CHAPTER 4 Structure of the Atom

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1 CHAPTER 4 Structure of the Atom 4.1 The Atomic Models of Thomson and Rutherford 4.2 Rutherford Scattering 4.3 The Classic Atomic Model 4.4 The Bohr Model of the Hydrogen Atom 4.5 Successes and Failures of the Bohr Model 4.6 Characteristic X-Ray Spectra and Atomic Number 4.7 Atomic Excitation by Electrons

2 4.1 The Atomic Models of Thomson and Rutherford Pieces of evidence that scientists had in 1900 to indicate that the atom was not a fundamental unit: 1) There seemed to be too many kinds of atoms, each belonging to a distinct chemical element. 2) Atoms and electromagnetic phenomena were intimately related. 3) The problem of valence ( 원자가 ). Certain elements combine with some elements but not with others, a characteristic that hinted at an internal atomic structure. 4) The discoveries of radioactivity, of x rays, and of the electron

3 Thomson s Atomic Model Thomson s plum-pudding model of the atom had the positive charges spread uniformly throughout a sphere the size of the atom with, the newly discovered negative electrons embedded in the uniform background. In Thomson s view, when the atom was heated, the electrons could vibrate about their equilibrium positions, thus producing electromagnetic radiation.

4 Experiments of Geiger and Marsden Under the supervision of Rutherford, Geiger and Marsden conceived a new technique for investigating the structure of matter by scattering particles (He nuclei, q = +2e) from atoms. Plum-pudding model would predict only small deflections. (Ex. 4-1) Geiger showed that many particles were scattered from thin gold-leaf targets at backward angles greater than 90. Rutherford: It was as if you fired a 15 inch shell at a tissue paper and it came back and hit you

5 Example 4.1 The maximum scattering angle corresponding to the maximum momentum change Maximum momentum change of the α particle is or Determine θ by letting p max be perpendicular to the direction of motion.

6 Multiple Scattering from Electrons? If an α particle were scattered by many electrons and N electrons results in The number of atoms across the thin gold layer of m: Assume the distance between atoms is and there are That gives

7 Rutherford s Atomic Model even if the α particle scattered from all 79 electrons in o o each atom of gold (0.016 ) 6.8 total The experimental results were not consistent with Thomson s atomic model. Rutherford proposed that an atom has a positively charged core (nucleus) surrounded by the negative electrons.

8 4.2 Rutherford Scattering The Assumptions 1. The scatterer is so massive that it does not recoil significantly; therefore the initial and final kinetic energies of the particle are practically equal. 2. The target is so thin that only a single scattering occurs. 3. The bombarding particle and target scatterer are so small that they may be treated as point masses and charges. 4. Only the Coulomb force is effective.

9 The Relationship Between the Impact Parameter b and the Scattering Angle There is a relationship between the impact parameter b and the scattering angle θ. When b is small, r gets small. Coulomb force gets large. θ can be large and the particle can be repelled backward. (p )

10 however, cannot pick impact parameter in an experiment. particles are incident at varied impact parameters all around the scatterer. Any particle inside a circle of area πb 02 will be scattered at angles greater than θ 0. Define cross section σ = πb 02 for scattering at an angle greater than θ 0. σ related to the probability for a particle being scattered by the nucleus. σ = πb 0 2

11 Rutherford Scattering equation The number of particles scattered per unit area is

12 The Important Points 1. The scattering is proportional to the square of the atomic number of both the incident particle (Z 1 ) and the target scatterer (Z 2 ). 2. The number of scattered particles is inversely proportional to the square of the kinetic energy of the incident particle. 3. For the scattering angle, the scattering is proportional to 4 th power of sin( /2). 4. The Scattering is proportional to the target thickness for thin targets.

13 4.3: The Classical Atomic Model As suggested by the Rutherford Model the atom consisted of a small, massive, positively charged nucleus surrounded by moving electrons. This then suggested consideration of a planetary model of the atom. Let s consider atoms as a planetary model.

14 The Planetary Model is Doomed From classical E&M theory, an accelerated electric charge radiates energy (electromagnetic radiation) which means total energy must decrease. Radius r must decrease!! Electron crashes into the nucleus!? Obviously most atoms are stable, so once again classical physics breaks down at small length scales. Physics had reached a turning point in 1900 with Planck s hypothesis of the quantum behavior of radiation.

15 4.4: The Bohr Model of the Hydrogen Atom In 1913 Neil s Bohr developed a structural model for the hydrogen atom consistent with the nuclear model of Rutherford and the observed spectral series. Believed quantum principles should govern more phenomena than just blackbody radiation and the photoelectric effect. Bohr assumed electron moved around a massive positively charged nucleus (mass of nucleus taken to be infinite). radius of the orbit >> radius of nucleus.

16 The Bohr Model of the Hydrogen Atom Bohr also made the following general assumptions: 1) Stationary states (orbiting electrons in atoms do not radiate energy). These states have definite total energy. 2) EM radiation emitted/absorbed when electrons make transitions between stationary states such that hf = E = E 1 E 2 3) Classical laws of physics do not apply to transitions between stationary states. 4) The mean kinetic energy of the electron-nucleus system is quantized such that K = n h f orb /2, where f orb is the frequency of rotation. As we shall see this is equivalent to quantizing the angular momentum of the stationary states in multiples of h/2π. Used these four assumptions to derive the Rydberg equation for characteristic spectra. Four assumptions keep as much as possible the ideas of classical physics.

17 Quantization of angular momentum

18 Bohr Radius Rearranging for radius, result is that the radii of the stationary states in the hydrogen atom are quantized: Where the Bohr radius is given by The smallest diameter of the hydrogen atom is plugging the radius into our previous expression in the planetary model for the energy of a stable electron orbit: n = 1 gives its lowest energy state (called the ground state)

19 The Hydrogen Atom The energies of the stationary states where E 0 = 13.6 ev Emission of light occurs when the atom is in an excited state and decays to a lower energy state (n u n l ). where f is the frequency of a photon. R is the Rydberg constant.

20 Transitions in the Hydrogen Atom Lyman series: n = 1 (invisible). Balmer series: n = 2 (visible). Paschen series: n = 3 (visible). It was clear Bohr s model predicts all the frequencies in atomic hydrogen.

21 Fine Structure Constant The electron s velocity in the Bohr model: On the ground state, v 1 = m/s ~ less than 1% of the speed of light The ratio of v 1 to c is the fine structure constant. α is a fundamental physical constant characterizing the strength of the electromagnetic interaction

22 The Correspondence Principle Classical electrodynamics + Bohr s atomic model Determine the properties of radiation Need a principle to relate the new modern results with classical ones. Bohr s correspondence principle In the limits where classical and quantum theories should agree, the quantum theory must reduce the classical result.

23 The Correspondence Principle The frequency of the radiation emitted f classical is equal to the orbital frequency f orb of the electron around the nucleus. The frequency of the transition from n + 1 to n is For large n, Substitute E 0 :

24 4.5: Successes and Failures of the Bohr Model The electron and hydrogen nucleus actually revolved about their mutual center of mass. The electron mass is replaced by its reduced mass. The Rydberg constant for infinite nuclear mass is replaced by R E, 0 R R 2 2 nl nu hc Discovery of Deuterium (1932) from slight splitting of Hα line between Hydrogen ( nm) and Deuterium ( nm)

25 Limitations of the Bohr Model The Bohr model was a great step of the new quantum theory, but it had its limitations. 1) Works only to single-electron atoms 2) Could not account for the intensities or the fine structure of the spectral lines 3) Could not explain the binding of atoms into molecules To overcome these limitations a full quantum mechanical treatment is required (Chapter 7).

26 4.6: Characteristic X-Ray Spectra and Atomic Number Discussed the production of X-rays previously (Section 3.7). The bremsstrahlung radiation is superimposed on peaks whose wavelength is element specific. Although strictly speaking the Bohr model only applies to one electron atoms it was believed that the Bohr-Rutherford atom would also apply to many-electron atoms. Bohr model suggests that the electrons occupy shells of radius r n characterized by the principle quantum number n.

27 Characteristic X-Ray Spectra and Atomic Number

28 Atomic Number L shell to K shell K α x ray M shell to K shell K β x ray Atomic number Z = number of protons in the nucleus Moseley found a relationship between the frequencies of the characteristic x ray and Z. This holds for the K α x ray

29 Moseley s Empirical Results The x ray is produced from n = 2 to n = 1 transition. From Bohr model: E 0 and thus R is prop to (Ze) 2. For electron making a transition into K-shell from L-shell it feels an effective charge Z-1 due to remaining electron in the K-shell therefore: Moseley s research clarified the importance of the electron shells for all the elements, not just for hydrogen. Also demonstrated Z determined ordering in the periodic table.

30 4.7: Atomic Excitation by Electrons Franck and Hertz studied the phenomenon of ionization via electron bombardment. Observations: Accelerating voltage is below 5 V electrons did not lose energy Accelerating voltage is above 5 V sudden drop in the current

31 Atomic Excitation by Electrons Ground state has E 0 to be zero. First excited state has E 1. The energy difference E 1 0 = E 1 is the excitation energy. Hg has an excitation energy of 4.88 ev in the first excited state No energy can be transferred to Hg below 4.88 ev because not enough energy is available to excite an electron to the next energy level Above 4.88 ev, the current drops because scattered electrons no longer reach the collector until the accelerating voltage reaches 9.8 ev and so on.

32 Atomic Excitation by Electrons The Franck-Hertz experiment provided further proof of the quantization of atomic energy levels. Bombarding electron kinetic energy can only change in discrete amounts which depend on the energy levels of the atoms in the vacuum tube.

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