Chapter 14: Dual Nature of Radiation and Matter
In 1887, a young German physicist named Heinrich Hertz was busy proving that light is a wave. His apparatus was simple in idea: two metal spheres separated by a small air gap, with a high voltage across them. When a spark jumped the gap, it sent out electromagnetic waves, and a second gap some distance away sparked in response. Hertz was confirming Maxwell’s prediction that electromagnetic waves exist and travel through space — a triumph for the wave picture of light.
While adjusting the set-up, Hertz noticed something he had not gone looking for. When ultraviolet light was allowed to fall on the metal electrodes, the spark jumped across far more easily. Shield the electrodes from that light, and the spark became reluctant again. Light shining on a metal was somehow helping electricity escape from it.
Hertz recorded the observation and moved on. He could not have guessed that this small side-effect would refuse to fit into the very wave theory he had just confirmed. Over the next twenty years it would force physicists to accept an uncomfortable idea: that light travels as a wave, but arrives as a particle — and that matter, in turn, hides a wave inside it.
Diagram
Fig. 14.0 – Hertz’s spark-gap set-up: two metal spheres with a spark across the gap, ultraviolet light falling on the electrodes, and electrons leaving the metal surface.
By the end of this chapter, you will be able to answer each of these questions yourself.
14.1 Introduction
The Wave Picture of Light
In earlier chapters you have studied several optical phenomena — reflection, refraction, interference, diffraction and polarization. Light is electromagnetic radiation, and almost all of these phenomena have been explained by treating light as a wave.
Two of them are worth pausing on. Interference happens when two light waves overlap and add up or cancel out, and diffraction happens when light bends around an obstacle or spreads after passing through a narrow slit. Only something that travels as a wave can behave this way, so these phenomena are strong evidence for the wave nature of light.
Visible light, however, is only a narrow band of a much larger family. You are already familiar with the wave nature of electromagnetic radiation in other regions of the spectrum as well — X-rays, γ-rays, infrared and ultraviolet radiation, and microwaves. All of these are the same kind of radiation; they differ only in frequency and wavelength.
Structurally, electromagnetic radiation consists of mutually perpendicular oscillating electric and magnetic fields, and both these fields are perpendicular to the direction in which the wave and its energy are travelling, as shown in Fig. 14.1a.
Diagram
Fig. 14.1a – An electromagnetic wave showing the oscillating electric field and magnetic field in mutually perpendicular planes, both perpendicular to the direction of propagation.
Where the Wave Picture Begins to Fail
In Chapter 3, on Kinetic Theory of Gases and Radiation, you came across the spectrum of black body radiation. A black body is an ideal body that absorbs all the radiation falling on it, and when heated it emits radiation whose spread over different wavelengths depends only on its temperature.
This spectrum cannot be explained using the wave nature of radiation. The wave theory predicts a curve of one shape; the measured curve has a quite different shape, as suggested in Fig. 14.1b.
Diagram
Fig. 14.1b – Black body radiation curve: intensity versus wavelength, with the experimentally observed curve and the (failing) prediction of wave theory drawn together.
Such phenomena appear during the interaction of radiation with matter, and they need quantum physics to explain them. That is the common thread running through this entire chapter: light behaves like a wave while it travels, but something else happens when it meets matter.
Planck’s Idea: Energy Comes in Packets
The idea of quantization of energy was first proposed by Planck to explain the black body spectrum. To quantize a quantity means to allow it only in whole-number multiples of a fixed basic amount, instead of in any amount you please.
Planck proposed a model that says (i) energy is emitted in packets, and (ii) at higher frequencies, the energy of a packet is large.
Planck assumed that atoms behave like tiny oscillators that emit electromagnetic radiation only in discrete packets,
\[E = nh\nu\]
where \(\nu\) is the frequency of the oscillator, \(h\) is a constant (its value is given in Section 14.2.4), \(n\) is a whole number, and \(E\) is the energy emitted. In SI units, \(E\) is in joule (J), \(\nu\) is in hertz (Hz, that is \(\mathrm{s^{-1}}\)), and \(h\) is in joule second (J s).
The emissions occur only when the oscillator makes a jump from one quantized level of energy to another of lower energy. The oscillator cannot slide down smoothly; it can only drop from one allowed level to a lower allowed level, releasing the energy difference as a packet of radiation.
Notice what the second part of Planck’s model implies: since a packet at higher frequency carries more energy, radiation of high frequency delivers its energy to matter in large lumps, while low-frequency radiation delivers it in small lumps. This single idea decides almost everything that follows in this chapter.
This model of Planck turned out to be the basis for Einstein’s theory to explain the observations of experiments on the photoelectric effect, which we will study in the following section.
In examination questions, this section usually appears indirectly: you are expected to know that the energy of a packet is proportional to frequency, so a statement such as “increasing the intensity increases the energy of each packet” is false. Intensity and frequency control two different things, and keeping them apart is the key skill for the whole chapter.
14.2 The Photoelectric Effect
How the Effect Was First Noticed
Heinrich Hertz discovered photoelectric emission in 1887, while he was working on the production of electromagnetic waves by spark discharge. He noticed that when ultraviolet light is incident on a metal electrode, a high voltage spark passes across the electrodes.
What was actually happening was this: electrons were being emitted from the metal surface. These free electrons in the gap made it easier for the spark to pass, which is why the spark improved whenever ultraviolet light fell on the electrode. Hertz saw the effect of the emission before anyone understood the emission itself.
A surface that emits electrons when it is illuminated with appropriate radiation is known as a photosensitive surface.
What the Effect Is
The phenomenon of emission of electrons from a metal surface, when radiation of appropriate frequency is incident on it, is known as the photoelectric effect. The process is shown in Fig. 14.1, where radiation arriving at the metal surface is followed by electrons leaving that surface.
Diagram
Fig. 14.1 – Process of photoelectric effect: ultraviolet radiation falling on a metal plate and electrons being ejected from its surface.
Read the definition carefully. It says radiation of appropriate frequency, not radiation of appropriate brightness. This is the first hint that frequency, and not intensity, is the deciding factor — a point that the experiments in the next sections will establish beyond doubt.
Which Metals Respond to Which Radiation
The radiation that works is not the same for every metal. For metals like zinc, cadmium and magnesium, ultraviolet radiation is necessary, while for alkali metals, even visible radiation is sufficient.
Alkali metals are the metals of the first group of the periodic table — sodium, potassium, caesium and their neighbours. They hold their outermost electrons rather loosely, which is why ordinary visible light is enough to knock electrons out of them, whereas zinc or cadmium demands the higher frequency of ultraviolet radiation.
In examinations, a common one-mark trap is a question asking whether red light can produce photoelectric emission from a given metal. The answer never depends on how bright the red light is; it depends only on whether that metal responds to visible radiation at all.
14.2.1 Experimental Set-up of Photoelectric Effect
The Apparatus
A typical laboratory experimental set-up for the photoelectric effect is shown in Fig. 14.2. It consists of an evacuated glass tube with a quartz window, containing a photosensitive metal plate — the emitter E — and another metal plate — the collector C.
Diagram
Fig. 14.2 – Schematic of experimental set-up for photoelectric effect: evacuated tube with quartz window, emitter E and collector C, variable voltage source with a voltmeter, ammeter in the circuit and light source S.
Two features of the tube deserve attention. The tube is evacuated so that the emitted electrons can travel from E to C without colliding with air molecules and losing energy on the way. The window is made of quartz rather than ordinary glass because ordinary glass absorbs ultraviolet radiation, while quartz allows it through to reach the emitter.
The emitter and collector are connected to a voltage source whose voltage can be changed, and to an ammeter to measure the current in the circuit. A potential difference of \(V\), as measured by the voltmeter, is maintained between the emitter E (the cathode) and the collector C (the anode), normally C being at a positive potential with respect to the emitter.
Controlling the Potential Difference
This potential difference can be varied, and C can even be at a negative potential with respect to E. The circuit is therefore able to do two opposite jobs, simply by reversing the polarity of the source, as indicated in Fig. 14.2a.
Diagram
Fig. 14.2a – The same circuit drawn twice, once with the collector positive (electrons pulled towards C) and once with the collector negative (electrons pushed back towards E).
When the anode potential \(V\) is positive, it accelerates the electrons, and it is therefore called the accelerating potential. When the anode potential \(V\) is negative, it retards the flow of electrons, and it is therefore known as the retarding potential.
The ability to reverse the polarity is what makes this apparatus so useful. With a positive collector, the experimenter can count how many electrons are being emitted, because eventually every emitted electron is collected. With a negative collector, the experimenter can measure how energetic the electrons are, because only those with enough kinetic energy manage to fight their way across.
The Light Source
A source S of monochromatic light (light corresponding to only one specific frequency) of sufficiently high frequency, that is short wavelength \(\lambda \le 10^{-7}\) m, is used.
Monochromatic light is essential here. If the beam contained many frequencies together, each frequency would produce electrons of a different energy, and it would be impossible to say which frequency produced which result. By fixing the frequency, the experimenter can change one quantity at a time — intensity, or frequency, or voltage — and watch what happens.
What Happens Inside
Light is made to fall on the surface of the metal plate E, and electrons are ejected from the metal through its surface. These electrons, called photoelectrons, are collected at the collector C.
We now know that free electrons are available in a metal plate. They are emitted if sufficient energy is supplied to them to overcome the barrier that keeps them inside the metal.
That word “barrier” is important. A free electron inside a metal can wander from atom to atom quite easily, which is why metals conduct, but it is still trapped inside the metal as a whole. Something must supply it with enough energy to cross the surface and get out. In this experiment, that something is the incident light.
In the late nineteenth century, these facts were not known. Scientists working on the photoelectric effect performed various experiments with set-ups like this one and simply noted down what they saw. Their observations are summarized in the next section, and we will then try to analyze the observations and their explanation.
14.2.2 Observations from Experiments on Photoelectric Effect
What the Experimenters Could Change
Before reading the observations, it helps to be clear about the three things an experimenter can adjust in the set-up of Fig. 14.2, because every observation below is the result of changing one of them and keeping the others fixed.
The first is the intensity of the incident radiation, which means the energy falling on unit area of the emitter in unit time — in everyday language, how bright the light is. The second is the frequency of the incident radiation, which is fixed by the source and decides the “colour” or the region of the spectrum. The third is the potential difference \(V\) applied between emitter and collector, which can be accelerating or retarding.
The quantity measured in every case is the photocurrent \(I\), read on the ammeter. Since each photoelectron carries the same charge, the photocurrent is a direct measure of the number of electrons reaching the collector per second.
Frequency Decides Whether Emission Happens at All
Observation 1. When ultraviolet radiation was incident on the emitter plate, current \(I\) was recorded even if the intensity of radiation was very low. Photocurrent \(I\) was observed only if the frequency of the incident radiation was more than some threshold frequency \(\nu_0\). This \(\nu_0\) was the same for a given metal and was different for different metals used as the emitter. For a given frequency \(\nu\) (\(> \nu_0\)) of the incident radiation, no matter how feeble the light was — meaning however small the intensity of radiation may be — electrons were always emitted.
This is a surprising result, and it is worth stating in the two directions separately. Below \(\nu_0\), no amount of brightness helps: the current stays at zero. Above \(\nu_0\), no amount of dimness hurts: some current always flows.
Observation 2. There was no time lag between the incidence of light and the emission of electrons. The photocurrent started instantaneously — within \(10^{-9}\) s — on shining the radiation, even if the intensity of radiation was low. As soon as the incident radiation was stopped, the flow of current stopped.
Notice that emission begins and ends with the light itself. The metal does not “store up” light energy and release electrons afterwards.
Intensity Decides How Many Electrons Come Out
Observation 3. Keeping the frequency \(\nu\) of the incident radiation and the accelerating potential \(V\) fixed, if the intensity was increased, the photocurrent increased linearly with intensity, as shown in Fig. 14.3.
Diagram
Fig. 14.3 – Photocurrent as a function of incident intensity for fixed incident frequency and accelerating potential: a straight line through the origin.
A straight line through the origin means simple proportionality: double the intensity and you double the photocurrent, that is, double the number of electrons emitted per second.
Observation 4. The photocurrent \(I\) could also be varied by changing the potential of the collector plate. For given incident radiation (intensity and frequency both fixed), \(I\) depended on the accelerating potential \(V\), that is, on the potential difference between the emitter and the collector. Initially the current increased with voltage, but then it remained constant. This constant value was termed the saturation current \(I_0\), as seen in Fig. 14.4. The superscripts 1, 2, 3 of \(I_0\) in the figure refer to different intensities of the incident light.
Diagram
Fig. 14.4 – Photocurrent as a function of accelerating potential for fixed incident frequency and three different incident intensities, showing saturation currents \(I_0^1\), \(I_0^2\), \(I_0^3\) and a common stopping potential \(-V_0\).
Observation 5. Keeping the accelerating voltage and the incident frequency fixed, if the intensity of incident radiation was increased, the value of the saturation current also increased proportionately. For example, if the intensity was doubled, the saturation current was also doubled.
What Decides the Energy of the Emitted Electrons
Observation 6. The maximum kinetic energy \(KE_{max}\) of the electrons, and hence their maximum velocity, depended on the potential \(V\) for a given metal used for the emitter plate and for a given frequency of the incident radiation. If the material was changed, or the frequency of the incident radiation was changed, \(KE_{max}\) changed. It did not depend on the intensity of the incident radiation. Thus, even for a very small incident intensity, if the frequency of incident radiation was larger than the threshold frequency \(\nu_0\), the value of \(KE_{max}\) from a given surface was always the same for a given incident frequency.
Here the word maximum matters. Not every emitted electron has the same energy; \(KE_{max}\) refers to the most energetic electron leaving the surface. The reason for this spread is taken up later in the chapter.
Observation 7. If increasingly negative potentials were applied to the collector, the photocurrent decreased, and for some typical value \(-V_0\) the photocurrent became zero. \(V_0\) was termed the cut-off or stopping potential. The fact that some current still flowed before this value was reached indicated that when the potential was retarding, the photoelectrons still had enough energy to overcome the retarding (opposing) electric field and reach the collector. The value of \(V_0\) was the same for any incident intensity as long as the incident frequency was the same, as seen in Fig. 14.4, but was different for different emitter materials.
Changing the Frequency Changes the Stopping Potential
Observation 8. If the frequency of incident radiation was changed keeping the intensity constant, then the saturation current remained the same but the stopping potential \(V_0\) changed. In Fig. 14.5, \(V_0^1\), \(V_0^2\) and \(V_0^3\) are the stopping potentials for incident frequencies \(\nu_1\), \(\nu_2\) and \(\nu_3\) respectively, with \(\nu_3 > \nu_2 > \nu_1\).
Diagram
Fig. 14.5 – Photocurrent as a function of accelerating potential for fixed incident intensity but three different incident frequencies on the same emitter material, showing a common saturation current and three different stopping potentials.
The stopping potential \(V_0\) varied linearly with \(\nu\), as shown in Fig. 14.6. For different metals, the slopes of such straight lines were the same, but the intercepts on the frequency axis and on the stopping potential axis were different.
Diagram
Fig. 14.6 – Stopping potential as a function of frequency of incident radiation for emitters made of different metals: parallel straight lines with different intercepts.
Read Fig. 14.6 carefully, because it carries two separate messages. Parallel lines mean that the rate at which stopping potential grows with frequency is a property of nature, the same for every metal. Different intercepts mean that where each line begins — the threshold frequency — is a property of the particular metal.
Observation 9. The photocurrent, and hence the number of electrons, depended on the intensity but not on the frequency of the incident radiation, as long as the incident frequency was larger than the threshold frequency \(\nu_0\) and the potential of the anode was higher than that of the cathode.
The Pattern Behind the Nine Observations
Taken together, these observations divide neatly into two groups, and this division is what examination questions test again and again.
Intensity controls the number of photoelectrons, and therefore the photocurrent and the saturation current. Frequency controls the energy of the photoelectrons, and therefore the stopping potential, and it also decides whether there is any emission at all.
Not one of these observations mentions any waiting time, and none of them allows brightness to substitute for frequency. As the next section shows, this is precisely where the wave theory of light runs into trouble.
14.2.3 Failure of Wave Theory to Explain the Observations from Experiments on Photoelectric Effect
The Problem of Instantaneous Emission
Most of the observations listed in the previous section could not be explained by the wave theory of electromagnetic radiation. First and foremost was the instantaneous emission of electrons on the incidence of light.
The wave picture would expect the metal surface to absorb the incident energy continuously, the way a beach absorbs the energy of waves arriving on it. All the electrons near the surface would go on absorbing energy little by little. The metal surface would require a reasonable time — of the order of a few minutes to a few hours — to accumulate sufficient energy to knock off electrons.
The same picture also predicts a clear dependence on brightness. The greater the intensity of incident radiation, the more will be the incident energy, and hence the shorter the expected time required to knock off the electrons. For small incident intensity, the energy incident on unit area in unit time will be small, and it will take longer to knock off the electrons.
These arguments were contradictory to the observations. In the actual experiment, emission began within \(10^{-9}\) s, and it began even when the light was extremely feeble.
Let us try to estimate the time that will be required for the photocurrent to start, so that we can see just how badly the wave picture fails. We need to define the term ‘work function’ of a metal for this exercise.
The Work Function of a Metal
We know that metals have free electrons. This fact makes metals good conductors of heat and electricity. These electrons are free to move inside the metal but are otherwise confined inside the metal. They cannot escape from the surface unless sufficient energy is supplied to them, as suggested in Fig. 14.6a.
Diagram
Fig. 14.6a – An electron free to move inside a metal but held back at the surface by an energy barrier, with the minimum energy needed to cross the barrier marked as the work function.
The minimum amount of energy required to be provided to an electron to pull it out of the metal from the surface is called the work function of the metal and is denoted by \(\phi_0\).
Work function depends on the properties of the metal and on the nature of its surface. Values of the work function of metals are generally expressed in a unit of energy called the electron volt (eV).
One electron volt is the energy gained by an electron when it is accelerated through a potential difference of one volt, so \(1\ \mathrm{eV} = 1.6 \times 10^{-19}\) J. This unit is used because the joule is far too large a unit for a single electron; a work function of a few electron volts is a comfortable number to write and to compare across metals.
Table 14.1 gives typical values of the work function for some common metals.
Table 14.1: Typical values of work function for some common metals.
| Metal | Work function (in eV) |
|---|---|
| Potassium | 2.3 |
| Sodium | 2.4 |
| Calcium | 2.9 |
| Zinc | 3.6 |
| Silver | 4.3 |
| Aluminum | 4.3 |
| Tungsten | 4.5 |
| Copper | 4.7 |
| Nickel | 5.0 |
| Gold | 5.1 |
Estimating the Waiting Time Predicted by Wave Theory
We can now put numbers to the wave theory’s prediction. The idea is simple: work out how much energy the incident light delivers to a single electron per second, and then find how long that electron must wait before it has collected an amount equal to the work function.
Two More Failures
Secondly, since a larger incident intensity implies larger energy, the electrons are expected to be emitted with larger kinetic energy. But the observation showed that the maximum kinetic energy did not depend on the incident intensity and depended instead on the incident frequency. According to wave theory, the frequency of incident radiation has no role at all in determining the kinetic energy of photoelectrons.
Moreover, wave theory expected photoelectrons to be emitted for any frequency, provided the intensity of radiation was large enough. But the observations indicated that for a given metal surface, some characteristic cut-off frequency \(\nu_0\) existed, below which no photoelectrons were emitted however intense the incident radiation was; and photoelectrons were always emitted if the incident frequency \(\nu\) was greater than \(\nu_0\), even if the intensity was low.
Three failures, then, and all of them of the same kind. The wave theory keeps insisting that energy is delivered smoothly and shared out among many electrons, while the experiment keeps insisting that energy is delivered to one electron at a time, in a lump whose size is fixed by the frequency. The next section takes that hint seriously.
14.2.4 Einstein’s Postulate of Quantization of Energy and the Photoelectric Equation
Einstein’s Bold Extension of Planck’s Idea
Planck’s hypothesis of energy quantization, which he had used to explain black body radiation, was extended by Einstein in 1905 to all types of electromagnetic radiation.
The step Einstein took was bigger than it first looks. Planck had quantized only the emission by the atomic oscillators; the radiation itself was still thought of as a spreading wave. Einstein proposed that the radiation stays in lumps while it travels, and delivers itself to matter in lumps as well.
Einstein proposed that under certain conditions, light behaves as if it was a particle, and its energy is released or absorbed in bundles or quanta. He named the quantum of energy of light a photon, with energy \(E = h\nu\), where \(\nu\) is the frequency of light and \(h\) is the constant defined by Planck in his model to explain black body radiation. It is now known as Planck’s constant and has a value \(6.626 \times 10^{-34}\) J s.
\[E = h\nu \qquad \text{--- (14.1)}\]
Here \(E\) is the energy of one photon in joule (J), \(\nu\) is the frequency of the radiation in hertz (Hz), and \(h\) is Planck’s constant in joule second (J s).
It may be noted that Equation (14.1) is a relation between a particle-like property, the energy \(E\), and a wave-like property, the frequency \(\nu\). Equation (14.1) is known as Einstein’s relation.
Energy, Frequency and Wavelength
Einstein’s relation (14.1) holds good for the entire electromagnetic spectrum. It says that the energy of electromagnetic radiation is directly proportional to the frequency, and, since \(\nu = c/\lambda\), it is inversely proportional to the wavelength. Here \(c\) is the speed of electromagnetic radiation in vacuum and \(\lambda\) is the wavelength in metre (m).
Hence high frequency radiation means high energy radiation. Alternatively, short wavelength radiation means high energy radiation.
This single sentence explains a great deal that you already know from elsewhere. γ-rays and X-rays, at the short-wavelength end, are the ones that can damage tissue and are handled with shielding; radio waves, at the long-wavelength end, pass through us all day without any effect at all. The difference is not how much total energy arrives, but how big each individual bundle is.
Compare the three answers and the pattern of the whole chapter appears in miniature. The ultraviolet photon carries more than twice the energy of the red photon, and the FM photon carries less than a millionth of either. It is easy to see why ultraviolet radiation can throw electrons out of metals while an FM transmitter, radiating thousands of watts, cannot throw out a single one.
Explaining the Observations, One by One
The explanation using Einstein’s postulate of quantization of energy for the observations mentioned in Section 14.2.2 is given below. The numbering follows the same order, so each explanation can be read against the observation it accounts for.
Explanation 1. Einstein argued that when a photon of ultraviolet radiation arrives at the metal surface and collides with an electron, it gives all of its energy \(h\nu\) to the electron. The energy is gained by the electron and the photon no longer exists. There is no question of half a photon being absorbed; the transaction is all or nothing.
If \(\phi_0\) is the work function of the material of the emitter plate, then electrons will be emitted if and only if the energy gained by the electron is more than or equal to the work function, that is,
\[h\nu \ge \phi_0\]
Thus a minimum or threshold frequency \(\nu_0\) is required to eject electrons from the metal surface, where
\[\nu_0 = \frac{\phi_0}{h}\]
If \(\nu < \nu_0\), the photon will not have enough energy to liberate an electron. As a result, no electron will be ejected, however intense the incident radiation is. Similarly, if \(\nu > \nu_0\), the energy will always be sufficient to eject an electron, however small the incident intensity is.
Explanation 2. Energy is given by the photon to the electron as soon as the radiation is incident on the surface. The exchange of energy between the photon and the electron is instantaneous. Hence there is no time lag between the incidence of light and the emission of electrons. Also, when the incident radiation is stopped, there are no photons to transfer the energy to electrons, and hence the photoemission stops immediately.
Explanation 3. According to Einstein’s proposition, if the intensity of incident radiation for a given wavelength is increased, there will be an increase in the number of energy quanta (photons) incident on unit area in unit time, the energy of each quantum being the same, equal to \(h\nu = hc/\lambda\).
Therefore larger intensity radiation will knock off a larger number of electrons from the surface, and hence the current will be larger, provided \(\nu > \nu_0\). Conversely, lower intensity implies a smaller number of incident photons, hence a smaller number of ejected electrons and therefore a lower current.
In the photon picture, then, intensity is simply a photon count per second, not a measure of how energetic each photon is. This is the sentence to remember for every question that mixes up intensity and frequency.
Explanation 4. Once the electron is emitted from the surface, if the collector is at a higher potential than the emitter, the electric field will accelerate the electrons towards the collector. The higher the accelerating potential, the greater will be the number of electrons reaching the collector. Hence the photocurrent \(I\) increases with the accelerating potential initially.
Moreover, since the intensity of incident radiation determines the number of photons incident on the metal surface on unit area in unit time, it determines the maximum number of electrons that can be knocked off by the incident radiation. Hence, for a given intensity, increasing the accelerating potential can increase the current only till all the knocked-off electrons have reached the collector. No increase can be seen in the current beyond this limit. This explains the saturation current \(I_0\).
Explanation 5. Increasing the incident intensity will increase the number of incident photons and eventually the saturation current, since more electrons are now available to be collected.
Explanation 6. If the frequency of incident radiation is more than the threshold frequency, then the energy \(\phi_0\) is used by the electron to escape from the metal surface, and the remaining energy of the photon becomes the kinetic energy of the electron.
Depending on the energy of the electron inside the metal and other processes such as collisions after emission from the surface, the maximum kinetic energy is equal to \((h\nu - \phi_0)\). Hence
\[KE_{max} = h\nu - \phi_0 \qquad \text{--- (14.2)}\]
Equation (14.2) is known as Einstein’s photoelectric equation. Here \(KE_{max}\) is the maximum kinetic energy of the emitted photoelectrons in joule (J), \(h\nu\) is the energy of the incident photon in joule (J), and \(\phi_0\) is the work function of the emitter material in joule (J).
Read the equation as a simple statement of energy accounting, as shown in Fig. 14.2c: the photon brings in \(h\nu\), the metal charges a fixed toll \(\phi_0\) at the surface, and whatever is left over appears as kinetic energy of the escaping electron.
Diagram
Fig. 14.2c – Energy accounting in photoemission: an incoming photon of energy \(h\nu\) split into the work function \(\phi_0\) spent at the surface and the remainder appearing as \(KE_{max}\) of the electron.
\(KE_{max}\) depends on the material of the emitter plate and varies linearly with the incident frequency \(\nu\); it is independent of the intensity of the incident radiation.
Explanation 7. The electrons that are emitted from the metal surface have different kinetic energies. The reasons for this are manifold: all the electrons in a solid do not possess the same energy; the electrons may be ejected from varying depths inside the metal surface; and electrons may suffer collisions before they come out of the metal surface and may lose their energy.
If \(V\) is the potential difference between the emitter and the collector, and the collector is at a lower potential, an electron will lose its kinetic energy in overcoming the retarding force. If the kinetic energy is not sufficient, the emitted electrons may not reach the collector and the photocurrent will be zero.
If \(KE_{max}\) is the energy of the most energetic electron at the emitter surface, where its potential energy is zero, and \(-V_0\) is the stopping potential, then this electron will fail to reach the collector if \(KE_{max} < eV_0\), where \(e\) is the electron charge in coulomb (C) and \(eV_0\) is the energy needed for the electron to overcome the retarding potential \(V_0\).
If the electron just fails to reach the collector, that is, if it has lost all its kinetic energy exactly at the collector, then \(KE_{max} = eV_0\) and the photocurrent becomes zero. Equation (14.2) then explains that the stopping potential \(V_0\) depends on the incident frequency and on the material of the emitter, and does not depend on the incident intensity.
Explanation 8. If the ejected electrons have kinetic energy more than \(eV_0\), electrons can reach the collector and hence current flows. When the kinetic energy of the electron is less than or equal to \(eV_0\), no current will flow. The photocurrent will become zero when \(KE_{max} = eV_0\).
Using \(KE_{max} = eV_0\), we can write Eq. (14.2) as
\[eV_0 = h\nu - \phi_0\]
or,
\[V_0 = \left(\frac{h}{e}\right)\nu - \frac{\phi_0}{e} \qquad \text{--- (14.3)}\]
The above equation tells us that \(V_0\) varies linearly with the incident frequency \(\nu\), and the slope of the straight line depends on the constants \(h\) and \(e\), while the intercept of the line depends on the material through \(\phi_0\).
Compare this with the standard straight-line form \(y = mx + c\): here \(V_0\) is \(y\), \(\nu\) is \(x\), the slope \(m\) is \(h/e\), and the intercept \(c\) is \(-\phi_0/e\). Since \(h\) and \(e\) are universal constants, every metal must give a line of the same slope.
Thus the slopes of the lines in Fig. 14.6 are the same and are independent of the material of the emitter, but the intercepts are different for different materials. This is exactly what was observed, and it also gives experimenters a beautiful method of measuring \(h\): plot \(V_0\) against \(\nu\), measure the slope, and multiply by \(e\).
Explanation 9. All the above arguments thus bring out the fact that the magnitude of the photocurrent depends on the incident intensity, through the number of emitted photoelectrons, and on the potential \(V\) of the collector, but not on the incident frequency \(\nu\) as long as \(\nu > \nu_0\).
How the Photon Idea Came to Be Accepted
Thus all the observations related to the experiments on photoelectric effect were explained by Einstein’s hypothesis of the existence of a photon, or treating light as bundles of energy.
Although Einstein gave his hypothesis in 1905, it was not widely accepted by the scientific community. The wave theory of light had a century of spectacular success behind it, and physicists were unwilling to give up any part of it on the strength of one idea.
In 1909, when Millikan measured the charge of an electron, and the value of \(h\) calculated from Eq. (14.3) matched with the value given by Planck, the hypothesis was accepted. The work function values \(\phi_0\) for some metals were also confirmed from Eq. (14.3). Einstein and Millikan received Nobel prizes for their respective discoveries in 1921 and 1923 respectively.
Why We Never See Individual Photons
According to Einstein, the energy of radiation of frequency \(\nu\) comes in bundles with magnitude \(h\nu\). Thus the energy of a light beam having \(n\) photons will be \(nh\nu\), where \(n\) can take only integral values.
Is it then possible to vary the incident energy continuously? Why do we not see individual photons? To understand this issue, let us consider the following example.
This is the resolution of an apparent paradox. Light really is granular, but the grains are so fine and so numerous that a beam looks perfectly smooth to us — much as a photograph looks continuous until you magnify it enough to see the individual dots.
The Two Pictures Side by Side
Table 14.2 places each observation against what the wave theory predicts and what the photon picture predicts. Reading down the last two columns is the quickest way to revise this entire section.
Table 14.2: Summary of analysis of observations from experiments on photoelectric effect.
| Observation | Wave theory | Photon picture |
|---|---|---|
| Electrons are emitted as soon as the light is incident on the metal surface. | Very intense light is needed for instantaneous emission of electrons. | Only one photon is needed to eject one electron from the metal surface, and energy exchange between electron and photon is instantaneous on collision. |
| Very low intensity of incident light is also sufficient to generate photocurrent. | Low intensity should not give photocurrent. | Low intensity of incident light means a smaller number of photons and not low energy photons. Hence a low current will be produced. |
| High intensity gives larger photocurrent, which means a higher rate of release of electrons. | High intensity means higher energy radiation and therefore more electrons are emitted. | Higher intensity means a larger number of photons incident in unit time, therefore a larger number of electrons are emitted in unit time and hence the photocurrent is larger. |
| Increasing the intensity has no effect on the electron energy. | Higher intensity should mean electrons emitted with higher energies. | Higher intensity means a higher number of incident photons per unit time. The energy of a photon is the same, as it does not depend on the intensity. |
| A minimum threshold frequency is needed for photocurrent to start. | Low frequency light should release electrons but would take more time. | A photon of low frequency light will not have sufficient energy to release an electron from the surface. |
| Increasing the frequency of incident light increases the maximum kinetic energy of electrons. | Increasing intensity should increase the maximum kinetic energy. Maximum kinetic energy should not depend on the incident frequency. | Increasing the frequency increases the energy of the photon. Therefore electrons receive more energy, which results in increasing the maximum kinetic energy. |
In numerical problems from this section, almost everything reduces to three relations used together: \(E = h\nu = hc/\lambda\) for the photon, \(KE_{max} = h\nu - \phi_0\) for the emitted electron, and \(KE_{max} = eV_0\) for the stopping potential. The commonest errors are forgetting to convert eV to joule, and using wavelength in Å without converting to metre.
14.3 Wave-Particle Duality of Electromagnetic Radiation
From Energy Packets to Momentum
In its interaction with matter, light behaves as if it is made up of packets of energy called quanta. That much was established by the photoelectric effect.
Later it was confirmed from other theoretical and experimental investigations that these light quanta can have associated momentum as well. This is a significant step forward. Energy alone can be argued about; momentum is the property we normally reserve for something that has mass and moves — something we would unhesitatingly call a particle.
Hence the question came up whether a particle can be associated with light, or with electromagnetic radiation in general. The particle nature was confirmed by Compton in 1924, in experiments on the scattering of X-rays due to the electrons of matter.
In 1923, A. H. Compton made a monochromatic beam of X-rays, of wavelength λ, incident on a graphite sheet and measured the intensity of the scattered rays in different directions as a function of wavelength. He found that although the incident beam consisted of a single wavelength λ, the scattered intensity was maximum at two wavelengths. One of these was the same as the incident wavelength, but the other, λ′, was larger by an amount Δλ. This Δλ is known as the Compton shift, and it depends on the scattering angle.
Compton explained his observations by considering the incidence of the X-ray beam on graphite as a collision of X-ray photons with the electrons of graphite, like a collision of billiard balls. Energy and momentum are transferred during the collision, and the scattered photons have lower energy than the incident photons. Therefore they have lower frequency, or higher wavelength.
The Compton shift is given by the relation
\[\Delta\lambda = \lambda' - \lambda = \frac{h}{m_e c}(1 - \cos\theta)\]
where θ is the scattering angle. The shift depends only on the scattering angle and not on the incident wavelength. This shift cannot be explained using wave theory. If we let Planck’s constant go to zero, we get the result expected from wave theory. This is the test to check whether the new picture is correct or not.
Diagram
Fig. 14.6d – Compton scattering: an X-ray photon of wavelength λ colliding with an electron in graphite, the scattered photon of longer wavelength λ′ leaving at angle θ and the recoiling electron moving away.
The Properties of a Photon
Compton showed that a photon has an associated momentum along with the energy it carries. All photons of electromagnetic radiation of a particular frequency have the same energy and the same momentum.
Photons are electrically neutral, and so they are not deflected by electric or magnetic fields. This is one clear way in which a photon differs from an electron; a beam of electrons can be steered with a magnet, a beam of light cannot.
Photons can have particle-like collisions with other particles such as electrons. In a photon–particle collision, the energy and momentum of the system are conserved, but the number of photons is not conserved.
That last point deserves care, because it is unlike anything in mechanics. Photons can be absorbed, or new photons can be created. Photons can transfer their energy and momentum during collisions with particles and simply disappear. When we turn on a light, photons are created. There is no conservation law that fixes how many photons exist.
A photon always moves with the speed of light; it is never at rest. The mass of a photon is not defined as we do for a particle in Newtonian mechanics. Its rest mass is zero, in all frames of reference.
When Each Nature Shows Itself
Effects of the wave nature of light were seen in experiments on interference or diffraction when the slit widths, or the separation between two slits, are smaller than or comparable to the wavelength of light. If the slit width is large, or the spacing between slits is more, the interference or diffraction patterns will not be the same, and the wave nature will not be so obvious.
This gives a useful rule of thumb. The wave nature announces itself when the obstacle or opening is of about the same size as the wavelength; when the obstacle is enormous by comparison, light simply travels in straight lines and the wave character stays hidden.
It was realized by scientists that some phenomena observed in experiments in the laboratory or in nature, such as interference and diffraction, can be explained by considering light in particular, and electromagnetic radiation in general, as a wave.
On the other hand, some other observations, such as the photoelectric effect and black body radiation, can be explained only if we consider electromagnetic radiation as consisting of photons with a definite quantum of energy, and of momentum as well, as is evident from the Compton scattering experiments.
Also, there are some phenomena which can be explained by both the theories.
The Dual Character of Radiation
It is therefore essential to consider that both the characters or behaviours hold good; one dominates in some situations and the other works in the rest of the situations. It is necessary to keep both the physical models to explain the careful experimental observations. There is thus a need to hypothesize the dual character of light.
Note carefully what is being claimed and what is not. Light is not sometimes a wave and at other times secretly something else; nor does it change its nature depending on who is watching. It is a single entity whose behaviour requires two complementary descriptions, and which of the two descriptions is useful depends on the kind of experiment being done.
Later it turned out that such a picture is required not only for light but for the whole electromagnetic spectrum. This phenomenon is termed as wave-particle duality of electromagnetic radiation.
In examinations, this idea is usually tested by asking you to place a given phenomenon in the correct column: interference, diffraction and polarization on the wave side; photoelectric effect, black body radiation and Compton scattering on the particle side. The reasoning to remember is that radiation travels as a wave but exchanges energy with matter as a particle.
14.4 Photo Cell
What a Photocell Is and How It Is Built
A photo cell is a device that makes use of the photoelectric effect and converts light energy into electrical energy. In other words, it is the photoelectric experiment of Section 14.2.1 turned into a practical component that can be wired into a circuit.
A schematic of a photocell is shown in Fig. 14.7. It consists of a semi-cylindrical photosensitive metal plate E, acting as a cathode, and a wire loop collector C, acting as an anode, both supported in an evacuated glass or quartz bulb.
Diagram
Fig. 14.7 – Schematic of a photocell: semi-cylindrical cathode E and wire loop anode C inside an evacuated bulb, connected through a high tension battery B and a microammeter, with incident light falling on the cathode.
The electrodes are connected to an external circuit having a high tension battery B and a microammeter μA. Instead of a photosensitive metal plate, the photosensitive material can be pasted in the form of a thin film on the inner walls of the glass bulb.
Each part of this design has a purpose. The cathode is made semi-cylindrical so that it presents a large curved surface to the incoming light, catching as much of the beam as possible. The anode is only a thin wire loop, so that it collects the electrons without standing in the way of the light. A microammeter is used because photocurrents are extremely small, of the order of microamperes.
How It Works
When light of suitable wavelength falls on the cathode, photoelectrons are emitted. These electrons are attracted towards the anode due to the applied electric field, and the generated photocurrent is noted from the microammeter.
The words “suitable wavelength” carry all the physics of the earlier sections. The material of the cathode must have a work function low enough that the light available in that situation crosses its threshold frequency. A photocell meant to work with visible light is therefore made with an alkali metal cathode, while a cell meant only for ultraviolet radiation can use a metal of higher work function.
Where Photocells Are Used
A photocell is used to operate control systems and in light measuring devices. Light meters in photographic cameras make use of a photocell to measure the intensity of light. A photocell can also be used to switch on or off street lights.
The same idea, drawn in Fig. 14.4a, lies behind the automatic doors at shop entrances and the safety beam across a lift door that keeps it from closing on a passenger.
Diagram
Fig. 14.4a – A light source and a photocell facing each other across a passage; a person walking through interrupts the beam and the photocurrent falls to zero, triggering a counter or alarm.
Notice how directly this application uses Observation 3 of Section 14.2.2. The photocurrent depends on the intensity of the light reaching the cathode, so anything that blocks the light produces an instant electrical signal. Because photoemission stops the moment the radiation stops, the response is effectively immediate — which is exactly what a counter or an alarm needs.
14.5 De Broglie Hypothesis
An Argument from Symmetry
In 1924, Prince Louis de Broglie (pronounced as “de broy”) proposed, on the basis of the symmetry existing in nature, that if radiation has a dual nature — sometimes wave nature dominating and sometimes particle nature — then matter may also possess a dual nature.
The reasoning deserves a moment’s thought, because it is unusually daring. There was, at that time, no experiment at all suggesting that matter behaves like a wave. De Broglie’s argument was that nature does not usually treat its two great categories, radiation and matter, by two different rules. If light, long believed to be purely a wave, had turned out to be partly a particle, then matter, long believed to be purely particles, might well turn out to be partly a wave.
Normally we talk about matter as being composed of particles, but are there situations where matter seems to show wave-like properties? This will become evident from the experiments on diffraction of electrons from nickel crystals, described later in this chapter.
Building the Relation
De Broglie used the properties frequency \(\nu\) and wavelength \(\lambda\) of a wave and proposed a relation to connect these with the particle properties, energy \(E\) and momentum \(p\).
He started from what was already known about the photon. The momentum \(p\) carried by a photon of energy \(E\) is given by the relation
\[p = \frac{E}{c} \qquad \text{--- (14.4)}\]
which is valid for a massless particle travelling with the speed of light \(c\), according to Einstein’s special theory of relativity. Here \(p\) is the momentum in kg m s\(^{-1}\), \(E\) is the energy in joule (J) and \(c\) is the speed of light in vacuum in m s\(^{-1}\). This is the promised relation that gives a photon a finite momentum despite its zero rest mass.
Using Einstein’s relation for \(E\), that is \(E = h\nu\),
\[p = \frac{E}{c} = \frac{h\nu}{c} = \frac{h}{\lambda} \qquad \text{--- (14.5)}\]
where \(\lambda\), the wavelength, is given by \(\lambda\nu = c\).
Look carefully at the last step in Eq. (14.5). It links the momentum of the photon — a particle property — directly to its wavelength, a wave property, through Planck’s constant. This is the equation de Broglie chose to carry over to matter.
De Broglie proposed that a moving material particle of total energy \(E\) and momentum \(p\) has associated with it a wave analogous to a photon. He then suggested that the wave and particle properties of matter can also be described by a relation similar to Eq. (14.5) for a photon.
Thus the frequency and wavelength of a wave associated with a material particle, of mass \(m\) moving with a velocity \(v\), are given as
\[\nu = \frac{E}{h} \quad \text{and} \quad \lambda = \frac{h}{p} = \frac{h}{mv} \qquad \text{--- (14.6)}\]
Here \(m\) is the mass of the particle in kilogram (kg), \(v\) is its velocity in m s\(^{-1}\), \(p = mv\) is its momentum in kg m s\(^{-1}\), and \(\lambda\) is the associated wavelength in metre (m).
He referred to these waves associated with material particles as matter waves. The wavelength of the matter waves, given by Eq. (14.6), is now known as the de Broglie wavelength. Greater the momentum, shorter is the wavelength. Equation (14.6) for the wavelength of matter waves is known as the de Broglie relation.
A moving particle is therefore to be pictured as in Fig. 14.5a — not as a hard little ball, but as a ball accompanied by a wave whose wavelength shrinks as the particle is made faster or heavier.
Diagram
Fig. 14.5a – A moving particle with its associated matter wave, drawn twice: a slow particle with a long wavelength and a fast particle with a short wavelength.
Useful Forms of the de Broglie Relation
In problems, the speed of the particle is usually not given directly. What is given is its kinetic energy, or the potential difference through which it has been accelerated. So it is worth rewriting Eq. (14.6) in those terms.
For a particle of mass \(m\) moving with a velocity \(v\), the kinetic energy is
\[E_K = \frac{1}{2}mv^2 \quad \text{or} \quad v = \sqrt{\frac{2E_K}{m}}\]
Thus,
\[\lambda = \frac{h}{mv} = \frac{h}{m}\sqrt{\frac{m}{2E_K}} = \frac{h}{\sqrt{2mE_K}}\]
For a charged particle of charge \(q\), accelerated from rest through a potential difference \(V\), the work done is \(qV\). This work provides the kinetic energy of the particle. Thus \(E_K = qV\), and therefore
\[\lambda = \frac{h}{\sqrt{2mE_K}} = \frac{h}{\sqrt{2mqV}}\]
Here \(q\) is the charge in coulomb (C) and \(V\) is the accelerating potential difference in volt (V).
This relation holds for any charged particle, like an electron, a proton, or even a charged ion, where \(m\) corresponds to the mass of the charged particle. Of course, when \(V\) is very large — say in kilovolts — so that the speed of the particle becomes close to the speed of light, such an equation will not be applicable. You will learn about the other effects that arise in such situations in higher classes.
The Working Formula for an Electron
For an electron moving through a potential difference of \(V\) (given in volts), substituting the mass and charge of the electron gives
\[\lambda = \frac{h}{\sqrt{2m_e eV}} = \frac{6.63 \times 10^{-34}\ \mathrm{J\,s}}{\sqrt{2 \times 9.11 \times 10^{-31}\ \mathrm{kg} \times 1.6 \times 10^{-19}\ \mathrm{C} \times V\ (\text{in volts})}} = \frac{1.228 \times 10^{-9}}{\sqrt{V\ (\text{in volts})}}\ \mathrm{m}\]
or,
\[\lambda\ (\text{in nm}) = \frac{1.228}{\sqrt{V\ (\text{in volts})}} \qquad \text{--- (14.7)}\]
Equation (14.7) is worth memorizing, but only along with its conditions: it applies to an electron, accelerated from rest, through a potential difference \(V\) expressed in volts, and it gives the answer directly in nanometre.
An answer of about a tenth of a nanometre, that is about 1 Å, is worth remembering as a rough guide. It is the size of an atom — and that fact is the key to the whole of the next section.
The Prediction Is Confirmed
Shortly after the existence of photons — particles associated with electromagnetic waves — was postulated, it was also experimentally found that sub-atomic particles like electrons, protons and neutrons, and atomic particles, also exhibit wave properties.
The wavelength associated with an electron of energy a few eV is of the order of a few Å. Therefore, to observe the wave nature of an electron, the slit width or the diffracting objects should be of the same order of magnitude, that is, a few Å.
This is the same condition that appeared at the end of Section 14.3 for light. No ordinary slit cut with a blade is anywhere near a few ångström wide, so an entirely different kind of “slit” had to be found.
The wave property of the electron was confirmed experimentally in 1927 by Davisson and Germer in America, and in 1928 by George P. Thomson in England, by diffraction of electrons by atoms in metals.
Knowing that the size of atoms and their spacing in crystals is of the order of a few Å, they anticipated that if electrons are scattered by atoms in a crystal, the associated matter waves will interfere and will show diffraction effects. A crystal, in other words, is a ready-made diffraction grating of exactly the right spacing, supplied by nature.
It turned out to be true in their experiments. Electrons showed constructive and destructive interference. No electrons were found in certain directions, due to destructive interference, while in other directions maximum numbers of electrons were seen, due to constructive interference.
14.6 Davisson and Germer Experiment
The Arrangement
A schematic of the experimental arrangement of the Davisson and Germer experiment is shown in Fig. 14.8. The whole set-up is enclosed in an evacuated chamber, for the same reason as in the photoelectric apparatus: a stray gas molecule in the path would scatter the electrons and destroy the pattern being looked for.
Diagram
Fig. 14.8 – Schematic of Davisson and Germer experiment: electron gun with filament F and battery B, cylindrical anode producing a collimated electron beam, nickel crystal target, movable electron detector on a circular scale, and galvanometer.
The set-up uses an electron gun — a device to produce electrons by heating a tungsten filament F using a battery B. A hot filament releases electrons from its surface, which is the simplest way of obtaining a supply of free electrons.
Electrons from the gun are accelerated through vacuum to a desired velocity by applying a suitable accelerating potential across a cylindrical anode, and are collimated into a focused beam. The accelerating potential is the crucial control here: as Eq. (14.7) shows, fixing the voltage fixes the de Broglie wavelength of the electrons in the beam.
This beam of electrons falls on a nickel crystal and is scattered in different directions by the atoms of the crystal. Thus, in the Davisson and Germer experiment, electrons were used in place of light waves, and a crystal was used in place of a diffraction grating.
Scattered electrons were detected by an electron detector, and the current was measured with the help of a galvanometer. The current measured is a direct count of how many electrons are arriving in that particular direction each second.
By moving the detector on a circular scale — that is, by changing the scattering angle \(\theta\) — the intensity of the scattered electron beam was measured for different values of the scattering angle.
What Was Observed
The scattered intensity was not found to be uniform in all directions, as had been predicted by classical theory. If electrons were simply small bullets bouncing off atoms, the scattered electrons would be spread out smoothly, with no direction sharply favoured over its neighbours.
Instead, the intensity pattern resembled a diffraction pattern, with peaks corresponding to constructive interference and troughs corresponding to regions of destructive interference.
Diffraction is a property of waves. Hence the above observations implied that the electrons formed a diffraction pattern on scattering, and that particles could show wave-like properties.
The Numerical Test
Davisson and Germer varied the accelerating potential from 44 V to 68 V and observed a peak in the intensity of the scattered electrons at a scattering angle of 50º for a potential of 54 V.
This peak was the result of constructive interference of the electrons scattered from different layers of the regularly spaced atoms of the nickel crystal — waves reflected from successive atomic layers arriving in step with one another, exactly as in the diffraction of light from a grating.
The real test, however, is quantitative. The de Broglie relation predicts a definite wavelength for 54 V electrons, and the diffraction pattern independently measures a wavelength. If the two numbers agree, the hypothesis stands.
From Eq. (14.7), for \(V = 54\) V, we get
\[\lambda = \frac{1.228}{\sqrt{54}} = 0.167\ \mathrm{nm} \qquad \text{--- (14.8)}\]
From the electron diffraction measurements, the wavelength of the matter waves associated with the electrons was found to be 0.165 nm.
The two values of \(\lambda\), one obtained from the experimental results and the other from the theoretical de Broglie relation, were in close agreement. The Davisson and Germer experiment thus substantiated de Broglie’s hypothesis of wave-particle duality and verified his relation.
Diagram
Fig. 14.6a – Electron diffraction tube: hot cathode and electron beam, thin graphite target, and diffraction rings of radius \(r\) formed on the phosphor screen at a distance L from the target.
In examinations, this section is most often asked as “state the importance of the Davisson and Germer experiment”. The expected answer has two parts: it demonstrated the diffraction of electrons, thereby establishing the wave nature of matter, and the wavelength it measured agreed with the value calculated from the de Broglie relation, thereby verifying that relation quantitatively.
14.7 Wave-Particle Duality of Matter
Two Vocabularies for One Reality
Material particles show wave-like nature under certain circumstances. This phenomenon is known as wave-particle duality of matter.
Frequency \(\omega\) and wave number \(k\) are used to describe waves in classical theories, while mass \(m\) and momentum \(p\) are used to describe particles. These are two separate sets of words, developed for two kinds of objects that were believed to have nothing in common.
Wave-particle duality implies that all moving particles have an associated frequency and an associated wave number, and all waves have an associated energy and an associated momentum. The two vocabularies, in other words, describe the same objects, and the de Broglie relation is the dictionary that translates between them.
We come across the wave-particle duality of matter, due to quantum behaviour, when we are dealing with microscopic objects, of sizes \(\le 10^{-6}\) m. The small order of magnitude of \(h\) sets the scale at which quantum phenomena manifest themselves.
That last sentence explains a great deal. Planck’s constant is about \(6.6 \times 10^{-34}\) J s, an extraordinarily small number, and it sits in the numerator of \(\lambda = h/p\). Any object with an everyday momentum therefore gets an absurdly small wavelength. If \(h\) had been a large number, the world around us would look thoroughly quantum.
Why We Never Notice the Wave Nature of Everyday Objects
If all the material objects in motion have an associated wavelength, and therefore an associated wave, why then do we not talk about the wavelength of a child running with speed \(v\) on a pathway 2 m wide, or of a car moving with speed \(v\) on a road 20 m wide? To understand this, let us try to calculate these quantities.
The wavelengths calculated in Example 14.5 are negligible compared to the size of the moving objects, as well as to the widths of the paths on which the objects are moving.
To feel how negligible, compare \(10^{-36}\) m with the size of an atomic nucleus, about \(10^{-15}\) m. The student’s wavelength is smaller than a nucleus by a factor of a billion billion. There is no obstacle anywhere in the universe narrow enough to diffract it.
Diagram
Fig. 14.7a – A scale line marked with the width of a footpath (metres), the size of an atom (\(10^{-10}\) m), the size of a nucleus (\(10^{-15}\) m) and the de Broglie wavelength of a running student (\(10^{-36}\) m), showing how far off the scale the last one lies.
Therefore the wavelengths associated with macroscopic particles do not play any significant role in our everyday life, and we need not consider their wave nature. Also, the wavelengths for macroscopic particles are so small that they cannot be measured.
On the other hand, if we try to estimate the associated de Broglie wavelength of a moving electron passing through a small aperture of size \(10^{-10}\) m, or of an oxygen molecule in air, we will find it to be significant, as can be seen in the following example.
The wavelength of the electron in the above example is comparable to the size of the hole through which the electron is passing. The wavelength associated with this electron is the same as the size of a helium atom, and more than double the size of a hydrogen atom.
That comparison is the whole point. For the running student, the wavelength was smaller than the path by a factor of \(10^{36}\); for this electron, the wavelength and the hole are of the same size. Under such conditions the electron cannot be treated as a tiny bullet — it will diffract on its way through.
When Newtonian Mechanics Is Still Enough
The photon picture allows transfer of energy and momentum in the same manner as in Newtonian mechanics. The wave nature does not modify that.
Whenever the wavelengths are small compared to the dimensions of slits or obstacles, or are not measurable, we can use Newtonian mechanics. This is a reassuring statement: quantum physics does not throw away the mechanics you have learnt, it restricts it to the domain where it works, which happens to include the whole of ordinary engineering and daily life.
In conclusion, for both electromagnetic radiation and atomic and sub-atomic particles, the particle nature is dominant during their interaction with matter. On the other hand, while travelling through space, particularly when their confinement is of the same order of magnitude as their associated wavelength, the wave nature is dominant.
Seeing with Electrons
You have learnt about the resolving power and the resolution of telescopes and microscopes that use ordinary visible light. The resolution of a microscope is limited by the wavelength of the light used.
The shorter the wavelength of the characterizing probe, the smaller is the limit of resolution of a microscope, that is, the better is the resolution of the microscope. Better resolution can therefore be attained by illuminating the objects to be seen by radiation of smaller wavelengths. No microscope can show clear detail much finer than the wavelength it uses to look, which is why an optical microscope cannot resolve objects much smaller than a few hundred nanometre.
We have seen that an electron can behave as a wave, and its wavelength is much smaller than the wavelength of visible light. The wavelength can be made much smaller still, as it depends on the velocity and the kinetic energy of the electron.
An electron beam accelerated to several keV of energy will correspond to a de Broglie wavelength much smaller than an angstrom, that is, \(\lambda_e \ll 1 \times 10^{-10}\) m. The resolution of such an electron microscope will be several hundred times higher than that obtainable with an optical microscope.
Other advantages of electron microscopes are that (i) electrons do not penetrate matter as visible light or X-rays do, (ii) electron beams can be more easily produced and controlled by electric and magnetic fields than electromagnetic waves, and (iii) electrons can be focused just as light is focused with lenses.
It was proposed in 1925 that atoms in solids can act as diffraction centres for electron waves and can give information about the geometry or structure of a solid, just as X-rays do on getting diffracted by solids. However, it took many years to realize an electron microscope for practical applications. The first electron microscope was developed by Ruska, in Berlin, Germany, in the year 1929.
Notice that the electron microscope is the practical pay-off of this entire chapter. It works only because electrons have a wavelength, only because that wavelength can be shortened by accelerating them, and only because a crystal can diffract them — three results established in Sections 14.5 and 14.6.
14.8 Summary
Quantization of Energy
- Electromagnetic radiation consists of mutually perpendicular oscillating electric and magnetic fields, both perpendicular to the direction in which the wave and its energy travel. Interference, diffraction and polarization are explained by this wave picture.
- The black body radiation spectrum could not be explained by the wave nature of radiation. Planck explained it by proposing that energy is emitted in packets, and that at higher frequencies the energy of a packet is large.
- Planck assumed that atoms behave like tiny oscillators that emit electromagnetic radiation only in discrete packets, \(E = nh\nu\), where \(\nu\) is the frequency of the oscillator and \(n\) is an integer. Emission occurs only when the oscillator jumps from one quantized energy level to another of lower energy.
The Photoelectric Effect
- The photoelectric effect is the emission of electrons from a metal surface when radiation of appropriate frequency is incident on it. A surface that emits electrons on illumination is called a photosensitive surface, and the emitted electrons are called photoelectrons.
- Ultraviolet radiation is necessary for metals like zinc, cadmium and magnesium, while for alkali metals even visible radiation is sufficient.
- The work function \(\phi_0\) of a metal is the minimum energy required to pull an electron out of the metal from its surface. It depends on the metal and on the nature of its surface, and is usually expressed in electron volt (eV).
- Key experimental observations: emission occurs only if \(\nu > \nu_0\), the threshold frequency, whatever the intensity; emission begins within \(10^{-9}\) s, with no time lag; photocurrent increases linearly with intensity; photocurrent saturates at \(I_0\) on increasing the accelerating potential; the saturation current increases proportionately with intensity; \(KE_{max}\) depends on the frequency and the emitter material but not on the intensity; the photocurrent falls to zero at the stopping potential \(-V_0\); and \(V_0\) varies linearly with \(\nu\), the lines for different metals having the same slope but different intercepts.
Failure of the Wave Theory
- The wave theory predicts that energy is absorbed continuously and shared among all surface electrons, so a waiting period of minutes to hours would be needed before emission — of the order of \(4.428 \times 10^{4}\) s, about half a day, for the case worked out in Example 14.1. Emission is in fact instantaneous.
- The wave theory predicts that higher intensity should give more energetic electrons and that any frequency should work if the light is bright enough. Both predictions are contradicted by experiment.
Einstein’s Photon Picture
- Einstein extended Planck’s quantization to all electromagnetic radiation in 1905, proposing that light energy is released or absorbed in bundles called photons, of energy \[E = h\nu \qquad \text{--- (14.1)}\] where \(h = 6.626 \times 10^{-34}\) J s is Planck’s constant. Since \(\nu = c/\lambda\), energy is directly proportional to frequency and inversely proportional to wavelength: \(E = hc/\lambda\).
- A photon gives its whole energy \(h\nu\) to a single electron in one instantaneous collision, and then ceases to exist. Emission occurs if and only if \(h\nu \ge \phi_0\), so the threshold frequency is \(\nu_0 = \phi_0/h\).
- Intensity fixes the number of photons arriving per unit area per unit time, and therefore the number of photoelectrons and the photocurrent. Frequency fixes the energy of each photon, and therefore the maximum kinetic energy of the photoelectrons.
- Einstein’s photoelectric equation: \[KE_{max} = h\nu - \phi_0 \qquad \text{--- (14.2)}\]
- At the stopping potential the most energetic electron just fails to reach the collector, so \(KE_{max} = eV_0\), which gives \[V_0 = \left(\frac{h}{e}\right)\nu - \frac{\phi_0}{e} \qquad \text{--- (14.3)}\] The slope \(h/e\) is the same for all metals; the intercept depends on \(\phi_0\) and hence on the material.
- Millikan’s measurements in 1909 confirmed the value of \(h\) obtained from Eq. (14.3), after which the photon hypothesis was accepted. Einstein and Millikan received Nobel Prizes in 1921 and 1923 respectively.
- Useful numerical checks: wavelength (in Å) × energy (in eV) ≈ 12500, and wavelength (in nm) × energy (in eV) ≈ 1250.
Wave-Particle Duality of Radiation
- Compton confirmed the particle nature of radiation through the scattering of X-rays by electrons. The scattered radiation contains a longer wavelength \(\lambda'\), and the Compton shift is \[\Delta\lambda = \lambda' - \lambda = \frac{h}{m_e c}(1 - \cos\theta)\] where \(\theta\) is the scattering angle. The shift depends only on \(\theta\) and not on the incident wavelength, and cannot be explained by wave theory.
- A photon carries momentum as well as energy, is electrically neutral, is not deflected by electric or magnetic fields, always moves with the speed of light, and has zero rest mass in all frames of reference. In photon–particle collisions, energy and momentum are conserved but the number of photons is not.
- Wave behaviour of light shows up when slit widths or slit separations are smaller than or comparable to the wavelength. Since some phenomena need the wave model and others need the photon model, both must be retained; one dominates in some situations and the other in the rest. This is the wave-particle duality of electromagnetic radiation.
Photocell
- A photocell uses the photoelectric effect to convert light energy into electrical energy. It has a semi-cylindrical photosensitive cathode and a wire loop anode in an evacuated glass or quartz bulb, connected to a high tension battery and a microammeter.
- Photocells are used in control systems, in light meters of photographic cameras, for switching street lights on and off, and — by interrupting a light beam — in counters and burglar alarms.
De Broglie Hypothesis
- For a photon, momentum and wavelength are related by \[p = \frac{E}{c} \qquad \text{--- (14.4)}\] \[p = \frac{E}{c} = \frac{h\nu}{c} = \frac{h}{\lambda} \qquad \text{--- (14.5)}\]
- De Broglie proposed in 1924 that a moving material particle also has a wave associated with it, with \[\nu = \frac{E}{h} \quad \text{and} \quad \lambda = \frac{h}{p} = \frac{h}{mv} \qquad \text{--- (14.6)}\] These are called matter waves, and \(\lambda\) is the de Broglie wavelength. The greater the momentum, the shorter the wavelength.
- In terms of kinetic energy, \(\lambda = \dfrac{h}{\sqrt{2mE_K}}\); and for a charge \(q\) accelerated from rest through a potential difference \(V\), since \(E_K = qV\), \(\lambda = \dfrac{h}{\sqrt{2mqV}}\). This holds for any charged particle, but not when \(V\) is so large that the speed approaches that of light.
- For an electron accelerated through \(V\) volts, \[\lambda\ (\text{in nm}) = \frac{1.228}{\sqrt{V\ (\text{in volts})}} \qquad \text{--- (14.7)}\]
Davisson and Germer Experiment
- An electron beam from an electron gun was scattered by a nickel crystal in an evacuated chamber, and the scattered intensity was measured as a function of the scattering angle \(\theta\).
- The intensity was not uniform in all directions but showed peaks and troughs, that is, a diffraction pattern — a property only of waves.
- On varying the accelerating potential from 44 V to 68 V, a peak was observed at a scattering angle of 50º for 54 V. From Eq. (14.7), for \(V = 54\) V, \[\lambda = \frac{1.228}{\sqrt{54}} = 0.167\ \mathrm{nm} \qquad \text{--- (14.8)}\] while the diffraction measurements gave 0.165 nm. The close agreement verified the de Broglie relation.
- De Broglie received the Nobel Prize in 1929; Davisson, Germer and Thomson shared it in 1937.
Wave-Particle Duality of Matter
- All moving particles have an associated frequency and wave number, and all waves have an associated energy and momentum. Quantum behaviour becomes noticeable for microscopic objects of size \(\le 10^{-6}\) m, the scale being set by the small magnitude of \(h\).
- For macroscopic objects the de Broglie wavelength is far too small to matter: about \(6.63 \times 10^{-36}\) m for a 45 kg student running at 8 km/hr, and about \(3.32 \times 10^{-38}\) m for a 1200 kg car at 60 km/hr.
- For an electron of kinetic energy 100 eV, the wavelength is \(1.23 \times 10^{-10}\) m, or 1.228 Å — comparable to atomic dimensions, so wave effects are significant.
- Whenever wavelengths are small compared with the dimensions of slits or obstacles, or are not measurable, Newtonian mechanics may be used.
- In general, particle nature dominates during interaction with matter, while wave nature dominates during travel through space, particularly when the confinement is of the same order as the associated wavelength.
- Electrons can be freed from a metal by thermionic emission (heating to about 2000 ºC), field emission (fields of about \(10^{6}\) V/m at a metal tip), or photo-electron emission (ultraviolet or visible radiation).
- Because resolution improves as the probe wavelength decreases, electron beams of several keV, with \(\lambda_e \ll 1 \times 10^{-10}\) m, give electron microscopes a resolution several hundred times better than optical microscopes.
14.9 Objective Questions
- Choose the correct answer.
- A photocell is used to automatically switch on the street lights in the evening when the sunlight is low in intensity. Thus it has to work with visible light. The material of the cathode of the photocell is
- zinc
- aluminum
- nickel
- potassium
- Polychromatic (containing many different frequencies) radiation is used in an experiment on photoelectric effect. The stopping potential
- will depend on the average wavelength
- will depend on the longest wavelength
- will depend on the shortest wavelength
- does not depend on the wavelength
- An electron, a proton, an α-particle and a hydrogen atom are moving with the same kinetic energy. The associated de Broglie wavelength will be longest for
- electron
- proton
- α-particle
- hydrogen atom
- If \(N_{Red}\) and \(N_{Blue}\) are the number of photons emitted by the respective sources of equal power and equal dimensions in unit time, then
- \(N_{Red} < N_{Blue}\)
- \(N_{Red} = N_{Blue}\)
- \(N_{Red} > N_{Blue}\)
- \(N_{Red} \approx N_{Blue}\)
- The equation \(E = pc\) is valid
- for all sub-atomic particles
- is valid for an electron but not for a photon
- is valid for a photon but not for an electron
- is valid for both an electron and a photon
14.10 Check Your Concepts
Why does a threshold frequency exist at all? Explain, using the photon picture, why radiation below this frequency fails to eject electrons no matter how long it is allowed to fall on the metal.
A certain metal shows photoelectric emission when blue light falls on it, but not when red light of much higher intensity is used. Predict what will happen if (a) ultraviolet radiation and (b) green light are used instead, and justify each prediction.
The wave theory of light predicts a waiting period of several hours before photoemission begins, while experiments show emission within \(10^{-9}\) s. Explain the reasoning behind the wave-theory prediction, and state exactly which assumption of that reasoning is abandoned in Einstein’s picture.
Why is the stopping potential in a photoelectric experiment independent of the intensity of the incident radiation, while the saturation current is not?
Graphs of stopping potential against incident frequency are plotted for three different emitter metals and are found to be parallel straight lines with different intercepts. What physical quantity does the common slope represent, and what information is carried by the differing intercepts?
The cathode of a photocell meant to work in visible light is made of an alkali metal rather than of tungsten or nickel. Explain the reason using the idea of work function.
Why is the window of a photoelectric tube made of quartz rather than of ordinary glass, and why is the tube evacuated?
Explain, in terms of photons and electrons, why the photocurrent first rises with the accelerating potential and then becomes constant. What decides the value at which it becomes constant?
Photons are not deflected by electric or magnetic fields, but photoelectrons are. Explain why, and describe one experimental situation in which this difference is put to use.
In Compton’s experiment, the scattered X-rays contain a wavelength longer than the incident one. Explain why the wave theory of radiation cannot account for this, and what the photon picture assumes instead.
If Planck’s constant is allowed to approach zero in the Compton shift relation, the result predicted by wave theory is recovered. Why is such a limiting check considered a good test of a new theory?
The rest mass of a photon is zero, yet it carries momentum. Explain how this is possible, and state whether the relation \(p = E/c\) may be applied to an electron.
De Broglie’s hypothesis could not be tested using an ordinary slit cut in a metal sheet. Explain why a crystal had to be used instead, and what property of crystals makes them suitable.
In an electron diffraction tube, the accelerating voltage between the cathode and the anode is increased. State whether the diffraction rings on the screen become larger or smaller, and justify your answer using the de Broglie relation.
Why can an electron microscope resolve much finer detail than an optical microscope? State two further advantages of using an electron beam as a probe.
A proton and an electron are accelerated from rest through the same potential difference. Without doing any calculation, state which of them will have the longer de Broglie wavelength, and explain your reasoning.
Every moving object has a de Broglie wavelength, yet a cricket ball never shows diffraction. Explain, referring both to the magnitude of Planck’s constant and to the condition under which diffraction becomes observable.
State the circumstances under which the particle nature of an entity dominates and those under which its wave nature dominates. Give one experiment from this chapter as evidence for each case.
A student says that making a beam of red light brighter increases the energy of each of its photons. Identify the error in this statement and explain what actually increases.
Explain why the photoelectrons emitted from a metal surface do not all have the same kinetic energy, even though every incident photon of a monochromatic beam carries exactly the same energy.
Answer in brief.
- What is photoelectric effect?
- Can microwaves be used in the experiment on photoelectric effect?
- Is it always possible to see photoelectric effect with red light?
- Using the values of work function given in Table 14.1, tell which metal will require the highest frequency of incident radiation to generate photocurrent.
- What do you understand by the term wave-particle duality? Where does it apply?
Explain the inverse linear dependence of stopping potential on the incident wavelength in a photoelectric effect experiment.
It is observed in an experiment on photoelectric effect that an increase in the intensity of the incident radiation does not change the maximum kinetic energy of the electrons. Where does the extra energy of the incident radiation go? Is it lost? State your answer with explanatory reasoning.
Explain what do you understand by the de Broglie wavelength of an electron. Will an electron at rest have an associated de Broglie wavelength? Justify your answer.
State the importance of Davisson and Germer experiment.
14.11 Practice with Numericals
(Take \(h = 6.63 \times 10^{-34}\) J s, \(c = 3 \times 10^{8}\) m s\(^{-1}\), \(e = 1.6 \times 10^{-19}\) C, mass of electron \(= 9.11 \times 10^{-31}\) kg unless stated otherwise.)
Calculate the energy of a photon of wavelength 5000 Å, expressing your answer both in joule and in electron volt.
A lamp radiates 60 W of light energy, all of wavelength 550 nm. How many photons does it emit per second?
Using the work function of calcium given in Table 14.1, calculate its threshold frequency and threshold wavelength.
Radiation of wavelength 3000 Å is incident on a sodium surface of work function 2.4 eV. Find the maximum kinetic energy of the emitted photoelectrons in eV and the stopping potential.
In a photoelectric experiment, light of wavelength 2500 Å produces a stopping potential of 1.5 V. Calculate the work function of the emitter material in eV.
The threshold frequency of a metal is \(5.5 \times 10^{14}\) Hz. Radiation of frequency \(7.5 \times 10^{14}\) Hz falls on its surface. Calculate the maximum kinetic energy of the photoelectrons and their maximum speed.
A graph of stopping potential against incident frequency for a certain metal cuts the frequency axis at \(4.5 \times 10^{14}\) Hz. Determine the work function of the metal in eV and the stopping potential for incident radiation of frequency \(9.0 \times 10^{14}\) Hz.
A photocell carries a saturation current of \(7.2\ \mu\)A. Calculate the number of photoelectrons reaching the anode per second.
X-rays are scattered by free electrons and observed at a scattering angle of \(60^\circ\). Calculate the Compton shift, taking the mass of the electron as \(9.11 \times 10^{-31}\) kg.
An X-ray photon has a wavelength of 1.0 Å. Calculate its energy in eV and its momentum.
An electron is accelerated from rest through a potential difference of 200 V. Calculate its de Broglie wavelength in nanometre.
An α-particle of mass \(6.64 \times 10^{-27}\) kg and charge \(+2e\) is accelerated from rest through a potential difference of 500 V. Calculate its de Broglie wavelength.
Calculate the speed of an electron whose de Broglie wavelength is 0.5 Å, and find the kinetic energy of this electron in eV.
A cricket ball of mass 0.15 kg moves with a speed of 30 m s\(^{-1}\). Calculate its de Broglie wavelength and comment on whether its wave nature can be detected.
A thermal neutron has kinetic energy 0.025 eV. Taking the mass of the neutron as \(1.67 \times 10^{-27}\) kg, calculate its de Broglie wavelength and state whether it is suitable for the diffraction study of crystals whose atomic spacing is about 2 Å.
A proton and an α-particle are accelerated from rest through the same potential difference. Taking the mass of the α-particle as four times that of the proton and its charge as twice that of the proton, find the ratio of their de Broglie wavelengths.
A photon and an electron both have a wavelength of 1.5 Å. Calculate the energy of the photon in eV and the kinetic energy of the electron in eV, and comment on the large difference between the two values.
Ultraviolet radiation of wavelength 2000 Å falls on a silver surface whose work function is 4.3 eV. Calculate the maximum kinetic energy of the photoelectrons in eV and the de Broglie wavelength associated with the fastest photoelectron.
What will be the energy of each photon in monochromatic light of frequency \(5 \times 10^{14}\) Hz?
Observations from an experiment on photoelectric effect for the stopping potential by varying the incident frequency were plotted. The slope of the linear curve was found to be approximately \(4.1 \times 10^{-15}\) V s. Given that the charge of an electron is \(1.6 \times 10^{-19}\) C, find the value of Planck’s constant \(h\).
The threshold wavelength of tungsten is \(2.76 \times 10^{-5}\) cm. (a) Explain why no photoelectrons are emitted when the wavelength is more than \(2.76 \times 10^{-5}\) cm. (b) What will be the maximum kinetic energy of electrons ejected in each of the following cases: (i) if ultraviolet radiation of wavelength \(\lambda = 1.80 \times 10^{-5}\) cm and (ii) radiation of frequency \(4 \times 10^{15}\) Hz is made incident on the tungsten surface.
Photocurrent recorded in the micro ammeter in an experimental set-up of photoelectric effect vanishes when the retarding potential is more than 0.8 V if the wavelength of incident radiation is 4950 Å. If the source of incident radiation is changed, the stopping potential turns out to be 1.2 V. Find the work function of the cathode material and the wavelength of the second source.
Radiation of wavelength 4500 Å is incident on a metal having work function 2.0 eV. Due to the presence of a magnetic field B, the most energetic photoelectrons emitted in a direction perpendicular to the field move along a circular path of radius 20 cm. What is the value of the magnetic field B?
Given the following data for incident wavelength and the stopping potential obtained from an experiment on photoelectric effect, estimate the value of Planck’s constant and the work function of the cathode material. What is the threshold frequency and corresponding wavelength? What is the most likely metal used for the emitter?
| Incident wavelength (in Å) | 2536 | 3650 |
|---|---|---|
| Stopping potential (in V) | 1.95 | 0.5 |
Calculate the wavelength associated with an electron, its momentum and speed (a) when it is accelerated through a potential of 54 V, (b) when it is moving with kinetic energy of 150 eV.
The de Broglie wavelengths associated with an electron and a proton are same. What will be the ratio of (i) their momenta (ii) their kinetic energies?
Two particles have the same de Broglie wavelength and one is moving four times as fast as the other. If the slower particle is an α-particle, what are the possibilities for the other particle?
What is the speed of a proton having de Broglie wavelength of 0.08 Å?
In nuclear reactors, neutrons travel with energies of \(5 \times 10^{-21}\) J. Find their speed and wavelength.
Find the ratio of the de Broglie wavelengths of an electron and a proton when both are moving with the (a) same speed, (b) same energy and (c) same momentum. State which of the two will have the longer wavelength in each case.