Experiment 1 Purpose The photoelectric effect is a key experiment in modern physics. In this experiment light is used to excite electrons that (given sufficient energy) can escape from a material producing a current proportional to the intensity of the light. The concept of the photon that light comes in discrete packets of energy (E = hf, where h is Planck s constant) is essential to understand the phenomena. The purpose of this experiment is to determine the maximum kinetic energy of electrons photo-emitted from a metal surface for several light frequencies. This data is then used to determine Planck s constant. I ph Figure 1: A schematic of the photoelectric effect. A photon absorbed by a metal can cause an electron to escape producing a photocurrent, I ph. 2 Theory Electrons are held within a material by Coulomb interactions, the same force that binds electrons and protons in hydrogen (and other atoms). Most materials are electrically neutral (or close to neutral, i.e. not charged). So if one were to remove an electron from a material, the material would become charged. A charged state takes energy to create; one needs to do work against the Coulomb forces, the attractive force between unlike charges. The minimum energy to remove one electron from a material is known as its work function, and given the symbol φ. Light can transfer energy and momentum to electrons in a material. If the energy transferred is larger than the material s work function (hf φ) then an electron can be liberated from the material, as shown schematically in Fig. 1. An electron energy diagram representing the metal and vacuum is shown in the figure below (Fig. 2). The photoemission process involves exciting an electron in the conduction band with energy E CB E E F to an energy (of at least) E F + φ. Electrons with the largest initial energy have the largest kinetic energy when they escape from the metal; these electrons have an initial energy equal to the Fermi energy. 1
E E F metal vacuum E CB 0 z Figure 2: Electron energy versus position. The metal surface is at z = 0. Electrons in a metal occupy a band of states called the conduction band, starting from E CB, the bottom the band to E F, the Fermi energy. The Fermi energy (or Fermi level) is the highest occupied electron energy state at zero temperature. An energy equal to the workfunction, φ, is needed to remove an electron at the Fermi level. E hf E F metal vacuum E CB 0 z Figure 3: Schematic of the photoemission process. An electron in the conduction band absorbs a photon and is excited to an energy larger than E F + φ, i.e., it acquires enough energy to overcome the work function, the barrier to escaping from the material. The kinetic energy of this electron in vacuum is KE = E i E F +hf φ, where E i is the electron s initial energy. As there is a distribution of electron initial energies there will be a distribution of electron kinetic energy. Electron s excited from the Fermi energy will have the largest kinetic energy: KE max = hf φ. The photoemission process is shown schematically in Fig. 3. A photon excites an electron in the conduction band from an initial energy E i to an energy E = E i + hf. The kinetic energy of the electron in vacuum is KE = E i E F + hf φ (Note: that it has this form because the work function is measured from the Fermi energy, E F.) The maximum kinetic energy, KE max is when the initial electron energy is E F : KE max = hf φ (1) Electrons with energy less than E F will have less kinetic energy; there will thus be a distribution of electron energies up to the maximum kinetic energy KE max. Exciting an electron leaves an empty electron state in the metal s conduction band, which is represented by the empty circle in Fig. 3. The absence of an electron is called a hole. This hole will be filled by another electron, from a current that results from the photoemission process, the photocurrent I ph. A current must flow in order for the metal to remain electrically neutral. 2
The circuit used in photoemission experiments is shown in Fig. 4. cathode UV light anode I ph A ammeter + - V ca Figure 4: Circuit used in photoemission experiments. The cathode is the metal from which electrons are emitted. The anode collects the photoemitted electrons. The photocurrent I ph is measured with a sensitive ammeter. A variable voltage between the cathode and anode V ca is used to accelerate (or decelerate) the photoemitted electrons. The metal surface from which electrons are emitted is called the cathode; the surface at which the electrons are collected is called the anode. Both surfaces are in vacuum so that electrons flow without collisions. An adjustable voltage V ca is applied between the cathode and anode, and the photocurrent I ph is measured by a sensitive ammeter. When V ca has the polarity shown in Fig. 4 it creates an electric field that tends to slow down the photoelectrons as they move from the cathode to the anode. For the reverse polarity photoelectrons are accelerated. For light of fixed intensity and frequency (i.e. monochromatic light), the photocurrent I ph will vary with the voltage as shown in Fig. 5. I ph V s V ca Figure 5: Variation of the photocurrent as a function of the voltage between the cathode and the anode. At a voltage V s, the stopping voltage, the photocurrent is zero. When V ca is negative, essentially all the photoemitted electrons are collected by the anode. However, if V ca is positive, electrons with kinetic energy less than ev ca cannot reach the anode (e = 1.6 10 19 C is the electron charge). The stopping voltage V s is the cathodeanode voltage that is just sufficient to prevent the most energetic photoemitted electrons, those with energy KE max, from reaching the anode. In the experiment one measures V s and from this measurement one can determine the photoelectron s maximum kinetic energy as ev s = KE max. Using Eqn. 1 one has V s = h e f φ e. (2) 3
The stopping voltage V s is determined for several different light frequencies f and then V s is plotted versus f. As can be seen from Eqn. 2 this is predicted to give a straight line with a slope of h/e, Planck s constant divided by the electron charge. The intercept of the line gives the work function of the metal, φ/e. Momentum conversation. One may wonder why we did not consider (or mention) momentum conservation on our analysis of the photoelectric effect. After all we know that photons carry energy and momentum p photon = hf/c. The electron s acquire a momentum opposite the direction of the incoming photon; that is, the photon s momentum would tend to push the electron into the material instead of out. The reason momentum conversation is not considered in the photoemission process is that electron s initial momentum typically is far larger than the photon s momentum. The momentum of an electron near the Fermi energy is of order p 2 electron 2m ee F, where m e is the electron s mass. The Fermi energy of a metal is about 4 ev. Show that the momentum of a UV photon is much smaller than the momentum of an electron at the Fermi level in a metal. Figure 6: A photon transfers momentum to an electron. Electron energy distribution. The Fermi energy separates occupied and unoccupied states at zero temperature. However, electrons can have energies greater than E F when the temperature is finite; when the temperature is finite there is no longer a sharp division between occupied and unoccupied electron states. This is illustrated in Fig. 7. The probability of an electron state being occupied f(e) is plotted versus energy. f(e) is called the Fermi function. The blue curve shows the electron state occupancy at zero temperature: states with energy less than E F are occupied and those with energy less than E F are empty. At finite temperature the boundary is not as sharp, as illustrated by the purple and red curves; some states with energies larger than E F are occupied. The occupancy goes from 1 to zero in an energy range 3.5kT, where k is Boltzmann s constant and T is the temperature. Show that the maximum kinetic energy of photoemitted electrons is much larger than 3.5kT in this experiment. 3 Additional Considerations f(e) (E EF )/kt Figure 7: Electron state occupancy versus energy. The above analysis has neglected two phenomena that can influence the experimental results. The first is associated with the fact that different metals have different work functions. (In fact, the work function can also depend on how the surface of a material is prepared; it can be different even for the same materials.) In this experiment the anode and cathode have different work functions. Typically the anode has a larger work function than the cathode. If the two materials were to be put in electrical contact (such as, by connecting a wire between them), electrons would flow from the lower work metal to the higher work function metal (why?) and a voltage V c would be established between the materials, known as the contact 4
E hf c a E F E CB metal cathode vacuum metal anode Figure 8: Contact potential: A difference in workfunction between the anode and the cathode leads to a larger energy barrier in the photoemission process. z potential; the higher work function metal would become negatively charged with respect to the lower work function at which point no there would be no further charge flow. The contact potential is ev c = φ a φ c. This potential adds to the potential barrier that electron s must be overcome by photoemitted electrons to reach the anode. As a result Eq. 2 becomes V s = h e f φ c e V c. (3) The slope of the line V s versus f is unchanged, but the intercept is different, it is no longer only the workfunction of metal from which electron s are emitted, The second effect is that when V ca is positive (with the sign convention of Fig. 4), electrons emitted by the anode will be accelerated to the cathode. We might expect that this current would be negligible, as the anode temperature is far too low for thermionic emission. However, some photoelectric emission is possible possible from the anode. Fig. 9 shows an example of the photocurrent as a function of the stopping potential. The solid curve shows I ph Forward current Reverse current V s V ca Figure 9: Forward (cathode to anode) and reverse (anode to cathode) photocurrent. The net photocurrent, indicated by the solid line, is the sum of the these two currents. the measured total photocurrent. This is the sum of the forward current (electrons going from cathode to anode) and the reverse current (electrons going from anode to cathode). 5
The reverse current saturates and becomes a straight line that is not quite horizontal. As indicated in Fig. 9, a reasonable place to choose the stopping potential is where the solid line stops having curvature and becomes straight. You should follow this procedure in analyzing your data. Determining the exact stopping potential is difficult. You should also take several sets of data and average your results. 4 Experimental Tasks 1. Calculate the light frequencies f dependence on the spectrometer angle. 2. Experimentally determine the stopping voltage V s for different light frequencies and plot it versus the light frequency f. 3. Determine Planck s constant from the dependence of stopping voltage V s on the light frequency f. 5 Experimental Procedures 5.1 Description The photoelectric effect is demonstrated in a photocell: a cell in which the cathode is irradiated with a beam of light of frequency f; a potentiometer is used to apply a voltage V ca on the anode, which is positive or negative with respect to the cathode; a voltmeter is used to measure this voltage; a micro-ammeter is used to measure the photoelectric current I ph. 5.1.1 Comments The amplifier input has a resistance of 10,000 Ohm. If the amplifier is set to a gain of 10 4 so that a 1 V output corresponds to 0.1 mv (10 4 V) at the input and thus a current of 10 na. Fig. 10 is an example reading the vernier scale at the turning knuckle. Figure 10: Example of reading the vernier scale: The next lowest mark near the zero mark is 15, the next marks that align occur are at 1.5, so the angle reads 16.5 6
5.1.2 Setup Set up the two optical benches with the turning knuckle so that the arrangement stands firmly on the table and the right optical bench can be turned. Position the lamp at 9.0 cm, slit at 34.0 cm and first 100 mm-lens at 44.0 cm from the left end of the left optical bench and turn on the lamp. Set the slit width to be the same as the photocell entrance slit width. Adjust the lamp holder to focus the light coming out of the lamp on to plane of the slit. Move the lens such that the light is collimated after the lens. Insert the grating holder in the center of turning knuckle. Align the vertical grating lines vertical by observing the spectra on index cards. Place the photocell to the right end of the right optical bench. Install the other 100 mm lens in front of the photocell and focus the light rays from the illuminating slit onto the entrance of the photocell. The angle at which all the non diffracted light enters the photocell is zero. Wire up the apparatus as illustrated in Fig.6. Note: The negative terminal of the DC power supply connects across the rheostat. Not to the adjustable resistance end of the rheostat. Set the measuring amplifier to low drift mode, and an amplification to 10 5 and time constant 0.3 s. With no connection on the input set the amplifier output voltage to zero with the zeroing button. Set the DC power supply on the potentiometer to 3 V, current to 1 A. Observe the amplifier output which is proportional to photo current in dependence on photocell bias voltage. Measure the bias voltage for zero current at different angles in the first order diffraction spectrum of the lamp for angles between 13 to 25. Above angles of 21, use the red filter to block second order UV light, which disturbs the measurements. 7
Figure 11: Circuit diagram for the experiment 5.2 Analysis Calculate the light frequency f on dependence of spectrometer angle. The frequency of light irradiating the photocell is determined using the following equations: d sin α = nλ (4) α = arcsin(λ/d) (5) α is the spectrometer angle, d is the slit separation of the grating (here: 1/600 mm), λ is the wavelength of emitted light and order of diffraction n is 1 in this case. The light frequency f can be calculated from wavelength λ by f = c/λ with speed of light c = 299, 792, 458 m/s. Determine the stopping voltage V s experimentally for different light frequencies and plot it versus the light frequencies f. Inside the photocell, a cathode with special low-work function coating is situated together with a metal anode in a vacuum tube. If a photon of frequency f strikes the cathode 8
with sufficient energy then an electron can be liberated from the cathode material (external photoelectric effect). If the emitted electrons reach the anode, they are absorbed by the anode work function and the result is a photocurrent. Figure 12: The photoelectric current intensity I ph as a function of the bias voltage at different frequencies of the irradiated light Figure 13: The photoelectric current intensity I ph as a function of the bias voltage at different light intensities (at fixed wavelength: 436 nm). Calculate Planck s constant from the dependence of stopping voltage on the light frequency. The calculated value may deviate ±25% from the literature value: h = 6.62 10 34 Js. 9
Figure 14: Stopping voltage V s as a function of the irradiated light frequencies. There is a negative current for higher bias voltages. As discussed above, this current is due to the photocurrent flowing from anode to cathode. The number of electrons flowing from the anode to the cathode depends on light frequency in a different way than for the cathode. However, it can be assumed, that the intensity and wavelength sensitivity of the reverse photoelectron current (anode to cathode) is different from the one of the larger cathode to anode electron current. So the zero point shift versus light intensity due to this effect is different for different wavelengths making the zero point of the I ph vs V ca characteristic curve of the photocell a not very reliable measure. The overall reverse current can nevertheless be regarded as small because of the lower work function of the cathode compared to the anode. 10