Astrophysics Lab - Electron Diffraction (Lab Report) PDF

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Astrophysics Lab - Electron Diffraction (Lab Report)...

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ELECTRON DIFFRACTION Experiment K

Thomas Bowen 18012374

Abstract Analysis of electron diffraction over a range of kV provides a set of data that can then be used to calculate an approximate value of Planck’s constant. Two concentric green rings were made by electron acceleration through a crystal lattice. This nature of electron diffraction was observed within a dark room in the lower laboratories of the Lennard-Jones building. The concentric rings were made via a diffraction apparatus and made visible on the surface of a glass projection screen. This was measured by three experimentalists each taking individual measurements of the rings’ diameter through increments of 0.2 kV from 2.0 to 5.0 kV. The average value of the diameter was taken from both rings at each interval and then an approximation for Planck’s constant was found using the de Broglie Hypothesis. The results from the smaller ring gave an approximation of Planck’s constant to be 2.399 x 10-35 m2 Kg/s; this value differs by 6.3861 x 10-34 m2 Kg/s from the literature value. The larger outer ring concluded with a value of 2.215 x 10-33 m2 Kg/s. This measurement is -1.5524 x 10-33 from the literature value of 6.626 x 10-34 m2 Kg/s (h).

Introduction The fundamental physics that underlies this experiment is the nature of light and its behaviour to be diffracted through a crystalline lattice with a grating of spacing d. Particles, (in this case, electrons) behave with wave like properties and should be diffracted through some angle θ given by equation 1. The experiment resulted in a diffraction pattern of two separate sized concentric rings projected onto a glass screen. The results arose from measuring their diameter and two independent angles that can be observed in fig.2. Equ. 1

𝜆 = 𝑑𝑠𝑖𝑛𝜃

Where λ is the wavelength of the light. For very small angles however, the wavelength is equal to: Equ. 2

𝜆 = 𝑑𝜃

The electrons in this experiment are said to have an effective wavelength which is inversely proportional to the electron’s momentum. This was first suggested by Louis de Broglie in 1926 in the thesis of the ‘de Broglie Hypothesis’ (Electron Diffraction and the Measurment of Planck's Constant, 2018-2019). The de Broglie Hypothesis equation is as follows: Equ. 3

𝜆 = ℎ⁄𝑚𝑣

Where h is Planck’s constant; this equation can be used to find a close approximation to Planck’s constant provided that the wavelength of the electrons and their corresponding velocities are known. The apparatus used in this experiment involved a crystal lattice with an approximate lattice spacing of 10-10 m. This is effective for the purposes of the experiment because the required spacing to diffract electrons is approximately equal to the lattice spacing used. Within the apparatus is an electron gun which accelerates the focused electrons through a graphite target. This is achieved by using a high voltage source to accelerate the electrons through a potential difference of up to 5kV. The electrons are then projected through the graphite target and are then diffracted through the spacing onto a luminous screen. The resultant diffraction pattern that is projected through the potential difference and shown on the screen are two separate concentric green rings of dissimilar diameter. The inner ring corresponds to a lattice spacing of d = 0.213 nm and the outer ring corresponds to d = 0.123 nm. (Electron Diffraction and the Measurment of Planck's Constant, 2018-2019) Measurements of the diameter of the two rings was taken by each experimentalist with ascending voltage between increments of 0.2 kV. This data, coupled with the de Broglie Hypothesis, subsequently leads to accurate findings of Planck’s constant, h. The measurement of Planck’s constant arises from the relationship between the rings’ diameter from the diffraction grating of the lattice and the corresponding potential difference associated with

the velocity of the electrons, given by equation 3. The length from the crystal lattice to the end of the electron diffraction tube is denoted as L (0.135m) and can be viewed in fig. 2. This length can be used alongside the diameter of each ring to calculate the small angle from the diffracted electrons. Geometrically: 𝜃 = 𝐷⁄2𝐿

Equ. 4

Therefore, substituting in equation 2, the final equation for calculating the wavelength which can be used to determine a value for Planck’s constant is as follows: 𝜆 = 𝑑𝐷⁄2𝐿

Equ. 5

Experimental Procedure The apparatus used in this experiment was located within a dark optic room in the lower laboratories of the Lennard-Jones building. The use of a dark room provided the experimentalists with a clear view of the luminescent screen and could therefore accurately measure the diameter of both diffraction rings using a 0.3m plastic ruler. The dark room becomes entirely dark with the ceiling light powered off. Additionally, the set up on the desk houses electrical equipment that exceeds 5 kV which could cause large extent of harm if used inappropriately. Working in these darkened conditions, it was imperative that the experimentalists move around the room and operate the equipment with care and treat the apparatus as possible harmful machinery, agreeing with the risk assessment form (Fig. 9). The experiment was thus conducted in a vigilant and cooperated fashion to ensure the set of results were safe, independent and as precise as possible. The apparatus shown in figure 1 consisted of an electron gun positioned on a stand and placed on the desk inside the dark optic room. The electron gun is concealed within an electron diffraction

tube and wired up to a potential difference converter which is connected to a 230V mains power supply within the room. The electron gun fires the electrons outward through the lattice which are then diffracted towards the observable screen housed within a glass chamber; the concentric rings are then made observable and were measured on the front facing luminescent screen concealed within the glass chamber as seen in figure 2. With the apparatus powered on, an initial reading was made at a potential difference of 2.0 kV. At this voltage, the two concentric rings were dim and small, this required careful measurements made by each experimentalist. A total of three separate measurements were made for each increment of 0.2 kV and the average value of the ring’s diameter from the experimentalists was taken from this data. The range of potential difference covered a total of 16 sets of three independent measurements, ranging from 2.0 Kv to 5.0 Kv. The experimentalists were careful not to exceed the voltage further than 5.0 kV due to any potential overload hazards. The total set of results were recorded on a table of figures in the laboratory booklets. The average values of each increment of 0.2 kV were taken for both D1 and D2 at each interval. This data was recorded and used in the final calculable approximation of Planck’s constant. The calculation is made using equation 5 as to approximate the closest value of Planck’s constant and arrive at a result close to the literature value of 6.62 x 10-34 m2 Kg/s. The two rings produced by the diffraction pattern are a result of having two functional grating spaces in the graphite lattice structure. The two separate concentric rings will thus have a separate result for the approximation of Planck’s constant as they both have dissimilar spacings, i.e. d1 = 0.213 nm and d2 = 0.123 nm.

Diagrams Fig. 1: Experimental Apparatus Set Up

Fig. 2: Side View: Electron Diffraction Through Crystal Lattice

Fig. 3: Front View: Diameter of the Two Visible Rings

Results Fig. 4: Measurements and Data for Inner Ring

INNER RING Voltage (Va) / kV 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0

R1

0.034 0.032 0.034 0.031 0.032 0.028 0.027 0.025 0.026 0.025 0.025 0.024 0.023 0.023

Ring Diameter (D) / m R2 R3 0.040 0.038 0.035 0.037 0.035 0.035 0.031 0.033 0.030 0.031 0.028 0.030 0.027 0.027 0.026 0.027 0.026 0.026 0.025 0.026 0.025 0.025 0.023 0.024 0.024 0.023 0.023 0.024 0.023 0.024

AVG 0.040 0.038 0.035 0.034 0.033 0.031 0.030 0.027 0.027 0.026 0.026 0.025 0.024 0.024 0.023 0.023

λ/m 3.15556E-11 2.99778E-11 2.78741E-11 2.68222E-11 2.57704E-11 2.41926E-11 2.36667E-11 2.1563E-11 2.1037E-11 2.02481E-11 2.02481E-11 1.97222E-11 1.89333E-11 1.86704E-11 1.84074E-11 1.84074E-11

σ(D) / m

ΔD / m

0.00153 0.00173 0.00153 0.00058 0.00200 0.00058 0.00058 0.00058 0.00058 0.00000 0.00100 0.00058 0.00058 0.00058

0.00088 0.00100 0.00088 0.00033 0.00115 0.00033 0.00033 0.00033 0.00033 0.00000 0.00058 0.00033 0.00033 0.00033

Δλ / m 0 0 6.96E-13 7.89E-13 6.96E-13 2.63E-13 9.11E-13 2.63E-13 2.63E-13 2.63E-13 2.63E-13 1.94E-27 4.55E-13 2.63E-13 2.63E-13 2.63E-13

1/Va / kV-1 0.5000 0.4545 0.4167 0.3846 0.3571 0.3333 0.3125 0.2941 0.2778 0.2632 0.2500 0.2381 0.2273 0.2174 0.2083 0.2000

Note: The first two results were measured singularly due to minor inaccuracies in the method.

Fig. 5: Inner Ring Wavelength Vs Potential Difference

Inner Ring λ 3.5E-11 3E-11

λ/m

2.5E-11 2E-11 1.5E-11 1E-11 5E-12 0 0.0000

0.1000

0.2000

0.3000

1/Va / V-1

0.4000

0.5000

0.6000

Fig. 6: Measurements and Data for Outer Ring Ring Diameter (D)

OUTER RING

1/Va / VVoltage (Va) / V

R1

R2

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4

σ(D) / m

ΔD / m

Δλ / m

1

R3

AVG

λ/m

0.058 0.055 0.054 0.053 0.049 0.047 0.048 0.044 0.044 0.043 0.042

0.063 0.061 0.060 0.059 0.054 0.050 0.049 0.049 0.046 0.046 0.043 0.044 0.042

0.060 0.058 0.055 0.051 0.049 0.048 0.047 0.045 0.044 0.043 0.042

0.063 0.061 0.059 0.057 0.054 0.051 0.049 0.048 0.047 0.045 0.044 0.043 0.042

2.87E-11 2.77889E-11 2.70296E-11 2.61185E-11 2.47519E-11 2.33852E-11 2.23222E-11 2.18667E-11 2.14111E-11 2.05E-11 1.98926E-11 1.97407E-11 1.91333E-11

0.00115 0.00208 0.00058 0.00153 0.00000 0.00100 0.00100 0.00100 0.00058 0.00058 0.00000

0.00067 0.00120 0.00033 0.00088 0.00000 0.00058 0.00058 0.00058 0.00033 0.00033 0.00000

3.04E-13 5.48E-13 1.52E-13 4.02E-13 2.24E-27 2.63E-13 2.63E-13 2.63E-13 1.52E-13 1.52E-13 0

0.5000 0.4545 0.4167 0.3846 0.3571 0.3333 0.3125 0.2941 0.2778 0.2632 0.2500 0.2381 0.2273

4.6 4.8

0.043 0.041

0.042 0.041

0.042 0.041

0.042 0.041

1.92852E-11 1.86778E-11

0.00058 0.00000

0.00033 0.00000

1.52E-13 0

0.2174 0.2083

5.0

0.040

0.040

0.040

0.040

1.82222E-11

0.00000

0.00000

0

0.2000

Note: The first two results were measured singularly due to minor inaccuracies in the method. Fig. 7: Outer Ring Wavelength Vs Potential Difference

Outer Ring λ 3.5E-11

3E-11

2.5E-11

λ/m

2E-11

1.5E-11

1E-11

5E-12

0 0.0000

0.1000

0.2000

0.3000

1/Va / V-1

0.4000

0.5000

0.6000

Fig. 8: Final Results Table.

INNER RING G=

4.44E-11

ΔG =

1.76E-12

2me =

2.91852E-49

h= Δh =

2.39864E-35 9.50811E-37

G=

4.1E-09

ΔG =

4.6E-18

2me =

2.91852E-49

h=

2.21496E-33

Δh =

2.48507E-42

OUTER RING

Fig. 9: Risk Assessment Form.

Discussion The final results, shown in figure 8, display the two final approximations of Planck’s constant. The first set of results, gathered from the data of the inner ring, shows the approximation value for Planck’s constant to be 2.399 x 10-35 m2 Kg/s. This value is 6.3861 x 10-34 m2 Kg/s from the literature value of 6.626 x 10-34 m2 Kg/s. The value gathered from data of the outer ring gives an approximation of Planck’s constant to be 2.215 x 10-33 m2 Kg/s. This set of data is -1.5524 x 10-33 from the literature value. Thus, the closest approximation arises from the outer ring; this may be due to the wider diameter of the ring and thus higher accuracy in the recording of the measurements. The experimentalists were careful in the direct measurements of each ring and each experimentalist were sure not to mislead any incorrect values in the conduction of the method. Tabulating and overviewing the results (primarily fig.5 and fig.7) made it clear that the experiment went as planned and successful readings were made. However, this method of finding an approximation for Planck’s constant was rather rudimentary and the value was slightly off the exact literature value. Accounting for inaccuracies in the experiment, the first two results, taken at 2.0 and 2.2kV, were recorded singularly and the following values were recorded in sets of three by each experimentalist. This was a slight mistake in the method, however the experiment proceeded on until the final value of 5.0kV was reached. This caused a slight error in the final results and without this minor mistake, the final two values for Planck’s constant may well have been slightly more precise. To attain a higher precision value of Planck’s constant, the experiment could have been conducted several times with many more readings. The sets of three could have been recorded as sets of six and the average value of each would have been closer to the true value of the diameter of the concentric rings. This would have narrowed down a more correct value but may have still not quite reached the literature value of Planck’s constant....