Electron Diffaction Lab Report PDF

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A lab report on Electron Diffraction. May not be 100% correct, but is a good template to work with....

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Ab Abstr str stract act Using a graphite chip as a target and the relevant equipment, the diameter of the diffraction rings on the phosphor screen were used successfully to measure the spacing between carbon atoms in the sample. The distance spacing was found to be 2.95∙10-10m ± 0.073∙10-10m by using all relevant equations given by de Broglie and Bragg. The accuracy of the measurements and the exhibition of diffraction rings show that in accordance with de Broglie’s hypothesis, electrons display both particle and wavelike behaviour, interacting with 𝒉 the sample in accordance to the equations 𝝀 = and 2dsin dsin(θ (θ (θ)=n )=n )=nλ. λ. For different voltages 𝒑

(momentum), different diffraction ring diameters were found showing that electron 1 wavelength is inversely proportional to the momentum, that is wavelength has a √𝑣

dependency, and by using Bragg’s law an accurate measurement of carbon atom spacing was found.

Int Intro ro rodu du ductio ctio ction n Einstein’s work on the photoelectric effect proved that light can have both wave-like and particle-like behaviour, and in 1927 Louis de Broglie developed equations relating the wavelength of light to the momentum of photons and hypothesised that these particles also displayed wave-like behaviour. De Broglie also suggested that the inverse proportionality between wavelength and momentum should hold true for all particles, including the lightest known particle at the time; the electron. The de Broglie wavelength is given by : 𝒉 𝝀 = 𝒑 , where 𝒑 = √𝟐𝒎𝑬 (1)

Where E is the kinetic energy of the electron, and m is the electron mass which is 9.109∙10-31 Kg. This gives the relation 𝒉 , given that h=6.626∙ 10-34 J∙s , when E=3keV=3000eV 𝝀= √𝟐𝒎𝑬

𝜆=

6.626 × 10−34

= 2.24 ∙ 10−11 𝑚 √2 × 9.109 × 10−31 × 3000 × 1.602 × 10−19 which is the x-ray part of the spectrum. These were the nature of waves dealt with within the experiment. This experiment seeks to analyse the wavelike nature of the electron; If the wavelength is inversely proportional to the momentum the minima and maxima diffraction pattern due to an electron beam being passed through a graphite screen will change in a regular way with momentum which is controlled by changing the energy of the beam with an applied voltage. Graphite in this form’s atoms are oriented in many random directions, so several angles of incidence can be measured without varying the direction of the electron beam or the orientation of the sample/screen. If the electron wavelength is of the order of the separation between atomic planes in the sample then the sample will act as a three dimensional grating, hence producing a diffraction pattern on our screen. To develop the equation that relates de Broglie’s wavelength equation to the measured diffraction pattern we can use Braggs 𝜶 equation for diffraction 2d 2dsin( sin( sin(θ)= θ)= θ)=nλ nλ where θ=𝟐 . To formulate the equation used for α, that is: we can use the below geometry:

α= 𝒕𝒂𝒏−𝟏(𝑳 𝑹

𝒔𝒊𝒏(𝝋)

) (2)

+𝒄𝒐𝒔(𝝋)−𝟏

𝐿

We can derive: 𝐷 𝑍 Sin(φ)= Cos(φ)= 𝑅

𝑍

2𝑅

𝐷 2

Sin(α)=

𝐷

2𝐿

𝑅

Cos(α)=

𝐿−𝑅+𝑧 𝐿

⇒𝑇𝑎𝑛(𝛼) = Cos(α) = Sin(α)

= =

𝐷 (2𝐿) 𝐿−𝑅+𝑧 ( 𝐿 ) 𝐷 (2𝑅) 𝐿−𝑅+𝑧 ( ) 𝑅

=𝐿

i

Fig 2: Geometry of the electron tube and graphite target

Sin(φ)

𝐿 𝑅 𝑧 − + 𝑅 𝑅 𝑟

Sin(φ)

−1+𝐶𝑜𝑠 (𝜑) 𝑅 sin(𝜑)

⇒ α= 𝑡𝑎𝑛 −1 ( 𝐿

= (𝐿

sin(𝜑)

+cos(𝜑)−1

𝑅

𝜶

+cos(𝜑)−1

𝑅

)

) As required to show.

𝜶

)=nλλ. From former Using the equation 2dsin 2dsin((θ)= θ)=nλ nλ where θ=𝟐 , now we have 2d 2dsin( sin(𝟐)=n

calculations we derived that 𝝀 =

𝒉

√𝟐𝒎𝑬

, so now we can say: 𝜶

2ds 2dsin( in(𝟐)=n

√𝟐𝒎𝑬 𝒏𝒉

2dsin sin sin(( 𝟐)= ⇒2d 𝜶

√𝟐𝒎𝒒𝑽 𝟏 𝒏𝒉 √𝑽 √𝟐𝒎𝒒 𝟏

⇒2d 2dsin sin sin(( 𝟐)= 𝜶

𝛼

Hence, a sin( 2 ) Vs

1

√𝑉

𝒉

α/2)) ∝ ( ) ∴ si sin( n( α/2 √𝑽

w wher her heree E= E=V Vxq

(3)

graph will supply a the slope (∆) required to solve the equation for

distance spacing between atoms,

𝒏𝒉

∆√𝟐𝒎𝒒

d=

. (4 (4))

assuming De Broglie’s hypothesis is correct, then the above derived formulae should hold true for any electron energy and so an accurate measurement of atom spacing in the sample can be made.

Met Metho ho hod d Ap Appa pa para ra ratus tus To investigate the properties of electrons and the orientation of carbon atoms in graphite the following equipment was required:  EHT  DC power supply  Vernier Callipers

Pro Proce ce cedu du dure re Before the experiment began it was ensured that all equipment was turned off and the EHT was set to zero. All connection leads were then checked and the EHT and DC power supply were both turned on. The EHT was left for a few minutes to warm up, by observing the glow of the heater at the back of the tube it was confirmed the equipment was working and ready. The lights were dimmed so that the diffraction rings on the phosphor screen could be better observed. The DC power supply was set to 4V, and the voltage on the EHT was slowly increased until the rings on the screen appeared. The minimum voltage was around 1.5-2 kilo volts before rings began to appear. Starting at 1.8kV the callipers were used to measure both the inner and outer rings, and increasing the voltage by 0.2 kV steps 8 measurements were taken. The process was then repeated to ensure that the results were consistent, and so that the more accurate measurements were insured.

Res Result ult ultss and eerr rr rror or aanaly naly nalysi si siss Initial experimental data collected: Table 1: Experiment 1 Voltage (V) D1 (m) D2 (m) 1800 0.03727 0.06892 000 0.03715 0.06098 2200 0.03462 0.0591 2400 0.03385 0.05923 2600 0.03188 0.05621 2800 0.03047 0.0539 3000 0.02963 0.05243 3200 0.02799 0.05004

Table 2: Experiment 2 Voltage (V) D1 (m) D2 (m) 1800 0.03828 0.06646 2000 2200 2400 2600 2800

0.03755 0.03596 0.03291 0.03144 0.03095

0.06236 0.06059 0.05785 0.05511 0.05382

3000 3200

0.02727 0.02708

0.04855 0.04732

Table 3:Average of Experiment 1 and 2 Voltage D1 D2 (V) (m±0.0005m) (m±0.0005m) 1800 0.038 0.068 2000 0.037 0.062 2200 0.035 0.060 2400 0.033 0.059 2600 0.032 0.056 2800 0.031 0.054 3000 0.028 0.050 3200 0.028 0.049

𝛼 Using the above measured distances values for Sin(α) and sin(2 ) were then tabulated and graphed:

Table 4: sin(α) calculations sin(alpha) Voltage (D1) error

𝜶

sin(alpha) (D2) Error

Table 5: sin( )calculations 𝟐 sin(alpha/2) Voltage (D1) error

sin(alpha/2) error (D2)

1800

0.134

0.001 0.239

0.003

1800

0.067

0.001 0.119

0.002

2000

0.133

0.001 0.218

0.002

2000

0.066

0.001 0.109

0.002

2200

0.126

0.001 0.212

0.002

2200

0.063

0.001 0.106

0.002

2400

0.119

0.001 0.207

0.002

2400

0.059

0.001 0.103

0.002

2600

0.113

0.001 0.197

0.002

2600

0.056

0.001 0.098

0.002

2800

0.109

0.001 0.191

0.002

2800

0.055

0.001 0.095

0.002

3000

0.101

0.001 0.179

0.002

3000

0.051

0.001 0.089

0.002

3200

0.098

0.001 0.173

0.002

3200

0.049

0.001 0.086

0.002

Graph 1: sin(α) vs voltage

sin(α α) Vs Voltage 0.300

0.250

Sin(α)

0.200 y = -4E-05x + 0.3109

0.150

D1 0.100

y = -3E-05x + 0.1861

D2

0.050 0.000 1500

1700

1900

2100

2300

2500

Voltage (V)

2700

2900

3100

3300

𝛼

Graph 2: sin( 2 ) vs Voltage

sin(α α/2) Vs Voltage

Sin(α/2)

0.140 0.120 0.100

y = -2E-05x + 0.155

0.080 0.060

D1

y = -1E-05x + 0.093

0.040

D2

0.020 0.000 1500

1700

1900

2100

2300

2500

2700

2900

3100

3300

Voltage (V)

Based on our theoretical calculations, sin(α) should have a

1

√𝑣

dependency (equation 3),

based on our results and graphs there appears to be a linear relationship between the two. If a wider range of voltages were to be used, the relationship viewed in the results shouldld produce a plot that looks like this:

Fig 1: Example of a 𝛼

Now, plotting sin( ) Vs 2

1

√𝑉

1

√𝑥

graph

will give a linear relationship, and thus the slope needed to apply

to the formula derived for distance spacing between atoms (equation 4):

Graph 3: sin(𝛼) Vs

1

√𝑉

sin(𝛼) Vs 1/√𝑉 0.300 0.250

y = 10.404x - 0.0088

Sin(α)

0.200 0.150

y = 6.5941x - 0.017

D1

0.100

D2

0.050 0.000 0.017

0.018

0.019

0.02

0.021

0.022

0.023

0.024

1/√𝑉

𝛼

1

Graph 4: sin( 2 ) Vs √𝑉

sin(𝛼/2) Vs 1/√𝑉 0.140 0.120

y = 5.2943x - 0.0057

sin(𝛼/2)

0.100 0.080

y = 3.3023x - 0.0085

0.060

D1 D2

0.040 0.020 0.000 0.017

0.018

0.019

0.02

0.021

0.022

0.023

0.024

1/√𝑉 𝛼

2

Using our derivation of sin(α) and voltage relationship, the slope of the sin( ) Vs should give the value

nh ∆=2𝑑 2𝑚𝑞 √

1

√𝑉

graph

(equation 4). From the trend-line for D1, the slope which we

will denote ∆1=3 can be used to solve for an approximate spacing between carbon atoms for n=1 for D1 This gives 3.3 = 2𝑑

For D1, d=2.63∙10-10 m

h

√2𝑚𝑞

, rearranging for d: d=

h

(3.3)√2𝑚𝑞

From the trend-line for D2, the slope which we will denote ∆2= 5.3 can be used to solve for h an approximate spacing between carbon atoms for n=2 for D2 This gives 5.3 =𝑑√2𝑚𝑞, h

rearranging for d: d=(5.3) For D2, d=3.27

∙10-10 m

√2𝑚𝑞

Taking the average of these gives an atomic spacing dAVG= 2.95∙10-10m. For the errors on d we 𝛼 can take the percentage error applied to sin(2 ) values and apply it to our distance. There are errors applied to each of the constants used in the formula, but they are so small they are negligible. 𝛼 %error(sin( 2 ))≈2.5% (approximated as greater than the average due to consideration of human error) hence,

d= 2.95∙10-10m ± 0.073∙10-10m

Erro Errorr ana analy ly lysi si siss ffor or th the e re resu su sults lts lts:: Information we were provided to assist with our resultsii:

Calliper associated error: ± 0.5mm

Erro Errorr calc calcul ul ulat at atio io ions ns ns::

Using the equation 𝛼 = 𝑡𝑎𝑛−1(𝐿

𝐶𝑜𝑠(𝜑) − 1 = 𝑏 and 𝜑 = 2𝑅 𝜎(𝐷) 2 ) 𝐷 𝛿sin(𝜑)

Then σφ= 𝜙√( σa=

𝛿𝜑

σb=√(

+(

𝜎(𝜑)

𝜕 cos(𝜑 ) 𝜕𝜑

𝐷

𝑅

sin(𝜑)

+𝐶𝑜𝑠(𝜑)−1

) (equation 2) Where sin(𝜑) = 𝑎 and + 𝑅

𝜎(𝑅) 𝑅

2

)2 𝐿 2

𝜎(𝜑)) + ( ) (( 𝑅

𝑎

𝜎(𝐿) 2 ) 𝐿

hence σα=√(𝑎2 +𝑏2 𝜎(𝑏))2 + (𝑎2 +𝑏2 )𝜎(𝑎))2 𝑏

𝐿

+(

𝜎(𝑅) 2 𝑅

) )

Which have been incorporated into the results tables and distance spacing calculations.

Discu Discussion ssion To rule out relativistic effects in our experiment we need to show that the electrons have not been accelerated to a velocity that leads to relativistic effects, for this we can use 𝑣 2

the momentum formula 𝑝 = 𝛾𝑚𝑣 where 𝛾 is the Lorentz factor 𝛾 = √1 − (𝑐 ) we know the 2𝐸

velocity 𝑣 = √ 𝑚 = √

2𝑞𝑉 𝑚

where q is the charge on an electron, q=1.602x10-19 C , V is the

voltage and m the mass of the electron, in this case we are considering a voltage of 5 keV, the maximum available voltage for use in our experiment. Hence we have:

𝑣=√

2𝑞𝑉 𝑚

=√

2(1.602x10−19 )(5000) = 9.109 x 10−31

4.19∙107 m∙s-1.

Using the afore-calculated velocity : 2

4.19x10 𝛾 = √1 − ( 2.998x108 ) =1.0099 ≃1, 7

Hence we can say that we need not take relativistic effects into consideration during this experiment. Now we can ensure a non-relativistic prediction about how α is related to the accelerating voltage. To make an estimate of the spacing between atoms in a piece of graphite we can take the fact that in 12 grams of carbon, there is 6.022∙1023 atoms and the density of graphite ρ 2.2 ρ=2.2 g∙cm-3 to find that the volume of 12 grams of carbon has 𝑉 = 𝑚 = = 0.183cm3= 12 1.83∙10-7m3 . We can now say that there are 6.022∙1023 atoms in 1.83∙10-7m3 of graphite. For a result relating the space between atoms, we can simply take the cube root of 𝑣𝑜𝑙𝑢𝑚𝑒 that is 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑎𝑡𝑜𝑚𝑠 𝑣𝑜𝑙𝑢𝑚𝑒

√ 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑎𝑡𝑜𝑚𝑠 = √ 6.022x1023 = 6.72∙10-11 m∙atom-1

3

3

1.83x10−7

or 6.72∙10-11 metres between atoms, which corresponds to a spacing of 0.067 nm, slightly smaller than the agreed upon spacing between atoms in graphiteiii, most likely due to the complex spacing between atoms, here we are assuming they are aligned in a perfectly cubic nature. Two possible path distances for atoms arranged in squares are D1 and D2 shown in the below diagram. From the simple geometry we can dee that D2 is given by the hypotenuse, while D1 is half the hypotenuse. Taking the simple case of d=1, the hypotenuse, D2= D2=√1 + 1 = √2 and D1 D1=1 =1 giving the ratio 𝐷2: 𝐷1 = √2: 1

d=1

Two possible path distances for atoms arranged hexagonally are D1 and D2 shown in the below diagram. Taking the simple case of d=1, we have D2= D2=tan tan tan(6 (6 (60°)= 0°)= 0°)=√𝟑 and D1 D1=1 =1 giving us the ratio 𝐷2: 𝐷1 = √3: 1

ΘE=60°

ΘI=120°

d=1

D1

D2

Using Voltage=2400V gave D1=0.033 and D2=0.059 giving the ratio D2:D1=0.059:0.033=1.78 which is just over √3, suggesting that the crystalline lattice in the sample used has a more complex structure than either of the two structures and ratios derived above.

Co Conc nc nclus lus lusion ion Our results gave a distance spacing between atoms of d= 2.95∙10-10m ± 0.073∙10-10m Which is 0.295nm this result is larger than that which we predicted using Avogadro’s number and the density of graphite but is closer to the agreed upon density of the material used.iv The errors included in the distance are a little over-stated due to consideration of human error in taking measurements, however two sets of data were taken to increase the accuracy of results and to make sure all measurements were consistent, which they were within a reasonable error margin. The diffraction pattern we witness as rings is because of the random crystalline structure of the graphite. There are many scattering paths for the electrons to take, some produce constructive interference and some produce deconstructive interference. If the path taken corresponds to an integer number of wavelengths then constructive interference is produced. The diffraction patterns we see are a superposition of the patterns produced by the randomly oriented atoms in the crystalline structure and hold true in accordance with all hypothesis and equations used throughout the experiment, this is also suggestive of the fact

that the spacing calculated is an average spacing based on the voltage range we used rather than the definite spacing between every atom in the sample. In conclusion it can be said that Einstein and de Broglie’s hypothesis held true; that the electrons behave both as particles and waves for all momenta values, and from this hypothesis and associated mathematical equations we could to within a degree of accuracy derive that the average distance spacing between atoms in the sample was 2.95∙10-10m ± 0.073∙10-10m.

i

Andrew Cole & David Hughes, 2017, KYA 211/374 Second Year Physics Laboratory manual Andrew Cole & David Hughes, 2017, KYA 211/374 Second Year Physics Laboratory manual iii Alice Warren-Gregory ,2001,The Physics Fact Book,”Distance Between Carbon Atoms”,http://hypertextbook.com/facts/2001/AliceWarrenGregory.shtml iv Alice Warren-Gregory ,2001,The Physics Fact Book,”Distance Between Carbon Atoms”,http://hypertextbook.com/facts/2001/AliceWarrenGregory.shtml ii...