Quantum II Notes

 2013 Yeow Kok Han Page 1 21 2 eam v eV ---- (Eq 1.1) QQuuaannttuumm PPhhyyssiiccss IIII 11 XX--rraayy SSppeeccttrraa SSeett--uupp The key parts of the x-ray tube are shown in Fig. 1.1 . The heated cathode emits electrons which are accelerated by a high voltage of tens to hundreds of kV. A small portion (1%) of the electrons’ kinetic energies is converted to x-rays when they collide with the tungsten atoms. T he bulk of the kinetic energies become heat that is conducted away by a good conductor or with the use of circulating cooling liquid. An electron reaching t he tungsten target gained KE at the expense of electric PE lost, where Va is the accelerating p.d, me and e are the electron mass and charge respectively. A typical x -ray spectrum can be seen in Fig. 1.2. It is actually a superposition of a continuous spectrum and a line spectrum . These are produced by two different mechanisms at the atomic level. The spikes from the line spectrum occur at two specific wavelengths that are characteristic of the material used as the target. MMeecchhaanniissmm 11 -- AAcccceelleerraattiioonn ooff CChhaarrggee The high speed s of the electrons allow them to penetrate the target atoms. The size of an atom is  10-10 m while the size of a nucleus is  10-15 m and the electron is even smaller. If the nucleus were 1 mm in size, then the atom would be about 100 m across . However, in the vast empty space inside the atom, the elect ric field of the nucleus attracts an incoming electron and curves it around. The strong force and therefore acceleration(centripetal plus linear) results in radiation of an energetic x-ray photon , slowing down the electron in the process. Charged particles are accelerated to high speeds before colliding with target atoms to produce x- rays. The KE gained ½ mv2 = eVa for electrons of charge e. The resulting x- ray spectrum is made up of two parts - a continuous spectrum and a discrete spectrum. The mechanism for the production of continuous spectrum is based on the fact that accelerated or decelerated charges radiate electromagnetic energy. Whenever a charge particle is accelerated or decelerated, it radiates electromagnetic energy. Also, greater acceleration leads to greater rate of radiation. High voltage  10 kV to 200 kV cathode tungsten anode + - copper to conduct heat away electron beam X-rays vacuum glass tube Fig. 1.1 K  relative intensity Fig. 1.2 electron nucleus X-ray photon Fig. 1.3

 2013 Yeow Kok Han Page 2 The x-rays produced in this way are called ‘Brehmsstrahlung’ in German for ‘braking radiation’. Each incoming electron can potentially produce a few such x-ray photons of different energies by interacting with a series of different atoms. Also, an incoming electron can approach nuclei with different proximity and thus experience different forces and emit photons of different energies. A beam of incoming electrons will thus produce photons with a continuous range of energies or frequencies resulting in a spectrum as shown in Fig. 1.4 The maximum photon energy Ep is equal to the maximum amount of KE(Eq. 1.1) an electron can lose at one go. Hence Ep = eVa but Ep = hfmax = hc/min hc/min = eVa min = hc/eVa where the minimum wavelen gth of the continuous spectrum gets smaller as the accelerating p.d. gets larger. In other words, the sp ectrum extends more to the left as va increases as shown in Fig. 1.5. MMeecchhaanniissmm 22 -- DDiissllooddggiinngg IInnnneerrmmoosstt EElleeccttrroonn An incoming electron can also knock o ut an electron from the inner orbits of a target atom. Th e vacancy will be quickly filled by an electron from one of the outer orbits with higher energy levels, thus emitting an x-ray photon. Recall from Quantum I that jumping of valence (outermost) electron from higher to lower energy levels give rise to discrete emission spectrum. Similarly, here the finite number of possible transitions leads to a discrete number of spectral lines as in Fig. 1.7 Sometimes the L & L lines may not be present because those transitions do not correspond to x -rays but other lower frequency part of the EM spectrum. There is a shortest wavelength for the continuous spectrum because the greatest frequency or energy of a photon occurs when all the KE of an incoming electron is given to that photon. The second mechanism for x-ray production is due to the knocking out of an electron from the innermost orbits. X-ray photons of discrete energies are then produced due to electrons from outer orbits jumping to fill up the vacancy. Energy gaps between inner orbits are greater than between outer orbits, thus accounting for higher photon energies.  vacancy     incoming electron knocks out an innermost electron.  dislodged electron.  L shell electron may jump in to fill the vacancy (give K spectral line) or  M shell or other outer shell electron may jump in to fill the vacancy. K L M Fig. 1.6 K K  relative intensity Fig. 1.4 min Fig. 1.5 25 kV 20 kV 10 kV  relative intensity Fig. 1.7  Intensity K K L L K L M N K K L L  

 2013 Yeow Kok Han Page 3 CCoommbbiinneedd DDiissccrreettee aanndd CCoonnttiinnuuoouuss SSppeeccttrruumm Fig. 1.8 shows that when accelerating p.d. is too low, the incoming electrons do not have enough energy to knock out the innermost electrons and so the spectrum only has the continuous part without the discrete contribution known as the characteristic x-rays. At a high eno ugh p.d. of 25 kV, the characteristic spikes are present. The characteristic radiation is so called because the exact wavelengths and spacing of the lines or spikes are characteristic of the specific kind of target atoms. The key difference between Fig. 1. 8 and Fig. 1.9 is that the horizontal axis is photon wavelength and energy respectively. Fig. 1.9 is similar to a plot against frequency (E = hf). 22 PPootteennttiiaall WWeellllss aanndd BBaarrrriieerrss Consider a depression in the ground (Fig. 2.1). If we take gravitational PE, UG, to be zero at ground level , then at any depth h, UG = mgh. Gravitational potential  = UG/m,   at each h is equal to gh. If we plot UG or  versus horizontal position x, the result is shown in Fig. 2.2 . The plot is ca lled a gravitational potential energy well or just potential well. Consider a roller coaster ride as shown in Fig. 2.3. If the car were to start at rest below point P instead of above, it would not be able to reach S. We call the section QR S a gravitational potential barrier . For the car to cross the potential barrier from the left side, it must have total energy (GPE + KE) greater than its GPE at R. Potential wells and barriers can also be electromagnetic instead of gravitational. It all depends on the t ype of force or field involved. In general, a potential barrier is The discrete spectrum is also known as characteristic spectrum because the exact wavelengths and spacing of the spikes are characteristic of the kind of target atoms. A potential barrier is a region, in a force field, with higher potential than its surrounding such that an object requires energy to pass through it. a region, in a force field, with higher potential than its surrounding such that an object requires energy to pass through it. Fig. 2.1 0 J h GPE = mgh UG or  x Fig. 2.2 0 40 80 120 photon energy / keV relative intensity 120 kV 100 kV 90 kV Fig. 1.9 25 kV 20 kV 10 kV 0 K 10 20 30  / nm 2 4 rela