1 Quantum Physics Objective: 1. To measure the wavelengths of visible light emitted by atomic hydrogen and verify the measured wavelengths against those predicted by quantum theory. To identify an unknown element through its emission spectra. To examine an absorption spectrum. 2. To observe and measure the energy levels and wavelengths of light fluoresced from Quantum Dots semiconductor material acting as an infinite square well (particle in a box). Apparatus: 1) Optical spectrometer, spectral tubes, power supply, incandescent lamp, bottles of dyed water, elevating jack or block. 2) Electronic spectrometer, quantum dots in suspension, 400nm LED light source, fiber-optic cable (if available), USB cable, Mac computer with Logger Pro software, graphing calculator (or Open Office spreadsheet application)
2 Theory Atomic Spectra Using a diffraction grating spectrometer to separate the colors of light emitted from hot, glowing hydrogen gas, you see bright emission lines and not a continuous spectrum (rainbow) of colors. Observed (inverse) wavelengths of the lines are given by: 1 =0.01097 1 n l 2 1 n u 2 nanometers 1 Equation 1 I n 4 3 2 1 The numerical constant, 0.01097, is called the Rydberg constant and the n's are principal quantum numbers. n l (lower level) is smaller than n u (upper level); both n s are integers equal to 1,2, 3... By trying various values for n u and n l, we find that the only transitions we see with our eyes have n l = 2 and n u = 3,4,5,---. This set of visible transitions with n l = 2 is known as the Balmer series. These are shown in the figure above. Each transition down to a lower energy level (orbit) emits a photon of energy equal to the difference between those of the upper and lower levels. Note that some Balmer lines lie in the near-ultraviolet beyond the eye's sensitivity limit at 380 nm. The theory of the discrete spectral lines and structure of the atom was developed by Niels Bohr (1913). He postulated that the negative electron moves with discrete integer multiples of angular momentum. Thus, the electron also has quantized orbital level energies and radii, and this explained the spectral lines. It was later realized from the work of de Broglie (1924) that the electron moves as a wave (with wavelength inverse to linear momentum), along an orbit around the positive nucleus. The electron wave can fit around the nucleus only as a whole multiple of a wavelength - the electron wave must constructively interfere with itself over the orbital path, and only certain wavelengths qualify. Any other wavelengths result in destructive interference. This constructive interference means that the angular momentum, and correspondingly the energy and radius of the orbiting electron,
3 are quantized, just as Bohr theorized. These discoveries were foundation stones of quantum mechanics. Quantum Dots Quantum dots are small semiconductor particles that can contain one electron and one hole (the absence of an electron). Just like in the semiconductors used to make Flash cards and microprocessors, these electrons and holes (effectively, absences of electrons) act like small particles which can move freely inside the semiconductor, but cannot get out - just like a particle in a box, or Infinite Square Well. We will now apply the Schroedinger Equation to that problem. The time-independent Schroedinger Equation in one dimension is written as: ħ2 2m 2 x V x =E x x 2 In one-dimension, the potential for an Infinite Square Well, also known as a particle in a box, is zero inside a box of length L and infinitely large outside it: V 0,if 0 x L x =,if 0 x or L x The general solution to this second-order differential equation is: x =Asin kx B cos kx where E= k 2 ħ 2 2m Applying the boundary conditions 0 = L =0, since the wave function (and probability) must disappear at the left and right borders of the box, yields: x =Asin kx where k is defined as k= n L ; n=1,2,3,... The probability of finding the particle inside the box is 100% (and 0% outside), so we can normalize the wave function by integrating its probability between x positions 0 and 1 as follows: x=l x 2 =1 x=0
4 This determines the value of A, which gives the wavefunction and energy levels: n x = 2 L sin n L x and E n = n2 2 ħ 2 2mL 2 ; n=1,2,3,... Equation 2 You can see from the above result that the energy levels are quantized, and also that there the ground state (lowest possible energy) is non-zero, which is different from the classical (non-quantum) case. Note also that the sinusoidal wavefunction goes to zero at certain positions, meaning that the particle cannot be found there, at x=0, L=0 and (when n>1) at other nodes between those two positions. This contradicts the classical result where a particle can be found with equal probability at all positions within the box. See below the wavefunctions for n=1, 2, 3 and 4: Do you remember how similar this is to the vibrating string activity you did in the Waves experiment you may have done during your first semester of introductory Physics? There you found that standing waves (where the waveform, or envelope, did not appear to change with time, unlike that of a traveling wave) occurred only at certain frequencies which were multiples of each other. By carefully observing Quantum Dots with different sizes, we can see the effect of changing the size of the box on the energy levels of the system, as evidenced by the wavelength of the re-radiated (fluoresced) light observed. Remember that the energy and frequency (or wavelength) of a photon is given by: E=h = hc ;note that hc is a useful quantity and equals 1240 ev-nm Equation 3 Quantum dots are made of semiconductor material, whose conductivity is between that of a conductor and an insulator (recall the Electrostatics I lab you may have performed previously). You may recall that in an insulator, the charges are not free to travel throughout the macroscopic material whereas with a conductor, they are. The modern view
5 of conduction, called Band Gap Theory, apportions solids into the three categories insulators, semiconductors and conductors, based on the energy of their electrons: In a conductor, the conduction band (the range of electron energies required to free an electron from its tom) and valence band (the highest range of electron energy at absolute zero) overlaps, so that conduction is likely because there is no gap. In an insulator, there is a large gap between those two bands. In a semiconductor, the gap is small enough such that thermal energy can give a small fraction of electrons enough of a boost to make conduction possible; doping (introducing impurities in the form of other element) can make a big difference in the electrical properties of a material by changing the number of charge carriers and the types of charge carriers electrons or holes. See below: In the experiment you will perform, the size of the the nanocrystal, or quantum dot, determines the size of the gap between valence and conduction bands because dots of different sizes correspond to different energy levels (See Equation 2 for dependence on box size L). When the dot is excited by incoming photons (from a light source), an exciton (an electron-hole pair) is formed by a valence electron making the jump to the conduction band, and leaving behind a hole. See below:
6 The recombination of electron and hole emits a photons of lower energy, whose wavelengths we measure using the electronic spectrometer. The energy of this photon is the sum of the band gap energy, the confinement (particle in a box energy) of both electron and hole, and the bound energy of the exciton (from the Coulomb attraction between negative electron and positive hole): E emitted photon =E bandgap E confinement E Exciton =E bandgap ħ2 2 2 m e L ħ 2 2 Equation 4 2 2m h L 2 where m e and m h are the effective masses (the masses used to solve the simplified two-body problem of the interaction between electron and hole). For the quantum dots material (InP, Indium Phosphate) in this experiment, the bandgap energy will be E bandgap =2.15 x 10 19 J while m e =7.29 x 10 32 kg and m h =5.47 x10 31 kg. Planck's constant is h=6.6 x10 34 J s and the Reduced Planck Constant is ħ= h 2 =1.1 x10 34 J s Procedure (Atomic Spectra) Using a diffraction grating spectrometer, you will measure some of the wavelengths of the visible members of the hydrogen spectrum Balmer series n l = 2) and compare the results with the predictions of the Bohr model of the hydrogen atom. The diffraction grating equation gives the wavelength λin terms of the measured deviation angle and the spacing, d, between adjacent grating lines (d = 1/(# of lines per meter)): Eq.2: Grating equation: Figure 2 m =d sin where m is integral. Equation 5 This implies that you may see more than one angle for a given wavelength or color (multiple orders of m), as long as sin 1. (Note that in this experiment, unlike when you did the Light Wave Interference lab, m is used in the grating equation to avoid confusion with the orbital momentum integer, n).
7 Figure 3 The essential elements of the spectrometer are shown in Fig. 3: - The slit entrance is pointed at the spectral lamp. The slit can be turned vertical (parallel to the grating lines), and its width is adjustable. This shapes and limits the light beam before it strikes the grating. - A fixed diffraction grating. Most diffraction gratings for this experiment have 6000 lines per centimeter ==> 6 x 10 5 per meter. This implies that the slit separation distance is d = 1/ (6x10 5 ) meter = 1.667 x 10-6 meters = 1667 nanometers. - A telescope tube with an eyepiece fitted with a crosshair for pointing. This assembly rotates about the grating center. The angle with respect to the slit barrel can be read to 0.1 o with a vernier scale. (The spectrometers you will use are calibrated in wavelength-but for a different grating spacing than usually used. Therefore, the wavelength scale is not valid. You must use the measured angle (see below) and the integer order number m to calculate wavelength from the grating equation (Eq.2)).
8 1. Hydrogen spectrum a. First, pull out the telescope from the viewing end. Look through and adjust the eyepiece to focus on the cross-hairs. The eyepiece slides in and out of its draw tube. Put the telescope assembly back into the spectrometer. Be careful not to disturb the focused eyepiece. b. Turn on the power supply that applies a high voltage to the tube containing hydrogen gas. 5000 VOLTS: BE CAREFUL ABOUT ELECTRIC SHOCKS. c. Align the entire spectrometer in a straight line. Look through the eyepiece and align the slit with the vertical crosshair. Loosen the calibration screw and set =0 0 on the base plate. This is the m = 0 maximum which occurs at =0 0 for all wavelengths; thus, you see a mix of all original colors. You may need to swivel the slit to get it upright and parallel with the grating lines. d. Increase by swinging the telescope part to the left, and locate the angular position of the first (m = 1) three or four lines (red, green-blue, and two (?) violet) in the visible spectrum of hydrogen. Ignore the fainter lines, which are from contamination (water and air). e. If the cross-hairs cannot be observed, center the slit image in the telescope eyepiece and average a few readings. Record the wavelengths of any other observed lines. f. Compute the wavelength of each of the three (four?) lines from the grating equation and give the differences from the wavelengths predicted from quantum theory (Eq.1) for n l = 2 and nu = 3, 4,.... If there is sufficient intensity, measure the second order angle (m = 2) to double check your measured 's. 2. Atomic spectra of unknown The spectrum of atomic hydrogen, an atom with just one electron, is simple and the energy levels are given by the simple relation of Eq. 1. Other elements with many orbiting electrons have more complicated atomic spectra; their electrons interact with each other,as well as with the dominant nuclear charge. Nevertheless, the spectral line pattern is unique for each element. There are three tubes of "unknown" elements, helium (He), neon (Ne), or mercury (Hg), one of which you must identify. Measure the wavelengths of the lines in the emission spectrum of JUST ONE UNKNOWN. Identify the element by referring to the list of wavelengths of prominent spectral lines in the table below. Use your wavelength measurements, not chart colors. Note that some lines will be brighter than others. Be careful when changing spectral tubes: they become hot. It is best to hold them at the larger ends, which are cooler.
9 3. Absorption spectrum An emission spectrum is caused by electrons dropping to lower atomic energy states; electromagnetic energy in the form of photons of light is given off in each transition downwards. The electrons got into the higher energy states by collisional excitation, namely atomic collisions with other energetic free electrons. Now, we are going to investigate the opposite effect. Imagine an electron in a low orbital energy state, and along comes a photon of energy that the electron absorbs. The energy is the correct amount to boost the electron into a higher energy state. The excited atom will give up this energy very quickly in any number of ways: re-emit the same energy photon but in a direction different than the original photon; emit two or more lower energy photons as the electron "cascades" down through two or more energy levels or, before the atom can emit a photon, interact with another electron or atom with de-excitation of the atom back to its original state. The net result is that photons of a particular energy are removed from a beam of light. If we are observing a continuous spectrum (think of a rainbow) with a spectroscope, we will see a zone of relative darkness, an absorption line, superposed on the continuous bright wavelength spectrum. Remember that energy is not destroyed, just converted to other wavelengths or even thermal energy. Align the spectrometer to view white light from a tungsten lamp. Describe what you observe. How is it different from the light emitted by hydrogen gas? Now place one bottle filled with colored liquid between the light and the spectrometer. What do you observe? Devise an explanation for this. Repeat for the other bottles of colored liquid. Devise a qualitative rule involving the color of the liquid and the colors that the dark band removes from the spectrum. Determine the central wavelength of the dark absorption band" by measuring the angles corresponding to the left and right fringes of the dark absorption band and averaging to estimate the center of the band. How to Read a Vernier Scale A vernier scale, shown below in the dashed box, is used to interpolate an extra digit of accuracy. The position of the zero is used to read the scale to the accuracy of the drawn or "ruled" lines. The extra vernier's digit will line up with the scale's ruled lines to indicate the next digit of accuracy. Below, the vernier zero lies between 23 and 24. Notice that the vernier's "6" lines up with a ruled line on the scale. This means the answer is 23.6.
10 SPECTRAL LINE WAVELENGTHS Wavelengths are in nanometers. The colors are approximate; they should not be used to identify the elements. Neon 489, 496 (blue-green), 534 (green), 585 (yellow) 622, 633, 638, 640, (orange-red) 651, 660, 693 (red) Helium 447 (blue) 471 (turquoise) 492 (blue-green) 501 (green) 587 (yellow) 667(red) Mercury 404 (violet) 435 (blue) 491 (green) 546 (green) 576 (yellow) 579 (yellow) Sodium 568, 569 (green), 589, 590 (yellow), 615, 616 (red)
11 Procedure (Quantum Dots) 1. Examine the apparatus and measuring instrument there will be a brown felt-covered box containing the four vials of nanocrystal liquid and light source (hidden under vials), and the Red Tide USB650 electronic spectrometer (see below): The red arrow points to the opening where the light to be analyzed enters. If you have a fiber-optic cable available, one end should be screwed into this and the other end should be pointed toward the light source (one of the four vials, when illuminated by light source). If you don't have a cable available, you can partially insert the spectrometer into the opening of the felt-covered box (upside down, with the Red Tide lettering facing down) and point the opening at the vial you are investigating. 2. Make sure one end of the AC adapter is plugged into the wall and the other end plugged inside the brown felt-covered box, to the right of the right-most vial. See the little black button next to the plug. Look at the four vials; they are all partially filled with the same semiconductor substance in the same suspension liquid, and will be illuminated by the same 400nm light source when you press the black button. What happens to their colors when you press the button? 3. Double-click on the USB-Spectrometer file in the same experiment folder where this lab manual resides; it will launch Logger Pro using a template. Click on Connect if the Sensor Confirmation window appears. When you are ready to take measurements, click on the green Collect button in the menu bar. Point the spectrometer's opening (or the fiber-optic cable) at the first vial and fine-tune the aim until the peak in the graph is large and smooth (not a series of jagged lines). When the peak is stable, position the cursor at the middle of the curve peak and read off the Wavelength in the top-left of the window. Record the experimental wavelength in hand-in sheet. Repeat for the remaining three vials. 4. Calculate the theoretical wavelengths for all four vials using Equations 3 and 4. You will also need the sizes (L) of the quantum dots suspended in the clear vial liquids; they are (left to right): 2.3, 2.5, 2.7 and 2.9nm. Compare the theoretical wavelengths to the experimental values and record in hand-in sheet.