p41_052 - Lithium to Magnesium , we nd all the theoretical...

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52. (a) The transition is from n =2to n = 1, so Eq. 41-26 combined with Eq. 41-24 yields f = µ m e e 4 8 ε 2 0 h 3 ¶µ 1 1 2 1 2 2 ( Z 1) 2 so that the constant in Eq. 41-27 is C = s 3 m e e 4 32 ε 2 0 h 3 =4 . 9673 × 10 7 Hz 1 / 2 using the values in the next-to-last column in the Table in Appendix B (but note that the power of ten is given in the middle column). (b) We are asked to compare the results of Eq. 41-27 (squared, then multiplied by the accurate values of h/e found in Appendix B to convert to x ray energies) with those in the table of K α energies (in eV) given at the end of the problem. We look up the corresponding atomic numbers in Appendix F. An example is shown below (for Nitrogen): E theory = h e C 2 ( Z 1) 2 = 6 . 6260688 × 10 34 J · s 1 . 6021765 × 10 19 J / eV ³ 4 . 9673 × 10 7 Hz 1 / 2 ´ 2 (7 1) 2 = 367 . 35 eV which is 6 . 4% lower than the experimental value of 392 . 4 eV. Progressing through the list, from
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Unformatted text preview: Lithium to Magnesium , we nd all the theoretical values are lower than the experimental ones by these percentages: 24 . 8%, 15 . 4%, 10 . 9%, 7 . 9%, 6 . 4%, 4 . 7%, 3 . 5%, 2 . 6%, 2 . 0%, and 1 . 5%. (c) The trend is clear from the list given above: the agreement between theory and experiment becomes better as Z increases. One might argue that the most questionable step in 41-10 is the replacement e 4 ( Z 1) 2 e 4 and ask why this could not equally well be e 4 ( Z . 9) 2 e 4 or e 4 ( Z . 8) 2 e 4 ? For large Z , these subtleties would not matter so much as they do for small Z , since Z Z for Z ....
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