Wavelength Measurement with a Grating (O4)

Wavelength Measurement with a Grating (O4)

Experiment Objective

• To determine the wavelengths of the most intense lines in the Hg (Mercury) spectrum and the visible lines in the H (Hydrogen) spectrum using a grating spectrometer.

• To additionally determine the Rydberg constant.

Theoretical Background

Diffraction at a Grating

• A transmission grating consists of an arrangement of $N$ narrow slits, each with a width $b$, separated by a distance $d$ (grating constant).

• A typical value for a grating used in practical experiments is $d = 10 \ \mu m$.

• When the grating is illuminated perpendicularly with parallel light rays of wavelength $\lambda$, the phenomena of diffraction (Huygens' Principle) and interference lead to an angle-dependent intensity distribution $I(\Theta)$ of the diffracted light behind the grating.

• This is conceptually similar to experiment O3 (Fraunhofer diffraction).

• The intensity distribution $I(\Theta)$ is given by: $I(\Theta) = I<em>0 f</em>{ES}(\Theta)f<em>P(\Theta) \quad (1)$- Where $f</em>{ES}(\Theta)$ corresponds to the intensity distribution from a single slit of width $b$, and $f_P(\Theta)$ corresponds to the intensity distribution from $N$ slits of width $b \to 0$ spaced at distance $d$.

  • $I_0$ is a constant.

• The components are:- $f_{ES}(\Theta) = \frac{\sin^2 \phi(\Theta)}{\phi^2(\Theta)} \quad (2)$- With $\phi(\Theta) = \frac{\pi b \sin \Theta}{\lambda}$

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  • $f_P(\Theta) = \frac{\sin^2(N\varphi(\Theta))}{\sin^2 \varphi(\Theta)} \quad (3)$- With $\varphi(\Theta) = \frac{\pi d \sin \Theta}{\lambda}$

• The diffraction angles $\Theta<em>n$ for the principal maxima of intensity (constructive interference) are given by: $d \sin \Theta</em>n = n \lambda \quad (4)$- With $n = 0, \pm 1, \pm 2, \pm 3, \dots$

  • $n$ is called the diffraction order. For example, $n=1$ refers to the first-order diffraction maximum.

• If the grating is illuminated with parallel non-monochromatic light, the intensity distribution behind the grating is an undisturbed superposition of the diffraction patterns of all individual wavelengths.

• Since the diffraction angle $\Theta_n$ of the intensity maxima is wavelength-dependent according to equation (4), a spectral decomposition of the light occurs.

• The intensity distribution belonging to a single diffraction order is called a spectrum.

• Equation (4) can be used to determine the wavelengths of a spectrum if the diffraction angle $\Theta_n$ is measured for a known diffraction order $n$ and grating constant $d$.

• Spectra are generally distinguished between continuous (e.g., incandescent lamp, sunlight) and discontinuous (e.g., mercury lamp) spectra.

• This experiment focuses only on the discontinuous line spectra of selected atomic species.

Atomic Emission Spectra

• The spectral lamps used in this experiment consist of a tube containing an atomic gas.

• An electrical voltage is applied to the tube, causing a gas discharge.

• Free electrons accelerate in the electric field, and during inelastic collisions with gas atoms, they transfer part of their kinetic energy to these atoms.

• The bound electron system of the excited gas atoms then enters excited electronic states.

• Due to the quantum mechanical nature of electrons bound to an atomic nucleus, only specific energy states are permitted for electrons in a given atom type.

• The lifetime of an electron in an excited state is very short (approximately $10^{-9} \ \text{s}$).

• When an electron leaves an excited state (initial state) and transitions to the ground state (final state), it spontaneously emits the corresponding energy as a photon:
  

  $hf = E<em>A - E</em>E$

• This process is called spontaneous emission.

• The line spectrum produced by a gas discharge allows for the identification of the atomic species present.

• For the hydrogen atom, the energies of a bound electron are given by the Rydberg formula: $E<em>m = -R</em>{\infty} \frac{hc}{m^2} \quad (5)$- Where:- $h$ is Planck's constant
  - $c$ is the speed of light in vacuum
  - $\epsilon<em>0$ is the electric field constant - $m</em>e$ is the rest mass of the electron
  - $e$ is the elementary charge
  - $R<em>{\infty}$ is the Rydberg constant - $m$ is the principal quantum number ( $m = 1, 2, 3, \dots$) - $R</em>{\infty} = \frac{m<em>e e^4}{8 \epsilon</em>0^2 h^3 c} \approx 1.0974 \times 10^7 \ \text{m}^{-1}$

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• The hydrogen atom's spectrum arises from all possible transitions between an initial state $m<em>2$ and a lower-energy final state $m</em>1$ (where m2 > m1 ).

• Transitions leading to the same final state $m<em>1$ appear as a series of lines in the spectrum.- Lyman series: $m</em>1 = 1$ (ultraviolet)

  • Balmer series: $m_1 = 2$ (visible)

  • Paschen series: $m_1 = 3$ (infrared)

  • Bracket series: $m_1 = 4$ (infrared)

• The lines in the visible spectral range belong to the final state $m_1 = 2$ and are called the Balmer series.

• For the Balmer series, the wavelength $\lambda<em>m$ is given by: $\lambda</em>m = \frac{hc}{E<em>m - E</em>2} = \frac{1}{R_{\infty}} \frac{m^2}{m^2 - 4} \quad (6)$- Where $m = 3, 4, 5, \dots$

Experimental Setup and Procedure

Spectrometer Setup

• The experiment uses the same spectrometer base unit as in experiment O2 (Prism Spectrometer).

• Light Sources: Mercury (Hg) lamp (specifically an energy-saving lamp, which also emits spectra of rare earths in the red range) and a Hydrogen (H) lamp (or He lamp).

• Illumination: The spectral lamp illuminates the entrance slit via a lens, which aims to provide uniform and intense illumination.

• Collimator: The slit is moved relative to the collimator lens until parallel light exits the collimator.- The slit width affects both light intensity and the spectrometer's resolving power.

• Grating: The parallel light beam then falls onto the grating, which is placed on the goniometer table.

• Telescope: The diffracted light is observed through a telescope.

• Goniometer Table: This table can be adjusted in position using three adjustment screws. Both the goniometer table and the telescope can be rotated independently around the spectrometer axis.

• Axes: The collimator axis and telescope axis are fixed and should be perpendicular to the spectrometer axis according to manufacturer specifications.

• Angle Measurement: The telescope's rotation can be precisely controlled with a fine adjustment drive. Relative rotation angles between the telescope and goniometer table are measured using a fixed scale disk on the goniometer table and a vernier scale on the telescope holder.

• Reference Point: The zero point of the angular scale is arbitrary, so only angle differences $\Delta\Theta$ are measured.

Adjustment of the Optical Arrangement

1. Spectrometer Adjustment:

   • Telescope Adjustment (to Infinity): Aim the telescope at a distant object and focus it by sliding the eyepiece. Light from distant objects is almost parallel. Since crosshairs are located exactly at the focal point of the eyepiece lens, these should appear sharp and parallax-free with the distant object. The eyepiece must not be moved after this adjustment.

   • Collimator Adjustment: Align the telescope parallel to the collimator axis. Search for the initially unfocused image of the entrance slit. Focus the slit image by sliding the slit. After this, the slit will be precisely at the focal point of the collimator lens.

2. Goniometer Table Adjustment:

   • Objective: To align the grating slits parallel to the spectrometer axis and thus perpendicular to the collimator and telescope axes.

   • Initial Setup: Position the grating by eye perpendicular to the connecting line of two adjustment screws (1 and 2) and perpendicular to the collimator axis. This ensures that adjustment screw 3 acts independently of screws 1 and 2.

   • Procedure:1. Observe a first-order diffraction maximum in Telescope Position I and a negative first-order diffraction maximum in Telescope Position II alternately.

     2. Using screw 3, first align the slit image parallel to the crosshairs.

     3. Then, using screws 1 and 2, bring the center of the slit image to the height of the crosshair center in both telescope positions.

     4. Finally, use screw 3 again to re-establish parallelism between the slit image and the crosshairs.

Determination of the Diffraction Angle

• Lock the goniometer table with a locking screw to prevent unintended rotation.

• Measure the angular settings of the first-order and negative first-order diffraction maxima (corresponding to Positions I and II in Figure 5) symmetrically to the collimator axis.

• The diffraction angle $\Theta<em>1$ is calculated from the angular difference $\Delta\Theta</em>1$:
  

  $\Theta<em>1 = \frac{|\Delta\Theta</em>1|}{2} \quad (7)$

• The same principle applies to higher diffraction orders.

Tasks

1. Perform the specified adjustment procedure for the spectrometer.

2. Determine the grating constant $d$ using equation (4).- Use an energy-saving lamp as the mercury spectral lamp. This lamp emits both the Hg spectrum (gas discharge) and the spectrum of some rare earths (fluorescent coating on the inner wall) in the red spectral range.

   • Select the strong blue Hg line.

   • Measure the diffraction angles up to the $\pm 3^{rd}$ order.

   • Plot the diffraction order $n$ against $\sin \Theta_n$. The resultant graph should be a straight line through the origin.

   • Determine the grating constant $d$ from the slope of this line.

3. Determine the wavelengths of three other Hg lines.- Compare the determined wavelengths with the tabulated values (see Table 1).

4. Determine the wavelengths of the three brightest visible hydrogen lines: H$\alpha$ (red, strong), H$\beta$ (blue-green, medium strong), and H$\gamma$ (violet, medium strong).- Assign the corresponding Balmer transitions (i.e., principal quantum numbers $m$ for the initial states) to these lines.

   • Verify the validity of equation (6).

   • To do this, plot the determined wavelengths $\lambda_m$ against $\frac{m^2}{m^2 - 4}$.

   • Determine the Rydberg constant $R_{\infty}$ from the slope of the resulting straight line.

Table 1: Spectral Lines of Mercury in the Visible Spectral Range