Chapt 5. Complex atoms; The Pauli exclusion principle

1. Chapt 5. Complex atoms; The Pauli exclusion principle

2. The principal quantum number--n. lIn schrodinger’s original theory of hydrogen, the electronic energy levels are completely determined by a principal quantum number, n, and satisfy the Bohr formula 5-1.Quantum numbers of hydrogen atoms: a review. lThe Bohr formula is only approximately correct. The more complete relativistic theory of hydrogen leads to energy levels that have a slight dependence on other quantum numbers.

3. The angular momentum quantum number, l. lIn solving schrodinger’s equation, it is found that the magnitude L of the total orbital angular momentum is related to a separate quantum number l by: (l=0,1,….n-1) (z 28.2)

4. lTwo important results should be noted. First the angular momentum may be zero for any value of n. ( A classical picture of an electron with zero angular momentum would be one which moves back and forth directly through the nucleus!) lThe second point is that for each value of n, l may be any integer up to and including n-1. lA somewhat archaic(陈旧的) nomenclature (命名) is still used to designate states with certain values of n and l. lTable (5-1-1) shows the letters assigned to states with l=0 through 6.

5. Table 5-1-1 Spectroscopic notation for the first seven angular momentum states in atoms. The sequence is alphabetic for l ≥ 3 L = 0 1 2 3 4 5 6 Notation s p d f g h i lWith this notation, the state with n=1, l=0 is said to be the 1s state, while n=2, l=0 is the 2s state, and n=1, l=1 is the 2p state. Other states are denoted in a similar way. The Z component of the angular momentum. lThe analysis of the experiments and schroedinger’s theory both show that , ml=-l, -l+1, ……, 0, l-1, l. (5-1-2) (z 28.5)

6. lThus ml, the quantum number denoting the z component ofcan have any of 2l+1 integer values for a given value of l. Spin quantum number lThe magnitude of the spin angular momentum S of a particle is found to be Where s is the spin quantum number. Z28.8

7. lFor electrons, s=1/2, as it is for protons and neutrons(中子), for photons, s=1. These particles are said to have spin one-half and spin one, respectively. lIn atomic structure, it is the z component of the spin angular momentum Sz that is important. For a spin one-half particle, Sz is given by (ms= 1/2,-1/2) (5-1-4) Z28.9

8. l Despite the fact that never points directly along the z axis, the two states are often referred to as spin up ( ) and spin down ( ). lThus, for every set of values of n, l, and ml, there are two allowed values of ms. lIn conclusion, we need four quantum numbers n, l, ml, ms to describe the motion of an electron in the hydrogen atom, and similar situation occurs also in complex atoms.

9. 5-2. The Pauli exclusion principle and atomic structure Bosons and Fermions (玻色子与费米子) lIt has been recognized that all elementary particles have an intrinsic (本征的) spin angular momentum, which plays a crucial rule in atomic and molecular structure and in the behavior of collections of particles. lParticles which have zero or integral spin are called bosons, whereas particles have half-integral spin are called fermions.

10. lBosons and fermions not only follow different statistical laws but also exhibit different properties. lFor example, the two isotopes of helium(氦), 3He, with two protons and a neutron in the nucleus, and 4He, with two protons and two neutrons, have very different properties at low temperature, because the net spins of the two nuclei are different. lIn 4He the two protons have equal but oppositely directed spins, as do the two neutrons. Thus the net spin angular momentum is zero and it belongs to bosons.

11. lThis substance, which at atmospheric pressure is a liquid even at the lowest temperature so far achieved, because a superfluid below about 2k. l3He on the other hand displays equally remarkable but very different behavior below about 5x10-3 K. Here the net nuclear spin is one-half due to the single unpaired neutron. The Pauli exclusion principle lIn 1925, W. Pauli discovered a rule, known as the Pauli exclusion principle, for determining the number of electrons in filled shells of an atom.

12. lThe Pauli exclusion principle claims that two or more indistinguishable or identical particles with spin one –half (i, e, fermions) can not be in the same quantum state. lThis means for example that in an atom with many electrons, no more than one electron can be in a level denoted by a given set of quantum numbers, n, l, ml andms. lBosons, on the other hand, can be put in any quantum state whether it is occupied or not. When a lot of Bosons are in the same quantum state, it is called Bose-Einstein condensation (BEC) ( 玻色-爱因斯坦凝聚).

13. lNotice that in quantum mechanics two particles of the same kind are “indistinguishable” or “identical”, since the uncertainly principle precludes an accurate determination of the position and momentum, so at different times we can not tell if we are seeing the original particle or another that looks just like it.

14. Atomic structure lWe now have sufficient information to gain some insight into atoms with more than one electrons. We will see that many of the physical and chemical properties of atoms can be explained by the atomic shell model, which is based on the idea that the electrons in atoms are in hydrogen like states. lIn the ground state of an atom, the electrons occupy the lowest states allowed by the Pauli exclusion principle. lWhen all the states corresponding to the same energy or nearly the same energy have been filled, an electron shell is said to be completed or closed.

15. lIn x-ray jargon (行话) the innermost shell with n=1 is called the k shell, the next with n=2 is call the L shell. After that come the M, N and O shells etc. lAll atoms with a large number of electrons would then have several filled shells, any remaining electrons being in the next outer shell. lIn the first shell, for which n=1,l and ml must be zero,but ms may be plus or minus 1/2.So there may be only two electrons, identical except that their spins are in opposite direction.

16. lFor the next shell (n=2), l may have the value 0 or 1. If l = 0, ml =0, and two electrons are permitted with these quantum numbers if they have spin +1/2 and –1/2, respectively. lThe remaining electrons for which l=1 may have values of ml =-1, 0, +1 and for any of these, spins may be –1/2 or +1/2. This permits six electrons with l=1. lThese together with the two for l=0 make eight electrons permitted in the second shell. lSimilarly, for the third shell 18 electrons will be permitted, for the fourth 32, for the fifth 50, and for the nth, 2n2.

17. lElectrons in the first shell are known as s-electrons since they have value of l=0. In the second shell there are two sub-groups: two s-electrons and six p-electrons. In the third shell, there would be 2s-, 6p- and 10 d-electrons. lFor this reason it is customary to speak of the s-, p- and d-sub shells (支壳层). lThe L shell, for instance, has two sub shells, s and p, the M shell has three sub shells, s, p and d. The N shell has four sub shells, s, p, d and f. lFor any completely filled shell the total angular momentum is zero, since for every electron there is one other electron with oppositely directed angular momentum and they cancel by pairs.

18. 5-3 Periodic table of the elements lThe Pauli exclusion principle gives the key to building the atoms and to their relations represented by the periodic table. (table 5-3-1) Table 5-3-1 Electron configurations lIn the first period of the periodic table only two different atoms representing two elements exist, since only two electrons are permitted in the first or k shell by the Pauli principle.

21. lEight atoms representing eight elements occur in the next period of the periodic table, since only eight electrons are permitted by the Pauli principle in the second or L shell. lThe third period of the periodic table also has eight atoms representing eight more elements formed by adding 2 s-electrons and the 6 p-electrons of the third shell according to the same principle. lHere, however, the first of several deviations from the regular order begins. Starting with potassium(K, 钾), instead of adding the d-electrons belonging to the third shell, the s-electrons of the next or fourth shell are added first.

22. lAlthough the Pauli principle defines the number of possible electron states it does not specify the order in which the vacancies will be filled. The rules is that any particular electron goes to the lowest available state and the order in which this happens can only be determined spectroscopically. lNotice that in table 5-1 the order of filling the shells is indicated. For instance, in the long period from potassium K to Krypton(氪) Kr it is indicated how 3d electrons are filled in “underneath” 4s electrons with deviations from the regular order at Cr, Mn, Fe and Cu. lIn the group from Rb to Pd the 4d electrons are filled in under 5s electrons with some irregularities.

23. lSince the outer electronic structure is the same for all of these, this accounts for their similar chemical properties. lIt is like wise seen that the group of similar elements from Z=57 to Z=72 known as the rare-earth elements represent the filling in of the 4f electrons beneath outer 6s and 5d electrons. lWe can also have an explanation of the so-called “noble” elements, which do not interact chemically at ordinary temperatures, because of filled shells. lThese elements, beginning with helium(氦) with 2 electrons, neon (氖) with 10 electrons, and so on, represent atoms with completely filled shells and subshells.

24. lFor many years it had been thought that because of the filled shells they could not form chemical compounds, but in 1962 much to the surprise of scientific world it was found that this was not true at high temperatures, and the first such compound to be so formed was xenon (氙) and fluorine(氟). lThe electronic structure of an atom may be represented in terms of filled or partly filled shells; for instance, an aluminum atom would have the following electronic structure, 1s2, 2s2, 2p6, 3s2,3p and the normal or ground state of the atom is said to be the P state.

25. lThe lowest or normal state of an atom is the state of the last valence electron (价电子).It is frequently designated(标示) by the j value as a subscript(下标) after the letter, and a superscript before the letter indicates whether the state issinglet, doublet, triplet etc, as follows: oxygen (3p2), sodium (钠) (2s1/2). For the aluminum atom just discussed, the symbol for the lowest state is 2p1/2.