We examine the emission spectrum of molecular nitrogen in the final spectroscopy experiment of the semester. We will use the same experimental set up as Lab #6, however we'll use molecular nitrogen discharge tubes. The homo-nuclear diatomic molecule is perhaps the simplest system in which to test the reach of quantum mechanics to describe collections of quantons; so instead of one atom, we study the quantum rules of engagement for two atoms interacting with each other, trapping each other, in what up until the advent of quantum mechanics was unexplainable, a 'simple' chemical bond. Quantum fluids and solids would be next (many atoms interacting), but such experiments are beyond the scope of this course.
Already with simple molecules we find quantum systems incredibly rich in beautiful phenomena. The most common gas on our planet is not a monatomic gas but diatomic one: molecular nitrogen. Molecular nitrogen is of great importance in atmospheric physics, and one which contributes to the beauty of spaces above the atmosphere (in the ionosphere, typically more than 90 km above the surface of the Earth) in form of the Auroras, shown below both schematically (left), and how they actually look in the wild (middle). The emission spectrum of molecular nitrogen shown bottom right is taken from a Nitrogen plasma in a discharge tube (below, right) such as we've been using in our lab. Aren't the folds of color glorious? Hey, why doesn't our raw data look like that?
Figure 1. (left) Earth's aurorae, from Discovery ezine, July/August 2017 issue, from the article ``Everything worth knowing about...Auroras''. Note the scribbled in annotation-most of the auroral light from oxygen comes from the atom, while most of the auroral light from nitrogen comes from the molecule. Molecular oxygen is relatively easily dissociated in the upper atmosphere where aurora's live, compared with the much more strongly bound nitrogen atom. (middle left)McMurdo Station, Antarctica, during summer in the northern hemisphere, spectacular auroras abound. Check out the video, 'Antarctica: A year on Ice'.(middleright)A picture of the emission spectrum of molecular nitrogen showing 'band spectra', that is, distinct stripes of color which has stripes within stripes that are too close together for the spectrometer to resolve. The bands themselves are vibrational and the blurry stuff within each band is a rotational spectrum. The different degrees of freedom, quantized, make for a right tapestry of colors, more complex (and beautiful) than atomic spectra!.(right)A picture of electronic states in molecules Rb-H, that create the trapping potential for the nuclei which vibrate in the nearly harmonic trap. The dotted lines show a perfect SHO potential, the solid lines, the actual potential. The horizontal dotted lines show the eigenstates of oscillation for the nuclei, and the wavefunctions are drawn (using the energy level as a proxy for an x-axis!). Each parabolic potential structure is labeled (much like spectroscopic terms in atoms) according to the angular momenta of the electrons, following the quantum rules of adding individual momenta to get totals for both orbital and spin momenta. See the references for a discussion of this!
Some of the potential energy curves that confine the nitrogen nuclei and their electrons are shown in Figure 2. We come to molecular spectroscopy expecting that the energy states are straight lines, as in partial Grotrian diagram. Instead we find that each electronic state, or spectroscopic term, is a curve. Almost parabolic, like that of a simple harmonic oscillator. But the vibrational states within each curve is level (monoenergetic). Each spectroscopic term, like that of its atomic counterparts is characterized by a kind of total spin, and orbital angular momentum (e.g. $C ^3 \Pi_u$, where the capital Greek letter stands, as for atomic spectroscopic terms, the orbital angular momenta...don't ask me about the English letter, but in the Greek, you can see a capital 'P' for a state with one unit of orbital angular momentum....the un-gerade and gerade subscripts and the rest of it are discussed in the footnote [1]). In Figure 2, one can see horizontal numbered notches on the right hand inside rim of the confining potential, indicating states of definite (allowed, characteristic, 'eigen') vibrational energies of the molecule within that particular term or electronic state. Note: within a given electronic energy state or term, there are many vibrational states, associated with the quantum number $\upsilon$. The shape of the confining potential is quadratic closest to its minimum, and so molecules furnish a realistic approximation of the ideal quantum simple harmonic oscillator, as mentioned above. Of course the real molecular potential departs from that of the simple harmonic oscillator for a variety reasons which we will not explore except for the first order corrections. One could make a heuristic, classical argument to make the asymmetry about the minimum plausible, but that would be a classical understanding. If we were to take that sort of reasoning seriously, we could not understand how a stable configuration of the atoms the atoms could occur. There is no classical understanding of the (stable) chemical bond.
So, what are the research questions for the experiment?
They are these:
How many transitions can you identify between the vibrational states of two spectroscopic terms, labeled $C ^3 \Pi_u$ and $B ^3 \Pi_g$ (again, see note [1] for a brief explanation of the nomenclature), in the range of wavelengths $350 < \lambda < 500 nm $? Defend, support, explain your identifications in two ways, 1) by crudely calculating the expected wavelength for two of the transitions that you can read right off of Figure 2, as an intermediate check on progress, and 2) by comparing the observed band head wavelengths with the known band head wavelengths tabulated in the literature [2,3]. This question is described with greater specificity in task #1.
What are the values of a) $\omega_e$, b) the effective spring constant associated with it, and c) $\omega_e x_e$, the ``anharmonicity term'' for the lower electronic state ($B ^3 \Pi_g$). The values specified in a) and c) we will find by fitting data to a model, described in more detail below. Part b) will be direct calculation based on the value $\omega_e$ and may be compared to a crude calculation based on the shape of the potential energy curve for the state.
Figure 2. Potential energy curves for some of the electronic eigen-energy states (spectroscopic terms of $N_2$, from Figure 1, in Lofthus and Krupenie's work.
7.2 Task #1: How many transitions can you identify between the vibrational states of two spectroscopic terms, labeled $C ^3 \Pi_u$ and $B ^3 \Pi_g$, in the range of wavelengths $350 < \lambda < 500 nm $?
Before getting into the details, let's just say that identification simply means you know how to assign the correct vibrational state quantum numbers for the two states of the transition, the $\upsilon'$ vibrational state quantum number for the upper term ($C ^3 \Pi_u$), $\upsilon''$ integer for the lower term ($B ^3 \Pi_g$). So, what to do?
Obtain a spectrum of $N_2$ in the 350-500nm range, being careful to magnify weak patches (if you need to) by various means available to you. Print these together in your lab notebook so as to form a complete spectrum in the range given. Print a copy of the spectrum or spectra, and affix (tape!). Record in detail the sequence of software commands and settings necessary to obtain and save the data, recording of course the full path names of all files created, with a brief note of what the files contain. This is the raw data. The next few tasks help us characterize it and help us answer this task's question. The peaks you'll see prominently in this range of wavelengths are called band heads. For atomic spectra, the peaks are skinny and symmetric. In molecular spectroscopy, there is an asymmetry about the peak; the intensity gently slopes down on one side and drops precipitously on the other. The gentle slope hides rotational transitions unresolved by the spectrometer we are using. We will not pursue the physics of the rotational spectra even through they are as as cool as they are, and we'll leave the topic there.
What to do with the plot you'll make of the spectrum (saved probably as a .txt file, but plotted with excel, or maybe if you are more daring, FITTEIA...in any case, produce a plot with correctly labeled axes and then...here's what wants to be done:
Calculate the expected wavelength for two of the possible $\upsilon'\rightarrow\upsilon''$ visible emission transitions between the 2 electronic states, $C ^3 \Pi_u - B ^3 \Pi_g$, simply by placing a ruler directly on Figure 2, and measuring the vertical separation in mm, say, and from the crude measurement, calculate the expected wavelength with the appropriate conversion factor (e.g., measuring the distance in mm between 0 and 10 eV on the ordinate). Estimate uncertainty for this method, and look to see in the raw data to see whether there is a band head (within the range of uncertainty) at that wavelength. Write down your analysis of this intermediate check on progress in your notebook. The record of what you learned here must be featured in lab notebook. To facilitate this work, see the partial energy diagram shown below. Note that there are tools in Adobe reader that permits you to do this too! But 'analog' rulers are actually even more handy. Annotate the band heads of you have confirmed in this manner. To aid in distinguishing this way of checking results from the next one, please put the vibrational quantum numbers above the identified band head in a box (so for example, over two of the band head peaks there should be something like $\fbox{1'-3''}$, and $\fbox{1'-5''}$, say.
Figure 3. On the left, An edited version of the potential energy curves shown above, highlighting the electronic states, $C ^3 \Pi_u $ and $B ^3 \Pi_g$. A grid has been added to the image to permit the counting of boxes to aid in the calculations. On the right, another image of the same thing, showing the term energy differences for some of the transitions, which for purposes of this lab experiment, we'll define as $T_{\upsilon',\upsilon''}$. For this task however, one can get the energy units directly from the ordinate ('y-axis') units.
Find as many transitions beginning from the upper level $\upsilon' = 1$ vibrational state that you can find in your acquired spectrum, annotating the spectrum itself, and producing a table of these transitions. Look up the known values of the band head wavelengths in Table 29 in the 'nitrogen bible' [2] to identify as many band heads arising from vibrational state transitions of the sort, $\left(1'-\upsilon''\right)$, for as many values of $\upsilon''$ as you can find.
Annotate you plot of the acquired spectrum using proper nomenclature above each identified band head.
Produce a table of results with (at least) 7 columns, identifying 1) the $\upsilon' - \upsilon''$ transition scheme (as is done in table 29 in Lofthus and Krupenie), 2) the accepted wavelength from table 29 (described in the lab handout), 3) the observed value (from the plotted spectrum), 4) discrepancy ($\Delta \lambda_D ?$), 5) experimental uncertainty, ($\Delta \lambda_U ?$), 6) the measured energy gap itself for two of the observed lines, noting that the energy of the transition is referred to as $T_{\upsilon', \upsilon''}$ on our public course web site, and 7) the calculated value of the wavelength from the gap energies (see calculation in the checklist item above. Recall that all column headings must include a meaningful descriptor, and units.
Interpret the degree of agreement between these three indications of band head wavelengths in the usual way, through quantitative comparisons of discrepancy and uncertainty (of measured quantities and their estimates). There is a column for this. Make comments about it.
7.3 Task #2: What are the values of $\omega_e$, $k_o$, and the anharmonicity term, $\omega_e x_e$, for the lower electronic state $B ^3 \Pi_g$?
Another way to understand this task is to ask the same questions in a more pictoral way. How does the spacing between vibrational energy levels in a given molecular potential trap change as the vibrational energy level quantum number increases? If the trap is perfectly harmonic, with zero anharmonicity, the steps are all the same, a perfect ladder in energy space. But if there is some anharmonicity, there would be a systematic difference between steps. What follows is a description of how to quantify the size of the steps and that systematic difference, the one related to the effective spring constant, and the other to magnitude of the anharmonicity. First, some nomenclature.
In Table 1 of reference [3], 'Improved fits....'. we find what is meant to be an explanatory master equation for the vibrational state energies for a particular (arbitrary?) spectroscopic term (electronic state), \begin{equation}
T_\upsilon= T_e + \omega_e (\upsilon+\frac{1}{2}) - \omega_e x_e(\upsilon+\frac{1}{2})^2 + \omega_e y_e(\upsilon+\frac{1}{2})^3 + \omega_e z_e(\upsilon+\frac{1}{2})^4 + \omega_e a_e(\upsilon+\frac{1}{2})^5.
\end{equation}
An arbitrary vibrational state energy, $T_{\upsilon}$, is often given as an energy in units of $cm^{-1}.$ Notice that the first term, $T_e,$ is the energy of the minimum of the potential well and the second term is the expression for the energy of the (ideal) quantum SHO levels ($E_{\upsilon} = \hbar \omega_o (\upsilon + \frac{1}{2}))$. Note that $\omega_e$ also has the units of inverse cm., and the $x_e, \: y_e, \: etc.$ terms are dimensionless constants used to characterize anharmonicity. But we will only care about the coefficient of the term quadratic in $(\upsilon + \frac{1}{2}).$ Note too that the gap in vibrational energy states in the lower energy potential trap ($B ^3 \Pi_g$) is given by $\Delta T_{\upsilon''}$. Below we will describe an analytical model for plot of $\Delta T_{\upsilon''}$ vs. \upsilon'', from which we will obtain best fit values for $\omega_e$ and $\omega_e x_e$. But for the moment, lets keep focusing our attention on the nomenclature. The gap between adjacent vibrational states in a given electronic state ( $\Delta T_{\upsilon}$) corresponds to $E_{\upsilon}/hc,$ where $\Delta E$ analytic expression of an idea familiar to us, an energy gap, here applied to the gap between adjacent vibrational states etc., etc., but where cgs units are used, i.e, $ c \approx 3 \times 10^{10} cm/sec$, with result that $\Delta T$ is in units of $cm^{-1}.$ An easy way to see this is by supposing that $\Delta E = \hbar \omega_0$, so that $\Delta E /hc = \omega_0/(2\pi c)$, which clearly has units of 1/cm. In fact in this way we can use a numerical value for $\omega_e$ to get a value for $k$ and vice versa, because, leaving out a small but important algebraic step, $\omega_e = \omega_0/(2\pi c)$, and \begin{equation}
\omega_0 = \sqrt{\frac{k_0}{\mu}},
\end{equation}
where $k_0$ is the effective force constant (spring constant) for the electronic state, and $\mu$ is the
reduced mass, which is just $m_1/2$ for a homonuclear diatomic molecule of mass $m_{total} = m_1 + m_1$. The reduced mass is familiar to you from Kepler's laws, and it's part of how one can take a
two-body problem (gravitational orbits, say) and turn it into a one body problem and a force law. To return to our problem, it's really easy to make mistakes of factors of four here! AND MIND THE UNITS!
Figure 4. Here the QM SHO solutions for the vibrational energy eigenvalues are plotted along with the energy eigenvalues for a more realistic model (the Morse Potential model, which we will not use, but it's depicted here for the purpose of illustration). In physical chemistry texts, $k$ is often referred to as a 'force constant'. The SHO potential is chosen here to have the spring constant that fits the bottom of the well 'well' for the hydrogen iodide (HI) molecule. The energy units used are $cm^{-1},$ in common use in the chemistry world.
Estimate both $\omega_e$ and $k_o$ of the electronic state $B ^3 \Pi_g$ by direct calculation using ruler measurements of the confining potential itself, again, from Figure 2. A sketch of the procedure is shown below. Find $k_o$ and then estimate $\omega_e$ expressed inverse cm. Given that uncertainties (in ruler measurements) of $x$ and $E$, what is the uncertainty in this estimate of $\omega_e$ and $k_o$?
To check, compare your result for $\omega_e$ with the value published in Table 1 of ref. [3], expressing the uncertainty and discrepancy in the same units, and using significant figures appropriately. This is a intermediate check on progress , and a means of becoming more familiar with quantum oscillators in way that links visual and analytical elements. For this step and for all such intermediate checks, note your assumptions and evaluations, calculations, and so on, in your lab notebook. Is there 'ball park' agreement, or are you WAY OFF? Discuss.....
Figure 5. Use ruler measurements made directly on a copy of the confining potential associated with the $B ^3 \Pi_g$ vibrational term, to estimate $k_o$.
Obtain an experimental estimate of the harmonicity and the anharmonicity of the $B ^3 \Pi_g$ state spectroscopically, directly from table 29 in the nitrogen bible:
From the band head wavelengths in table 29, make your own table in your lab notebook, with at least 5 columns, with headings, $\upsilon' - \upsilon''$, $\lambda_{\upsilon',\upsilon''}$, $T_{\upsilon',\upsilon''}$, $\Delta T_{\upsilon''}$, and the uncertainty in $\Delta T_{\upsilon''}$.
Explain your estimate of the uncertainty of those energies and exhibit your calculation of them, at least of an example. Adopt suitable nomenclature for the uncertainty in that little gap energy! A question: is there an easy way to get $T_{\upsilon',\upsilon''}$ from $\lambda_{\upsilon',\upsilon''}$?
Plot gap energy vs. vibrational quantum number, using FITTEIA, using tabular elements composed above, including of course the uncertainty in gap energy in the usual $x,$ $y,$ and $\Delta y$, format wanted by Fitteia.
Create a numerical model for fitting the gap energies as a function of the vibrational quantum number. To arrive at the model, calculate the gap energy directly from Eq. 1, ignoring terms greater than quadratic in upsilon, doing the algebra, etc., obtaining
\begin{equation}
\Delta T_{\upsilon''} \equiv T_{\upsilon',\upsilon''} - T_{\upsilon',(\upsilon''+1)} = \omega_e - 2 \omega_e x_e(\upsilon''+1).
\end{equation}
What arguments in references [3] are offered about the validity of ignoring terms higher than the quadratic one in equation 1? Do you accept them? Explain. Use equation 3 to fit the data in the table described in part 2b).
Create the computational model from the analytical model found in the step above, showing your work (to yourself and future readers of your lab notebook, one of whom is certainly the instructor of the course!), and using correct C coding. Create the standard 'alignment matrix'.
Obtain in the Goldilocks fashion, fits for both $\omega_e$ and $\omega_e x_e$ from the experimental data. Compare these to the accepted values published in ref. [3], presenting your results as always, using significant figures appropriately for both uncertainty and discrepancy. How good is the fit? Explain in terms of the reduced chi-squared values, and
using significant figures appropriately.
Now, at the very end of your lab notebook entry for this lab, write your abstract. Make sure it answers the two questions posed for this lab in as few words as possible without leaving
anything important out. Be sure that the abstract posits answers to each research question that are justified (briefly) by summary results (noting appropriate discrepancies and uncertainties along the way, using significant figures correctly, etc., and so on) and be sure the methodology of the measurement techniques would make sense to someone who hasn't read the document you are reading now.
References:
The naming scheme for the electronic states differs a little
bit from the scheme used for atoms as expected because of
the increased complexity of the system. The ground state
in historical usage is called an $X$ state. The next excited
states are not denoted by 2, 3, 4, etc., but rather by A, B,
C, with perhaps, an $a$ and $a'$, $b$ and $b'$ thrown in. Then,
since the nuclei are in different locations, the neutralizing
electrons do not move in a spherically symmetric potential,
and so there is no conservation of angular momentum in
general. There is instead conservation of the projection of
angular momentum along the axis along which the nuclei
vibrate. These axial components of the angular momenta
for a single electron are designated as before, lz, and sz,
for the orbital and spin momenta, respectively. Delineating
the quantum numbers for angular momenta gets a little
tricky. Capital Greek letters are used to designate the quantum
number number for (the projection of the total) orbital angular momenta. The same sequence of s, p, d, for 0, 1, 2, is used.
The total orbital angular momentum
${\bf L}$ is added to the total spin angular momentum
${\bf S} $ to give the total angular momentum ${\bf J}$, where again,
there are multiple ways in which this needs to be done,
and only the component along the axis along the nuclei
matters, and one sees $L_z = \Lambda \hbar$ and $S_z = \Sigma \hbar$ (and one has to get used to the essential ambiguity in
reference to spin and orbital angular momenta occasioned by the use capital Greek letters, one simply needs to know the context to know which momentum is being discussed). Happily, the total angular momenta is not typically displayed! The total spin multiplicity is also the same as in atomic spectroscopic terms, $2S+1$.
Further, and finally, there are two designation for the spatial symmetry of the wave function that describes the electronic state (the non-spin part), and instead of a superscript there is a subscript, indicating the two kinds of symmetry. Shown below is Figure 9.17 of Demtroder's book Atoms, Molecules, and Photons, (Springer, Berlin, 2005), and is helpful for picturing these symmetries and their designation.
If the electronic state possesses
reflection symmetry about a plane passing through
the axis along which the nuclei vibrate, whether even (+)
or odd (-), this is denoted (sometimes explicitly) by a plus or minus sign as a
superscript, and, further, for the special case of homonuclear
diatomic molecules, there can be inversion symmetry
about the origin (center of mass) which is the same concept as 'parity' in atomic spectroscopic terms. The inversion is simply swapping the sign of all 3 coordinates, or rotating the radial vector by 180 degrees in both polar and azimuthal angles, sort of (see the figure above). States 'even' in this inversion are
said to be 'gerade' and the 'odd' states, 'ungerade', after
the German, for those words, and they appear as subscripts.
This is how spectroscopic terms are composed in diatomic molecules,
Okay? Got it? Easy as $\Pi$.
Also note: in their paper from with Fig1(right) was drawn, the authors Gaydon and Pearse do not label the states! But the modern labeling scheme has a evolved since they published and the foregoing notes attempt to summaritze current practice!
