The Franck Hertz Experiment

Andy Chmilenko, 20310799

Instructor: Tan Dinh

Section 1

(Dated: 2:30 pm Wednesday June 26, 2013)

I. PURPOSE

The purpose of this experiment is to study the quantization of energy levels in Hg atoms in a low pressure gas using electrons, specifically using a technique known as electron energy loss spectroscopy (EELS) to determine the energy for the lowest excited state for Hg and relate this to that of the ground state and energy level diagram for Hg.

II. ANALYSIS

FIG. 1: Measured current versus the accelerating potential of the Franck-Hertz unit operating on a Franck-Hertz tube filled with Hg Vapour at 405.65^{∘}K, 383.15^{∘}K, and 325.55^{∘}K, in red, green, and blue respectively. Note the Frank-Hertz signature is lost at low temperatures

Temperature Maxima δ Adjacent Minima δ Adjacent
of F-H tube  (V) Maxima (V) (V) Minima (V)
(^{∘}K)  ± 0.05V ± 0.07V ± 0.05V ± 0.07V
405.65 4.4, 8.8, 4.4, 4.6, 6.5, 11.2, 4.7, 4.7,
13.4, 18.5, 5.1, 5.1, 15.9, 21.0, 5.1, 5.1,
23.6 , 29.2  5.6  26.1, 31.1  5.0
383.15  4.4, 8.6,  4.2, 5.2, 6.2, 11.0, 4.8, 5.0,
13.8, 19.0  5.2  16.0, 21.1  5.1

TABLE I: Approximate local minima and maxima for the curves created by the data points in Fig 1, and the differences between adjacent local minima and maxima.

Sample Calculations for δ Adjacent Maxima using Table I row 1, between the two points, 4.4 V and 8.8

V

ΔV = V_{2} − V_{1} ΔV = 8.8 − 4.4

ΔV = 4.4V

Sample Calculations for average change in local maxima using Table I, column 3

\overline{V} = ^{\frac{∑_{i}^{n} δ_{maxima_{i}}}{n}} \overline{V} = \frac{4.4+4.6+5.1+5.1+5.6+4.2+5.2+5.2}{8}

\overline{V} = 4.925V

Sample Calculations for uncertainty in \overline{V}, assuming an error of measurement of 0.05V

Δ\overline{V} = ^{\frac{\sqrt{∑_{i}^{n} δ_{m_{i}}^{2}}}{n}}
Δ\overline{V} = ^{\frac{\sqrt{n·\sqrt{δ_{m1}^{2}+δ_{m2}^{2}}^{2}}}{n}}
Δ\overline{V} = ^{\frac{\sqrt{8·(0.05^{2}+0.05^{2})}}{8}}
Δ\overline{V} = \frac{0.2}{8}
Δ\overline{V} = 0.025V

By analysing the data in the curves made by the data points as seen in Fig.1, several maxima and minima can be found as seen in Table I, and the value between adjacent local minima and maxima can be calculated, as well as the uncertainty in this calculated value assuming we take half of the least significant digit to be 0.05V. The average potential change between local maxima was found to be when the accelerating potential 4.9V ± 2.5×10^{−2}V, meaning the electrons equivalently had that amount of kinetic energy in eV, and similarly the average potential change between local minima was found to be 4.9V ± 2.5×10^{−2}V as well. Measuring the difference between local maxima and minima yield the same result in finding the excitation energy of Hg, which in this case is approximately 4.9eV ± 2.5×10^{−2}V.

In this case, it’s the 6s6p^{3}P_{0,1,2} energy levels that are getting excited in collisions with electrons, since these 3 levels are close together around the 4.9eV line.

ε = w_{0} = \frac{1}{2}m_{e}v^{2}  (1)

The first maxima was found to be around 4.4eV which is the electron’s work function on an Hg atom, the minimum amount of energy needed for an electron, and using Eq.1 we can calculate the speed of electrons at this kinetic energy.

Sample Calculations for speed of an electron with a kinetic energy of 4.4eV

w_{0} = \frac{1}{2}m_{e}v^{2}

e × 4.4 = \frac{1}{2}9.11 × 10^{−31}v^{2}

v^{2} = \frac{2e×4.4}{9.11×10^{−31}}
v = \sqrt{1.55 × 10^{12}}
v = 1.24 × 10^{6} \frac{m}{s}

Since the kinetic energy of the electron isn’t too high and the velocity is small relative to the speed of light, we don’t need to really account for relativistic effects, so the velocity of the electron should be quite close to 1.24×10^{6} \frac{m}{s}.

P V = NkT  (2)

MF P = (σn)^{−1} = (πr^{2}n)^{−1}  (3)

Sample Calculations for number density of Hg at 405.65^{∘}K with a Vapour pressure of 181.87 Pa

P V = NkT \frac{N}{V} = n = \frac{P}{kT}

n = \frac{181.87}{1.38×10^{−23}·405.65} n = 3.25 × 10^{22} \frac{atoms}{m^{−3}}

Sample Calculations for the mean free path of electrons in an Hg vapour that have an atomic radius of approximately 150pm with a number

density of 3.25 × 10^{22} \frac{atoms}{m^{−3}}

MF P = (πr^{2}n)^{−1}

MF P = (π(1.5 × 10^{−10})^{2} · 3.25 × 10^{22})^{−1} MF P = (π(1.5 × 10^{−10})^{2} · 3.25 × 10^{22})^{−1}

MF P = 4.35 × 10^{−4} \frac{m}{collision} = 4.35 × 10^{−1} \frac{mm}{collision}

Sample Calculations collision rate of electrons travelling with a velocity of 1.24×10^{6} \frac{m}{s}, and a

calculated MFP of 4.35 × 10^{−4} m

Rate = \frac{v}{MF P} Rate = \frac{1.24×10^{6}}{4.35×10^{−4}}

Rate = 2.85 × 10^{9} \frac{collision}{sec}

Sample Calculations uncertainty in vapour pressure using row 1 from Table II

\begin{matrix}Δρ = \sqrt{\frac{ΔT}{T} ^{2} + 0.01^{2}} \\ Δρ = \sqrt{(\frac{2+405.65· \frac{0.02}{273.15}}{405.65} )^{2} + 0.01^{2}}Δρ = \sqrt{(\frac{2+405.65· \frac{0.02}{273.15}}{405.65} )^{2} + 0.01^{2}} \end{matrix}

Temperature Vapour Number Mean Free Collision
of F-H tube Pressure Density n Path  Rate
(^{∘}K)  (Pa) ( \frac{atoms}{m^{−3}} ) ( \frac{mm}{collision}) ( \frac{collision}{sec})

±(2^{∘}K+2%)

405.65  181.87 3.25×10^{22} 4.35×10^{−1} 2.85×10^{9}
383.15  62.09 1.17×10^{22} 1.21 1.02×10^{9}
325.55  2.12 4.72×10^{20} 30.0 4.13×10^{7}

TABLE II: Approximate local minima and maxima for the curves created by the data points in Fig 1, and the differences between adjacent local minima and maxima.

Δ  Δ  Δ  Δ
Temperature Vapour Number Mean Free Collision
of F-H tube Pressure Density n Path  Rate
(^{∘}K)  (Pa) ( \frac{atoms}{m^{−3}} ) ( \frac{mm}{collision}) ( \frac{collision}{sec})

±(2^{∘}K+ \frac{2}{273.15} %) 405.65 383.15 325.55

TABLE III: Uncertainty in values from Table II

Using the table of Vapour pressures at specific temperatures as outlined in the CRC handbook the Vapour pressures were found and tabulated into Table II. Using Eq.2 the number density of the Hg in the F-H tube was also calculated, as well as the mean free path with Eq.3.

The calculated values and results are in line with the dimensions of the tube, with higher temperatures, like the 405.65^{∘}K and 383.15^{∘}K cases, where there is more Hg atoms, the MFP is higher and consequently we see a higher signal on the collector electrode. However for the low temperature situation at 325.55^{∘}K where the MFP was 30mm, which was much larger than the distance to the collector electrode the signal disappeared since there were little to no collisions between electrons and Hg atoms in the tube. The MFP isn’t constant with different temperatures, as the vapour density is dependent on the temperature of the tube. The primary uncertainty in this calculation is from the temperature reading, as the pressure is also dependent on the temperature. The Protek TM-1300k Thermometer is rated to having an accuracy of 2^{∘}C + 2% within the range of temperatures the F-H tube was at, and the CRC has a 1% uncertainty with the vapour pressures of mercury.

FIG. 2: Measured current versus the accelerating potential of the Franck-Hertz unit operating on a Franck-Hertz tube filled with Hg Vapour at 405.65^{∘}K with the subtracted background current.

By subtracting the background current where I ∝ V^{\frac{3}{2}} , we can get a better reading for maxima on our data set, as seen in Fig.2. No evidence in this instance can be seen for excitations for more than one Hg energy level, perhaps at higher temperatures electron energies the other excitation energies may be more evident.

III. CONCLUSION

4

Appendices