Understanding Light Theory: Wave-Particle Duality and Matter Wave Discovery, Lecture notes of Physics

Download Understanding Light Theory: Wave-Particle Duality and Matter Wave Discovery and more Lecture notes Physics in PDF only on Docsity! 1 Teaching guide: Turning points in physics This teaching guide aims to provide background material for teachers preparing students for the Turning Points in physics option of our A-level Physics specification (7408). It gives teachers more detail on specification topics they may not be familiar with and should be used alongside the specification. This guide is not designed to be used as a comprehensive set of teaching notes. Contents Page Introduction 2 Section 1 The discovery of the electron (specification reference 3.12.1) 3 a) Cathode rays 3 b) The specific charge of the electron 5 c) Millikan’s determination of the charge of the electron 9 Section 2 Wave particle duality (specification reference 3.12.2) 10 a) Theories of light 10 b) Matter waves 17 c) Electron microscopes 18 i) the transmission electron microscope 18 ii) the scanning tunneling microscope 20 Section 3 Special relativity (specification reference 3.12.3) 22 a) Frames of reference 23 b) The Michelson-Morley experiment 23 c) Time dilation and proper time 25 d) Length contraction and proper length 27 e) Evidence for time dilation and length contraction 28 f) Mass and energy 30 Appendix 34 A Suggested experiments and demonstrations 34 2 Introduction Turning Points in physics is intended to enable key developments in physics to be studied in depth so that students can appreciate, from a historical viewpoint, the significance of major conceptual shifts in the subject, both in terms of the understanding of the subject and in terms of its experimental basis. Each of the three sections within the course represents a different approach to the subject. The discovery of the electron and the determination of e/m and e took the subject forward through important experimental work by Thomson and Millikan. Further work on electron beams led to support for relativity and the discovery of electron diffraction. Recent 'Millikan' experiments have unsuccessfully sought evidence for fractional charge. The section on wave particle duality follows the changes in the theory of light arising from experimental discoveries over two centuries from Newton to Einstein, and links up to the prediction and discovery of matter waves. Recent important developments in electron microscopy illustrate the unpredicted benefits of fundamental research. Einstein's theory of special relativity overthrew the concept of absolute motion. The null result of the Michelson-Morley experiment presented a challenge to accepted ideas of space and time which was not solved until Einstein put forward his revolutionary theory of special relativity. Each section offers opportunities to consider the relationship between experimental and theoretical physics. The nature of scientific proof can be developed in each section in line with the philosophy of science; no amount of experimentation can ever prove a theory fully, yet one experiment can be sufficient to overthrow it. 5 Since the work done on each electron increases its kinetic energy from a negligible value at the cathode, then the speed, v, of each electron leaving the anode is given by 1 2 𝑚𝑣2 = 𝑒𝑉 For the above equation to apply, the speed of the electrons must be much less than the speed of light in free space, c. Students will not be expected in this section to use the relativistic expression for kinetic energy. b) The specific charge of the electron Students should know what is meant by the specific charge of the electron and should be able to understand the principles underlying its determination through the use of the relevant equations in the Data and Formulae Booklet, namely: 𝐹 = 𝑒𝑉 𝑑 𝐹 = 𝐵𝑒𝑣 𝑟 = 𝑚𝑣 𝐵𝑒 1 2 𝑚𝑣2 = 𝑒𝑉 As outlined below, given a diagram and description of suitable apparatus and sufficient relevant data, they should be able to use one of more of the above equations to calculate the speed of the electrons in a beam and/or determine the specific charge of the electron from the given data. They should also be able to describe one method, including the data to be collected, to determine the specific charge of the electron. They should appreciate why electron tubes in general need to be evacuated and why the tube in Figure 3 needs to contain a gas at low pressure. They should also appreciate that the specific charge of the electron was found by Thomson to be much larger than the previous largest specific charge, namely that of the hydrogen ion. (i) Using a magnetic field to deflect the beam (a) The radius of curvature r of the beam in a uniform magnetic field of flux density B may be measured using the arrangement shown in Figure 3 or using a ‘Teltron tube’ arrangement in which a straight beam enters the field and is deflected on a circular arc by a magnetic field. Details of how to measure the radius of curvature in each case are not required. 6 Figure 3 Deflection by a magnetic field Because the magnetic force on each electron (=Bev) provides the centripetal force (= mv2/r), then Bev = mv 2 /r which gives r = mv/Be. To determine e/m, the anode pd VA must also be measured. By combining r = mv/Be and 1 2 𝑚𝑣2 = eVA (where VA is the anode pd), the following equation for e/m is obtained 22 A 2 = rB V m e Note students should be able to explain why no work is done on the electrons by the magnetic field. (ii) Using a magnetic field to deflect the beam and an electric field to balance the deflection a) The radius of curvature r of the beam in a uniform magnetic field of flux density B is measured. As before, r = mv/Be from Bev = mv 2 /r. b) The speed, v, of the electrons is measured directly using an electric field E that is perpendicular to the beam and to the magnetic field B. This 7 arrangement of the fields is referred to as ‘crossed fields’. When the beam is straightened out (ie undeflected), the forces due to the crossed fields are balanced. Figure 4 Balanced forces due to crossed E and B fields As the magnetic force (Bev) on each electron is equal and opposite to the electric force (eE), the speed of the electrons passing through undeflected is given by the equation v = E/B from eE = Bev. Note that E = pd between the deflecting plates plate separation The measured values of v, B and r can then be substituted into the equation r = mv/Be and the value of e/m calculated. 10 (ii) A droplet falling vertically with no electric field present Figure 7 An oil droplet falling at terminal speed The droplet falls at constant speed because the drag force on it acts vertically upwards and is equal and opposite to its weight. Using Stokes’ Law for the drag force FD = 6πηrv therefore gives 6πηrv = mg. Assuming the droplet is spherical, its volume = 4 r 3 /3 hence its mass m = its density  × its volume = 4   r 3 /3. Hence 6πηrv = 4   r 3 /3 which gives r 2 = gρ vη 2 9 . Thus the radius can be calculated if the values of the speed v, the oil density  and the viscosity of air  are known. Section 2 Wave particle duality a) Theories of light Students should be able to use Newton’s corpuscular theory to explain reflection and refraction in terms of the velocity or momentum components of the corpuscles parallel and perpendicular to the reflecting surface or the refractive boundary. They should be able to explain reflection and refraction using wave theory in outline. Proof of Snell’s law or the law of reflection is not expected. Newton’s ideas about refraction may be demonstrated by rolling a marble down and across an inclined board which has a horizontal boundary where the incline becomes steeper. They should know why Newton’s theory was preferred to Huygens’ theory. They should also be able to describe and explain Young’s fringes using Huygens’ wave theory and recognise that interference cannot be explained using Newton’s theory of light as it predicts the formation of two fringes corresponding to the two slits. In addition, they should know that Huygens explained refraction by assuming that light travels slower in a transparent substance than in air, in contrast with Newton’s assumption that its speed is faster in a transparent substance. They should appreciate that Newton’s theory of light was only rejected in favour of wave theory long after Young’s discovery of interference when the speed of light in water was measured and found to be less than the speed in air 11 Measuring the speed of light. Early scientists thought that light travelled at an infinite speed. However, the earliest reliable experiments involving astronomical observations (Römer, 1676) suggested that light had a finite speed. In 1849, Fizeau obtained a value that differed by only 5% from that now accepted using a terrestrial method. The arrangement used is shown in Figure 8. Figure 8 Fizeau’s experiment for determining speed of light The principle was to determine the time taken for a beam of light to travel to a mirror and back to the observer. The time was measured using a toothed wheel that was rotated at high speed. Pulses of light were transmitted through the gaps in the wheel. At low speeds of rotation, light from the source passed through a gap and then passed through the same gap on its return so the observer could see the light. As the speed of rotation increased there came a time when the returning beam found its path blocked by the adjacent tooth. As this was the same for all the gaps the observer did not now see any reflected beam. For example, no light would have been seen when light that passes through gap 0 finds its path blocked by tooth a on return, light passing through gap 1 would be blocked by tooth b and so on. When the speed was doubled, the light passing through 0 could pass though gap 1 on return so the reflected beam was once again observable. 12 If there are n teeth and n gaps then a tooth replaces a gap after 1 2n of a revolution. If the frequency is f then the time for 1 revolution is 1 f and a tooth replaces a gap after 1 2nf seconds. If the distance from F to M is d the speed of light is given by 2d 1 2nf =4dnf In Fizeau’s arrangement, the distance from the point of origin of the light F to the mirror M was 8.6 km. Fizeau’s wheel had 720 teeth and 720 gaps. He found that the first time that the light disappeared was at a speed of 12.6 revolutions per second. This gave the speed of light to be 3.13 × 10 8 m s -1 . Students are expected to understand the physics principles that underlie the Fizeau experiment, including how the equation for the speed of light, c = 4dnf, is derived. Further experiments on the speed of light Fizeau and other contemporary scientists refined the experiments and showed the speed of light in water to be lower than that in air, which disproved Newton’s corpuscular theory. They also conducted experiments using interference effects to measure the speed in moving water and found that light travelled quicker when emitted in the direction of flow than against the flow. This was known as the ‘ether(aether) drift’. However, they found that the speed was lower than the expected simple sum of the speed of light in water plus the speed of the fluid which they could not explain. Their values were later found to be consistent with those predicted by the addition of speeds using Einstein’s theory of relativity and was supporting evidence for the theory. Maxwell and Hertz In considering Maxwell’s theory of electromagnetic waves, students should recognise that Maxwell predicted electromagnetic waves in terms of oscillating electric and magnetic fields before there was any experimental evidence for electromagnetic waves. In addition to being able to describe the nature of an electromagnetic wave, students should know that Maxwell derived the equation 𝑐 = 1 √0𝜀0 for their speed and used it to show that the speed of an electromagnetic wave in a vacuum is the same as the speed of light in free space. In this way, he showed that light is an electromagnetic wave and infra- red and ultraviolet radiations beyond the visible spectrum are also electromagnetic waves. The subsequent later separate discoveries of X-rays and of radio waves confirmed the correctness of Maxwell’s predictions. Maxwell’s 15 differences between the photon theory and corpuscular theory. They should know that metals emit electrons when supplied with sufficient energy and that thermionic emission involves supplying the required energy by heating the metal, whereas photoelectric emission involves supplying energy by illuminating the metal with light above a certain frequency. Photoelectric emission was first discovered by Hertz when he was investigating radio waves using a spark gap detector. He observed that the sparks were much stronger when ultraviolet radiation was directed at the spark gap contacts. Investigations showed that for any given metal:  photoelectric emission does not occur if the frequency of the incident light is below a threshold frequency  photoelectric emission occurs at the instant that light of a suitably high frequency is incident on the metal surface  the photoelectrons have a range of kinetic energies from zero up to a maximum value that depends on the type of metal and the frequency of the incident light. The number of photoelectrons emitted from the metal surface per second is proportional to the intensity of the incident radiation (ie the light energy per second incident on the surface). Students should know why the wave theory of light fails to explain the threshold frequency and the instantaneous emissions and they should be able to use the photon theory to explain these observations. According to wave theory, light of any frequency should cause photoelectric emission. Wave theory predicted that the lower the frequency of the light, the longer the time taken by electrons in the metal to gain sufficient kinetic energy to escape from the metal. So the wave theory could not account for the existence of the threshold frequency and it could not explain the instant emission of photoelectrons or their maximum kinetic energy. To explain the existence of a threshold frequency of light for each metal, students should know that in order for a conduction electron to escape, it needs to absorb a single photon and thereby gain energy hf and that the electron uses energy equal to the work function  of the metal to escape. They should be able to explain and use the equation EK = hf -  and they should be able to explain why the threshold frequency of the incident radiation, f0 = /h. They should also be aware that the mean kinetic energy of a conduction electron in a metal at room temperature is negligible compared with the work function of the metal so that the electron can only escape if the energy it gains from a photon is greater than or equal to the work function of the metal. Students should know that photoelectrons need to do extra work to move away from the metal surface if it is positively charged (relative to a collecting electrode) and that the number of photoelectrons emitted per second decreases as the potential of the metal is made increasingly positive. They should know that at a certain potential, referred to as the stopping potential, VS, photoelectric 16 emission is stopped because the maximum kinetic energy has been reduced to zero and they should be able to recall and explain:  why the stopping potential is given by eVS = hf -   why the graph of VS against f is a straight line with a gradient and intercepts as shown in Figure 10. Figure 10 Stopping potential v frequency They should appreciate how the stopping potential may be measured using a potential divider and a photocell and how the measurements may be plotted to enable the value of h and the value of  to be determined. The first measurements were obtained by RA Millikan and gave results and a graph as shown in Figure 10 above, that confirmed the correctness of Einstein’s explanation and thus confirmed Einstein’s photon theory of light. Einstein thus showed that light consists of photons which are wavepackets of electromagnetic radiation, each carrying energy hf, where f is the frequency of the radiation. Einstein was awarded the 1921 Nobel Prize for physics for the photon theory of light which he put forward in 1905, although it was not confirmed experimentally until 10 years later. Students should know that the photon is the least quantity or ‘quantum’ of electromagnetic radiation and may be considered as a massless particle. It has a dual ‘wave particle’ nature in that its particle-like nature is observed in the photoelectric effect and its wave-like nature is observed in diffraction and interference experiments such as Young’s double slits experiment. 17 b) Matter waves Students should know from their AS course that de Broglie put forward the hypothesis that all matter particles have a wave-like nature as well as a particle- like nature and that the particle momentum mv is linked to its wavelength by the equation: m v ×  = h where h is the Planck constant. De Broglie arrived at this equation after successfully explaining one of the laws of thermal radiation by using the idea of photons as ‘atoms of light’. Although photons are massless, in his explanation he supposed a photon of energy hf to have an equivalent mass m given by mc 2 = hf and therefore a momentum mc = hf/c = h/ where  is its wavelength. De Broglie’s theory of matter waves and equation ‘momentum × wavelength = h’ remained a hypothesis for several years until the experimental discovery that electrons in a beam were diffracted when they pass through a very thin metal foil. Figure 11 shows an arrangement. Figure 11 Diffraction of electrons Photographs of the diffraction pattern showed concentric rings, similar to those obtained using X-rays. Since X-ray diffraction was already a well-established experimental technique for investigating crystal structures, it was realised that similar observations with electrons instead of X-rays meant that electrons can also be diffracted and therefore they have a wave-like nature. So de Broglie’s hypothesis was thus confirmed by experiment. Matter particles do have a wave- like nature. The correctness of de Broglie’s equation was also confirmed as the angles of diffraction were observed to increase (or decrease) when the speed of the electrons was decreased (or increased). 20 They should know that the amount of detail in the image is determined by the resolving power which increases as the wavelength of the electrons decreases. They should also know that the wavelength becomes smaller at higher electron speeds so that raising the anode potential in the microscope gives a more detailed image. In addition, they should know that the amount of detail possible is limited by lens aberrations (because the lenses are unable to focus electrons from each point on the sample to a point on the screen since some electrons are moving slightly faster than others) and sample thickness (because the passage of electrons through the sample causes a slight loss of speed of the electrons which means that their wavelength is slightly increased, thus reducing the detail of the image). For a given anode potential, as outlined earlier, students should be able to calculate the de Broglie wavelength of the electrons using the de Broglie equation ) 2 ( = m e V h λ ; for example, the electrons in the beam of a TEM operating at 80 kV would have a de Broglie wavelength of about 0.004 nm. In addition, students should appreciate that:  in theory, electrons of such a small wavelength ought to be able to resolve atoms less than 0.1 nm in diameter  in practice in most TEMs, electrons of such a small wavelength do not resolve such small objects for the reasons outlined above. Note, the TEAM 0.5 electron microscope at the US Lawrence Berkeley Laboratory is the most powerful electron microscope in the world. Aberration correctors developed at the laboratory are fitted in the 80 kV microscope, enabling individual atoms to be seen. ii) The scanning tunneling microscope The scanning tunneling microscope (STM), invented in 1981, gives images of individual rows of atoms. Students should know that the STM is based on a fine- tipped probe that scans across a small area of a surface and that the probe's scanning movement is controlled to within 0.001 nm by piezoelectric transducers. They should be aware that the probe is at a small constant potential, with the tip held at a fixed height of no more than 1 nm above the surface so that electrons 'tunnel' across the gap. They should also know that if the probe tip moves near a raised atom or across a dip in the surface, the tunneling current increases or decreases respectively due to a respective decrease or increase of the gap width. They should also know that:  in constant height mode, the change of current is used to generate an image of the surface provided the probe’s vertical position is unchanged 21  in constant current mode, the change of current is used to move the probe vertically upwards or downwards respectively until the gap width and the current is the same as before. The vertical resolution is of the order of 0.001 nm, much smaller than the size of the smallest atom. The principle of tunneling is based on the wave nature of particles. Light can be seen through a thin metal film because the amplitude of the light waves is not reduced to zero by the passage of the light in the film. In the same way, the amplitude of matter waves in a barrier does not become zero if the barrier is sufficiently narrow. This process is referred to as 'tunneling'. Figure 13A Quantum tunnelling The 'de Broglie wavelength' of an electron depends on its momentum, and equals about 1 nm for electrons in a metal at room temperature. Hence tunneling is possible for gaps of the order of 1 nm or so and the tunneling current is sensitive to changes of the gap width as little as 0.001 nm. 22 Figure 13B Scanning tunneling microscope An example of an atomic resolution STM image is shown below where the dimensions of this image are 10.5 × 7.1 nm. The image is of one particular reconstruction of a 111-V semiconductor surface, namely the (100) surface of indium antimonide (InSb), and shows pairs of Sb atoms (dimers) arranged in groups of three in a brickwall-like structure. This image is an example of the best spatial resolution achievable from this class of materials. Note that, although some of these groups are incomplete, each bright dot represents the position of a single Sb atom. Section 3 Special relativity The notes below are intended to indicate the depth of study expected of students. Proofs for the formulae in the specification are not required and will not be examined. 25 Another conclusion from the null result is that the speed of light was not affected by the Earth’s motion, (ie the speed of light is invariant in free space). This was not explained until 1905 when Einstein put forward the theory of Special Relativity. Students are expected to understand the physics of the interferometer and outline how it is used as a means of attempting to detect absolute motion. In addition, students should be able to explain the significance of the null result in terms of the invariance of the speed of light and Einstein’s theory of Special Relativity. c) Time dilation and proper time A consequence of the invariance of the speed of light in free space is that 'moving clocks run slow' or time runs slower when you are moving. Consider an observer sitting in a moving train with a clock which is used to time a light pulse reflected between two horizontal mirrors in the carriage, one directly above the other at distance L apart, as shown in Figure 15. The train is travelling along a track parallel to a platform with a second observer watching. 26 Figure 15 Time dilation The moving observer times how long a light pulse takes to travel from one mirror to the other mirror, and back again. Since the distance travelled by the light pulse is 2L, the time taken is 2L/c where c is the speed of light in free space. This is the proper time t0. The platform observer sees the light pulse travel a distance 2 s where s =          4 22 2 tv L . According to this observer, the light pulse takes time t which is greater than t0 because the pulse travels a greater distance. Distance travelled, s, is given by s = ct = 2          4 22 2 tv L Rearranging this equation gives t = 2 1 )( 2 22 vc L  = t0 2 1 2 2 1         c v The observer in the train measures the proper time, t0, since they are stationary with respect to the light clock. The time interval, t, according to the platform observer is greater than the proper time t0. Time runs more slowly for the moving observer. 27 d) Length contraction and proper length Having seen that moving clocks run slow, another consequence of special relativity is that the length of a moving rod is found to be different when measured by an observer moving parallel to the rod and an observer at rest relative to the rod. The proper length, L0, of the rod is its length as measured by an observer at rest relative to the rod. To understand why the length measured by a moving observer is less, consider the situation as seen by each observer in turn. Figure 16 Length contraction  Observer O1 measures the time taken, t0, for the rod to pass by when it is moving at velocity, v. Hence the length of the rod according to this observer is L = vt0.  The time taken, t, for observer O1, moving at velocity, v, to pass the rod as measured by observer O2 is equal to t = 2 1 2 2 0 1          c v t since O1 is moving at velocity v relative to O2. This is a consequence of time dilation. Hence, the length of the rod according to O2, its proper length, L0 is equal to vt. Therefore L0 = v × 2 1 2 2 0 1        c v t = 2 1 2 2 1        c v L which gives l = l0 2 1 2 2 1        c v 30 Figure 17 Relativistic mass v speed f) Mass and energy In his theory of special relativity, Einstein proved that transferring energy in any form:  to an object increases its mass.  from an object decreases its mass. He showed that energy E and mass m are equivalent (ie interchangeable) on a scale given by the equation E = m c 2 Since the value of c = 3.0 × 10 8 m s –1 , then 1 kg of mass is equivalent to 9.0 × 10 16 J (= 1 × (3.0 × 10 8 ) 2 ). In terms of the rest mass m0 of an object, the above equation may be written as E =        2 2 2 0 1 c v cm At zero speed, v = 0, E = m0 c 2 which represents the rest energy E0 of the object. At speed v, the difference between its total energy E and its rest energy E0 represents its energy due to its speed (ie its kinetic energy). Therefore, its kinetic energy Ek = m c 2 – m0 c 2 31 For example, if an object is travelling at a speed v = 0.99 c, the relativistic mass formula gives its mass m = )99.01( 2 0  m = 7.1 m0 so its kinetic energy Ek = 7.1 m0 c 2 – m0 c 2 = 6.1 m0 c 2 . Bertozzi’s experiment This experiment set out to determine the variation of the kinetic energy of an electron with velocity based on direct measurements. When a charged particle of charge Q is accelerated from rest through a potential difference V to a certain speed, the work done on it is W = Q V. Its kinetic energy after being accelerated is therefore equal to Q V and its total energy E = m c 2 = m0 c 2 + QV. In previous experiments the kinetic energy had been determined indirectly using EK = VQ. Bertozzi (1962) used the apparatus shown schematically in Figure 18. Figure 18 Bertozzi’s experiment The speeds of bunches of electrons that had been accelerated in a particle accelerator were measured directly for five different accelerating voltages. The speeds were measured over a distance of 8.4 m using a ‘time of flight’ method and agreed with those expected from the principle of relativity. For each speed, the kinetic energy was measured directly at the end of its flight by a calorimetric method. The electrons gave up their energy to an aluminium plate of mass m and specific heat capacity c. The temperature rise  of the plate was measured for a known number n of electrons hitting the plate. The 32 kinetic energy EK of an electron was then mc /n where m was the mass of the aluminium plate. The graph in Figure 19 shows how the expected variation of kinetic energy varies with the velocity of the electron beam special relativity,        2 2 2 0 1 c v cm − mo c 2 . Figure 19 Variation of kinetic energy with velocity using relativity For each electron, kinetic energy used by Bertozzi, the velocity of the electron beam was found to be within 10% of the values expected by the special relativity formula. The results confirmed that although the kinetic energy continues to increase, the speed of the electrons approaches a limiting value. Using classical physics E = mv 2 /2 for the same range of kinetic energies, the electron speeds would vary as shown in Figure 20. However, this only agrees with the practical results for the variation of EK with v for very low speeds (v<<c).