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Experiment O6 The Diffraction Grating Spectrometer

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					                              Lab II Experiment O6 “Diffraction Grating Spectrometer”: session 2006-7




                Department of Physics & Astronomy
                     Second Year Laboratory

                   Courses: 2B30, 2B40, 2B41, 2B42




                            Experiment O6

  The Diffraction Grating Spectrometer
Experiment Objectives and planning:
   To align and calibrate a grating spectrometer using lines of known wavelength.
   To determine the number of rulings per metre on the grating.
   To determine the wavelengths of the first 3 lines in the Balmer series for Hydrogen.
   To use these values to determine a value for Rydberg’s constant and calculate the
   necessary parameters to draw an energy level diagram for Hydrogen.
   To measure the wavelengths of the 4 brightest lines in the visible Helium spectrum.

Read through the script and make an outline plan of how you will proceed and the
measurements you will make. Include briefly how you will analyse the data to determine the
quantities you are interested in. Discuss this plan with a demonstrator before you actually
begin the experiment.

Relevant Lecture Courses:

1B24 Waves, Optics and Acoustics
1B23 Modern Physics, Astronomy and Cosmology
2B22 Quantum Physics

Abstract
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                                Lab II Experiment O6 “Diffraction Grating Spectrometer”: session 2006-7




A diffraction grating spectrometer is calibrated using a known wavelength source (a sodium
lamp) and is then used to measure the wavelengths of the first three lines in the Balmer
Series of atomic hydrogen and other bright lines in the atomic spectrum of Helium. A value
of Rydberg’s constant is calculated from the determined wavelengths of the Balmer lines.


1 The Diffraction Grating Spectrometer
1.1 Theory

This experiment uses a standard spectrometer consisting of a slit, a collimator, a rotatable
table and a rotatable telescope. A transmission grating, consisting of a mask with a large
number of evenly spaced slits, is positioned on the table with the slits vertical. Parallel light
from the collimator is diffracted by the slits and the diffracted beams are combined to form
an image of the collimator slit at the telescope focus.




                                θi
                      θi
                                                     θp




                                                                                 1/N


                                                     D




                                            Figure 1

For the diffracted image to have a maximum intensity the path difference between adjacent
slits must contain an integral number of wavelengths, i.e.:

                                       1
                                         (sin θ p − sin θ i ) = pλ                                (A)
                                       N

where θi and θp are the incident and diffracted angles respectively (measured clockwise with
respect to the grating normal) (See Fig.1). N is the number of lines/unit length of the grating
and p is an integer known as the order of interference.


The angular deviation, D, of the diffracted beam will be given by

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                                Lab II Experiment O6 “Diffraction Grating Spectrometer”: session 2006-7


                                       D = θ p− θ i                                               (B)

       The undiffracted beam for which D = 0 will correspond to the zeroth order, p = 0, and
the orders extending out from this will be considered positive or negative orders depending
on whether D is positive or negative. There are two cases of interest, as follows.

1.2 Grating placed normal to incident beam

       In this case, θi = 0 and hence θp = D. The condition for a maximum in the diffracted
image is then

                                       sinθp = sinD = pNλ                                         (C)

1.3 Grating positioned for minimum deviation

        There is a position of the grating which for any given λ and p will give the minimum
deviation of the diffracted beam. It can easily be shown (by differentiating equations (A) and
(B) and substituting) that this occurs when θp = -θi (i.e. when the grating bisects the angle
between telescope and collimator). The deviation is then 2θp and the condition for a
maximum in the diffracted image is, from (A)

                                                   D
                               2 sin θ p = 2 sin     = pNλ                                        (D)
                                                   2

This condition is only mentioned for completeness here and will not be used in the
experiment. The alignment procedure, however, does depend upon the minimum deviation
condition being achieved with a prism and is used.

1.4 The Blaze of a grating

        For a simple transmission grating formed by uniform slits, the intensity of the light
diffracted into the different orders is determined by the diffraction pattern of a single slit and
thus is a maximum for the zeroth order. To use the incident light more effectively it is
common practice to form the individual grating elements as prisms (or, in the case of the
more commonly used reflection gratings, as angled facets) so that the peak of the single slit
pattern is shifted by refraction (or reflection) into the order in which it is desired to use the
grating. Such a "blazed" grating will show brighter images on one side of the zeroth order to
the other side. The grating used in this experiment may be blazed.




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                                Lab II Experiment O6 “Diffraction Grating Spectrometer”: session 2006-7




2. EXPERIMENT
2.1 Arrangement of apparatus

Before commencing any measurements it is worth making a rough trial with the grating and
sodium source in place to check that any visible orders on both sides of the zero order image
can be observed with the telescope and that the telescope and turntable verniers are in
positions where they can be conveniently read, i.e. will not bump into each other during the
experiment. (Some care of thought is needed here.)

2.2 Adjustment of the spectrometer

       To make accurate measurements the spectrometer must be correctly adjusted and
aligned. For these adjustments use a sodium lamp source and a narrow slit.

i)      Adjust the telescope eyepiece lens so that the cross-wires are in sharp focus without
eyestrain, i.e. the eye relaxed. Rotate the telescope cross-wires or the collimator slit so that
one of the wires is parallel to the slit and vertical.

ii) To focus both the telescope and the collimator for parallel rays Schuster’s method is used,
    as follows.
   •   Remove the diffraction grating from the turntable and set the equilateral glass prism
       centrally on the turntable.
   •   Search for the refracted image of the collimator slit through the prism with the
       telescope by moving either or both of the telescope and prism turntables if necessary.
   •   Rotate the prism turntable in small increments and track the image with the telescope
       until the position of minimum deviation is located. (The image of the slit will reverse
       its direction of motion in your field of view when minimum deviation is reached and
       then passed.)
   •   Set the system to minimum deviation and then increase the telescope angle by about 6o
       greater than the minimum deviation position and lock the telescope so that it cannot be
       accidentally rotated.
   •   There are now two positions of the prism, reached by rotating the prism turntable, for
       which the image is centred in the telescope.
   •   Choose the prism rotation which corresponds to the greater angle of incidence of the
       beam from the collimator onto the prism and focus the image by adjusting the
       telescope.
   •   Next move the prism table to the other position for the slit on the cross-wires and
       focus the image by adjusting the collimator.
   •   Repeat these adjustments alternately until the image is in focus for both prism
       positions.
   •   Both telescope and collimator are then in focus for parallel light. Only a few iterations
       between the two positions should be required before this situation is reached.

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                                 Lab II Experiment O6 “Diffraction Grating Spectrometer”: session 2006-7


iii)    Generally the principal axes of the telescope and collimator must be made
perpendicular to their common rotation axis. On the spectrometer provided this adjustment is
preset and need not be altered.

iv)     The prism should now be removed and replaced by the grating. The plane of the
grating must be made perpendicular to the horizontal plane containing the incident and
diffracted beams. For the first set of measurements the grating must also be set normal to the
incident beam. It is convenient to carry out both of these adjustments together.

          Set the telescope on the undeviated image of the collimator slit (i.e., the zero order
image). Read off its angular position and then rotate it through exactly 90o to make the optic
axes of the collimator and telescope perpendicular. Rotate the grating so that the image of the
slit is reflected from its front face and is centred in the telescope. This will make the angle of
the grating face 45o to the collimator axis. Now define the vertical centre of the collimator
slit by either fixing a thin wire across it at half height or, if the design of the spectrograph
allows, reducing the length of the slit to a small element about the centre. Observe the
reflected image in the telescope. Now tilt the grating using the appropriate levelling screw, to
make the slit centre coincident with the horizontal cross-wire. Check that the image is still
centred on the vertical cross wire and then read off the grating turntable position and rotate it
through 45o in the correct direction to make it perpendicular to the collimator axis.

v)      The grating rulings must next be made parallel to the spectrometer axis so that the
grating dispersion is in the rotation plane of the spectrometer. If this is so, on rotating the
telescope through the various diffracting images, the image of the slit centre will remain
fixed relative to the horizontal cross wire. If not the grating must be rotated in its own plane
until this condition is achieved. This is done using the grating levelling screws. If one screw
is in the plane of the grating, the adjustment is made by raising or lowering this one; if the
screws are positioned so that the line joining two of them is parallel to the grating then
adjustment must be made using equal and opposite rotations of these two screws.

iv)    The collimator slit must be made parallel to the grating rulings. On some laboratory
spectrometers the slit cannot be rotated. Otherwise turn the slit (opened up to its full length)
until one of the diffracted images is as sharp as possible. This position should be in
alignment with the vertical cross wire.

2.3 Calibration of the grating

Measure the diffraction angle θp (equal to the deviation from the straight through image) for
one of the sodium lines in as many orders (i.e. values of p) as possible on both sides of the
normal (the zero order position). Plot a graph of sinθp against p and from the slope of the
best fit straight line use equation (C) to determine N, the number of line/unit length of the
grating. (Lsfit2 or SigmaPlot on the lab PCs should be used for this fitting process as it
provides an uncertainty for the slope of the line. This will enable other errors in the
experiment to be calculated later.)
         The wavelengths of the sodium doublet lines are 589.0 and 589.6 nm. Use a narrow
slit to maximise the resolution.

2.4 Measurement of the Balmer series

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                                Lab II Experiment O6 “Diffraction Grating Spectrometer”: session 2006-7


Choose one of the two types of Hydrogen lamp available for this experiment. One hydrogen
source is of low intensity and mainly emits the molecular hydrogen spectrum as well as the
atomic hydrogen spectrum. In this lamp molecular hydrogen is dissociated into atomic
hydrogen. The second lamp is a water vapour filled discharge tube. The water vapour is
dissociated into atomic hydrogen and OH. This produces a ‘cleaner’ atomic hydrogen
spectrum, not being contaminated by a molecular hydrogen component.

        The Balmer series of lines from atomic hydrogen in both lamps are produced by a
dissociation process to produce atomic hydrogen and then exciting the resulting atoms. Both
of these processes occur through electron collisions in the gas discharge. The dissociation is
balanced by recombination to re-form molecules so that both molecules and atoms exist in
the source at the same time.
        The capillary of the discharge tube should be placed close to the slit and its position
optimised by observing the brightness of the straight-through image. The three visible
Balmer lines are violet, green/blue and red. Measure as many orders as possible for each
wavelength on both sides of the zero order image. You may find it helpful to widen the slit
slightly and to shield the telescope from stray light using a cloth or a shield. Be aware that
the positions of lines of a higher order may be mixed in with lines of a lower order, (i.e.,
overlap of orders) so you must be careful to sort out which lines belong to which order by
reasoning. Also be aware that other lines will be present which are not of interest here. You
will probably notice large variations in the intensities of the different orders.

        Plot sinθB against p for each line and calculate the line wavelengths using the value of
N obtained above. Again Lsfit2 or SigmaPlot are useful here. Use your values of λ to
calculate the Rydberg constant (see Appendix). Calculate uncertainties for all your values.


2.5 Energy level diagram

        Using your best estimate of the Rydberg constant, draw to scale an energy level
diagram for the hydrogen atom, stating clearly the energy units. Show on it the transitions
giving the observed spectral lines and indicate their wavelengths. (See Figure 2.)




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                                Lab II Experiment O6 “Diffraction Grating Spectrometer”: session 2006-7




                                       Figure 2

2.6 The Helium spectrum

Replace the Hydrogen lamp with a Helium lamp and measure the wavelengths of the 4
brightest lines in the spectrum which you can see. Use the same method as you used above
with the Hydrogen spectrum measuring the angular position of the various lines in the orders
each side of the zeroth order.

                                         Appendix

THEORY
        According to wave mechanical ideas the atom is confined, at any particular time, to
one of a set of discrete states of internal energy. Each of these energy states is characterised
by a set of 'quantum' numbers which can only have integer values. In this experiment no fine
structure of the spectral lines can be resolved and only one quantum number, n, the principal
quantum number is involved, the energy of the atom corresponding to the nth quantum state
being:

                                                         2
                                     m' Z 2   ⎛ q e2     ⎞ 1
                              Wn = −          ⎜          ⎟ 2                                      (1)
                                     2h 2     ⎜ 4πε      ⎟ n
                                              ⎝      o   ⎠

where

                                       mM
                               m' =            (reduced mass)                                     (2)
                                      m+M
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                                 Lab II Experiment O6 “Diffraction Grating Spectrometer”: session 2006-7




m is electronic mass (kg)
M is nuclear mass (kg)
Z is atomic number
h is the reduced Planck constant, h/2π
qe. is the electronic charge (C)
εo is the permittivity of free space

       The negative sign arises because zero energy is defined as the internal energy of a
nucleus - electron system in which the electron is at rest at infinity relative to the nucleus i.e.
energy must be supplied to the normal atom to achieve this ‘ionised’ state.


        The reduced mass appears, rather than just the electron mass, to allow for the centre
of the nucleus - electron system being not precisely centred on the nucleus. The ‘position’ of
the electron in the atom cannot be precisely defined although the classical Bohr Theory of
the hydrogen atom confined the electron to occupy an orbit of definite radius about the
proton, For the hydrogen atom in its ground state this orbit was circular, of radius ao, given
by:

                                       h 2 4πε o
                                ao =      .                                                        (3)
                                       m' q e2

       The wave mechanical picture is of the electronic charge occupying a space but its
density distributed so that, for a given quantum state, the distribution has a maximum (or
maxima) at a definite distance (or distances) from the nucleus. For the ground state of the
hydrogen atom it turns out that this distribution is spherically symmetrical and has its
maximum exactly at the radius given by ao above.

       The energy quantum can be represented on an energy level diagram The transition of
an electron from the ith to the jth energy level changes the total energy of the atom by an
amount

                                                                  2
                                          m' Z 2       ⎛ q e2     ⎞ ⎡1    1⎤
                               Wi − W j =              ⎜          ⎟ ⎢ 2 − 2⎥                       (4)
                                          2h 2         ⎜ 4πε      ⎟ j
                                                       ⎝      o   ⎠ ⎣    i ⎦

       If j > i, then this is the amount of energy that must be supplied to ‘excite’ the atom
from the ith to the jth state.

        If j < i, then the above energy will be emitted by the atom as a quantum (photon) hνij
where νij is the frequency of the emitted radiation. The wavelength of the emitted radiation
will therefore be given by

                                 1          ⎡1    1⎤
                                     = RZ 2 ⎢ 2 − 2 ⎥                                               (5)
                                λ ij        ⎣j   i ⎦

where
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                                Lab II Experiment O6 “Diffraction Grating Spectrometer”: session 2006-7



                                                   2
                             m'     ⎛ q e2     ⎞
                        R=          ⎜          ⎟       (Rydberg’s constant)                        (6)
                           4πch 3   ⎜ 4πε      ⎟
                                    ⎝      o   ⎠

        For any value of j, the final quantum number, there will be observed a series of
spectral lines corresponding to values of i = j +1, j + 2, j + 3 etc. When j = 1 (i.e. the final
state is the ground state of the H atom), the spectral series obtained is called the Lyman
series. This lies completely in the ultra-violet. The series characterised by j = 2 is called the
Balmer series. Its first three lines are in the visible, and can be observed in this experiment.
Other series are j = 3 - Paschen series, j = 4 - Brackett series (see Figure 2).

        The units in which Wn is expressed depends largely on the type of physics one is
familiar with:

absolute (S.I.) notation –
‘joules’ given by substituting S.I. unit throughout.



atomic physicists’ notation –
‘electron volts’ where one electron volt (eV) is the gain in kinetic energy, in joules, of an
electron when it falls through a potential difference of one volt, i.e. 1eV = 1.6022 x 10-19J.

Spectroscopists’ notation –
‘term values’ and ‘wave numbers’ where the term value of an energy state, Wn is defined as
Tn = -Wn/hc, and is usually expressed in units of cm-1 For atomic hydrogen, with Z = 1, the
term values are given by R/n2. The ‘wave number’ of a spectral line is the reciprocal of its
wavelength, usually denoted by n. Thus the wave number of any spectral line can be obtained
directly by subtracting the term values of the initial and final states. Atomic energy level
tables usually list term values.




ALA 01/08/06
I.F. & A.C.H.S
22/11/99
10/07/05




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