This paper shows my findings for determining the grating constant of a diffraction grating, the wavelengths of each line of the spectrum of hydrogen, and experimentally calculating the Rydberg constant.

1. Atomic Spectra
University of California, Santa Cruz
Physics 133 - Gregory Kaminsky
Ryan Dudschus
09 August 2016
Lab Partner: James Lindsey
Abstract
By examining the angles of the spectral lines produced by a helium lamp, we are able to determine
the spacing of the lines in a diffraction grating, which is known as a grating constant. With knowing the
grating constant, we are able to calculate the wavelengths of the spectrum produced by a hydrogen lamp. The
values of the wavelengths produced by the hydrogen lamp allows the Rydeberg constant to be experimentally
calculated. We also compare the spectrum of a helium-neon laser to the spectrum of a neon lamp, as well
as to the spectrum of a helium lamp.

2. 1 Introduction
Electromagnetic radiation from a specific light source of an element contains a certain number of different
wavelengths, which contribute to the visible color of the light source. When the emitted electromagnetic
radiation from the light source strikes a prism or a diffraction grating, each individual wavelength gets
diffracted to a certain angle. Each element has its own unique spectrum. The study of spectroscopy uses
this knowledge to determine which elements make up an object, such as stars.
1.1 Helium
By using a diffraction grating that was perpendicular to the incident beam emitted from a helium lamp, the
beam of the incident light is diffracted into certain angles, each depending on its wavelength. The relationship
between the angle and wavelength when the incident beam passes through the diffraction gratin is given by
mλ = d(sin(θ1) + sin(φ)) and mλ = d(sin(θ2) − sin(φ)), where θ1 and θ2 are the angles of diffraction, λ
is the wavelength, m is the order, d is the grating constant, and φ is the angle the light will be incident on
the grating(see figure 1). However, if we call the angles α1, α2, α0, where α0 is the angle the light will be
incident on the grating, then the angles of diffraction is measured by θ1 = |α1 − α0| and θ2 = |α2 − α0|.
Knowing this, we may define θ = 1
2 (θ1 +θ2) and ∆ = 1
2 (θ1 −θ2). Then mλ = d(sin(θ)(cos(∆)). If ∆ is small
enough, to within a fraction of a degree, we may neglect cos(∆) since , by a Taylor expansion, cos(∆) will
be approximately one. In this experiment, we will determine the grating constant d by using the equation
d = mλ
sin(θ)cos(∆) , or d = mλ
sin(θ) , if our ∆ is small enough.
Figure 1: This figure shows an incident beam of light hitting a diffraction grating at an angle φ,which is
normal to the diffraction grating, and refracted at two angles θ1 and θ2. However, the incident angle φ is
the same as our measured incident angle α0. The angles θ1 and θ2 is the same as our measured angles α1
and α2, respectively. However, for this experiment, we made the incident angle α0,or φ, as small as possible.
This results in the incident beam being perpendicular to the diffraction grating (Figure 5.2 from Physics 133
reader for University of California, Santa Cruz).
1

3. 1.2 Hydrogen and Rydberg Constant
By using the same diffraction grating that was used in the experiment involved with the helium lamp, the
light emitted from a hydrogen lamp will also be diffracted at certain angels, each depending on its wavelength
(seen in figure 1). When an electron in an atom transfers from a higher energy state to a lower energy state
∆E, a photon is released with the same change in energy ∆E. Since the energy of a photon is related to
its related to its wavelength, ∆E = hc
λ , each energy transition will have a unique wavelength that will be
emitted, where h is Planck’s constant, c is the speed of light, and λ is the wavelength. The quantum theory of
the hydrogen atom describes the energy levels as En = − µe4
8 2
0h2
1
n2 , where µ is the reduced mass 1
µ = 1
m + 1
M ,
where m is the mass of the electron and M is the mass of the proton, is the permittivity of free space, e is
the electron charge, and n is a positive integer called the principal quantum number. The lowest energy state
is n = 1, and as n increases, the electron becomes further from the proton, with the separation increasing
as n2
. Therefore the change in energy of a hydrogen atom is given by ∆E
hc = En1−En2
hc = µe4
8 2
0h3c
( 1
n2
2
− 1
n2
1
),
where the constants µe4
8 2
0h3c
is known as Rydberg’s constant RH. Since we know that ∆E = hc
λ , we can say
that 1
λ = ∆E
hc , and thus 1
λ = (RH)( 1
n2
2
− 1
n2
1
). In this experiment, we will determine the wavelengths of the
diffracted light of the hydrogen spectrum by measuring the angles at which they are refracted and with the
known grating constant d. With knowing those values for the wavelength, we will experimentally calculate
the Rydberg constant RH.
1.3 Helium-Neon Laser
When you create a potential difference in a helium-neon laser, the helium atoms will get excited. The energy
of an excited helium atom may transfer its energy to a neon atom, which now puts the neon atom into an
exited state and able to emit a photon. These photon emissions of a helium-neon laser happen at three
wavelengths. Two of the wavelength are in the infrared spectrum, at 3.39 microns and 1.15 microns, while
one wavelength occurs in the visible spectrum at 6328 Angstroms, which is a red light. Since we are unable
to see the two wavelengths in the infrared spectrum, the visible wavelength will be of importance. In this
experiment, we compare the visible 6328 Angstrom red light of the helium-neon laser to the spectrum of a
neon lamp and to the spectrum of a helium lamp.
2 Procedure
2.1 Alignment of the Spencer Spectrometer
A Spencer Spectrometer was used to view and measure the light spectrum of the lamps used (see figure 2).
In order to obtain accurate measurements, the Spencer Spectrometer must be properly aligned. First, the
cross-hairs in the telescope were brought to focus. The helium lamp was then placed in the power source, so
that the helium lamp will start to emit photons whenever there is a voltage applied to it. The lamp was put
behind the entrance slit of the collimator, which guided the light emitted from the lamp to the diffraction
grating (which is not in the grating hold yet). The lamp must be placed directly in front of the slit so that as
much light as possible may pass through the slit. Furthermore, the telescope was rotated so that was it was
nearly parallel to the collimator, with the light emitted from the helium lamp visible while peering through
the eyepiece of the telescope. This light will look like a vertical line resembling the same color as the light
emitted from helium lamp. The helium lamp needs to be in focus and not blurry, so we focused both the
collimator and the telescope in order to achieve this. Also, the slit width must be as small as possible so
that more accurate measurements are taken, while keeping in mind that enough light needs to go through
the entrance slit of the collimator so that the emitted light will be visible. If the slit width is too small, not
enough light will get through and the spectrum lines will be too dim to see. If the slit width is too large, the
spectrum lines will be too large to get an accurate measurement of the angle. Subsequently, the cross-hairs
in the telescope must be in the middle of the light that was passing through the collimators entrance slit.
This means that the collimator and the telescope are parallel to each other.
2

4. After the collimator and the telescope are parallel to each other with the emitted light from the helium
lamp is focused, we needed to get the diffraction grating perpendicular to the collimator. This will, in turn,
also be perpendicular to the telescope, since the telescope is parallel to the collimator. To do this, a mirror
was placed in the diffraction grating slot and a battery was attached to the telescope, which cast a shadow
of the cross-hairs on the mirror. The grating table was then leveled and rotated with the adjustment screws
on the grating table until the shadow of the cross-hairs was superimposed on the actual cross-hairs. This
means that the mirror, or soon-to-be diffraction grating, is now perpendicular to the telescope and collimator.
The mirror was replaced with the diffraction grating and the cross-hairs was focused on the undeflected light.
The vernier scale on the Spencer Spectrometer was calibrated so that it measured zero degrees. However,
this would create a challenge with reading the vernier scare. To make the readings easier, we set our zero mark
to 90◦
. When we measured this angle, which is our α0, it came to be 90◦
+ 2 , with an error in measurement
of 5 (this measurement was converted to radians by multiplying it by π
180 for the calculations). Since the
diffraction grating is nearly perpendicular to the incident light beam, the light will be refracted left and right
at almost the same angles. If these angles are not close to each other, then the Spencer Spectrometer is not
properly aligned. Repeat the process of alignment until these angles have nearly the same value.
Figure 2: This figure shows a model of a Spencer Spectrometer. A: The slit where the light emitted from a
lamp enters; B: The objective lens of the collimator; C: This is the diffraction grating; D: The objective lens
of the telescope; E: This is where the cross-hairs are located in the telescope; L: This is where the eyepiece
of the telescope is located; O: This is where the grating table leveling screws are located; I: This is the screw
that is used to lock the telescope into place so that it will not move if pushed or hit. Since the Spencer
Spectrometer is not exactly like the one shown here, the remaining points in the figure are not important
(Figure 5.1 from Physics 133 reader for University of California, Santa Cruz).
2.2 Helium
The helium lamp has a known spectrum, which means that we know the wavelengths of the spectral lines
helium. We do not know the grating constant d of the diffraction grating this is used. By measuring the
angles of the spectral lines of helium, α1 and α2, and knowing the wavelengths of helium’s spectrum, we
are able to find the grating constant d. Since we know the incident angle of the beam of light α0, we may
find the angles θ1 and θ2 by θ1 = |α1 − α0| and θ2 = |α2 − α0|, respectively. Since the beam of light is not
perfectly perpendicular on the diffraction grating, the diffraction angle of the spectrum will not be exactly
the same on each side of α0, but will be approximately the same if the Spencer Spectrometer is properly
aligned. By the theory of diffraction gratings, θ and ∆ are found by θ = 1
2 (θ1 + θ2) and ∆ = 1
2 (θ1 − θ2).
Then the grating constant d may be calculated by d = mλ
sin(θ)cos(∆) , where m is the order of the spectrum
and λ is the wavelength, which is found in the table in the appendix. Although a Spencer Spectrometer
that is perfectly aligned will have a ∆ equal to zero, by the Taylor expansion of cos(x), if ∆ is small enough,
cos(∆) will approximately be one. Therefore you may omit the cos(∆) and the grating constant d may be
determined by d = mλ
sin(θ) . Each measured spectral line of helium will correspond to a grating constant d.
Since there will be error in the measurements, it is needed to propagate the error through the calculations
3

5. to show that the error carries through the equations and effects the final results. Also, the weighted mean
of the grating constant must be found, since there are multiple measurements of the grating constant d.
2.3 Hydrogen
With the average grating constant known, the hydrogen spectral lines may be measured and calculated.
This is be done by taking the equation used to find the grating constant and using algebra to solve for the
wavelength λ, which turns out to be λ = dsin(θ)cos(∆)
m . If ∆ is small enough, cos(∆) ≈ 1, so then λ = dsin(θ)
m .
While viewing the hydrogen spectrum, we made sure to only measure the Balmer series. The hydrogen lamp
may emit a few spectral lines that do not correspond to the Balmer series, which are emitted by molecular
hydrogen.
2.4 Determine the Rydberg Constant
With the wavelengths of the hydrogen spectrum now known, we were able to experimentally calculate the
Rydberg Constant RH. The wavelengths of the spectral lines of hydrogen is related to the Rydberg constant
by the equation 1
λ = RH( 1
n2
2
− 1
n2
1
). Since we want to calculate the Rydberg constant, by doing algebra to the
equation above, RH = 1
λ ( 1
n2
2
− 1
n2
1
)−1
. Each of the hydrogen spectrum lines will result in a calculated value
for the Rydberg constant, so the weighted mean of the calculated values will be used as the experimentally
determined Rydberg constant.
2.5 Helium-Neon Laser
In order to compare the neon lamp with the helium-neon laser, we placed a piece of cardboard in front of
the bottom half of the collimator’s entrance slit. This will obstruct some of the neon lamp’s light resulting
in neon’s spectrum to show in the bottom half of the telescope’s eyepiece. We then aimed the helium-neon
laser so that the laser dot is on the cardboard in front of the collimator’s entrance slit. This will scatter
the emitted light into the collimator (see figure 3). This will result in the neon’s spectrum to show in the
bottom half of the telescope’s eyepiece, while the spectrum of the helium-neon laser will show in the upper
half of the telescope’s eyepiece. The spectrum of the helium-neon laser will be one visible line of light, while
the spectrum of neon will have many visible lines of light. Compare these two spectra with the eye and take
measurements of the angles. We repeated this process with the helium lamp.
Figure 3: This figure shows what the set up of the helium-neon laser and the neon lamp, or helium lamp,
should look like. Place the cardboard, or any material that will obstruct the light emitted from the neon,
or helium, lamp. Aim the helium-neon laser dot on the obstruction and in front of the collimator’s entrance
slit so that the emitted light will scatter into the collimator’s entrance slit.
4

6. Figure 4: This figure resembles our model of what is shown in figure 3. Cardboard was placed between the
neon lamp and the bottom half of the entrance slit of the collimator. Then the helium-neon laser red dot
was place on the cardboard in front of the entrance slit of the collimator so that the light emitted from the
helium-neon laser will scatter into the collimator’s entrace slit.
3 Results
3.1 Error Propagation
Recall that an experimental result is expresed in the form ¯x ± σ¯x, where ¯x is the measurement and σ¯x
represents the error. Examples of this may be found in table 1 in the α1 and α2 columns, where we have our
measurement with an error of 5 minutes. Error propagation for a function f with a variable x that has an
uncertainty is defined to be σ = |∂f
∂x σx|. Suppose there is a function f with multiple independent variables,
such as a = f(x, y, ...). The error propagation is found through the equation
σ2
a = (
∂f
∂x
)2
σ2
x + (
∂f
∂y
)2
σ2
y + . . . (1)
. Since we have error in our measurements, we need to propagate the error through our calculations. By
using equation 1, the error propagation of the function θ1 = |α1 − α0| is
σθ1 = (σα1 )2 + (σα0 )2 (2)
. The same is done to find θ2 and the corresponding error propagation. However, since the error for α0, α1,
and α2 are the same, set σα = σα0 = σα1 = σα2 . Then equation 2 becomes
σθ1 = (σα)2 + (σα)2 = 2(σα)2. Also, this means that σθ1 = 2(σα)2 = σθ2 .
Given that θ = 1
2 (θ1 + θ2), the error propagation of θ by equation one is
σθ = (
σθ1
2
)2 + (
σθ2
2
)2 (3)
. However, since σθ1
= σθ2
= 2(σα)2, by replacing σθ1
and σθ2
in equation 3 with 2(σα)2, it turns out
that σθ = σα.
The equation used to calculate the grating constant is d = mλ
sin(θ) . Our cos(∆) may be ignored since, by the
Taylor expansion of cosine, cos(∆) is approximately one. The error propagation is given by
σd = mλcot(θ)csc(θ)σθ (4)
5

7. Taking the inverse of the equation used to find the value of the grating constant d results in obtaining
the grating lines per distance. In this experiment, we will use lines per millimeter. Therefore we obtain
1
d = sin(θ)
mλ , with its error propagation resulting in
σ1
d
=
cos(θ)
mλ
σθ (5)
. Since each wavelength and its corresponding angle measurement results in its own grating constant, along
with error propagation, the weighted mean of the grating constant values must be found. The weighted
mean ¯x and its error propagation σ¯x is defined by
¯x =
Σn
k=1
xk
σ2
k
Σn
k=1
1
σ2
k
σ¯x =
1
Σn
k=1
1
σ2
k
(6)
Now that the grating constant is know, the wavelength of the hydrogen spectrum may be calculated by the
equation λ = dsin(θ)
m , with the error propagation being found by
σλ = (
dcos(θ)σθ
m
)2 + (
sin(θ)
m
σd)2 (7)
Finally, the Rydberg constant RH is found by RH = λ−1 1
( 1
n2
2
− 1
n2
1
)
. By using equation 1, the error propagation
of the Rydberg constant is define to be
σRH
=
σλ
λ2
(
1
n2
2
−
1
n2
1
) (8)
Since each wavelength will result in a value for the Rydberg constant, use equation 6 to find the weighted
mean and error propagation of the weighted mean. This will be the experimentally determined Rydberg
constant.
3.2 Helium and Grating Constant
Before we took measurements, we looked to see how many spectral lines of the helium lamp that were visible
and, we saw eight spectral lines. We measured our incident beam angle α0 to be 90◦
+ 2 ± 5 , where the
±5 is our error in measurement. The measurements of the angles of the spectral lines of helium are shown
in table 1, and each had an uncertainty in error of 5 minutes. Each of the measured angles, α1 and α2,
were not the exact same value, as seen by the θ1 and θ2 values. This is due to an error in measurement
from reading the vernier scale on the Spencer Spectrometer and from the cross-hairs not being exactly in the
middle of the light from the helium spectrum, and that our Spencer Spectrometer was not perfectly aligned.
However,by the Taylor expansion of cos(x), the measurements result in cos(∆) being close enough to one
that cos(∆) may be ignored in calculations. Also, all of the values in table 1 are in degrees, so they were
converted into radians for the calculations by multiplying the value of degrees by π
180 .
Using the values for θ in table 1 and the known wavelengths of the helium spectrum, we calculated the
grating constants d. By using 1
d = sin(θ)
mλ and equation 5, we also calculated the lines per millimeter for the
diffraction grating. These values are shown in table 2. Since There are multiple values for both d and 1/d,
equation 6 was used to find the weighted mean and the weighted error propagation(see table 3).
6

8. Table 1: Helium Spectrum Measurements
Color Order (m) α1 ±5 α2 ±5 θ1 θ2 θ ∆ λ(Angstroms)
Violet 1 97.4◦
+ 15 82.5◦
+ 5 7.62◦
7.45◦
7.53◦
0.08◦
4471.479
Blue 1 98.2◦
+ 11 82.0◦
+ 4 8.35◦
7.97◦
8.16◦
0.19◦
4713.146
Green (1) 1 98.5◦
+ 4 81.6◦
+ 16 8.53◦
8.17◦
8.35◦
0.18◦
4921.931
Green (2) 1 98.6◦
+ 3 81.5◦
+ 2 8.62◦
8.50◦
8.56◦
0.06◦
5015.678
Green (3) 1 98.6◦
+ 11 81.5◦
+ 0 8.75◦
8.53◦
8.64◦
0.11◦
5047.74
Yellow-Orange 1 100.1◦
+ 4 80.1◦
+ 4 10.13◦
9.87◦
10.00◦
0.13◦
5875.62
Red (1) 1 101.5◦
+ 3 78.7◦
+ 11 11.52◦
11.15◦
11.33◦
0.18◦
6678.15
Red (2) 1 102.2◦
+ 10 78.0◦
+ 3 12.33◦
11.98◦
12.16◦
0.18◦
7065.19
Violet 2 105.5◦
+ 0 74.8◦
+ 13 15.47◦
15.02◦
15.24◦
0.22◦
4471.479
Blue 2 106.4◦
+ 20 73.9◦
+ 23 16.70◦
15.75◦
16.22◦
0.48◦
4713.146
Green (1) 2 107.1◦
+ 5 73.1◦
+ 10 17.15◦
16.77◦
16.96◦
0.19◦
4921.931
Green (2) 2 107.4◦
+ 22 72.9◦
+ 16 17.73◦
16.87◦
17.35◦
0.46◦
5015.678
Green (3) 2 107.4◦
+ 28 72.6◦
+ 9 17.83◦
17.28◦
17.61◦
0.43◦
5047.74
Yellow-Orange 2 110.5◦
+ 2 69.7◦
+ 12 20.50◦
20.13◦
20.32◦
0.28◦
5875.62
Red (1) 2 113.5◦
+ 2 66.7◦
+ 17 23.50◦
23.05◦
23.28◦
0.22◦
6678.15
Violet 3 113.5◦
+ 3 66.6◦
+ 12 23.52◦
23.23◦
23.38◦
0.14◦
4471.479
Blue 3 114.9◦
+ 27 65.4◦
+ 25 25.32◦
24.27◦
24.77◦
0.55◦
4713.146
Green (1) 3 116.1◦
+ 5 64.2◦
+ 12 26.15◦
25.63◦
25.89◦
0.26◦
4921.931
Green (2) 3 116.5◦
+ 1 63.6◦
+ 10 26.48◦
26.27◦
26.38◦
0.11◦
5015.678
Green (3) 3 116.7◦
+ 11 63.4◦
+ 3 26.85◦
26.58◦
26.72◦
0.13◦
5047.74
Yellow-Orange 3 121.6◦
+ 9 58.6◦
+ 9 31.72◦
31.28◦
31.50◦
0.21◦
5875.62
Table 2: Grating Constants
Color Order (m)
Wavelength
(Angstroms)
Grating Constant
in millimeters (d)
Grating Constant Error
in millimeters (σd)
Lines per millimeter (1/d)
Lines per
Millimeter Error
(σ1/d)
Violet 1 4471.479 0.003 3.751x10−5 293.198 3.225
2 4471.479 0.003 1.816x10−5 293.964 1.569
3 4471.479 0.003 1.138x10−5 295.762 0.995
Blue 1 4713.146 0.003 3.367x10−5 301.092 3.055
2 4713.146 0.003 1.686x10−5 296.416 1.482
3 4713.146 0.003 1.064x10−5 296.280 0.934
Green (1) 1 4921.931 0.003 3.358x10−5 295.046 2.924
2 4921.931 0.003 1.610x10−5 296.302 1.413
3 4921.931 0.003 1.013x10−5 295.731 0.886
Green (2) 1 5015.678 0.003 3.257x10−5 296.702 2.867
2 5015.678 0.003 1.566x10−5 297.276 1.384
3 5015.678 0.003 9.935x10−6 295.237 0.866
Green (3) 1 5047.74 0.003 3.215x10−5 297.666 2.849
2 5047.74 0.003 1.529x10−5 299.647 1.373
3 5047.74 0.003 9.734x10−6 296.884 0.858
Yellow-
Orange
1 5875.62 0.003 2.791x10−5 295.540 2.438
2 5875.62 0.003 1.330x10−5 295.465 1.161
3 5875.62 0.003 8.007x10−6 296.422 0.704
Red (1) 1 6678.15 0.003 2.466x10−5 294.268 2.135
2 6678.15 0.003 1.143x10−5 295.85 1.000
Red (2) 1 7065.19 0.003 2.265x10−5 298.101 2.012
7

9. Table 3: Weighted Mean of Grating Constant
Grating Constant (d) Lines per Millimeter
Weighted Mean (3.376 ± 0.003)x10−3
296.212 ± 0.272
3.3 Hydrogen
The recorded angles for the hydrogen spectrum are shown in table 4. Our recorded data was measured in
degrees. The data shown in table 4 was converted from degrees to radians for the calculations by multiplying
the value of degrees by π
180 . Again, our ∆ calculations are small enough to omit cos(∆) from our calculations.
Since the grating constant d is known, the wavelength of the lines found in the hydrogen spectrum were
calculated by λ = dsin(θ)
m , along with the corresponding error propagation (see table 5).
Table 4: Hydrogen Spectrum Measurements
Color Order (m) α1(±5 ) α2(±5 ) θ1 θ2 θ ∆
Purple-
Blue
1 97.5◦
+ 1 82.6◦
+ 13 7.48◦
7.22◦
7.35◦
0.13◦
Cyan 1 98.4◦
+ 25 81.9◦
+ 25 8.78◦
7.72◦
8.25◦
0.53◦
Red 1 101.3◦
+ 21 78.9◦
+ 26 11.62◦
10.70◦
11.15◦
0.46◦
Purple-
Blue
2 105.0◦
+ 0 75.3◦
+ 12 14.97◦
14.53◦
14.75◦
0.22◦
Cyan 2 106.9◦
+ 21 73.4◦
+ 26 17.22◦
16.20◦
16.71◦
0.51◦
Red 2 113.0◦
+ 1 67.4◦
+ 16 22.98◦
22.37◦
22.67◦
0.31◦
Purple-
Blue
3 112.9◦
+ 21 67.4◦
+ 25 23.22◦
22.21◦
22.72◦
0.50◦
Cyan 3 115.7◦
+ 15 64.5◦
+ 1 25.92◦
25.52◦
25.72◦
0.20◦
Red 3 126.0◦
+ 3 54.5◦
+ 1 35.52◦
35.52◦
35.77◦
0.25◦
Since the spectrum of hydrogen was observed up to the third order, equation 6 was used to find the
weighted mean for each of the wavelengths, which is also shown in table 5.
Table 5: Wavelengths of Hydrogen Spectrum
Color Order (m) Wavelength (Angstroms)
Purple-
Blue
1 4318.68 ± 48.86
2 4297.44 ± 24.07
3 4345.50 ± 15.62
Weighted
Purple-Blue
4330.42 ± 12.65
Cyan 1 4844.04 ± 48.79
2 4852.72 ± 23.93
3 4882.79 ± 15.41
Weighted
Cyan
4872.00 ± 12.52
Red 1 6532.90 ± 48.54
2 6506.93 ± 23.43
3 6577.04 ± 14.59
Weighted
Red
6555.95 ± 12.00
We compared the weighted mean of the calculated wavelengths of the hydrogen spectrum to the actual
8

10. wavelength of the hydrogen spectrum, which is given in figure 5 in the Appendix. Our calculated wavelengths
are within its standard deviation from the accepted values of the wavelengths of the spectral lines of hydrogen,
which are found in figure 5 in the appendix. Also ,the resulting percent errors, as seen in table 6, tells us
that our calculated results for the wavelength of the hydrogen spectrum are close to the accepted values to
within less than a percent.
Table 6: Wavelength Comparison for Hydrogen Spectrum
Color
Calculated Wavelength
(Angstroms)
Actual Wavelength
(Angstroms)
Percent Error
Purple 4330.42 ± 12.65 4340.47 0.23 ± (6.7x10−7
)
Cyan 4872.00 ± 12.52 4861.33 0.21 ± (5.3x10−7
Red 6555.95 ± 12.00 6562.852 0.10 ± (2.8x10−7
)
3.4 Calculating the Rydberg Constant
The values for the Rydberg constant were calculated from each wavelength found for the spectral lines of
the hydrogen lamp by the equation RH = λ−1 1
( 1
n2
2
− 1
n2
1
)
. The values of the calculated Rydberg constant and
their weighted mean are shown in table 7. Since we measured only the Balmer series, n2 = 2 and the values
of n1 were estimated to be 3, 4, and 5, each for a wavelength of the hydrogen spectrum. The experimental
value we calculated, which is the weighted mean value, is within its uncertainty, or standard deviation, of
the accepted value of the Rydberg constant, which is 10973731.6 1
m . Our experimental value is around 0.04%
off from the accepted value of the Rydberg constant. This means that our calculated value of the Rydberg
constant is close to the accepted value.
Table 7: Rydberg Constant
Color
Wavelength
(Angstrom)
n 2 Value Rydberg Constant (m−1
)
Purple-
Blue
4330.42 ± 12.65 5 10996401.20 ± 1416.94
Cyan 4872.00 ± 12.52 4 10946903.79 ± 989.30
Red 6555.95 ± 12.00 3 10982387.80 ± 387.79
Weighted Mean 10978804.29 ± 349.86
3.5 Helium-Neon Laser
The spectrum of the neon lamp contained many lines - mainly of red, orange and green - that were very
close to each other. When the spectrum produced by the laser was compared to the spectrum emitted by
the neon lamp, only one line from the neon spectrum aligned with the spectrum of the laser. The color of
the aligned lines was red. When we measured the angles of the lines of both spectra that were aligned, it
was found that both resulted in the same angle, as shown in table 8.
Table 8: Measurements of the Neon Lamp and Helium-Neon Laser
Color Order (m) α1 (±5 ) α2 (±5 ) θ1 θ2 θ ∆
Neon (Red) 1 101.0 + 2 79.4 + 22 11.00 10.27 10.63 0.37
Helium-Neon
Laser (Red)
1 101.0 + 2 79.4 + 22 11.00 10.27 10.63 0.37
Neon (Red) 2 112.4 + 21 68.3 + 19 22.72 21.42 22.07 0.65
Helium-Neon
Laser (Red)
2 112.4 + 21 68.3 + 19 22.72 21.42 22.07 0.65
9

11. Since the measurements are in degrees, we had to convert them to radians by multiplying the values of
degrees by π
180 . The grating constant d is already known to be (3.376 ± 0.003)x10−3
. We then calculated
the wavelength of the line produced from the helium-neon laser, as well as the line from the spectrum of
the neon lamp that it aligns with, by using the equation λ = dsin(θ)
m , with the error propagation able to be
found with use of equation 7. Since the angle of the emitted line of the spectra was measured to the second
order, equation 6 was used to find the weighted mean and error propagation of each wavelength. The results
of the calculations can be seen in table 9. The neon lamp was then replaced with the helium lamp, while
Table 9: Wavelengths of the Neon Lamp and the Helium-Neon Laser
Color Order (m)
Wavelength
(Angstroms)
Neon (Red 1 6229.15 ± 48.59
Helium-Neon
Laser (Red)
1 6229.15 ± 48.59
Neon (Red) 2 6341.21 ± 23.48
Helium-Neon
Laser (Red)
2 6341.21 ± 23.48
Weighted
Average
6319.99 ± 21.14
the orientation with the laser, shown in figure 3 and figure 4, was maintained. When the spectrum produced
by the helium lamp was compared to the spectrum emitted by the helium-neon laser, it was found that no
lines from each spectrum aligned with each other. The spectra do not align since it is the neon atoms in the
helium-neon laser that emit the photons. Thus, the spectrum of the helium-neon laser is aligned with one of
the lines produced by the spectrum of the neon lamp, which confirms that it is indeed the neon atoms that
are emitting photons and not the helium atoms.
4 Conclusion
From using a helium lamp with the wavelengths of the spectrum of helium known, we determined the
grating constant d to be (3.376 ± 0.003)x10−3
millimeters. With the grating constant known, we were
able to calculate the wavelengths of the spectrum that is emitted by the hydrogen lamp. Using the found
wavelengths of the hydrogen spectrum, we experimentally determined the Rydberg constant and found it to
be 10, 978, 804.29±349.86 1
m . Since the grating constant of the diffraction grating is known, the wavelength of
the spectrum of a helium-neon laser was able to be calculated. We compared the spectrum of the helium-neon
laser to the spectrum emitted by a neon lamp and also by a helium lamp. The spectrum of the helium-neon
laser matched a line of the spectrum of the neon lamp, while the helium-neon laser’s spectrum did not match
up with any lines from the helium lamp.
10

12. 5 Appendix
Figure 5: This figure shows the intensity and wavelength, in Angstroms, of the spectral lines of hydrogen,
helium, and mercury. The right column shows the intensity, while the middle column shows the wavelength
in angstroms. In this experiment, the known wavelengths given in this figure for helium were used to help
calculate the grating constant of a diffraction grating (Figure 5.5 from Physics 133 reader for University of
California, Santa Cruz).
11

13. References
• Physics 133 Reader for University of California, Santa Cruz
12