Optical Spectroscopy of Hydrogenic Atoms
Prashanth S. Venkataram
MIT Department of Physics: 8.13
(Dated: November 4, 2012)
The quantum mechanical nature of an electron bound to a nucleus is well-shown by the discrete
emission lines observed in spectroscopy of hydrogenic atoms. We determine the calibration of the
spectroscopic apparatus used by measuring the emission lines of mercury. We then determine the
value of the hydrogen Rydberg constant by measuring the Balmer emission lines of hydrogen along
with the ratio of the deuteron mass to the proton mass by observing the isotopic shifts in the Balmer
emission lines of deuterium. We finally characterize the properties of the emission lines of sodium.
Classically, a particle in a potential should be able to
sample any energy given an appropriate energy input.
Over a century ago, however, it was found that atoms
emit energy in the form of electromagnetic waves only
at discrete wavelengths when energetically excited. Mercury has been particularly well-studied over the years,
and it has a multitude of bright emission lines over a
large range of wavelengths in the ultraviolet and visible
spectra, so it is ideal in forming a calibration curve for
the particular spectroscopic apparatus used.
Hydrogen is the only element whose quantum dynamics can be fully analytically described. Deuterium can
also be analytically described because it only has one
electron, but because its nucleus is more massive than
that of hydrogen, it exhibits slightly different quantum
dynamics than does hydrogen; such atoms with only one
electron are called hydrogen-like. Sodium cannot be analytically described as it has more than one electron, but it
has only one electron in the outermost energy level which
acts much like the single electron in a hydrogen-like atom,
so it is well-approximated as a hydrogen-like atom; furthermore, relativistic and other corrections yield further
insights into the fine structure of the sodium atom.
I.
THEORETICAL BASIS
The Balmer series of hydrogen consists of emission lines
whose wavelengths are almost all in the visible spectrum.
For an emission, the starting electron energy level n1 > 3
[1] (so as to distinguish n1 = 1 as being in the separate
Lyman series), where n1 is an integer, while the ending
energy level n2 = 2 in all cases. As the general equation
for calculating the wavelength of a hydrogen emission
emission line is given by
1
1
1
= RH
− 2
(1)
λ
n22
n1
then the specific equation for the Balmer series is
1
1
1
= RH
−
.
λ
4 n2
from the measured Balmer series wavelengths of hydrogen; me and mP are the masses of the electron and proton, respectively, while R∞ is the fundamental Rydberg
constant which is related to the mass of the electron along
with other physical quantities that were not measured
here.
e −1
R∞
In general, the Rydberg constant Rm = (1+ m
m )
for a nucleus of mass m. This means that as deuterium, being an isotope of hydrogen, has a different
mass than hydrogen, it will have a different Rydberg constant than that of hydrogen, so its Balmer series emission wavelengths will be shifted compared to the corresponding wavelengths for hydrogen Balmer series emissions. Where the letters ‘P’ and ‘D’ indicate the proton
and deuteron, respectively, and where me , R∞ , and n
are as they were previously, the isotopic wavelength shift
depends on the ratio of the deuteron to proton masses
through
(λP − λD )R∞ = (1 −
Here, the hydrogen Rydberg constant RH = (1 +
me −1
R∞ is a propotionality constant to be determined
mP )
1
n2
.
(3)
In all hydrogen-like atoms, it can be shown that in
the frame of the electron [2], the nucleus is orbiting and
producing a magnetic field proportional to the orbital
angular momentum in that frame. Accounting for the
non-inertial nature of the reference frame, the Hamiltonian for the electron is
H = βL · S
(4)
where β is a proportionality constant, L is the orbital
angular momentum of the electron around the nucleus,
and S is the intrinsic electron spin. Now, only the total
angular momentum is conserved. This means that where
previously different values of orbital angular momentum
could exist for the same energy level, the previous statement is now only true with regard to total angular momentum. Combined with relativistic corrections of order
H=
(2)
mP
me
)
mD mP 14 −
p2
+ γp4
2m
(5)
where γ is a proportionality constant, the states of different orbital angular momenta may no longer be degenerate
due to the differences in electron spin, and this degeneracy splitting is observable as an emission line doublet.
2
II.
APPARATUS AND METHOD
ode to register a signal. This signal is what is recognized
as a photon count rate in LABVIEW.
FIG. 2. Sodium emission line around 5867.84 Å
FIG. 1. Schematic of the Spectroscope (adapted from the
experiment guide[1])
This experiment is done through a Jobin-Yvon 1250M
monochromator, as shown in 1, in conjunction with a
grating containing 1800 grooves per millimeter, along
with a photomultiplier tube. The apparatus is controlled
by a computer terminal using the National Instruments
LABVIEW software. The emission lamp to be tested
is set at one end of the monochromator. No UV filters
were used as it was found that the glass on the lamps
absorbed almost all of the UV light. As the lamp emits
light, the grating in the monochromator causes all light
that is not around a certain wavelength to diffract away
from a series of mirrors. Combined with the grating, the
mirrors then amplify the spreading of the light through
reflection until the light reaches a detector at the other
end, by which point light that does not have a wavelength
at a given grating position within the resolution of the
apparatus has spread out and will not be detected. The
light that is detected passes through a photomultiplier
tube, where it first strikes the cathode to eject electrons
through the photoelectric effect, and these electrons pass
through more electrodes called dynodes to eject more
electrons until all these electrons pass through the an-
There are several ways to achieve a clean line, like the
one shown in 2. One is to increase the integration time,
which allows more light to be captured by the detector
through a longer exposure. This was not changed from
100 milliseconds due to time constraints. Another is to
decrease the width of the slits allowing light to pass in
and out of the monochromator. This was not changed
from 100 micrometers because otherwise the peak intensity of the light was found to be very low compared to
the ambient noise. The final way is to increase the resolution of the apparatus. This was done such that every
measurement of an emission line was performed over a
range of 2.00 Å with a step size of 0.01 Å.
Once a clean line is achieved, there are many differents
methods by which it is possible to define the center and
uncertainty of an emission line. We chose the smallest
number of points around the center of the peak where
the distribution of counts is roughly uniform; the chosen
center is then the center of that uniform region, and√the
uncertainty is the width of the region divided by 12,
which is the standard deviation of a uniform continuous
probaility distribution of that width.
III.
MERCURY: CALIBRATION
Mercury is used for the calibration because it has been
very well-studied over the last several decades. We find
on a purely phenomenological basis that the the relation between the measured wavelengths and the accepted
wavelengths is a polynomial calibration curve. Furthermore, a quadratic calibration formula minimizes the errors on the calibrated wavelengths, while a cubic calibration is the only other polynomial where the errors
on the calibrated wavelengths do not become significant
compared to properties like the wavelength separation of
emission doublets. Where λo is the observed wavelength
and λc is the calibrated wavelength, the quadratic cali-
3
As shown in 3, the calibrated wavelengths were fitted
against the energy levels according to
bration equation is
λo = −a1 λ2c + a2 λc + a3
(6)
a1 = (1.423 ± 0.015) × 10
a2 = 1.0016 ± 0.0002
−6
Å
−1
a3 = 3.7462 ± 0.0094 Å
(7)
(8)
(9)
while the cubic calibration is
λo = a0 λ3c − a1 λ2c + a2 λc + a3
a0 = (3.42 ± 0.13) × 10
−9
a1 = (3.86 ± 0.14) × 10
a2 = 0.99516 ± 0.00038
−6
(10)
Å
−2
(11)
Å
−1
(12)
(13)
a3 = −1.73 ± 0.32 Å.
(14)
Through the propagation of errors formula, it can be
seen that the cubic calibration tends to overestimate the
errors in the calibrated wavelengths. However, both of
these calibration models are phenomenological and there
is no a priori way of knowing which is more correct.
Thus, both are taken into account when analyzing hydrogen and deuterium.
IV.
HYDROGEN AND DEUTERIUM
Two sets of calibrated wavelengths were found for hydrogen: one was for the quadratic calibration, and the
other was for the cubic calibration. Furthermore, two
sets of data were found for hydrogen: one was from a
purpose-made hydrogen lamp, while the other was from
the deuterium lamp, and the upper wavelengths of the
emission doublets in deuterium corresponded to hydrogen emission lines within one standard error of the mean
of each emission wavelength. The presence of hydrogen
alongside deuterium is to be expected, as it is typically
difficult to fully purify deuterium from hydrogen. Hence,
there were four possible sets of wavelengths to be used to
calculate the hydrogen Rydberg constant RH .
FIG. 3. Balmer Series Hydrogen Fitting: One of Four Fits
λ=
RH
1
1
−
4 n2
−1
.
(15)
Some of these fits had values of χ2ν closer to 0 than others; these corresponded to the cubic calibration, as that
overestimated the errors on the wavelengths compared
to the quadratic calibration. To minimize the impact of
the cubic calibration overestimation of error on the determination of the hydrogen Rydberg constant, a weighted
average was performed so that the new mean value would
be [3]
P −2
j σj µj
µ = P −2
(16)
j σj
and the new uncertainty would be
−1

X
σ=
σj−2 
(17)
j
where each uncertainty σj was found through the propagation of errors. This would minimize the impact of
terms associated with high uncertainty, as smaller uncertainties would contribute a larger weight.
The systematic error caused by the uncertainty in the
calibration model itself along with the differences between the lamps is best estimated by the maximum difference between two values of RH ; in this case there were
four values of RH possible due to the existence of two
models each applied to two data sets. Hence, the value
of RH is given by
RH = (1.097096±0.000009stat ±0.000019sys )×10−3 Å−1 .
(18)
This differs from the accepted value of RH by 11 standard
deviations.
A similar set of calculations was performed on deuterium to find its Rydberg constant RD . However, there
was only one data set for deuterium, so only two sets of
wavelengths could be used, corresponding to the two different calibration models. This yielded for RD the value
RD = (1.097422±0.000013stat ±0.000011sys )×10−3 Å−1 .
(19)
This differs from the accepted value of RD by 15 standard
deviations.
In addition to calculating the Rydberg constants for
hydrogen and deuterium, the ratio of the deuteron mass
to the proton mass was calculated through the measured
deuterium and hydrogen wavelengths, the calculated hydrogen Rydberg constant, and the known masses of the
electron and proton. As there were four sets of wavelengths possible for hydrogen and two sets for deuterium,
there were eight possible combinations of wavelengths
yielding eight possible values for the deuteron to proton
4
mass ratio. These again were combined as a weighted
average and their uncertainties combined as given above.
This yielded a value of the mass ratio given by
mD
= 2.25 ± 0.11stat ± 0.23sys .
mP
(20)
This differs from the accepted value by 1 standard deviation.
V.
SODIUM
The wavelengths of sodium are tabulated [4] and compared [5] to existing transitions or other elements in I.
Typically, sodium lamps also contain noble gases such as
neon and argon for stability. The smallest and largest
pairs of wavelengths could not be matched to sodium
transitions but instead matched to argon transitions
to within one standard error for each [6]. All of the
other wavelengths matched to their respective transitions
within one standard error as given.
TABLE I. Wavelengths and Transitions of Sodium
λc (Å)
4510.61 ±0.14
4522.14 ±0.14
5682.55 ±0.17
5688.13 ±0.17
5889.79 ±0.18
5895.86 ±0.18
6153.94 ±0.19
6160.68 ±0.19
6169.99 ±0.20
6173.01 ±0.20
Based on relativistic and spin-orbit coupling corrections for a hydrogen-like atom, the doublet separations
should vary as n−3 for doublets that do not share a common starting or ending energy level n. As all the doublets
measured start or end in the 3P state, the differences of
λ−1 within doublets should be the same across doublets.
Indeed, none of the separations vary by more than 5%.
This indicates that sodium does behave as a hydrogenlike atom when relativistic and spin-orbit coupling corrections are applied.
Transition/Element
argon
argon
3P1/2 → 4D3/2
3P3/2 → 4D3/2
3S1/2 → 3P3/2
3S1/2 → 3P1/2
3P1/2 → 5S1/2
3P3/2 → 5S1/2
argon
argon
Doublet Separation
(N/A)
(N/A)
1.73 × 10−7 Å−1
1.73 × 10−7 Å−1
1.75 × 10−7 Å−1
1.75 × 10−7 Å−1
1.78 × 10−7 Å−1
1.78 × 10−7 Å−1
(N/A)
(N/A)
[1] M. I. T. D. of Physics, “Optical spectroscopy of hydrogenic
atoms,” (2012).
[2] D. Griffiths, Introduction to Quantum Mechanics, Second
Edition (Prentice Hall, 2005).
[3] P. Bevington and D. Robinson, Data Reduction and Error
Analysis for the Physical Sciences (McGraw-Hill, 2003).
[4] J. E. Sansonetti, J. Phys. Chem. Ref. Data 37, 1659
(2008).
[5] A. Melissinos, Experiments in Modern Physics (Academic
Press, 1966).
[6] W. Haynes, Handbook of Chemistry and Physics (CRC).
ACKNOWLEDGMENTS
I would like to thank my lab partner Max Zimet for all
the hours put into performing this experiment and doing
VI.
CONCLUDING REMARKS
The mercury calibration model could be either
quadratic or cubic; it appears that the quadratic model
minimizes the errors on the wavelengths but makes no
attempt to hide further systematic differences between
the measured and accepted wavelengths. Even accounting for the systematic errors caused by the differences
in calibration models and lamps, the calculated value
of the hydrogen Rydberg constant differed from the accepted value by 11 times the total uncertainty; similarly,
the calculated value of the deuterium Rydberg constant
differed from the accepted value by 15 times the total
uncertainty, and in both cases that is because the uncertainty is very small compared to the value itself, so there
is a systematic shift at play. Only the deuteron to proton
mass ratio was within 1 uncertainty of its corresponding
accepted value. Finally, the measured sodium doublets
can be matched to transitions that all begin or end at a
3P state except for two pairs of wavelengths which can
be attributed to the presence of argon in the lamp.
the data analysis with me. I would also like to thank
Dr. Sean Robinson for allowing me to go over 4 pages if
necessary, though that appears to not have been so.