L-5: Spectrometer and the H Balmer Series

NOTE: Consult your instructor for assignment of parts suitable for a single lab, e.g. do Part I, II (A or B), Part IV or Part V).

OBJECTIVES:

To become familiar with a precision spectrometer and some of its uses.

APPARATUS:

Precision spectrometer; mercury and hydrogen discharge tubes; prism; diffraction grating; light source for Gaussian eyepiece & its 12V power supply (L-2); ring stand & achromatic lens.

Figure 1: The spectrometer (side view).

INTRODUCTION and ADJUSTMENTS

PART I - A

1. Become familiar with the clamping and fine adjustment controls for telescope and prism table angles. Never force a motion - you may damage the instrument. If at the end of a fine adjustment, loosen and reclamp! The prism table clamp sets table elevation: no fine adjustment is necessary.
2. Note that 180^o apart are two angle scale reading ports and verniers, I and J. The verniers have 30 divisions per half degree. Hence each division is (1/30)x(1/2) = (1/60)^o or one minute (') of arc.
3. The knurled ring about the collimator, B, controls the slit opening.
4. Staff have already focussed the collimator for parallel light and have set collimator and telescope $\perp$ to spectrometer's rotation axis.

REMINDER ON READING A VERNIER

$${\frac{{1}}{{2}}} {\frac{{1}}{{2}}} \theta$$

PART I - B: FOCUS THE TELESCOPE FOR PARALLEL RAYS

1.

i)

Slide telescope eyepiece in or out until crosshairs are in sharp

ii)

Sight telescope at a distant object (thru an open window); then focus telescope by rotating its focus ring (Fig. 1) until the object's clear image falls on the cross hairs. The test for proper focus is absence of parallax between the image and cross hairs (appendix 4). The telescope, now focused for parallel rays, will stay so as long as the focus ring is unmoved; but one may still adjust the eyepiece to suit the observer.

2.

An alternative method to focus for parallel light is to use the Gaussian eyepiece + light source as described below in PART II B.

3.

With collimator and telescope both properly focused one should get a sharp image of the slit and no parallax between slit image and cross hairs. If you still get parallax, recheck your telescope focus and/or consult instructor.

4.

Optimizing light thru the collimator: A properly located short focal length lens can gather a large fraction of the light from a source and redirect it

thru the collimator thereby facilitating detection of weak spectral lines. For strong lines it permits narrowing the slit width and thus improving resolution. Use such a lens (mounted on a ring stand) to find an arrangement which fills the collimator with light.

Figure 3: Proper, a), and improper, b) prism locations.

PART II-A: MEASUREMENT OF PRISM ANGLE

1. Mount the prism (in holder) on the dowel pins (in the prism table) so that the prism is far enough from the collimator that beams B₁ and B₂ are centered with respect to the spectrometer axis; otherwise much of the reflected light may miss the telescope. Turn the table so that the apex angle A (Fig. 3) splits the beam into nearly equal parts.
2. Using the prism table levelling screws adjust the plane of the prism table so it is close to horizontal.
3. With the prism table clamped at the proper elevation, set the telescope to receive beam B₁ and form an image of the slit on the intersection of the telescope cross hairs. Record both VERNIER readings (in minutes). Average the two vernier readings (to eliminate any systematic error from misalignment of the circle scale with respect to bearing axis), and add the result to one of the angle scale readings. Note that the angle scale reads in half-degree units and the vernier in minutes (not in decimal degrees!). The zero position on the vernier determines the angle to the nearest half-degree (30 minutes) and the vernier reads the number of minutes past the half-degree mark.
4. Then with telescope in the 2nd position, set on the slit image. The change in reading of the same angle scale should be the angle D. (See Fig. 3.) Be sure you don't get the angle scales mixed up and subtract the first reading of angle scale 1 from a reading of angle scale 2. Also some students incorrectly handle the subtraction when the scale passes through 360^o or 0^o.
5. OPTIONAL: Try to prove that angle D is twice the prism angle A.

PART II-B: ALTERNATE method for prism angle measurement using Gaussian eyepiece

Figure 4: Spectrometer telescope

Introduction: The Gaussian eyepiece (Fig. 4) has a partially reflecting glass plate G set at 45^o to telescope axis so that light from the lamp reflects down the telescope tube, past the cross hairs and out the objective. If in front of the objective you place a reflecting surface perpendicular ($\perp$) to the telescope axis, the light will reflect back into the objective and form a real image of the cross hairs. If the cross hairs are in the focal plane of the objective, their reflected image will form in the same plane. Thus both the cross hairs and the image will be in focus through the eyepiece. When one orients the reflecting surface so that the image coincides with the cross hairs, the reflector is accurately $\bot$ to telescope axis.

Thus the Gaussian eyepiece permits both focusing the telescope for parallel rays and setting a reflecting surface $\bot$ to the telescope axis.

1.

To focus the telescope for parallel light:

a)

Rotate the Gaussian eyepiece to open the hole between reflecting plate G and lamp. Adjust eyepiece to give a sharp image of the cross hairs, but don't turn the eyepiece to block the light hole. Next, with prism on the table, adjust the prism table screws to make the table nearly horizontal.

b)

Now rotate the prism table until the prism face is approximately $\perp$ to telescope axis. Clamp the telescope and mount a light on the Gaussian eyepiece. (Check that the light hole is still open). With no illumination on the prism except that from the telescope, next rotate the prism table back and forth a few degrees until maximum reflected light appears in the eyepiece. Clamp the table in this position, and then focus the telescope (by turning the focus ring) until the reflected image of the cross hairs appears and shows no parallax with respect to the cross hairs. The telescope is now focused for parallel rays. Fine adjustments of the prism table leveling screws may help the images coincide and thus set telescope axis accurately $\perp$ to the reflecting surface. See Part VI, Sec. 3.

2.

To find the prism angle set the telescope $\bot$ to first one prism face and then the other. The angle between these two positions is the supplement of the prism angle. See Part IIA for detail about angle readings.

PART III: INDEX OF REFRACTION

Introduction: When the path thru a prism is symmetric, the deviation is a minimum. At this angle of minimum deviation the index of refraction, n, is

$${\frac{{\sin \left[ ( A + d)/\delta\right]}}{{\sin \left( A/2\right) }}}$$

$$\delta$$ n = refractive index of the prism.

Devise your own methodology and determine the prism's refractive index for one or more lines of the Hg spectrum.

PART IV: CALIBRATION OF PRISM SPECTROSCOPE

1. Set the prism for minimum deviation for a green line in the Hg spectrum.
2. Determine the angle readings for the yellow, green, blue-violet, and deep-violet Hg lines. Repeat this for the red line from a hydrogen discharge.
3. Plot telescope angle vs accepted $\lambda$'s for these lines.
4. For the same setting of the prism table, find the angles for the blue-green and two violet lines of the hydrogen spectrum.
5. Use your calibration curve to determine $\lambda$ of the blue-green and violet lines in the atomic hydrogen spectrum. Compare results to accepted values.

OPTIONAL: The calibration curve is very non-linear. Since a more linear plot facilitates interpolation of an unknown $\lambda$, try plotting deviation vs 1/ $\lambda^{2}_{}$.

TABLE OF WAVELENGTHS:

Mercury lines Hydrogen lines

$$\lambda \lambda^{2}_{} \lambda \lambda^{2}_{}$$

$$\left\rceil\vphantom{ }\right.$$ Yellow₁ x 10⁶(nm)^-2 x 10⁶(nm)^-2

$$\left\rfloor\vphantom{ }\right.$$ Yellow₂ 2.993 Red 656.3 2.322

Green 546.1 3.353 Blue-green 486.1 4.232

Blue 496.0 ^* 4.065

49l.6 ^* 4.138 Violet 434.0 5.309

Blue 435.8 5.265

  -violet 434.7 ^* Deep Violet 410.2 5.943

Deep- 433.9 ^*

   violet 407.8 ^* 6.013 NOTE: ^* implies

404.7 ^* 6.106       usually faint

PART V: DIFFRACTION GRATING

1. Grating constant (number of lines per cm): The lines/cm marked on these replica gratings is only approximate because the plastic often changes dimensions when it is stripped from the master grating. To achieve quantitative results calibration of the grating constant is desirable and can be performed using the known $\lambda$, 546.1 nm, of the Hg green line.
   

   CALIBRATION PROCEDURE:

   a)

   Align telescope with collimator so the slit image falls exactly on the vertical cross hair.

   b)

   Mount grating on the dowel pins in the prism table. Adjust and clamp the table so the grating is approximately $\perp$ to telescope-collimator axis.

   c)

   Then adjust the table so that the front grating face is accurately $\perp$ to telescope. Use the Gaussian eyepiece (see Part II - B) for the final adjustment: the reflected image of the cross hairs should fall on the cross hairs. (Two reflected images can result if the sides of the grating glass are not parallel. If so, test which image comes from the grating side by wetting the other side.) Clamp prism table in this position and record telescope direction.

   d)

   Next set telescope on the first and second orders of the Hg green line. Use both $\pm$ angles. The $\pm$ angles should of course agree. Use them and $\lambda_{{\mbox{\small green}}}^{}$ = 546.1 nm to calculate the grating spacing d from

   n$$\lambda$$ = d sin$$\theta$$.

2. Measurement of the wavelengths in the Balmer series in hydrogen:

   a)

   Use the calibrated grating spectroscope to measure $\lambda$ for three or four atomic hydrogen spectral lines.

   b)

   Compare these $\lambda$'s to those from the Balmer formula

   $${\frac{{1}}{{\lambda }}}$$  = R$$\left[\vphantom{ \frac{1}{2^2}~-~\frac{1}{n^2}}\right.$$$${\frac{{1}}{{2^2}}}$$  - $${\frac{{1}}{{n^2}}}$$$$\left.\vphantom{ \frac{1}{2^2}~-~\frac{1}{n^2}}\right]$$    where  R = 10, 967, 758 m^-1

   or calculate the frequencies of these lines and compare to

   f = cR$$\left[\vphantom{ \frac{1}{2^2} - \frac{1}{n^2}}\right.$$$${\frac{{1}}{{2^2}}}$$ - $${\frac{{1}}{{n^2}}}$$$$\left.\vphantom{ \frac{1}{2^2} - \frac{1}{n^2}}\right]$$  where  R = 10, 967, 758 m^-1.

   c)

   Use Planck's relation, $\Delta$E = hf, to calculate the energies in eV of the photons for each frequency and, therefore, the energy change in the hydrogen atom associated with each frequency.

   d)

   Assuming that each observed $\Delta$E leaves hydrogen in the n = 2 state, show the observed transitions on a hydrogen energy level diagram (drawn to scale; consult textbook).

PART VI: ADJUSTMENT OF A SPECTROMETER

1.

Use Gaussian eyepiece and prism face method to focus telescope for parallel rays. If necessary adjust prism table to make reflected image of the cross hairs coincide with the cross hairs.

2.

Remove prism from the table and align telescope and collimator approximately. Focus the collimator until there is no parallax between the image of the slit and the cross hairs. Adjust the ring on the collimator focusing sleeve so that the collimator is in focus with the ``V'' in the slot. Clamp the telescope in line with the collimator so that the image of the slit (vertical) falls on the vertical cross hairs. Now rotate the slit through 90<sup>o</sup> as permitted by the ``V'' projection and slot. Adjust the level of the telescope so that the image of the slit falls on the horizontal cross hair. The axes of telescope and collimator are now in line but not necessarily perpendicular to the axis of rotation of the instrument.

3.

$$\perp \perp$$

Bring the telescope to within < 90^o of the collimator and use a face of the prism to reflect light from the collimator down the telescope tube. Adjust the prism face so that the center of the slit image falls on the intersection of the cross hairs. The prism face is now parallel to the instrument axis.

4.

Set the telescope $\perp$ to the adjusted prism face (use Gaussian eyepiece method). Return to part 3 setup and adjust the collimator so that the center of the slit image falls on the cross hair's intersection. Telescope and collimator are now properly focused and are perp to the instrument axis.

5.

To adjust the second prism face $\perp$ to the axis of the telescope (again use the Gaussian eyepiece method). Turn only the proper screw on the prism table: otherwise one disturbs the adjustment of the first face. Recheck the first face and, if necessary, readjust it. Continue this process until both faces are parallel to the axis of the instrument. Turn collimator slit to vertical position. The spectrometer should now be in adjustment.