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1、Shanxi University Atomic Physics,Books,The teaching book:C.J.Foot,Atomic physics,Oxford University Press,1st Edition,2005Reference:杨福家，原子物理学，北京：高等教育出版社，第三版，2000年7月,Shanxi University Atomic Physics,Mark Proportion:Final Exam:45%Midsemester:30%Homework:15%Attending class:10%,Shanxi University Atomic P
2、hysics,Main Contents,1 Early atomic physics2 The hydrogen atom3 Helium4 The alkalis5 The LS-coupling scheme6 Hyperfine structure and isotope shift7 The interaction of atoms with radiation8 Doppler-free laser spectroscopy9 Laser cooling and trapping10(1)Magnetic trapping10(2)Evaporative cooling and B
3、ose-Einstein condensation11 Atom interferometry12 Ion traps 13 Quantum computing,Shanxi University Atomic Physics,Chapter 1 Early atomic physics,1.1 Introduction 1.2 Spectrum of atomic hydrogen 1.3 Bohrs theory 1.4 Relativistic effects 1.5 Moseley and the atomic number 1.6 Radiative decay 1.7 Einste
4、in A and B coeffecients 1.8 The Zeeman effect 1.8.1 Experimental observation of the Zeeman effect 1.9 Summary of atomic unitsExercises,Shanxi University Atomic Physics,1.1 Introduction,The origins of atomic physics:quantum mechanics Bohr model of the HThis introductory chapter surveys some of the ea
5、rly ideas:Spectrum of atomic H and Bohr Theory Einsteins treatment of interaction of atom with lightthe Zeeman effectRutherford scattering And so on,Shanxi University Atomic Physics,1.1 Introduction,Before describing the theory of an atom with one electron,some experimental facts are presented.This
6、ordering of experiment followed by explanation reflects the authors opinion that atomic physics should not be presented as applied quantum mechanics,but it should be motivated by the desire to understand experiments.This represents what really happens in research where most advances come about throu
7、gh the interplay of theory and experiment.,Shanxi University Atomic Physics,1.2 Spectrum of atomic hydrogen_2,n and n:whole numbers;R:Rydberg constant.Balmer series:spectral lines for which n=2 and n=3;4;The first line at 656 nm is called the Balmer-(or H)line,the characteristic spectrum for atoms i
8、s composed of discrete lines that are the fingerprint of the element.,In 1888,the Swedish professor J.Rydberg found that the spectral lines in hydrogen obey the following mathematical formula:,Shanxi University Atomic Physics,1.2 Spectrum of atomic hydrogen_3,Wavenumbers may seem rather old-fashione
9、d but they are very useful in atomic physics,Wavenumber,In practice,the units used for a given quantity are related to the method used to measure it,e.g.spectroscopes and spectrographs are calibrated in terms of wavelength.,Ritz combination principle:the wavenumbers of certain lines in the spectrum
10、can be expressed as sums.,Shanxi University Atomic Physics,1.2 Spectrum of atomic hydrogen_4,Examination of the sums and differences of the wavenumbers of transitions gives clues that enable the underlying structure to be deduced,rather like a crossword Puzzle-some examples of this are given in late
11、r chapters.The observed spectral lines in hydrogen can all be expressed as differences between energy levels.Which as shown in Fig.1.1,where the energies are proportional to 1/n2.The following section looks at how these spectra can be explained theoretically.,Shanxi University Atomic Physics,Lyman s
12、eries:n=2;3;4;n=1.Balmer(n=2),Paschen series:(n=3),Brackett(n=4)and Pfund(n=5),Shanxi University Atomic Physics,Bohr assumed that each electron orbits the nucleus in a circle,whose radius r is determined by the balance between centripetal acceleration and the Coulomb attraction towards the proton.Fo
13、r electrons of mass me and speed v this gives(1.3)In SI units the strength of the electrostatic interaction between two charges of magnitude e is characterised by the combination of constants e2/40.This leads to the following relation between the angular frequency=v/r and the radius:(1.4)This is equ
14、ivalent to Keplers laws for planetary orbits relating the square of the period 2=!to the cube of the radius(as expected since all steps have been purely classical mechanics).The total energy of an electron in such an orbit is the sum of its kinetic and potential energies:(1.5),1.3 Bohrs theory-2,Sha
15、nxi University Atomic Physics,1.3 Bohrs theory-3,Using eqn 1.3 we nd that the kinetic energy has a magnitude equal to half the potential energy(an example of the virial theorem).Taking into account the opposite signs of kinetic and potential energy,we find(1.6)This total energy is negative because t
16、he electron is bound to the proton and energy must be supplied to remove it.To go further Bohr made the following assumption.Assumption I:There are certain allowed orbits for which the electron has a xed energy.The electron loses energy only when it jumps between the allowed orbits and the atom emit
17、s this energy as light of a given wavelength.,Shanxi University Atomic Physics,That electrons in the allowed orbits do not radiate energy is contrary to classical electrodynamics|a charged particle in circular motion undergoes acceleration and hence radiates electromagnetic waves.Bohrs model does no
18、t explain why the electron does not radiate but simply takes this as an assumption that turns out to agree with the experimental data.We now need to determine which out of all the possible classical orbits are the allowed ones.There are various ways of doing this and we follow the standard method,us
19、ed in many elementary texts,that assumes quantisation of the angular momentum in integral multiples of(Plancks constant over 2):Mevr=n(1.7)where n is an integer.Combining this with eqn 1.3 gives the radii of theallowed orbits as r=a0n2(1.8)where the Bohr radius a0 is given by(1.9),1.3 Bohrs theory-4
20、,Shanxi University Atomic Physics,1.3 Bohrs theory-5,This is the natural unit of length in atomic physics.Equations 1.6 and 1.8 combine to give the famous Bohr formula:(1.10)The positive integer n is called the principal quantum number.Bohrs formula predicts that in the transitions between these ene
21、rgy levels the atoms emit light with a wavenumber given by(1.11)This equation fits very closely to the observed spectrum of atomic hydrogen described by eqn 1.1.The Rydberg constant R in eqn 1.11 is defined by(1.12),Shanxi University Atomic Physics,1.3 Bohrs theory-6,The factor of hc multiplying the
22、 Rydberg constant is the conversion factor between energy and wavenumbers since the value of R is given in units of m(or cm in commonly-used units).The measurement of the spectrum of atomic hydrogen using laser techniques has given an extremely accurate value for the Rydberg constant R=10 973 731.56
23、8 525 m-1.However,there is a subtle difference between the Rydberg constant calculated for an electron orbiting a xed nucleus R and the constant for real hydrogen atoms in eqn 1.1(we originally wrote R without a subscript but more strictly we should specify that it is the constant for hydrogen RH).T
24、he theoretical treatment above has assumed an infinitely massive nucleus,hence the subscript.In reality both the electron and proton move around the centre of mass of the system.For a nucleus of finite mass M the equations are modied by replacing the electron mass me by its reduced mass(1.13),Shanxi
25、 University Atomic Physics,1.3 Bohrs theory-7,For hydrogen(1.14)where the electron-to-proton mass ratio is me/Mp 11/1836.This reduced-mass correction is not the same for different isotopes of an element,e.g.hydrogen and deuterium.This leads to a small but readily observable difference in the frequen
26、cy of the light emitted by the atoms of different isotopes;this is called the isotope shift(see Exercises 1.1 and 1.2).,Shanxi University Atomic Physics,Complementarity:,Shanxi University Atomic Physics,Large size:2 n2 a0 1 m(n=100)Large dipole moments:n2 ea0 10000 D(H2O 2.6 D)Long lifetime:n3 1 msE
27、asy to ionize Eion n-4 10 V/cm,Rydberg atoms:“Semi-classical atoms”,Rydberg system(atom,molecule,)=Big n Hydrogen atom,Large size:2 n2 a0 1 m(n=100)Large dipole moments:n2 ea0 10000 D(H2O 2.6 D)Long lifetime:n3 1 msEasy to ionize Eion n-4 10 V/cm,Shanxi University Atomic Physics,1.4 Relativistic eff
28、ects-1,Applying this quantisation rule to momentum around a circular orbit gives the equivalent of eqn 1.7:mev2r=nh(1.15),Bohrs theory was a great breakthrough.However assumption of circular orbits is too much of an over-simplication.,Sommerfeld produced a quantum mechanical theory of electrons in e
29、lliptical orbits.introduced quantisation through a general rule that stated the integral of the momentum associated with a coordinate around one period of the motion associated with that coordinate is an integral multiple of Plancks constant.,Shanxi University Atomic Physics,1.4 Relativistic effects
30、-2,Sommerfeld also considered quantisation of the radial degree of freedom r.AND found that some of the elliptical orbits expected for a potential proportional to 1=r are also stationary states.,In special relativity a particle of rest mass m moving at speed v has an energy E(v)=mc2(1.16),Much eort
31、was put into complicated schemes based on classical orbits with quantisation,and by incorporating special relativity this old quantum theory could explain accurately the ne structure of spectral lines,Shanxi University Atomic Physics,1.4 Relativistic effects-3,where the gamma factor is.The kinetic e
32、nergy of the moving particle is E=E(v)-E(0)=(-1)mec2.Thus relativistic effect produce a fractional change in energy:(1.17)This leads to energy dierences between the various elliptical orbits of the same gross energy because the speed varies in dierent ways around the elliptical orbits,e.g.for a circ
33、ular orbit and a highly elliptical orbit of the same gross energy.From eqns 1.3 and 1.7 we nd that the ratio of the speed in the orbit to the speed of light is(1.18)Where the fine-structure constant is given by(1.19),Shanxi University Atomic Physics,1.4 Relativistic effects-4,The idea of elliptical
34、orbits provides a connection with our intuition based on classical mechanics and we often retain some traces of this simple picture of electron orbits in our minds.However,for atoms with more than one electron,e.g.helium,classical models do not work and we must think in terms of wavefunctions.,This
35、fundamental constant plays an important role throughout atomic physics.Numerically its value is approximately=137,It is not necessary to go into all the renements of Sommerfelds relativistic theory that gave the energy levels in hydrogen very precisely,by imposing quantisation rules on classical orb
36、its,since ultimately a paradigm shift was necessary.,Shanxi University Atomic Physics,1.5 Moseley and the atomic number_1,At the same time as Bohr was working on his model of the hydrogen atom,H.G.J.Moseley measured the X-ray spectra of many elements.Moseley established that the square root of the f
37、requency of the emitted lines is proportional to the atomic number Z(that he dened as the position of the atom in the periodic table,starting counting at Z=1 for hydrogen),i.e.(1.20)Moseleys original plot is shown in Fig.1.2.As we shall see,this equation is a considerable simplication of the actual
38、situation but it was remarkably powerful at the time.By ordering the elements using Z rather than relative atomic mass,as was done previously,several inconsistencies in the periodic table were resolved.There were still gaps that were later lled by the discovery of new elements.In particular,for the
39、rare-earth elements that have similar chemical properties and are therefore diffcult to distinguish,it was said in an afternoon,Moseley could solve the problem that had baed chemists for many decades and establish the true number of possible rare earths(Segre 1980).Moseleys observations can be expla
40、ined by a relatively simple model for atoms that extends Bohrs model for hydrogen.,Shanxi University Atomic Physics,1.5 Moseley and the atomic number_2,A natural way to extend Bohrs atomic model to heavier atoms is to suppose that the electrons ll up the allowed orbits starting from the bottom.Each
41、energy level only has room for a certain number of electrons so they cannot all go into the lowest level and they arrange themselves in shells,labelled by the principal quantum number,around the nucleus.This shell structure arises because of the Pauli exclusion principle and the electron spin,but fo
42、r now let us simply consider it as an empirical fact that the maximum number of electrons in the n=1 shell is 2,the n=2 shell has 8 and the n=3 shell has 18,etc.For historical reasons,X-ray spectroscopists do not use the principal quantum number but label the shells by letters:K for n=1,L for n=2,M
43、for n=3 and so on alphabetically.This concept of electronic shells explains the emission of X-rays from atoms in the following way.Moseley produced X-rays by bombarding samples of the given element with electrons that had been accelerated to a high voltage in a vacuum tube.These fast electrons knock
44、 an electron out of an atom in the sample leaving a vacancy or hole in one of its shells.This allows an electron from a higher-lying shell to fall down to ll this hole emitting radiation of a wavelength corresponding to the dierence in energy between the shells.,Shanxi University Atomic Physics,Shan
45、xi University Atomic Physics,Fig.1.2 Moseleys plot of the square root of the frequency of X-ray lines of elements against their atomic number.Moseleys work established the atomic number Z as a more fundamental quantity than the atomic weight(now called relative atomic mass).Following modern conventi
46、on the units of the horizontal scales would be(108(Hz)1/2)at the bottom and(10-10m)for the log scale at the top.,Shanxi University Atomic Physics,1.5 Moseley and the atomic number_3,To explain Moseleys observations quantitatively we need to modify the equations in Section 1.3,on Bohrs theory,to acco
47、unt for the effect of a nucleus of charge greater than the+1e of the proton.For a nuclear charge Ze we replace e2=40 by Ze2=40 in all the equations,resulting in a formula for the energies like that of Balmer but multiplied by a factor of Z2.This dependence on the square of the atomic number means th
48、at,for all but the lightest elements,transitions between low-lying shells lead to emission of radiation in the X-ray region of the spectrum.Scaling the Bohr theory result is accurate for hydrogenic ions,i.e.systems with one electron around a nucleus of charge Ze.In neutral atoms the other electrons(
49、that do not jump)are not simply passive spectators but partly screen the nuclear charge;for a given X-ray line,say the K-to L-shell transition,a more accurate formula is(1.21),Shanxi University Atomic Physics,1.5 Moseley and the atomic number_4,The screening factors K and L are not entirely independ
50、ent of Z and the values of these screening factors for each shell vary slightly(see the exercises at the end of this chapter).For large atomic numbers this formula tends to eqn 1.20(see Exercise 1.4).This simple approach does not explain why the screening factor for a shell can exceed the number of
