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Atomic Physics Chapter 1 Atomic Physics Chapter 1 Basic Properties of Atom 11/20/1309/03/2021 Jinniu Hu 1 1 What is an atom An atom is the smallest unchangeable component of a chemical element. 1. Unchangeable means in this case by chemical means 2. Moderate temperatures: kT < eV Mass range: 1.67×10−27 to 4.52×10−25 kg Electric charge: zero (neutral), or ion charge Diameter range: 62 pm (He) to 520 pm (Cs) Components: Electrons and compact nucleus of protons and neutrons 09/03/2021 Jinniu Hu The mass of an atom Atomic mass unit (AMU): 1u: 1/12 of the mass of a neutral carbon atom with nuclear charge 6 and mass number 12 Mass number (A): The total number of protons and neutrons in nucleus Mole (mol): 1 mol is the quantity of a substance that contains the same number of particles (atoms or molecules) as 0.012 kg of carbon 12C. 1 mol of atoms or molecules with atomic mass number A AMU has a mass of A grams. 09/03/2021 Jinniu Hu The mass of an atom The relation between 1u and NA 1 27 1u = =1.660539040(20) 10− kg NA ⇥ Electronvolt 19 1 eV = 1.602176565(35) 10− C 1V ⇥ ⇥ 19 =1.602176565(35) 10− J Mass–energy equivalence ⇥ E = mc2 1u transfer to eV 1 u = 931.478 106 eV/c2 ⇥ = 931.478 MeV/c2 09/03/2021 Jinniu Hu The mass of an atom The mass of electron: 31 m =9.10938356(11) 10− kg e ⇥ 4 =5.48579909070(16) 10− u ⇥ =0.5109989461(31) MeV The mass of proton: 27 m =1.672621898(21) 10− kg p ⇥ =1.007276466879(91) u = 938.2720813(58) MeV The mass of neutron: 27 m =1.674927471(21) 10− kg n ⇥ =1.00866491588(49) u = 939.5654133(58) MeV 09/03/2021 Jinniu Hu Avogadro’s Number Avogadro’s number is a Bridge from macroscopic to microscopic physics. 1 mole of any substance contains the same number (NA ) of atoms (molecules) Mass of 1 mole of the substance N = A Mass of an atom 23 1 =6.02214078(18) 10 mol− ⇥ 1. The Faraday constant and elementary charge F = NAe 2. Gas constant and Boltzmann constant R = kBNA 3. Molar volume and atomic volume Vm = VatomNA 09/03/2021 Jinniu Hu 14 2 The Concept of the Atom 14 2 The Concept of the Atom c) Determination of Avogadro’s Constant from c)Electrolysis Determination of Avogadro’s Constant from ElectrolysisAnother method for the determination of NA is based on Fara- Avogadro’sAnotherday’s method law Number for for electrolytic the measurements determination processes. of ItN statesA is based that the on Fara- electric day’scharge law for electrolytic processes. It states that the electric Thecharge Faraday’s constant F NA e 96, 485.3383(83) C/mol (2.15) = · = F NA e 96, 485.3383(83) C/mol (2.15) is the electric= charge· = transported to the electrode in a is transported to the electrode in an electrolytic cell, when a an electrolytic cell, when 1 mol of singly charged ions is transported1molofsinglychargedionswithmass to the electrode in an electrolyticmX cell,and elemen-when with1molofsinglychargedionswithmass masstary chargemX ande haselementary been deposited charge at the e electrode.hasmX andbeen elemen- Therefore, a/2 tary charge e has been deposited at the electrode. Therefore, depositedweighing at the the masselectrode. increase ∆m of the electrode after a charge a/2 weighingQ has thebeen mass transferred, increase yields:∆m of the electrode after a charge Therefore, weighing the mass increase m of the Q has been transferred, yields: electrode after a charge Q hasQ beenQ transferred,MX Fig. 2.9 Elementary cell of a cubic crystal ∆m Q mX Q M yields: = e = e XNA Fig. 2.9 Elementary cell of a cubic crystal ∆m mX = e Q M=X e NA NA (2.16a) Phase ⇒ =Q Me X∆m NA (2.16a) Phaseplanes ⇒ = e ∆m planes 09/03/2021where MX is the molarJinniu mass Hu of the ions with mass mX. where MX is the molar mass of the ions with mass mX. ϑ Example ϑ ExampleIn the electrolytic process In the electrolytic process d ϑ Crystal d Crystalplanes AgNO3 Ag+ NO3− d ϑ ↔ + d planes AgNO3 Ag+ NO3− of silver nitrate the transport↔ of+ charge F means a simulta- of silverneous nitrate deposition the transport of the molar of charge mass FMmeansNA a simulta-m(Ag) at = · neousthe deposition negative electrode, of the molar which mass can beM measuredNA bym(Ag weighing) at = · Fig. 2.10 Bragg-reflection of X-rays by two crystal planes thethe negative cathode electrode, before and which after can the be charge measured transport. by weighing With the theatomic cathode mass before number and after of silver the AM charge(Ag transport.) 107.89 With AMU the the Fig. 2.10 Bragg-reflection of X-rays by two crystal planes = wavelength λ,theinterferenceisconstructiveandtheampli- atomicAvogadro mass number number of silver AM(Ag) 107.89 AMU the = tude of the different partial waves add up. This is expressed Avogadro number wavelength λ,theinterferenceisconstructiveandtheampli- 107.89 AMU Q tudeby of the the Bragg different condition partial waves add up. This is expressed NA (2.16b) 107= .89 AMU∆m Q· e by the Bragg condition NA (2.16b) 2d sin ϑ m λ. (2.17) = ∆m · e · = · is obtained from the measured mass increase ∆m of the elec- 2d sin ϑ m λ. (2.17) is obtainedtrode and from the transported the measured charge massQ increase(∆m∆/mMof)N theA e elec-. At a given wavelength·λ one obtains= · maxima of intensity I (ϑ) = · trode and the transported charge Q (∆m/M)NA e. Atof a given the scattered wavelength radiationλ one only obtains for maxima those inclination of intensity anglesI (ϑ)ϑ, = · offor the which scattered (2.17 radiation)isfulfilled. only for those inclination angles ϑ, d) Determination of N from X-Ray Diffraction A for whichOne sees(2.17 from)isfulfilled. (2.17)thatform > 0thewavelength The most accurate method for the determination of N is d) Determination of NA from X-Ray Diffraction A One sees from (2.17)thatform > 0thewavelength based on X-ray diffraction or X-ray interferometry, which 2d The most accurate method for the determination of NA is λ sin ϑ < 2d basedare on used X-ray to measure diffraction the distances or X-ray between interferometry, atoms in awhich regular =2dm λ sin ϑ < 2d arecrystal used to [10 measure]. This theyields distances the total between number atoms of atoms in a per regular volume has to be smaller than= twicem the distance d between adjacent crystalif the [10 crystal]. This structure yields the is total known. number of atoms per volume crystal planes. For visible light λ d,butforX-raysof if the crystalLet us consider structure a is cubic known. crystal, where the atoms sit at the has to be smaller than twice the distance$ d between adjacent corners of small cubes with sidelength a (Fig. 2.9). When a crystalsufficient planes. energy Forλ visible< 2d can light beλ achievedd,butforX-raysof (see Sect. 7.6). Let us consider a cubic crystal, where the atoms sit at the $ cornersplane of wave small with cubes wavelength with sidelengthλ is incidenta (Fig. on the2.9 crystal). When under a sufficient energy λ < 2d can be achieved (see Sect. 7.6). Note planean waveangle withϑ against wavelength a crystalλ planeis incident (Fig. 2.10 on the)thepartialwaves crystal under In (2.17) ϑ is the angle of the incident radiation against the an anglescatteredϑ against by the a different crystal plane atoms (Fig. of adjacent2.10)thepartialwaves planes with dis- Note crystal planes not against the normal to the planes, different scatteredtance d byinterfere the different with each atoms other. of adjacent In the direction planes withϑ,which dis- In (2.17) ϑ is the angle of the incident radiation against the − from the conventional definition in optics. tancecorrespondsd interfere to with the direction each other. of In specular the direction reflection,ϑ,which their path crystal planes not against the normal to the planes, different difference is ∆s 2d sin ϑ.If∆s equals an integer− m of the from the conventional definition in optics. corresponds to the direction= of specular reflection, their path difference is ∆s 2d sin ϑ.If∆s equals an integer m of the = 2.2 Experimental and Theoretical Proofs for the Existence of Atoms 13 where M is the molar mass and κ Cp/Cv [9]. Since Fg (m "L Vp)g (2.11a) = = − · the2.2 acoustic Experimental losses and of Theoretical the spherical Proofs resonator for the Existence are low, of Atoms the 13 resonances are very sharp and the resonance frequencies is just compensated by the friction force f can be determined with high accuracy. Tuning the 0,nwhere M is the molar mass and κ Cp/Cv [9]. Since Fg (m "L Vp)g (2.11a) = = − · loudspeaker-frequencythe acoustic losses of through the spherical several resonator resonances are low, from the Ff 6πηrv, (2.11b) 2 = − n 1 to higher values the squares f0,n of the resonance is just compensated by the friction force =resonances are very sharp and the resonance frequencies frequencies can be plotted against n2 which gives a straight f0,n can be determined with high accuracy. Tuning the 2 which spherical particles of radius r experience when they fall lineloudspeaker-frequency with the slope RκT/(Mr through0 ).SinceMand several resonancesκ are known from Ff 6πηrv, (2.11b) 2 with the velocity v in a medium= − with viscosity η,thenetforce andn T and1 r to0 can higher be accurately values the measured, squares f0 the,n gasof the constant resonance R = canfrequencies be determined can [ be9].
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