Fundamental Constants

Laboratory & Computational Physics 2

Last compiled August 22, 2017

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Contents

1 Introduction 31.1 Prelab questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Experiment 1 - The Planck constant, ~ 52.1 Background theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Diffraction from a lamp . . . . . . . . . . . . . . . . . . . . . . . 62.2 The Planck constant - Equipment . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 The mercury vapour lamp . . . . . . . . . . . . . . . . . . . . . . 72.2.2 Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 The Planck constant - Procedure . . . . . . . . . . . . . . . . . . . . . . . 92.3.1 Measuring the stopping potential of a single spectral line . . . . . . 92.3.2 Stopping time ts . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3.3 Variation of stopping potential with frequency . . . . . . . . . . . . 10

3 Experiment 2 - The charge to mass ratio of the electron, e/me 123.1 Background theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 The charge to mass ratio of the electron - Equipment . . . . . . . . . . . . 13

3.2.1 The cathode ray tube . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 The charge to mass ratio of the electron - Procedure . . . . . . . . . . . . . 15

3.3.1 Equipment overview . . . . . . . . . . . . . . . . . . . . . . . . . 153.3.2 Measuring your radii . . . . . . . . . . . . . . . . . . . . . . . . . 153.3.3 Measuring e/m and plotting e/m directly . . . . . . . . . . . . . . 163.3.4 Calculating e/m using a field variable . . . . . . . . . . . . . . . . 163.3.5 A closer look at energy loss . . . . . . . . . . . . . . . . . . . . . 173.3.6 Combining your measured values to estimate other fundamental con-

stants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Useful data 18

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1 Introduction

Energy

Electron

Over the course of this experiment you will investigate two very important fundamentalconstants in physics. We are looking for experimentally determined values of both Planck’sconstant and the charge to mass ratio of the electron.

Planck’s constant is fundamental for many reasons. Expressing it most simply, Planck’sconstant relates the frequency of a photon to its energy. You may have also heard of thePlanck length and Planck time - these both involve Planck’s constant and define lower boundsfor space and time in our universe, respectively. We will be determining a value for Planck’sconstant using the photoelectric effect.

The charge to mass ratio of the electron meanwhile is extremely useful as the mass or thecharge of the electron independently can be quite difficult to determine. It was in fact theratio of the charge to mass of the electron that differentiated electrons (in the form of cathoderays) from charged atoms1.

1.1 Prelab questions

The Planck constant.

1. What are the accepted values for the Planck constant, Planck length, and Planck time?

2. How did classical theory fail to account for the observed features of the photoelectriceffect? A table may be useful.

3. Why did we decide to use a mercury lamp for our measurements? Why not a sodiumor neon lamp?

4. Platinum is the metal with the highest work function. What is its work function, and

1https://www.aip.org/history/electron/jj1897.htm

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what wavelength photon would be required to liberate an electron? Where in the elec-tromagnetic spectrum would this photon be located?

The charge to mass ratio of the electron.

1. What is the accepted value for the charge to mass ratio of the electron, e/me?

2. Draw and describe the motion of and forces acting on an electron travelling perpen-dicularly to a magnetic field.

3. How would the forces and motion change if protons were used instead of electrons?

4. Using the equations in the theory section determine a formula for the ratio e/me interms of the applied field strength, B, the radius of the circle, r, and the acceleratingpotential, V .

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2 Experiment 1 - The Planck constant, ~

2.1 Background theory

The Planck constant, (h, where ~ = h/2π) is used in many equations. Examples include theuncertainty principle,

∆x∆p ≥ ~2

(1)

the rest mass of the electron,

me =2R∞h

c0α2(2)

the Planck time,

tP =

√~Gc5

(3)

the quanta of magnetic flux in a superconducting loop,

Φ = nh

2e(4)

and so on... It’s very important to know the Planck constant extremely accurately. Ourmethod, while perhaps not the exceedingly most accurate, does also let us explore anotherimportant concept in physics: The photoelectric effect.

The wave theory of light was the prevailing understanding of light throughout the 19th cen-tury. However, in 1887, Heinrich Hertz noticed that a spark induced in a circuit was strongerwhen he deliberately illuminated the detector with UV light. This was the first observationof the liberation of electrons from the clean surfaces of metals under the action of radiantenergy. This is known as the photoelectric effect. His colleague, Philipp Lenard, later con-firmed that the carriers emitted from the surface were negatively charged. But it wasn’t untilEinstein’s revolutionary postulate in 1905 that all the observed features of the photoelectriceffect were reconciled with mainstream theory.

The failure of classical theory and Einstein’s postulatesExperiment showed that electrons ejected from the surface had small but finite speeds rang-ing from zero to some maximum value. By making the collecting plate negatively chargedwith respect to the illuminated plate, they could measure the force required to stop the mostenergetic of electrons. This is how the stopping potential, V0, is defined.

Einstein extended the quantum theory of Planck to the radiation field itself, hypothesisingthat the light existed in quanta of energy, with magnitude proportional to the Planck constantmultiplied by the light’s frequency, or:

E = hν. (5)

But we can’t directly use this to find h just yet... The mechanism of the photoelectric effect isas follows: an electron absorbs a photon of energy hν and attains enough energy to escape thesurface. Energy not used to overcome the binding of the electron to the atom becomes kineticenergy. Thus we can write the following, known as Einstein’s photoelectric equation:

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Ek = eVs = hν − φ0 (6)

where Ek is the kinetic energy of the photoelectrons and φ0 is the work function particular tothe material that is illuminated. The work function is the energy required for the electron toescape the surface.

Question 1 If we plot a graph of stopping voltage versus light frequency, what physicalproperties be the gradient and intercept?

You should from this have an inkling as to how we’re going to determine a value for thePlanck constant.

2.1.1 Diffraction from a lamp

You should notice the frequency, ν in the equation above. We want to look at specific fre-quencies of light in dealing with this equation. We can’t use white light, for example, asit contains all frequencies in a smooth spectrum. We could use an elemental lamp - say,sodium - but this would still contain more than one frequency. What we need to introduce isa diffraction grating.

Light from a lamp will be dispersed into its constituent colours using a transmission diffrac-tion grating. The grating equation is

a sin θm = mλ (7)

where a is the distance between the ruled grooves, m is the order of interference and θm isthe angle through which the order is deflected. Note that we should see the spectral linesrepeat as the order (m in the grating equation) increases. A diagram illustrates this below.

a

Diffractionangle

θb

1st order (m=1)

0th order (m=0)

1st order (m=-1)

Incidentlight

Transmissiongrating

Figure 1: The transmission diffraction grating with the orders of interference shown.

This means that different frequencies/colours will be spread out at different angles, allowingus to choose specific, identifiable frequencies to have incident on our detector.

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2.2 The Planck constant - Equipment

A representation of the apparatus is given in figure 2. Note that the light source and detectorare connected - you’ll have to be careful in aligning the system. Before you begin, becomefamiliar with the various components labelled.

Removablelight shield

Whitereflective mask

Transmissiongrating and lens

Mercury lamp

Figure 2: The apparatus for the observation of the photoelectric effect.

2.2.1 The mercury vapour lamp

The spectral lamp you will use is a high intensity (100 W) mercury vapour lamp. Atoms areionised by passing a current through the mercury vapour. The recombination of electronsand mercury ions produces a light spectrum composed of discrete wavelengths.

The diffraction gratingTwo important things to note about the image you will see coming from the lamp throughthe diffraction grating:

• You will see the colours repeat in the second (and possibly third) order groups outfrom the central maximum. Does it matter which order you choose?

• The blue colours can be hard to tell apart. You’ll need to stop and think about howyou’ll tell the lines apart and which colour corresponds to which frequency. Diagramscan be useful for labelling which line you thought was each colour, so you can go backand re-assign frequencies later if you weren’t sure to begin with.

The photoelectric cell

You are supplied with a photoelectric cell for measuring the stopping potential. A schematicof the photoelectric cell is shown in figure 3. Monochromatic light from the mercury lampstrikes the surface of the metal, and if the photons have sufficient energy they will liberateelectrons from the metal surface, which will then be collected by the detector. Each electroncollected will increase the overall negative charge at the detector, and hence make it harderfor electrons to jump the gap. Eventually, the number of electrons will reach a maximum.

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hf Ek

light electrons

metal

detector

ϕ

Figure 3: Light is incident on the metal surface, which liberates electrons, which are thencollected by the detector.

This will take a finite amount of time (called the stopping time, ts) and will have an associatedstopping voltage Vs (as in equation 6). In this experiment, both Vs and ts will be measured.The red discharge button on the photoelectric cell grounds the detector plate, ready for anew measurement.

2.2.2 Filters

Three filters are required for this experiment:

• A filter with 5 strips of various transmission strengths (from 20% to 100%, in steps of20%)

• A yellow filter to prevent ambient light sources (UV from overhead fluorescent lightsand also the violet line from the 3rd order spectrum) from interfering with measure-ments of the stopping potential for the yellow line

• A green filter for use with the green line for similar reasons to those cited above.

• Note that there isn’t a blue filter. Can you think why this is?

The filters have magnetic strips for attachment to the photocell, but you may need blutac aswell.

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2.3 The Planck constant - Procedure

2.3.1 Measuring the stopping potential of a single spectral line

First, switch the lamp on and allow it to warm up for ten minutes.

While it’s warming up, perform a battery voltage check of the detector. The photocell willnot function properly if the battery voltage falls below 6 volts. To check the batteries, usea voltmeter to measure between the OUTPUT ground terminal and each BATTERY TESTterminal (-6V MIN and +6V MIN). If the voltage is too low, ask your demonstrator to changethe battery.

Once you can see the spectral lines on the wall, move the lens and grating back and forth toproduce a nice and sharp image.

Question 2 Observe the diffracted light against a sheet of white paper. What do you ob-serve? How many orders? Are both sides equally bright?

Question 3 Look up the precise values of the spectral frequencies for a mercury lamp.Which frequencies do you observe here?

1. Turn the photocell cover away and focus a spectral line of the 1st order onto the whitereflective mask of the photocell. The mask is made of a fluorescent material that allowsyou to see the ultraviolet line as blue. It also makes the violet line appear more blue.

2. Align the light directly on to the black squares on the photocell. These are the detectorapertures and your results will vary if you aren’t consistent in alignment.

3. Place the transmission filter on the front of the photocell to allow 100% of the light toenter the photocell. If you have chosen the yellow or the green spectral lines rememberto use the appropriate filters.

4. Connect the voltmeter to the output of the photocell and switch the photocell on.

5. Zero the photocell by pushing the red discharge button. During this zeroing processthe anode voltage will reach zero but the output of the built-in amplifier will float andtherefore the output voltage will fluctuate.

6. Note which spectral line you have chosen and then record the stopping potential. Re-member to wait a minute or so for the end-point to be reached.

It is quite important to draw diagrams of which lines you’re choosing for any given measure-ment. Don’t just write the frequency as you may be mistaken!

Question 4 How important is it that we minimise the effects of other light sources? Do younotice an effect on your results with lights on?

2.3.2 Stopping time ts

Okay, so you know how to measure the stopping potential. Soon, you will repeat this processfor different frequencies to determine your value for the Planck constant. But while we have

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this equipment, let’s take a brief detour and look at determining the stopping time, ts, thetime it takes for the detector to reach its maximum value.

1. For a chosen spectral line measure the approximate time it takes for the photocell toreach the stopping potential at 100% transmission.

2. You will note it’s difficult and time consuming to determine exactly when it hits 100%.Calculate a value that is 85% of the stopping potential, so that you can collect data forthe different intensity filters without taking several minutes on each.

3. Repeat the measurement procedure at 85% for all of the values of the transmissionfilter. Make sure to take 2-3 measurements of each, and estimate errors.

4. Perform the same procedure with one other spectral line with all transmission filters.

5. Graph the stopping time vs. percentage transmission (choose your axes carefully) andcomment on your findings.

6. Estimate the times you would measure for 500%, 75%, 25%, 1% and 0% transmission.What does this tell you about your plot?

Question 5 Is it necessary to measure the stopping potential for all transmission filters?Why/why not?

Okay, so you should be convinced now that the intensity affects the stopping time, but notstopping voltage, and we can disregard the transmission filter for the rest of the experiment.

2.3.3 Variation of stopping potential with frequency

Now let’s look at the stopping potential with different frequencies, and use these values todetermine a value for both Planck’s constant, h, and, incidentally, the work function, φ0.

1. Make a note of how many different colours you can see on each side in each mode.Draw a diagram. You should be able to see at least to second order.

2. Before you begin measuring, consider also the effects of other light sources, for in-stance, overhead lighting.

3. Remove the transmission filter.

4. Using appropriate colour filters, measure the stopping potential of each colour in asingle order of your choosing.

5. Now perform the same measurements on two other orders: one of a different number(1 or 2) and one of a different sign (opposite side of zeroth order).

6. For each order, fit your data with Einstein’s photoelectric equation. What are you plot-ting against what? Use the values you looked up for the spectral colour frequencies.

7. Determine h/e and φ0 from the parameters of each fit.

Question 6 Do the second order set of spectral lines give a different result for the Planckconstant? Are there any differences between the first and second order lines?

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Question 7 Which varies more - your value for h or your value for φ0? Why? How couldyou reduce these variations?

8. Compare your result with the accepted value of h or h/e (what is the difference?)

9. How do the values from each of the orders compare?

10. Now take the average of each frequency and re-plot your results. Does your resultimprove? Why/why not?

Question 8 How would you change this experiment to improve your value for Planck’s con-stant? e.g.: Would you use a different metal? A different light source?

Summarise what you’ve learnt and note your value for the Planck constant (with errors!)and start preparing for the second part of this experiment, the charge to mass ratio of theelectron.

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3 Experiment 2 - The charge to mass ratio of theelectron, e/me

3.1 Background theory

In 1833 Faraday demonstrated the quantisation of charge through a careful series of elec-trolysis experiments. In 1897 Thomson ended a heated debate among physicists when hewas able to identify the cathode rays seen in low pressure gas discharges as caused by neg-atively charged particles. These particles, electrons, were accepted as being fundamentalconstituents of matter (we’d refer to them as point particles), firmly banishing the notionthat the atom was indivisible and fundamental.

Thomson was able to measure the charge to mass ratio of this new particle with an experimentthat made use of the same principles as the one you are about to perform. The value of e/me

that Thomson determined was 1.0 × 1011 C/kg, somewhat close to the currently acceptedvalue. Since the properties of the electron depend almost solely on these two quantities weseek an accurate determination of their value. It would be of interest to compare the valueof e/me that you calculate with Thomson’s first effort!

If a beam of electrons is accelerated through a known potential (between charged plates, inour case) then their kinetic energy is given by the expression

Ek =1

2mev

2 = eV (8)

where me is the mass of the electron, e is the charge of the electron and V is the acceleratingpotential.

If a magnetic field is then applied perpendicular to the velocity of the electrons, they willexperience a Lorentz force mutually perpendicular to both the field, B, and the electrondirection of motion, v. The magnitude of this Lorentz force is

F = evB. (9)

This force is always perpendicular to the electron direction of motion, so the electrons movein a circle. By travelling in a circle, the electrons experience a centripetal force, a forcepointing towards the centre of the circle, with a magnitude given by

F =mev

2

r(10)

where r is the radius of the circle described by the electrons’ path.

Using these equations you are able to derive a formula for the e/me ratio, as instructed in theprelab questions. It is this equation, e/me, you will be investigating in this experiment.

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3.2 The charge to mass ratio of the electron - Equipment

Deflectingvoltage

Coil currentVoltagecontrol

+ve

-ve

CW

CCWV A

Helmholtz coils

Ruler

Electron gun

Cathode ray tube(glass bulb)

Electron beam

Figure 4: The apparatus used to determine e/me. Note that you should use the multimetersto measure the accelerating voltage and coil current, not the in-built displays.

Helmholtz coils

The magnetic field for this experiment is generated by a pair of Helmholtz coils with thefield generated at a volume in their centre given by

B =

(4

5

)3/2Nµ0I

a(11)

where

• N is the number of turns in each coil, 140

• µ0 = 4π × 10−7 T m A−1 is the permeability of free space,

• I is the current through the coils and

• a is the radius of the coils, 150 mm.

We use two Helmoholtz coils to produce a region of uniform strength between the coils. Ifwe used a single coil the strength would vary with distance to the coil and equation 11 wouldbecome dependent on r, for something likeB(r) ∝ 1/r2, with the field not filling the volumeuniformly perpendicularly to the electron path.

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3.2.1 The cathode ray tube

The cathode ray tube is a 150 mm diameter helium-filled glass bulb. This contains at itsbase the electron gun consisting of a heated cathode which produces electrons that are thenaccelerated towards the anode. The anode is made of a wire grill so that the electrons maypass through it and enter the body of the tube. The helium gas is retained at a pressureof 10−2 Torr and is ionised by the electrons as they pass through it. As the helium ionsrecombine with their valence electrons to form atoms, they emit light, allowing the radius ofthe electron beam to be measured.

It is at this point you might ask: what wavelength does the light we see correspond to?

Question 9 Look up helium spectral lines and convert the colour you see into an energy.What does this energy correspond to? A helium energy level or something else?

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3.3 The charge to mass ratio of the electron - Procedure

3.3.1 Equipment overview

1. With everything turned off, identify each component of the equipment. Find the con-trols for both the Helmholtz coils and accelerating voltage, and examine as best youcan the apparatus inside the bulb.

2. Identify all of the dials and switches on the front of the e/m coil apparatus. Make sureyou know what each of these connections controls or what they output.

3. You will control the Helmholtz current during the experiment using the ‘coil current’dial on the e/m apparatus.

4. Turn this dial to the right before turning on the accelerating potential (voltage) to theelectron gun: this ensures a magnetic field is present and the electron beam is notincident on the glass.

5. Now turn on the power supply to the electron gun. Set the voltage to around 200 Vwhile you wait for the gun to heat up for a couple of minutes.

6. When the beam is visible, bring the accelerating voltage up to 300 V and try to form acircle. You will need to change the coil current, too.

7. If you can’t see any glow you may need to turn the lights off or block out light moreeffectively. In which direction are the electrons being fired?

Question 10 What effect does changing the direction of the deflecting voltage have? Shouldyou use it during your experiment?

3.3.2 Measuring your radii

It is now very important you know how to take measurements accurately using this equip-ment. Your goal is to maintain the beam radius at a constant value while you changeother variables. Note: you should choose one radius (left or right) to be the one you keepconstant. The other will vary, do not adjust for this, just keep one side constant throughoutyour measurements. Choose your side carefully!

Question 11 Why are you attempting to keep the radius constant rather than other vari-ables? Consider both the measurement itself and the equations you’re using to describe thephysics.

1. Use the dials to form a circle of electrons with its centre in the middle of the ruler. Itis up to you to determine the accelerating voltage and coil current, but do not exceed500 V on the accelerating voltage.

2. What we want to do is avoid parallax in our measurements.

3. Move your head in line with the radial point you want to measure. Do not measurewith your head in the centre of the circle!

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Question 12 What is your value for the right radius, given the electron beam circle shouldbe centred on the middle of the ruler?

Question 13 Draw a diagram of how you used the scale to record the radius of the beam.Make it as clear as possible!

When you are convinced your electron beam is properly centred, write down the Helmholtzcurrent, the accelerating voltage and both the left and right radii of the circle. Have yourpartner perform this measurement too and discuss errors before continuing.

3.3.3 Measuring e/m and plotting e/m directly

1. From your starting voltage, decrease the accelerating voltage by an appropriate stepsize.

2. You should notice a change in your electon beam radius. In response to this, adjust thecoil current to return the radius to your earlier selected value.

3. As before, record the coil current, the accelerating voltage and the left and right radiiof the circle.

4. Make sure you note the methods you are using to ensure accurate results. Are youmaking sure you’re consistent in your measurements?

5. Keep stepping down in accelerating voltage until you have a reasonable number ofpoints, about ten.

6. If you can’t find a reasonable number of points by stepping down in voltage, try pointsslightly higher than your original value.

7. Enter your data into excel and calculate e/me (using your answer to prelab question3) for each accelerating voltage. Plot your values of e/me against the acceleratingvoltage V for both the left and right radii.

8. Comment on and explain the shape of these graphs.

Question 14 What differences did you notice between the two radii measurements? Was onesize more consistent, did they both decrease at the same rate? Explain why by referring tothe equipment.

3.3.4 Calculating e/m using a field variable

An alternative method for determining the value of e/me is to calculate the field variable, χ.χ is a measure of the energy loss of the electrons due to their interactions with the gas.

We may fit the function

V =

(e

me

× 10−11)χ+ δ (12)

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where e/me and δ are parameters and V and χ = [(Br)2/2]×1011 are variables. This allowsa unique value of e/me to be determined while allowing for the effective potential loss dueto the presence of the gas with the parameter δ.

1. Plot V against χ (using your values for B) on another graph and overlay a line of bestfit.

2. Determine a value for e/me and compare it to your value from the first measurementmethod and the accepted value.

3. Discuss your methods for obtaining a value for e/me and comment on which methodyou think is more rigorous. Keep in mind a more accurate result in one case doesn’tnecessarily mean that method is more rigorous.

Question 15 What physical quantities are represented by the gradient and axes intercepts?

Question 16 The ionisation energy of He is 24.58 eV. How many electron-He collisionsoccur during the electron path through the tube on average?

3.3.5 A closer look at energy loss

Let’s turn back to the actual setup of the experiment for a moment. We began by talking aboutthe Lorentz force felt by electrons in a magnetic field, specifically including the acceleratingpotential, V . We set this potential at the base of the tube - so the start of the circle. But wemeasured the circle radius further along the beam. Is our value for the accelerating voltagethus inaccurate?

What then could we change in our equations to better model the system we have?

Try to estimate how much energy is lost between the left and right radii. Given that this is180◦ of the circle, can you estimate how much energy is lost per degree of the circle? Usingjust these two values, you’re only able to assume a linear decrease in energy with angle, thatis, Eloss ∝ θ, not θ2 or other powers.

But, we can check (roughly) whether the function is simply linear. Using your known valuesfor the initial accelerating potential, and the energy loss per degree you just obtained, can youestimate values for your left and right radii? Do these estimates match with your recordedvalues? What does this tell you?

3.3.6 Combining your measured values to estimate other fundamentalconstants

The third fundamental constant is introduced in the electron spin resonance experiment andis the Bohr magneton

µb = e~/2me (13)

The value of this constant has been measured as (9.274096± 0.000065)× 10−24 J T−1.

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Now, use your results for h/e, e/m, and the result for the Bohr magneton cited, to calculatevalues for

• h,

• me

• and e.

Determine errors for each of these quantities so that you can effect a meaningful comparisonwith the accepted values of these constants.

4 Useful data

Quantity Valueh 6.6261 ×10−34 J se 1.6022 ×10−19 Cme 9.1096× 10−34 kge/m (1.7588028± 0.0000054)× 1011 C kg−1

h/e (4.135708± 0.000014)× 10−15 J s C−1

µ0 4π × 10−7 T m A−1

µb (9.274096± 0.000065)× 10−24 J T−1

He ionisation, IHe 24.58 e V

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