Download Note to 8.13 students

Note to 8.13 students:
Feel free to look at this paper for some suggestions about the lab, but please
reference/acknowledge me as if you had read my report or spoken to me in
person. Also note that this is only one way to do the lab and data analysis, and
there are nearly an infinite number of other things to do that would be better.
I made some mistakes doing this lab. Here are a couple I found (and some
more tips):
• Use a smaller step and get more precise data than what we had. Then fit
a curve, don’t just pick out a peak.
• The Hg calibration is actually something like a sin wave instead of a cubic.
Also don’t follow the old version of Melissionos word for word. They use
an optical system with a prism oppose to a diffraction grating.
• We took data strategically so we could use unweighted fits
• The mystery tube was actually N, whoops my bad. We later found a
drawer full of tubes and compared the colors. It was obviously N. Again
whoops, it was our first experiment okay?!
1
Optical Spectroscopy
Rachel Bowens-Rubin∗
MIT Department of Physics
(Dated: July 12, 2010)
The spectra of mercury, hydrogen, deuterium, sodium, and an unidentified mystery tube were
measured using a monochromator to determine diffrent properties of their spectra. The spectrum of
mercury was used to calibrate for error due to optics in the monochronomator. The measurements of
the Balmer series lines were used to determine the Rydberg constants for hydrogen and deuterium,
Rh = 1.0966 × 107 ± 0.0002 × 107 m−1 and Rd = 1.0970 × 107 ± 0.0003 × 107 m−1 . Also, the mass
ratio between the deuteron and proton was calculated to be 2.35 ± 0.45. The energy separation
between the sodium doublet peaks was measured, ∆E = 2.1 × 10−3 ± 0.3 × 10−3 eV. The prominent
lines and the overall shape of the spectrum of the mystery tube were used to identify that the tube
contains neon. Overall, the results were consistent with quantum mechanical theories .
I.
INTRODUCTION
The measurements of spectra of hydrogen and other
one electron atoms were essential building blocks for
modern atomic theory. In 1885, Johann Balmer noticed
a relationship between the emitted wavelengths of light
from hydrogen. Around five years later, Johannes Rydberg built off Balmer’s work by generalizing a relationship for the emitted wavelengths of light for particles
other than hydrogen. In 1913, Niels Bohr published a
theory describing the quantized nature of the atom, inspired by and incorporating Rydberg’s work.
During the development of quantum mechanics a more
accurate theory of the atom was created which could explain other properties of emission spectra. These theories take into account not only an electron’s principal
quantum number, but also its angular momentum and
its spin, which can be used to explain spectral properties
like doublets.
transitions between a higher energy level and nf = 2 are
known as the Balmer series.
By knowing this relationship between an atom’s rest
mass and its spectra, the mass of the nucleus can calculated from its spectral data. In this lab, the measured
spectra of hydrogen and deuterium and the known mass
of a hydrogen nucleus (Mp = 1amu) is used to calculate the mass of a deuteron. From the Rydberg formula
(Equation 1), the mass of the deuteron is given by
Md =
A.
THEORY
The Rydberg Formula Related to Hydrogen
When an electron moves from a higher energy state to
a lower energy state, a photon of a specific wavelength
is emitted. The Rydberg formula relates the change in
energy levels of this transitioning electron to the wavelength of the emitted photon for a given particle:
1
1
= µR∞ (
)
2
λ
1/nf − 1/n20
(1)
where λ is the wavelength, µ = Mnucleus /(Mnucleus +
Melectrons ) is the rest mass, R∞ = 1.097373 × 107 m−1
is the Rydberg constant, and n0 and nf are the initial
and final energy states of the electron. For hydrogen, the
∗ Electronic
address: [email protected]
(2)
where Md is the mass of the deuteron, Me is the mass
of an electron, λH is the measured hydrogen line, and
∆λ is the difference between the measured hydrogen and
deuterium line.
B.
II.
Me λ H
Me λH − ∆λ − ∆λMe
Sodium Fine Structure
Within the same principal energy level in an atom,
electrons can have different amounts of energy depending upon their angular quantum number (l). This number gives information about the shape of the electron’s
orbit and is often represented by the letters and numbers s=0, p=1, d=2, f=3. The energy difference between
these states is caused by the spin of the electron interacting with the atom’s internal magnetic field. If the spin
is parallel to the orbital dipole, the energy state will be
lower than if it is anti-paralell. In this lab we calculated
the energy splitting in the sodium atom between the two
3P states using the equation
∆E =
hc hc
−
λ1
λ2
(3)
where ∆E is the difference in energy, h is Plank’s constant, c is the speed of light, and λ1 and λ2 are the doublet wavelengths.
Due to the conservation of angular momentum, when
a photon is emitted, it possesses a quantized, non-zero
2
value of angular momentum. This means that the angular momentum quantum number can only change by ±1,
so the transitions that can occur are restricted. Figure
1 shows the allowed transitions between energy states in
the sodium atom.
monochromator was 1800 grooves/mm, the entrance and
exit slit widths were 10.0 micrometers, and the PMT
voltage was kept at 900 Volts.
The step size of the monochromator could be varied
to obtain!"#$%&'()&'*+"+,)-+./(+different levels of detail. For this experiment,
most measurements were taken using a step size of 0.05
Angstroms.
*+"+,)-+./(+8+",/9&'*$--+-';
45$('23$(
@)+(+.13($03$&-'
A1B&
6-/($"7
!"01('23$(
'
<$7)('$"'+='./">'
?/9&3&"7()#
8+",/9&'*$--+-':
!
!
FIG. 2: Inside the Monochromator: The instrument works
by first focusing the light from a lamp onto the input slit.
The light passes through the input slit and hits a concave
mirror where it is collimated. The collimated light is reflected
and dispersed after it hits a grating, which can be turned at
different angles. In this figure, the grating is turned a large
amount, causing the longer wavelength light to be reflected
into the photomultiplier tube.
B.
FIG. 1: Energy Levels in Sodium. Allowed transitions between energy states in sodium are represented by lines connecting the energy levels involved in the transition. The numbers are equal to the corresponding wavelength emitted in the
transition from higher to lower energy.
III.
EXPERIMENTAL SETUP
A.
Apparatus
The spectrum of different gasses were measured using a research grade monochromator. The path of light
through the monochromator is represented in Figure 2.
Wavelengths from the incoming light are separated using
spherical concave mirrors and a reflection grating. Certain wavelengths will be directed toward the next concave
mirror, depending on the orientation of the grating.
The light that is directed toward the second concave
mirror is focused onto the exit slit and then measured by
a photomultiplier tube (PMT). The reading is then sent
to the computer where the measurement is displayed in
LabVIEW. The path of light through the monochromator
is represented in Figure 2. The grating density in the
Mercury Calibration
In order to account for the systematic error in the optics of our setup, the emission spectrum of an Oriel 65130
mercury lamp was measured. The measured spectral
lines for mercury were extracted by locating the wavelengths of maximum intensity. The measurement error
in our determination of the peak was equal to one-half of
our step size.
Eleven mercury lines were measured in the range between 2974.85 and 7037.35 Angstroms, which were compared to the established values listed in the CRC Handbook for Chemistry and Physics. A fit was then created
from this comparison to interpolate for other wavelengths
not yet measured. Many different fits were tried including linear through seventh degree polynomials, sine, and
exponential. The Harttman method, a technique traditionally used to account for systematic error in optical
systems involving prisms, was also tested. The fit which
minimized the difference between the measured and established values was
y = −51+1.033x−8.165×10−6 x2 +7.708×10−10 x3 (4)
where x represents the measured lines and y represents
the expected lines. Figure 3 plots the measured versus
expected lines, showing the obtained data and calculated
3
fit. The reduced chi squared of the fit was χ2r = 0.99998,
with degrees of freedom ν = 7 which leads to a probability of P=0.57. The standard error of the fit was σ =
1.97.
FIG. 4: Example Peak from Deuterium Lamp (Intensity vs
Measured Wavelength): The peak on left is the delta line
measured from Deuterium, and the peak on the right is the
same line from the hydrogen in the tube. The calibrated
wavelength is measured in Angstroms.
FIG. 3: Mercury Data with Fit (Expected Wavelength vs
Measured Wavelength): The data is plotted by the blue
points, and the calculated fit is red. The error bars are nearly
invisible due to their small size.
IV.
A.
RESULTS AND DISCUSSION
Balmer Series for Hydrogen and Deuterium
Isotope Shift
In order to calculate the Rydberg constants for hydrogen and deuterium, measurements of the six Balmer lines
were obtained for both lamps. Figure 4 shows an example of the raw data obtained for the Delta line (n0 =6)
using the deuterium lamp. Once the data were obtained,
they were calibrated using the cubic fit obtained from
the mercury data (Equation 4). The calibrated data was
then multiplied by the refractive index of air to obtain
the value of the wavelength had it been measured in a
vacuum.
The Rydberg formula (Equation 1) was applied to the
data and graphed to find the Rydberg constant. Figure
5 shows this graph for hydrogen. The fit was generated
using the method of least squares.
The calculated value of the Rydberg constant for hydrogen was Rh = 1.0966 × 107 ± 0.0002 × 107 m−1 compared to the expected value of Rh = 1.0968 × 107 m−1 .
The calculated value of the Rydberg constant for deuterium was Rd = 1.0970 × 107 ± 0.0003 × 107 m−1 compared to the expected value of Rd = 1.0971 × 107 m−1 .
Both of these measurements were within one standard
deviation of the expected value.
1
FIG. 5: Hydrogen Balmer Lines with Fit ( λ1 vs 1/4−1/n
2 ):
0
The slope of the line of best fit is equal to the Rydberg constant for Hydrogen. The error bars are nearly invisible due
to their small size. A similar analysis was performed for Deuterium.
B.
Sodium Fine Structure
Eight peaks in the sodium spectrum were measured
and calibrated. Figure 6 is a plot of the measured sodium
spectrum. Table 2 shows the calibrated wavelengths, differences in wavelengths, and energy separation for each
transition that was used to determine the energy differ-
4
ence between the two 3P states.
In addition to the sodium peaks, many extra lines were
measured in the spectral data. These extra lines closely
matched the CRC values for argon.
Overall, the measured value for the difference in energy
for the 3P level of sodium was ∆ E = 2.1×10−3 ± 0.3 ×
10−3 eV. This value is within one standard deviation of
the expected value found of 2.1 × 10−3 eV.
FIG. 7: Mystery Spectrum (Intensity vs Calibrated Wavelength): Measurements were taken between 200.0 and
630.0nm of an unknown lamp in order to identify its contents. The mystery spectrum has two regions of higher intensity within the data set. In each of these higher intensity
regions, the most pronounced lines were found and used to
compare to known emission spectra.
FIG. 6: Sodium Spectrum (Intensity vs Measured Wavelength): The dots mark the sodium doublets used to determine the energy separation. Many of the unmarked lines are
due to the argon from the lamp. The wavelength was measured in angstroms.
Table 1: Energy Change in Sodium Doublets:
Trans λ1 (A)* λ2 (A)* ∆λ(A)** ∆ Energy (eV)
3p5s 6160.10 6166.62
6.53
(2.1 ± 0.06) × 10−3
3s3p 5893.96 5899.86
5.90
(2.1 ± 0.07) × 10−3
3p4d 5683.84 5685.49
5.68
(2.2 ± 0.08) × 10−3
3p6s 5149.00 5150.49
4.44
(2.1 ± 0.09) × 10−3
3p5d 4978.81 4983.05
4.24
(2.1 ± 0.10) × 10−3
3p7s 4748.36 4752.19
3.83
(2.1 ± 0.11) × 10−3
3p6d 4664.96 4669.00
4.03
(2.3 ± 0.11) × 10−3
3p7d 4495.06 4498.68
3.62
(2.2 ± 0.12) × 10−3
*Error in λ1 (A) & λ2 = 1.97 A
**Error in ∆λ = 2.78A
C.
Mystery Lamp
In addition to hydrogen, deuterium, and sodium, the
spectrum of an unlabeled lamp was taken in order to
identify which element it contains. Data were obtained
from 2000.0 to 6300.0 Angstroms, and then calibrated
using the fit created from the mercury data. Figure 7
shows the measured mystery spectrum data. The most
prominent peaks and the overall shape of the spectrum
were used to determine which element was in the tube.
To help identify the tube, several factors were considered to narrow the search. First, the visible color of the
mystery tube was similiar to the color of hydrogen and
deuterium but was slightly redder. Second, the company
from which Junior Lab orders the mercury calibration
tubes only makes four others of the same style according
to their website and costumer service agent: argon, krypton, neon and xenon. Although these factors narrowed
the the search to four elements, other elements were considered to account for the possibility that Junior Lab
recieved the tube from another source.
The spectrum was compared to the values listed for
prominent peaks of different elements listed in the CRC
Handbook for Chemistry and Physics and the Typical
Spectra of Oriel Spectral Calibration Lamps provided by
the company. It was noticed after some time that the
spectrum shared some basic characteristics to neon. Two
of the four lines were within one standard deviation from
the CRC values for neon, and the other two were also
close. The overall spectrum shape also matched the spectrum of neon. The two high intensity regions between
330nm to 360nm and 600nm to 630nm and the low intensity region between 460nm to 600nm are also present
for neon.
Table 2: Mystery Peaks and Known Values of Neon
Peaks
Measured λ (nm) CRC (nm) Oriel (nm)
337.5 ± 0.2
337.8
337.0
354.0 ± 0.2
354.2
352.1
358.0 ± 0.2
357.4
359.4
607.3 ± 0.2
607.4
607.4
Even though most of the spectrum of the mystery lamp
can be explained by the contents of neon, the range between the wavelengths 365nm to 450nm can not. In fact,
this region can not be explained by any of the lamps made
by Oriel. More data and searching is required to deter-
5
mine what is causing this spectral pattern. The most
probable cause would be another element besides neon
in the tube.
V.
SUMMARY
The measured values for the Rydberg constant for hydrogen and deuterium, the ratio of the mass of a deutron to proton, and energy separation between the two
3P states of sodium were found. Table 3 summarizes
these results. The four determined values were within
one standard deviation of the expected value, indicating
a consistency with quantum theory.
Table 3: Optical Spectroscopy Summary
Measured λ
Expected
Rh
(1.0966 ± 0.0002) × 107 m−1 1.0967 × 107 m−1
Rd
(1.0970 ± 0.0003) × 107 m−1 1.0971 × 107 m−1
Md /Mp
2.35 ± 0.45
2.00
∆E
(2.1 ± 0.3) × 10−3 eV
2.1 × 10−3 eV
using a smaller step size for measurements and obtaining
more points. The peak could also have been determined
more precisely by fitting the peak using the CauchyLorentz distribution with the Cauchy probably density
function.
The spectrum of the mystery tube resembles the spectrum of neon in the location of prominent peaks and in
overall shape. One region was not similar to neon, which
indicates there may be another gas in the tube in addition
to neon. Because the resemblance was not perfect, it is
my recommendation that more data should be obtained
before labeling the tube.
Acknowledgments
Most of our error was due to an imperfect calibration
using mercury. Two ways to reduce this error include
I would like to acknowledge Kathryn Decker French for
taking data with me, Burak Alver for spending his time
to teach me more about error analysis, and Sid Creutz
for proof reading my paper.
Melissionos, Experiments in Modern Physics (Academic
Press, 1966).
MIT Physics Department, Junior lab written report notes
(2007).
French and Taylor. An Introduction to Quantum
Physics (Norton, 1978).
Georgia
State
University.
Hyperphysicshttp://hyperphysics.phyastr.gsu.edu/hbase/quantum/
CRC. Handbook for Chemistry and Physics.