Atomic Structure and Atomic Spectra：原子结构与原子光谱

1、.Atomic Structure and Atomic SpectraEquipment used:Incandescent lamp, H, He, Hg, Na & K discharge tubes, hand-held spectrometers and diffraction gratings, 1.5 m spectrograph (Rm 1), prism spectrometer (Rm 42), ccd-based optical multichannel analyzer (OMA), PC & spreadsheet.Objectives:A) visu

2、ally observe blackbody and atomic spectraB) record the visible-near UV spectrum of atomic hydrogen using an OMA and relate the observed Balmer series emission lines to the predicted energy level diagram of the H atom - develop spectroscopic notation.C) measure the spin-orbit splitting of sodium and

3、potassium "D-lines and determine the sizes of the magnetic fields experienced by the "optical electrons" in the upper levels for those those transitions.References:R. Harris, Nonclassical Physics: Beyond Newton's View, Addison Wesley, Menlo Park, California (1998).R. Serway, C. Mo

4、ses, and C. Moyer, Modern Physics, Second Edition, Saunders College Publishing, Fort Worth, Texas (1997).K. Krane, Modern Physics, Second Edition, John Wiley & Sons, New York (1996).Spectra - a brief historyBlackbody, atomic and molecular spectra (intensity as a function of wavelength or frequen

5、cy) have been the subjects of intensive examination for nearly 200 years (Fraunhofers observations of dark lines in the solar spectrum, including the “D” lines, were made in 1814!). For nearly half that time, the study of the optical spectra associated with heated solids and atoms and molecules in t

6、he gas (or plasma) phase was of essentially an empirical nature. For the continuous blackbody spectra, the empirical relations developed by Wein (Weins displacement law) and by Stefan and Boltzmann during the latter part of the nineteenth century lacked a theoretical foundation. On a Sunday evening

7、in early October of 1900, Max Planck discovered the blackbody formula that allowed an exact prediction of the shapes of blackbody spectra. Plancks formula allowed the derivation of Weins law and the Stephan-Boltzmann law, and provided the source of the empirical constants contained within them. To d

8、erive his formula Planck had to assume that electromagnetic radiation was quantized. Quantum physics and the photon had been invented!A similar situation obtained with respect to the discrete line spectra of atoms and the associated band spectra of molecules. Accurate measurements of the four visibl

9、e emission lines of atomic hydrogen had been made by Angstrom (who labeled them alpha, beta, gamma and delta). In 1885 Balmer published a paper entitled “Notice Concerning the Spectral Lines of Hydrogen,” which included an empirical formula for the wavelengths of the lines measured by Angstrom. Balm

10、er's formula waswhere C2 = 3645.6 x 10-8 cm and n = 3, 4, 5. Although only 4 lines were known by Balmer when he started the paper, 10 other lines in the violet and ultra-violet had been measured by the time he finished its writing!It 1913, Niels Bohr announced the first theoretical explanation o

11、f the spectrum of the hydrogen atom (although his basic ideas are applicable to other atomic systems as well). Bohr assumed that the angular momentum of the electron in its "orbital" motion about the proton was quantized (which had the effect of quantizing the electron's orbital radius

12、 and energy). He further assumed that when an electron undergoes a transition from one allowed orbital state to another, it can do so by emitting or absorbing radiation whose energy equaled the energy difference between the orbital states involved. According to Bohr, the allowed energies of an elect

13、ron bound to the proton in a hydrogen atom were given by,where n = 1, 2, 3. Thus, Balmer's empirical formula gained a theoretical foundation. Note that Bohr's theory requires only a single quantum number (the principal quantum number) to explain the observed spectrum.The Modern ViewBohrs the

14、ory included an explanation for the lack of radiation that classical electromagnetic theory predicted should arise from orbiting electrons: the electrons were in “stationary states.” In other words he said they didnt radiate when they occupied an allowed orbit, never mind that they were undergoing c

15、entripetal acceleration due to their orbital motion! Needless to say, the argument was less than convincing to many.In an effort (successful!) to explain Bohrs angular momentum assumption, DeBroglie (1923) interpreted Bohrs postulate in terms of wave resonance. The result was the DeBroglie relation,

16、Three years later, Schrodinger introduced his wave equation, whose solutions are the wavefunctions for a particle trapped in a potential well. Since the wave function contains all that is knowable about a particle that is behaving as a wave, when Schrodinger's equation is applied to an electron

17、bound to a single proton (the hydrogen atom), Bohr's formula for the allowed energy levels for the electron must necessarily be contained in the resulting solutions.The solution of Schrodinger's equation for the problem of the H-atom is far too involved to reproduce here (see references). In

18、deed Bohr's relation for the electronic energy levels in the hydrogen atom is a result of that solution. However, such a full quantum mechanical treatment also results in two other quantum numbers in addition to Bohr's principal quantum number n. That is, the energy for a hydrogen atom elect

19、ron is determined by the principal quantum number, but for each n, there are also other quantum numbers required to fully describe the allowed energy levels of the H atom. Those quantum numbers are the electronic orbital-angular-momentum quantum number l, and the magnetic quantum number m (which qua

20、ntizes the z-component of the electron's orbital angular momentum). Further results of the Schrodinger equation for the hydrogen atom place restrictions on the allowed values of m and l. Those restrictions are:1. For each value of the principal quantum number n, the possible values for the angul

21、ar momentum quantum number l are l = 0,1,2.(n-1).2. For each value of l, there are 2l + 1 possible values of m ranging from -l to +l (in integer steps of course, and including zero)That set of restrictions together implies n2 different wavefunctions that describe n2 different possible states for the

22、 electron for each principal quantum number n. This seriously complicates the energy level diagram for the electronic states of the hydrogen atom over the rather simple case suggested by Bohr's work.Although it is not a result of any solution to Schrodinger's equation, it is nevertheless a f

23、act that electrons possess an intrinsic spin that is also quantized. The allowed values of the electron spin quantum number s are + 1/2 and - 1/2 (or spin up, and spin down). Thus, the number of possible unique wave functions for each principal quantum number n described in the previous paragraph mu

24、st be multiplied by 2!The net result is that 4 quantum numbers (n, l, m & s) are required to fully specify any given allowed state of an electron bound to a hydrogen atom. When this fact is coupled with the Pauli exclusion principle (no two electrons bound to an atom can possess an identical set

25、 of quantum numbers) and generalized to other atoms, the periodic table of the elements is explained!Spectroscopic Notation and Transition Selection RulesA shorthand notation for the possible electronic energy levels in an atom involves specifying the principal quantum number of the electron, the to

26、tal electronic angular momentum due to spin, and the total electronic orbital angular momentum, and the total electronic angular momentum i.e. n, L = l1 + l2 + l3 , S = s1 + s2 + s3 , and J = L + S (yes, they are vectors!). The notation is known as L-S coupling notation since the total electronic an

27、gular momentum is simply the vector sum of the electronic orbital and spin angular momenta. The notation is written as, n 2S + 1LJ,where 2S + 1 is known as the multiplicity of the state (the terminology is 1 = singlet, 2 = doublet, 3 = triplet, and so on). For historical reasons, L = 0 is referred t

28、o as an S-state, L = 1 is a P-state, L = 2 is a D-state, L =3 is an F-state and the alphabet is followed thereafter. As an example of an application of this notation, the ground state of the hydrogen atom is 12S1/2.Since a photon carries energy and angular momentum (1 unit), any electronic transitio

29、n from one allowed energy level to another must conserve energy and angular momentum for the atom-photon system. Thus, restrictions on the allowed changes in multiplicity, orbital and total angular momentum can be partially understood. These restrictions are known as selection rules and they are,wit

30、h J = 0 to J = 0 forbidden. Any allowed transition between two electronic states must obey these selection rules. Notice that the magnetic quantum number does not appear in the spectroscopic notation. The reason for this is that effects on electronic energies in atoms due to quantum number m are onl

31、y realized if the atom is placed in an external magnetic field (Zeeman effect). Nevertheless, the selection rule still applies and is written,Spin-Orbit Interaction Given that an electron spins and that it is charged, it should come as no surprise that it possesses a magnetic dipole moment (i.e. it

32、is a tiny compass needle). Also given that an electron bound to an atom is in some sense in motion relative to the nucleus and any other bound electrons, it experiences a magnetic field due to that motion. Given further that an electron's spin is quantized (it can take on only 2 values), the int

33、eraction between the electron's magnetic moment and the magnetic field it experiences due to it's orbital motion should result in an energy difference between otherwise similar electronic levels if the electron's spin is aligned parallel or anti-parallel to the magnetic field. It turns o

34、ut that the energy difference between spin-up and spin-down states is justWhere B is the magnetic field due to the orbital motion and mu is the Bohr magneton (i.e. the electron's magnetic moment). By measuring energy differences between spin-orbit split levels in atoms, the magnetic field experi

35、enced by the electron due to its orbital motion about the nucleus can be easily determined. This effect is not easily observable in the spectrum of the hydrogen atom, but is easily observed in the spectra of sodium and potassium atoms. Measurements of the visible-near UV spectrum of the hydrogen ato

36、m (Balmer series), and portions of the spectra of Na & K atoms (the "D" lines) atoms are the experimental goals of this lab.Pre-lab questions.1. How would you justify the layout of the first three rows of the periodic table based upon the information supplied above?2. Determine the spe

37、ctroscopic notation for the ground states of H, He, Li, Be, Na, K, Rb and Cs.3. Construct a complete energy level diagram for the H atom based upon Bohr's explanation of the H atom only.4. Construct an energy level diagram up through n = 3 for the H atom, based upon the results of application of

38、 Schrodinger's equation to the problem of the H atom. Determine all possible transitions between allowed levels in the H atom that give rise to H-alpha radiationPage: 5.The ExperimentThree spectroscopic measurement tools (spectrometers) will be employed to accomplish the goals of this laboratory

39、 exercise. Two are antiques that have the wonderful advantage of allowing you to actually see the spectra of interest with your own eyes. In addition, some hand-held spectroscopic instruments which can be used to observe the spectra of various light sources will be available for use. The key quantit

40、ative spectroscopic measurement tool available is a ccd-based optical multichannel analyzer that will allow quantitative measurements of the spectra of interest (H, Na & K). Spectra will be recorded onto floppy disks in a format that can easily be imported into a spreadsheet for analysis and pro

41、duction of plots of the spectra.1) H-atom (Balmer series)Balmer suggested that his empirical formula for the wavelenghts of the emission lines of atomic hydrogen in the visible-near UV portion of the spectrum might be a special case of a more general expression that would applicable to the spectra o

42、f other elements. Indeed, Rydberg and Ritz discovered just such a formula (Rydberg-Ritz formula), expressing the reciprocal wavelength as,Where l and u refer to lower- and upper-level principal quantum numbers, respectively, Z is the atomic number (here Z=1) and R = 1.097373 x 107 m-1 is the Rydberg

43、 constant. We will measure the Balmer series of spectral lines from a hydrogen discharge tube, determine the wavelengths of the Balmer lines experimentally, and, using the above formula, experimentally determine the value of the Rydberg constant. It is left to the student to determine how this shoul

44、d be done, given the wavelenghts of the Balmer lines (and the attendant uncertainties in wavlenght) derived from the recorded spectra, and the Rydberg-Ritz formula shown above.Bohr's model for the H atom predicts the value of the Rydberg constant. It is an interesting combination of fundamental

45、constants (find it in your favorite physics text!). Calculate Bohr's prediction for the Rydberg constant and compare it to your experimentally derived value. You should also use your experimentally derived value for the Rydberg constant to determine the value of the ionization potential of the H

46、 atom. That too is an interesting combination of fundamental constants (find it too in your favorite physics text). Compare your measured value to the prediction of Bohr.2) Spin-orbit interaction (magnetic fields in atoms, Na & K "D" lines), spectroscopic notation, and selection rules.

47、A brief discussion of the spin-orbit interaction in atoms, spectroscopic notation and the selection rules for allowed (dipole) transitions between stationary states in an atom is found above. We will measure the spectra of Na and K in the vicinity of their respective "D" lines. The "D

48、" lines in alkali atoms are transtions from the first excited states (spin-orbit-split states) to the ground state. As noted above, the splitting is due to the interaction between the magnetic dipole moment of the electron (due to its intrinsic spin, a constant), and the magnetic field it experiences due to its "orbital motion" relative to the nucleus (and inner