Download ATOMIC SPECTRA Theory

ATOMIC SPECTRA
Measurement of the wavelength of light emitted by excited atoms
The purpose of this laboratory exercise is to measure the wavelength of light emitted when hydrogen and other atoms make transitions between various energy states. These wavelengths are to
be compared to those predicted by the Bohr Model of atoms. The limitations of the spectrometer
used in these measurements will also be studied.
J
J
d
d
Figure 1: Constructive interference depends on the diffraction angle.
Theory
Light wavelengths are usually measured either by using a dispersion system like a prism or an
interference device such as a diffraction grating. Gratings are used in this experiment because
the diffraction maxima can be calculated absolutely. The prism spectrometer must be calibrated
with a series of known spectral lines. The term ’lines’ is used because one observes the line
image of a narrow slit at the entrance to the spectrometer.
A spectrometer using a diffraction grating does not have to be calibrated since the grating equation for normally incident light relates the wavelength λ to the geometry as shown in Fig. 1 and
is given by:
d · sin ϑ = nλ
(1)
where d is the spacing of the grating, n (integer) is the order of the interference maximum and ϑ
is the angle for constructive interference.
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The resolution A is the smallest (relative) wavelength difference that produces separate (resolved) images:
λ
A =
= nN
(2)
∆λ
where ∆λ is the wavelength difference and N the number of lines of the grating. The dispersion
is dϑ/dλ = n/(d · cos ϑ), derived by differentiating Equation 1.
In general, two signals are said to be resolvable if their maxima are separated by more then the
FWHM (full width (at) half maximum) of the signals. In this case, the sum of the signals will
still show a ’dip’ between the two signals.
According to quantum theory, atoms that are excited above their lowest energy level emit photons
as they return to a lower energy level. The energy difference between the initial state i and the
final state f of the atom is Ei − Ef = hfif . Since the energy states of the atom are quantized,
one observes that photon emission occurs only with a discrete set of frequencies fif rather than
with a continuous spectrum of frequencies.
Bohrs energy levels of hydrogen corrected for the motion of the proton-electron system about its
center of mass are
1
R
µe4
(3)
En = − 2 2 · 2 = − 2
80 h n
n
where µ is the reduced mass mp me /(mp + me ) of the proton-electron system. R is the Rydberg
constant, equal to 13.6eV , and n is any positive integer.
Figure 2: Two (equally shaped) signals can be resolved, if they are separated by more then the FWHM.
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diffraction grating
slit
condenser
J
teleskop
source
Figure 3: Schematic Diagram of the Spectrometer.
Apparatus
Light sources
As light sources various discharge tubes will be used in this lab [H, He, Na, Cd, Hg/Cd, Ti, Ne].
When connected to a high voltage power supply, these discharge tubes emit their characteristic
spectra. The spectra of hydrogen (H) and helium (He) will be studied qualitatively and quantitatively. The sodium arc lamp (Na) will show the characteristic D-doublet. Examination the
doublet in different orders will give an estimate for the resolution of the apparatus. The other
discharge tubes are used for qualitative studies only.
Diffraction Grating
————————————————————————————
Do not touch the grating.
Do not attempt to clean the diffraction grating; it is easily damaged.
————————————————————————————-
The diffraction grating has many parallel light absorbing lines on its surface separated by distances of the order of light wavelengths. When monochromatic light strikes the grating one
observes a series of maxima (of the same wavelength) at different scattering angles. When light
of various wavelengths strikes the grating each wavelength (color) shows a series of spectral lines
emerging at different angles. A spectrometer is a device used to measure these scattering angles.
The grating is protected by glass on each side which helps when adjusting the grating (see below).
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Spectrometer
A spectrometer with a diffraction grating is used to measure the wavelength of the atomic spectra.
Fig. 3 shows a schematic diagram of the spectrometer.
Light enters through a collimator (a small slit) and passes through a collimating lens (condenser) that produces a parallel beam of light. The grating diffracts the parallel beam and directs
some light into the telescope. Constructive interference occurs under the proper conditions (Eq.
1), which can be observed by rotating the telescope through various positions. Within the telescope a cross wire is placed which helps to find the position of the diffraction maxima. Due
to the cross wire and the very exact scale it is possible to measure angles with an accuracy of
≈ 10 = 1/60deg.
Procedure
Spectrometer Adjustments
Before using the spectrometer a proper adjustment is necessary. Ask your tutor, if you encounter
any problems.
Insert the diffraction grating into its holder on the center platform. It must be set perpendicular
to an imaginated line from the slit to the grating itself. Turn on any lamp and watch the 0th -order.
Get familiar with the apparatus and it’s very sensitive adjustment devices. Align the cross wire
to the observed image.
At the ’ocular’ a lamp is provided. The light from this lamp passes a semi transparent mirror and
gives an image of the cross wire and simultaneously cross wires reflection in the glass covering
the grating.
Adjust the apparatus so that at least the vertical wire and its reflection coincides.
Lamps
——————————————————————————————— WARNING: HIGH VOLTAGE !! —
You must NOT touch the discharge tubes of the lamps!
Turn power off if lamp is not needed!
Turn power off before changing tubes!
——————————————————————————————The lamps are found in proper storing devices. Set up the desired lamps carefully in front of the
slit. For every lamp adjust the collimating slit in front of the lamp to a good compromise between
slit width an light intensity. Never adjust the slit during a measurement series. It takes the lamps
3-5 minutes to warm up. Rotate the telescope and note that indeed different colors are found at
different angles. Note that this behavior is left-right-symmetric.
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Measurements
As the spectrometer is being rotated one sees the entire spectrum of the incident light. After the
last spectral line, the pattern repeats itself. A full set of spectral lines is an order of diffraction.
Take angular measurements for as many spectra and orders of diffraction as you can observe for
the light sources H, He and Na. Measure the diffraction angle by measuring each line’s angular
displacement from either side of the un-diffracted light. Use the average of the two values for
analysis. These two values should not differ greatly. In case values can be obtained just from one
side, consider only those, but be sure to note this in your data sheet.
The measurement series should be done three times for H, He and Na lamps. In order to make
three independent measurements change lamps after each measurement series.
Hydrogen
Measure the angle for each hydrogen spectral line on the right and left of the un-diffracted light.
Do this for as many orders as can be observed. At the end of this series, measure the 0th -order
again and check if the cross wire and its reflection are still aligned. If the apparatus has been
displaced during the experiment, this part of the measurement has to be repeated.
Sodium
Try to measure the separation of the sodium D lines. Measure the angular position for each
sodium spectral line for as many orders as can be observed. At the end of this series, measure
the 0th -order again and check if the cross wire and its reflection are still aligned. If the apparatus
has been displaced during the experiment, this part of the measurement has to be repeated.
Helium
Observe the helium spectrum and record the angles of all lines for at least one order of diffraction.
Measure the angular position for each helium spectral line for as many orders as can be observed.
At the end of this series, measure the 0th -order again and check if the cross wire and its reflection
are still aligned. If the apparatus has been displaced during the experiment, this part of the
measurement has to be repeated.
Other discharge lamps : Cd, Hg/Cd, Ti, Ne
Observer the spectrum of the Cd, Hg/Cd, Ti, Ne lamps and take notes in order to describe the
spectra in a qualitative way.
Analysis
1. Calculate the average position for each spectral line you measured and calculate each wavelength from Equation 1. Include angles to the right and left and a description of the color
seen.
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2. Compare the measured wavelengths to the predicted wavelengths for hydrogen. Which
energy levels were involved? On the assumption that the lower level is the same for all the
lines observed in the hydrogen spectrum, construct an energy level diagram for hydrogen.
Label the energy axis in electron volts (eV ).
3. Sodium D lines are at λ = 5889.95Å and 5895.92Å. What is the angle between them in
the first order spectrum? What angle has been measured? What is the angle between them
in the second order spectrum? What angle has been measured?
4. Describe the observed spectra of Cd, Hg/Cd, Ti, Ne in a qualitative way.
singlet states
1
24.6
20.6
19.8
noo
S
1
P
triplet states
1
3
D
n S
5
4
3
5
4
2
2
3
P
3
D
3
(parahelium)
(orthohelium)
Figure 4: Excitation Energies of Helium. The spectroscopic notation 2S+1 L is used where the upper
affix indicates the spin multiplicity (singlet or triplet) and the letters S,P,D refer to the quantum
number of angular momentum L=0,1,2 respectively.
Questions
1. Derive Equation 1 by using Huygen’s principle.
2. Calculate the resolution of a grating (N = 9000) for first order maxima (n = 1).
3. Calculate the energies and wavelengths of photons emitted in transition from levels ni =
3, 4, 5, 6, 7 to level nf = 2 and ni = 2, 3, 4, 5 to level nf = 1.
4. Which of the wavelengths from Question 3 should be visible? (Wavelengths in the 400 to
700nm range are visible.)
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Yellow
Green
Blue
4026
4388
4922
4713
4471
5015
5876
6678
7065
Red
Violett
Figure 5: Spectrum of Helium.
5. For the visible photons of question 3, predict the angles ϑ for the first order maxima. The
grating has d = 1/651mm. Compare with measured angles.
6. A heavy hydrogen (deuterium) has a mass twice that of a hydrogen. Hydrogen and deuterium are ’isotopes’, which means that they have the same number of protons but different
numbers of neutrons in the nucleus of the atom. Calculate the difference in wavelength
produced by the difference in reduced mass µ for two or three of the visible lines. Predict the angle between these hydrogen and deuterium lines. Can they be resolved with the
spectrometer?
7. The analysis for helium is more difficult. Helium has two electrons, so one expects its
spectrum to differ from hydrogen. However, for one electron in the ground state and the
other excited to a higher orbit, the effective charge seen by the excited electron is still just
e+ . One should therefore observe transitions between states with similar energies as those
seen in hydrogen. Since each electron has a spin the two electrons can be in states with
total spin quantum number
S = s1 − s2 = 0(singlet) or S = s1 + s2 = 1(triplet).
(4)
The difference in binding energy for the singlet and triplet states for each value of n causes
the energy level for n to split into two separate sets of states (Fig. 4). Only transitions
indicated by arrows in Fig.4 are allowed by selection rules, namely ∆L = ±1. Does your
data concur with the assumption that the He transitions are all to the nf = 2 level, using
the formula
1
1
Ephoton = Em − En=2 = R( 2 − 2 )
(5)
2
m
similar to the formula for hydrogen?
8. Based on the observations of sodium, was the resolution of the spectrometer only limited
by the natural line width? What else does limit resolution.
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