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Ultraviolet light of 0.1 um wavelength is incident upon a metal. Which of the metals listed in table 6.2 will emit electrons in response to the input light? Show your reasoning..

Ultraviolet light of 0.1 um wavelength is incident upon a metal. Which of the metals listed in table 6.2 will emit electrons in response to the input light? Show your reasoning..

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Ultraviolet light of 0.1 um wavelength is incident upon a metal. Which of the metals listed in table 6.2 will emit electrons in response to the input light? Show your reasoning..

 

88 Table 6.2 Work functions of metals Metal Li Na K Cs Cu Ag Au Mg Ca Ba Al Cr Mo Ta W Co Ni Pt Work function (eV) 2.48 2.3 2.2 1.9 4.45 4.46 4.9 3.6 3.2 2.5 4.2 4.6 4.2 4.2 4.5 **** 668 4.9 5.3 The free electron theory of metals term in the denominator. We are left then with some Gaussian functions, whose integrals between 100 can be found in the better integral tables (you can derive them for yourself if you are fond of doing integrals). This leads us to 4xmkTx/kTep/2mT dpx- N(px)dpx= (6.36) Substituting eqn (6.36) into (6.30) and assuming that r(px) = r is independent of px, which is not true but gives a good enough approximation, the integration can be easily performed, leading to J = Ap(1-r)T²e/KT where 4лemk² h³ (6.37) Ao = = 1.2 x 106 Am ² K-². This is known as the Richardson (Nobel Prize, 1928) equation. The most important factor in eqn (6.37) is exp(-/kT), which is strongly dependent both on temperature and on the actual value of the work function. Take, for example, tungsten (the work functions for a number of metals are given in Table 6.2), for which 4.5 eV and take T = 2500 K. Then, a 10% change in the work function or temperature changes the emission by a factor of 8. The main merit of eqn (6.37) is to show the exponential dependence on temperature, which is well borne out by experimental results. The actual numerical values are usually below those predicted by the equation, but this is not very surprising in view of the many simplifications we had to introduce. In a real crystal, is a function of temperature, of the surface conditions, and of the directions of the crystallographic axes, which our simple model did not take into account. (6.38) There is one more thing I would like to discuss, which is really so trivial that most textbooks do not even bother to mention it. Our analysis was one for a piece of metal in isolation. The electron current obtained in eqn (6.37) is the current that would start to flow if the sample were suddenly heated to a temperature T. But this current would not flow for long because, as electrons leave the metal, it becomes positively charged, making it more difficult for further electrons to leave. Thus, our formulae are valid only if we have some means of replenishing the electrons lost by emission. That is, we need an electric circuit like the one in Fig. 6.3(a). As soon as an electron is emitted from our piece of metal, another electron will enter from the circuit. The current flowing can be measured by an ammeter. A disadvantage of this scheme is that the electrons travelling to the electrode will be scattered by air; we should really evacuate the place between the emitter and the receiving electrode, making up the usual cathode-anode configuration of a vacuum tube. This is denoted in Fig. 6.3(b) by the envelope shown. The electrons are now free to reach the anode but also free to accumulate in the vicinity of the cathode. This is bad again because by their negative charge they will compel many of their fellow electrons to interrupt their planned journey to the anode and return instead to the emitter. So again we do not measure the 'natural' current.
Transcribed Image Text:88 Table 6.2 Work functions of metals Metal Li Na K Cs Cu Ag Au Mg Ca Ba Al Cr Mo Ta W Co Ni Pt Work function (eV) 2.48 2.3 2.2 1.9 4.45 4.46 4.9 3.6 3.2 2.5 4.2 4.6 4.2 4.2 4.5 **** 668 4.9 5.3 The free electron theory of metals term in the denominator. We are left then with some Gaussian functions, whose integrals between 100 can be found in the better integral tables (you can derive them for yourself if you are fond of doing integrals). This leads us to 4xmkTx/kTep/2mT dpx- N(px)dpx= (6.36) Substituting eqn (6.36) into (6.30) and assuming that r(px) = r is independent of px, which is not true but gives a good enough approximation, the integration can be easily performed, leading to J = Ap(1-r)T²e/KT where 4лemk² h³ (6.37) Ao = = 1.2 x 106 Am ² K-². This is known as the Richardson (Nobel Prize, 1928) equation. The most important factor in eqn (6.37) is exp(-/kT), which is strongly dependent both on temperature and on the actual value of the work function. Take, for example, tungsten (the work functions for a number of metals are given in Table 6.2), for which 4.5 eV and take T = 2500 K. Then, a 10% change in the work function or temperature changes the emission by a factor of 8. The main merit of eqn (6.37) is to show the exponential dependence on temperature, which is well borne out by experimental results. The actual numerical values are usually below those predicted by the equation, but this is not very surprising in view of the many simplifications we had to introduce. In a real crystal, is a function of temperature, of the surface conditions, and of the directions of the crystallographic axes, which our simple model did not take into account. (6.38) There is one more thing I would like to discuss, which is really so trivial that most textbooks do not even bother to mention it. Our analysis was one for a piece of metal in isolation. The electron current obtained in eqn (6.37) is the current that would start to flow if the sample were suddenly heated to a temperature T. But this current would not flow for long because, as electrons leave the metal, it becomes positively charged, making it more difficult for further electrons to leave. Thus, our formulae are valid only if we have some means of replenishing the electrons lost by emission. That is, we need an electric circuit like the one in Fig. 6.3(a). As soon as an electron is emitted from our piece of metal, another electron will enter from the circuit. The current flowing can be measured by an ammeter. A disadvantage of this scheme is that the electrons travelling to the electrode will be scattered by air; we should really evacuate the place between the emitter and the receiving electrode, making up the usual cathode-anode configuration of a vacuum tube. This is denoted in Fig. 6.3(b) by the envelope shown. The electrons are now free to reach the anode but also free to accumulate in the vicinity of the cathode. This is bad again because by their negative charge they will compel many of their fellow electrons to interrupt their planned journey to the anode and return instead to the emitter. So again we do not measure the 'natural' current.
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