube when re-pressurizing after stopping the vacuum pump) is closed
  • Ensure that the clamps on the water and heavy-water vials are closed (clamp-1 and clamp-2 respectively)
  • Ensure that the valves connecting the water and heavy-water vials to the Wood’s tube are closed (valve-1 and valve-2 respectively)
  • Activate the vacuum pump
  • Completely open the valve to the Wood’s tube (main-valve) to attain vacuum in the Wood’s tube
  • Wait till the pump becomes less noisy than when initialized. This ensures a good enough vacuum inside the Wood’s tube for discharge
  • Partially close the main-valve
  • For observing the H spectrum, loosen clamp-1 and slowly turn valve-1 to allow water vapor inside the Wood’s tube.
  • Now apply the high voltage across the Wood’s tube for discharge. Slowly adjust valve-1 to regulate the flow of water vapor through the Wood’s tube. Recall that the flow is being created due to the vacuum pump. A good balance between the inflow and outflow is required to attain an optimal discharge. It is noted that both the main-valve and valve-1 are only partially open for an attained optimal discharge.
  • The pocket-spectrometer helps a great deal to preliminarily gauge the intensity of the H_{\alpha} lines so that adjusting valve-1 is greatly convenienced. Put it to use! (After you have the CCD set up, use it directly to observe the live image and thus adjust valve-1/ valve-2 to get the optimum intensity.)
  • When switching over to D, open clamp-2 and valve-2 as described above before closing valve-1 and clamp-1. This ensures that the Wood’s tube remains in discharge throughout. Doing it in the reverse order may cause the vacuum to be so ‘good’ that the discharge doesn’t occur.
  • When we want to close off the experiment, unplug the high discharge voltage first. Then ensure that the clamps 1 and 2 and valves 1 and 2 are closed. Also ensure that the main-valve is still open, at least partially (It should be!) (We want to re-pressurize the entire Wood’s tube, so it is crucial that it is open.) Now unplug the vacuum pump. And finally, open the air-valve till done re-pressurizing. The whistling flow of air into the Wood’s tube will decay as the pressures inside and outside the Wood’s tube equate. Close the air-valve in preparation for the next experiment run.

A illustration of the discharge attained is shown below.

2

Discharge attained in the Wood’s tube for observing the D spectrum. The green blob to the left in the background is the Hg arc-lamp.


Capturing Data - Using the CCD Edit

Note that the freedom that the eyepiece affords in its motion along the optical-bench axis is crucial for a great focus. With the CCD camera in the place of the eyepiece/ CCD holder, the CCD ‘chip’ has no such corresponding motion along the optical-bench axis, and as such, the focusing lenses have to be made use of, as described in detail below. In spite of the same, the best focus obtained with the CCD camera is not as great as the focus attained with the eyepiece. That's okay.

The steps to capture the data on the computer are as follows.

  • Connect the CCD USB cable to the computer. The driver should already be installed thereon. Open the ‘Autostar’ program by clicking on the desktop icon[14]. Click on the ‘Imgaing’ menu and select DSI Imaging.
  • In the new window that opens, in the settings and preferences menu, change the folder in which we want to save the images. Also choose the type of file in which we want to save the images. We prefer the .tif(f) format because of the higher possible pixel values that it allows. Remember to Save the settings.
  • In the drop-down options for the mode of data acquisition, choose ‘Planet’ / ‘Moon’ / ‘Terrestrial’.
  • If not already checked, check the ‘Live’ box so that before initializing data capture, we preview the data as it shall be acquired. Note that it is easy, as mentioned above, to focus the interference image by observing on the large computer screen. Utilize the ease afforded!
  • It may happen that the image observed through the eyepiece does not show up on the CCD. This implies that the eyepiece is not centered in its horizontal freedom. Rotate the prism slowly till we get the image on the CCD. (Caution: By default, the live acquisition occurs at 1 Hz. But in case you have changed this to .5 Hz, as later we shall, change it back to 1 Hz during all adjustments done using the live imaging.)
  • As always with focusing, first move the collimator objective to achieve as sharp an image as possible. Only then move the f-lens. It is noted that the best focus is attained when the f-lens is closest to the eye-piece/ CCD holder. You may attempt to change the position of the eye-piece/ CCD holder to move it close toward the prism blackbox. But the optical bench apparatus would have to be significantly altered to achieve that, the luxury of the time required for which you probably cannot afford. Its effect on the error in the experiment is detailed later.
  • Uncheck the option ‘combine images’
  • Adjust the exposure time to 2.0 seconds
  • Click ‘Start’ to start acquiring images and click ‘Stop’ when a single image has been captured.

A couple of images so got are reproduced below.

  • Note the varying line widths for each of H, D and Hg
  • Note the contrast in the peaks and troughs of H as compared to D
R2 H 3 images - Copy - Copy

H_{\alpha} higher order fringes as captured by the CCD camera. The Fabry-Perot interferometry resolution is along the vertical. The prism resolves the spectrum along the horizontal.

R2 - Copy

D_{\alpha} higher order fringes as captured by the CCD camera. The Fabry-Perot interferometry resolution is along the vertical. The prism resolves the spectrum along the horizontal.

R2 Hg - Copy - Copy

Hg higher order fringes as captured by the CCD camera. The Fabry-Perot interferometry resolution is along the vertical. The prism resolves the spectrum along the horizontal. The prism had been rotated to bring the green line in the CCD field of vision as compared to the position of the prism used to get the H/ D lines.

Analysis Edit

As always, it is advisable to perform a preliminary analysis using a pilot data. You can thus ensure that the data taken is usable.

In all the analyses that follow, t=7.00(0)\pm(1)mm is the etalon thickness.

We need to plot the interference pattern from the Fabry-Perot etalon as intensity as a function of the vertical coordinate (ordinate) of the screen [15]. The ordinate unit in our case is a pixel. The intensity is measured in units of pixel value (which is that for RGB each). It may be aesthetically pleasing to normalize these. However, the analyses remains completely unaffected by the same and hence, is relegated to if leisure time permits.

We consider the interference profile obtained to be the convolution of the spectral and broadening profiles we discussed about above. Recall that we decided to apply Equation 7 under the assumption of neglecting the natural line width and approximating the apparatus-broadening by a Gaussian profile.

We obtain an intensity profile as a function of the ordinate by analyzing the pilot images in MATLAB (or any other data processing software). A step-by-step procedural algorithm for obtaining the same and fitting it to a Gaussian (as discussed above) is not given here. Do work it out. Here is presented a representative plot obtained as such.

4.6 D odd

Intensity spectrum of D_{\alpha} line. The numbers in the box are the fitting parameters of the Gaussians. Note that each pixel stands for a fringe order, even if a partial order. Error bars are standard deviations for the intensity observed for that fringe order. Also note that the Gaussians are fitted with a weight given by \dfrac{1}{\sigma^{2}}, where \sigma is the error bar as described.


Also, note the following. It can be derived to a first approximation that the ordinate and the spectral wavenumber [16] are linear in between adjacent peaks/ troughs of the interference pattern. A straightforward derivation of the same is succinctly given in Section 5 by [12]. It has thereby been shown that the ordinate spacing between two successive peaks (say X pixels) corresponds to a wavenumber-difference \Delta\tilde{\nu}=\dfrac{1}{2t} which in our case is given by

\Delta\tilde{\nu}_{adjacent-orders}=\dfrac{1}{2\times7.000mm}=0.714(3)\pm(2)cm^{-1} (Equation 10)

As such, if we measure the line width to be y pixels, then the line width in terms of wavenumber is given by

\Delta\tilde{\nu}=0.714(3)\pm(2)cm^{-1}\times\dfrac{y}{X} (Equation 11)

where X is measured for that particular peak.

This constitutes the working equation for Aim 1.

Note that we can replace the line width y by the Gaussian standard deviation x. In the entire analysis that follows, we never are interested in the line width per se. We need at most the ratio of line widths, and in that sense, dealing with the Gaussian standard deviation is preferred. After all, x and y are linearly related, because of which the considerations are independent of our choice of variable.

We only need to change the LHS of Equation 11 to reflect this -

\sigma_{\tilde{\nu}}=0.714(3)\pm(2)cm^{-1}\times\dfrac{x}{X} (Equation 12)

Now, we may proceed with the preliminary analysis, remembering that we have chosen to analyze in terms of x.

In all the analyses that follow, \sigma stands for the standard deviation and x stands for the mean. We continue to use X for the separation between two adjacent peaks. Consider that in a given image, we have several peaks. Then we have a set of \sigma_{i},x_{i},X_{i}.

We define

X_{1}=x_{1}-x_{2}

By Equation 12,

(\sigma_{\tilde{\nu}})_{1}=\dfrac{\sigma_{1}}{X_{1}}\times0.714(3)cm^{-1}

Errors are calculated as follows

\Delta(X_{1}=x_{1}-x_{2})=\sqrt{(\Delta x_{1})^{2}+(\Delta x_{2})^{2}}

\Delta(\dfrac{\sigma_{1}}{X_{1}})=\dfrac{\sigma_{1}}{X_{1}}\sqrt{\dfrac{(\Delta\sigma_{1})^{2}}{\sigma_{1}}+\dfrac{(\Delta X_{1})^{2}}{X_{1}}}

\Delta(\sigma_{\tilde{\nu}})_{1}=(\sigma_{\tilde{\nu}})_{1}\sqrt{\dfrac{\Delta^{2}(\dfrac{\sigma_{1}}{X_{1}})}{(\dfrac{\sigma_{1}}{X_{1}})^{2}}+\dfrac{.0002^{2}}{0.7143^{2}}}

=(\sigma_{\tilde{\nu}})_{1}\sqrt{\dfrac{(\Delta\sigma_{1})^{2}}{\sigma_{1}}+\dfrac{(\Delta X_{1})^{2}}{X_{1}}+\dfrac{.0002^{2}}{0.7143^{2}}} (Equation 13)

You shall probably see that in Equation 13, the latter two terms inside the square-root can be consistently deemed to be negligible compared to the former.

These calculations can be done in any software.

Another important point to note is that all rounding is done at the last step so that we do minimize the propagating error. In all the analyses that follow, use this same convention.

Results Edit

The `formulae' needed for obtaining `all' results are given above.

We can evaluate \dfrac{m_{D}}{m_{H}} and \alpha with them. The Doppler line widths (and thus standard deviation of the Doppler profile) are related as given in Section 21.1, pp 420 by [2]. We thus have \dfrac{\sigma_{Doppler}^{(H)}}{\sigma_{Doppler}^{(D)}}=\dfrac{f_{Hydrogen}}{f_{Deuterium}}\sqrt{\dfrac{m_{D}}{m_{H}}}\approx\sqrt{\dfrac{m_{D}}{m_{H}}} where f stands for the frequency of emission, which we assume to be the same because our experimental resolution is only so high that we cannot differentiate the error introduced as such. Strictly speaking, of course, f is different for H and D.

A more important part is making sense of the measured values. A common occurrence is that the measured values, even with experimental error, do not lie in the currently accepted range. The common reason for this is either systematic error or an underestimation of error, both leading to skewed values for the measurements. Try to justify, if required, such an occurrence. You may also include how these observations and measured values support the quantum theory, etc. Or you may have a different take on this experiment altogether!

And with this, tada! Congrats on having done this revolutionary experiment! :)

ReferencesEdit

  1. 1.0 1.1 1.2 1.3 Quantum Mechanics (2 vol. set), Claude Cohen-Tannoudji and Bernard Diu and Frank Laloe, Wiley-Interscience, 2006, ISBN 9780471569527
  2. 2.0 2.1 2.2 2.3 2.4 2.5 Introduction To Atomic Spectra (International Series in Physics), Harvey Elliott Ph.D. White, New York, NY McGraw-Hill Book Co, 1934
  3. 3.0 3.1 T. W. HANSCH,I. S. SHAHIN {\&} A. L. SCHAWLOW, Optical Resolution of the Lamb Shift in Atomic Hydrogen by Laser Saturation Spectroscopy, nature physical science, 1972, 235, 63
  4. Update: In spite of this, in the analysis, we make the approximation that all observed peaks are Gaussian.
  5. Lasers, Anthony E. Siegman, University Science Books, 1986, ISBN 9780935702118
  6. In our case, instead of a ‘gas’ in the strictest sense, we have H and D in the form of water vapor and heavy-water vapor, respectively.
  7. For an ideal gas, the Maxwell-Boltzmann distribution reduces to an exactly Gaussian profile. In our case, it would essentially be Gaussian.
  8. 8.0 8.1 8.2 8.3 Fundamentals of Optics, Francis Jenkins and Harvey White, McGraw-Hill Science/Engineering/Math, 2001, ISBN 9780072561913
  9. Optics: Principles and Applications, Kailash K. Sharma, Academic Press, 2006, ISBN 9780123706119
  10. This too is strictly not correct, as can be seen from back of the envelope calculations. If we assume as mentioned before that the apparatus-broadening is dominated by the etalon response function, then as the wavelength of the observed spectral line decreases, \delta varies faster with x, or equivalently \delta varies faster with \nu, and thus, the response function as a function of \nu gets scaled. This means that \Delta_{apparatus}^{(Hg)}\ne\Delta_{apparatus}^{(H/D)}. We can show that the line width due to the etalon-broadening is \propto\sqrt{\lambda}\Rightarrow\dfrac{Delta_{etalon}^{(Hg)}}{Delta_{etalon}^{(H)}}=\sqrt{\dfrac{546.1}{656.3}}=0.912(2)\pm(2). (This discussion is of no consequence to the analysis because of an assumption we later make - that the apparatus-broadening is sufficiently approximated by a Gaussian profile!)
  11. 11.0 11.1 {Van de Hulst, H.C. and Reesinck, J.J.M., Line Breadths and Voigt Profiles, Astrophysical Journal, 1947, 106, 121, doi:10.1086/144944
  12. 12.0 12.1 Karl Wilh. Meissner, J. Opt. Soc. Am., 6, 405, OSA, Interference spectroscopy. Part I, 31, 1941, doi:10.1364/JOSA.31.000405
  13. W H J Childs, ={The Fabry and Perot parallel plate Etalon, Journal of Scientific Instruments, 1926, 3, 4, pp. 97, doi:10.1088/0950-7671/3/4/301
  14. Note that this is a CCD camera meant for imaging a telescope. But it doesn’t really matter because all we want is an ‘intensity’ profile of the spectrum.
  15. This is because the resolution of the Fabry-Perot interferometer is along the vertical.
  16. defined as \dfrac{1}{\lambda}

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