Lecture notes by Prof. Andrea Di Cicco. Di Cicco UniCam Italy 2009, rev. 2010

Transcription

1 Modern Physics constituents of atter Lecture notes by Prof. Andrea Di Cicco Di Cicco UniCa Italy 009, rev

2 Background The idea of atter as constituted by individual particles (called atos, indivisibles) is introduced in odern science first by looking at cheical investigations (reactions that ust be associated with well defined proportions of atos or olecules, Prout, Dalton, Avogadro) and afterwards in electricity looking at the aount of charge (first by Faraday in electrolysis). It was only at the end of the 19th century that becae evident that atos were coposed by particles of different charges, and the first optical spectra showing the characteristics of each eleent in absorbing and eitting light were discovered. The central role of the hydrogen ato as the siplest atoic syste was soon discovered, both by cheical and physical ethods. Typical atoic asses as ultiples of the hydrogen ass were established. The typical atoic sizes were deterined by various ethods, by collision techniques with atoic beas (scattering cross section), therodynaics (co volue in Van der Waals equation of state) and by x ray diffraction. Evidence for the existence of tiny negative particles, the cathod rays later called electrons were obtained by Thoson, Lenard and any others. It was shown that electrons could penetrate deeply in condensed and gaseous atter and that atos could be seen like a sall planetary syste. The realization that sall nuclei (10 15 ) contained the entire positive charge and alost all the ass of the atos is due to Rutherford who investigated the scattering of a particles (+ charged 4He nuclei) eitted by radioactive substances (5 MeV). Di Cicco UniCa Italy 009, rev. 010

3 The electron A fundaental step in our understanding of the atoic structure has been the discovery of the electron, a subatoic particle with a quantu of negative charge e. The concept was introduced in cheistry in early 1800 but it was only when Crookes evacuated tubes were available that direct proof of existence of these eleentary charges was obtained. The key experients for the electron discovery were perfored by J. J. Thoson (Nobel prize, 1906) following the work of any previous scientists (Hittorf, Goldstein, Schuster, Lenard in Gerany and Crookes in England) indicating that the cathod rays were actually coposed by charged particles. The basic ideas for the experient were 1) assessing that the charged cloud eitted by the cathode could not be separated by other effects (luinescence of the glass) possibly due to other kind of rays; ) assessing whether this cloud can be deflected by an electric field (not only agnetic as it was known); 3) easuring the ratio e/ of the particles coposing the cloud. Crookes (Maltese) tube during a discharge using a high voltage Ruhkorff coil. The profile of the cross shaped target is projected against the tube face at right by a bea of electrons Di Cicco UniCa Italy 009, rev

4 Thoson's experient (I) Thoson and collaborators devised a series of experients particularly suited for their ais, using quite sophisticated glass tube asseblies, electrical equipents and puping systes for the tie. In his first experient, he investigated whether or not the negative charge could be separated fro the cathode rays by eans of agnetis. He constructed a cathode ray tube ending in a pair of cylinders with slits in the. These slits were in turn connected to a gold leaf electroeter. Thoson found that if the rays were agnetically bent such that they could not enter the slit, the electroeter registered little charge. Whenever the cathode rays were forced to enter the slits the charging was iediately visible (1 sec to alter the potential of a 1.5 F capacity to 0 V). The charge was seen to be negative by charging the inner cylinder before observing the effect of the cathod rays, either with positive or negative charges. Thoson concluded without any doubt that the negative charge was inseparable fro the cathode rays and that the objection about the Nature of cathod rays as ethereal radiation (raised ainly by Geran Di Ciccoby UniCa Italy particles 009, rev. 010 Physicist), soeties accopanied charged was not true. 4

5 Thoson's experient(ii) In his second experient, he investigated whether or not the rays could be deflected by an electric field (soething that is characteristic of charged particles). Previous experienters had failed to observe this, but Thoson believed their experients were flawed because they contained trace aounts of gas. Thoson constructed a cathode ray tube with a very good vacuu, and coated one end with phosphorescent paint. Thoson found that the rays did indeed bend under the influence of an electric field, in a direction indicating a negative charge. Thoson and collaborators defined a precise bea of electrons with slits at the anode exit and observed the deflection with a scale depicted at the end of the glass tube at soe distance of the plates. Voltage could be raised up to about 00 V causing large deflections, before the residual gas begin to ionize reducing the effective field in between the plates to zero. When a discharge between the plate was observed the deflection was suddenly eliinated. Di Cicco UniCa Italy 009, rev

6 Thoson's experient(iii) In his third experient, Thoson easured for the first tie the ass to charge ratio of the cathode rays by easuring how uch they were deflected by a agnetic field and how uch energy they carried. 1 Q W v v N v =W ; N = v= Ne=Q ; =e v H =rh e e Q r e W W v e r H /= v= e Q rhq W W W W rh Q v = = = = e Q e rhq Q e Q W /QrH e W Looking at these equations we conclude that we can easure /e by easuring r, H, Q, and W. The radius of curvature was easured by the distances travelled by the electron (naely the collision position with the glass wall) and soe uncertainty was due to the spread observed in those positions (segent instead of punctual bea). The unifor agnetic field was provided by two large circular coils and easured by the current circulating in the coils. Q is easured as in experient (I) (two coaxial cylinders) keeping as short as possible the easureents to avoid discharging in the residual gas. W was easured with a theroelectric couple Cu Fe behind the slit of the inner cylinder with known heat capacity ( cal/k). Di Cicco UniCa Italy 009, rev

7 Thoson's experient (results) The first quantitative results of the 3rd Thoson experients, perfored using three different configurations indicated a /e ratio for the electron ranging between 0.3x10 8 g/c to about 1x 10 8 g/c (.3 1x10 7 g/eu) depending on the particular set up used but was not found to depend on the gases present and on the nature of the electrodes.. The current accepted value is /e = x 10 8/ 1.60 x = x 10 8 g/c An iportant iproveent of the experiental set up (II) described by Thoson and used to easure again /e in an independent way is the cobination of the application of an electric field and of a agnetic field acting on the sae distance path. This technique is fundaental for further developents in physics (ass spectroetry) and applications (oscilloscope, cathodic tube for tv sets...). ee ee l t l=v x t ; v y = vx eh z v x e Hz v x =cost ; v y = t l=v x t ; v y = l v x =cost ; v y = Applying an electric field: Applying a agnetic field: v y ee l = v x v x v y e Hz l tg = = vx vx tg = Tuning E, H fields to obtain sae deviation angles we obtain: tg =tg = ee l = e H zl E = H z v v v H H H So: x x x e z e z l z tg = E Hz l= E l e = tg l E Those easureents gave results in agreeent witht the first ethod (~1x10 8 g/c). Di Cicco UniCa Italy 009, rev

8 Thoson's conclusions Thoson's conclusions were bold: cathode rays were indeed ade of particles which he called "corpuscles", and these corpuscles cae fro within the atos of the electrodes theselves, eaning that atos are in fact divisible. The "corpuscles" discovered by Thoson are identified with the electrons which had been proposed by G. Johnstone Stoney. He conducted this experient in He found that the ass to charge ratio was over a thousand ties lower than that of a hydrogen ion (H+), suggesting either that the particles were very light or very highly charged. The /e ratio of hydrogen ions was estiated by electrolysis by easuring the aount of hydrogen produced by a current (1 Faraday = kcoulob per ole). By using an atoic weigth (1 ole = g) it was known that (H+)/e = g/ C = x 10 5 g/c So assuing the charge was the sae, the ratio between the asses turned to be about ~1830. Thoson iagined the ato as being ade up of these corpuscles swaring in a sea of positive charge; this was his plu pudding odel. This odel was later proved incorrect when Ernest Rutherford showed that the positive charge is concentrated in the nucleus of the ato. Di Cicco UniCa Italy 009, rev

9 e/ ratio and ass spectroetry Thoson's ethods and results were used and iproved afterwards for deterining charges and asses of atoic and subatoic particles. An elegant ethod was developed by Classen (1907) as illustrated in the figure. The parabola ethod (E and B parallel) introduced by Thoson (1913) was used and iproved for years and is still used for siple ass spectroetry of ions. It allowed to discover the isotopes (Aston 190). r =e v ] x :[ F B = F c = v ]; y :[ F =e E r e 1 ee e l y = E y= t = E v ebv ebv ebv l ebl x = x= t = = v v Di Cicco UniCa Italy 009, rev. 010 v B ] F = e[ E ev v v =e V v= ; =e v H r e V = r B y= E x l B e 9

10 ass spectroetry An exaple of the application of the parabola ethod is shown in the figure. Aston, using a ore sophisticated (focussed) set up, deterined the coposition of naturally occurring Ne isotopes (ass weight nubers A: 0, 1,, Z= 10) and suggesting the existence of neutrons. In the Aston spectroeter E and B are perpendicular, and tuning the B field one can focus ions with different velocity (but sae /e) in the sae point. Modern ass spectrograph using various focussing techniques to iprove signal and ass resolution. Quadrupole (electric) high frequency ass filters provides better separation of asses. Di Cicco UniCa Italy 009, rev

11 Electron charge (Millikan) Knowledge of the e/ ratio by Thoson's experients called for independent easureents of the fact that a quantu of charge e actually exists.* Millikan (1911) accoplished this task in a faous series of experient on the dynaics of oil droplets under the action of gravity, of an electric field and of the bouyancy into the fluid (air) and friction in air (Nobel prize 193). The experient needed to be perfored in a very accurate way, taking into account all of the source of possible errors. The oil droplets were naturally charged (by friction for exaple) in a first version of the experient, while charges could be induced by x rays in a second one. *At that tie ainly inferred by electrolysis through the Faraday constant F: q =F/N, F=96.5 kc*ole. Di Cicco UniCa Italy 009, rev

12 Electron charge II Observation of the falling spherical oil droplets, eitted by an atoizer and passing through a hole into a capacitor (voltage 3 8 kv) was perfored using a short distance telescope allowing a precise easureents of tiing over short distances (fraction of ). Tiing obviously depended on charges deposited onto the droplets, ranging usually fro 1 to hundreds, and sudden variation of state of otion was recorded upon increasing or decreasing the charge by (ultiples of) e. The chaber was filled by dust free clean air under controlled pressure (1 15 atosphere) and teperature conditions (engine oil bath 40 l). The easured eleentary charge was 4.93x10 10 esu (statcoulob), later corrected by Millikan into esu. The ain source of error was the viscosity of air (calculated assuing a Stokes law F = 6 rv is the viscosity coefficient, r is the radius of the oil drop, but the value of had to be reconsidered and the Stokes law was found inaccurate for sall r). The current accepted value is (1statC= e= esu = C. C) Di Cicco UniCa Italy 009, rev

13 Atoic structure At the beginning of the 0th century our understanding of the atoic structure was about to clarify, ainly due to the discovery of the electrons (negatively charge) as individual particles and of the nuclei (positive) studied by Rutherford (1911) by ion scattering. The electron is considered a structureless point like particle (as shown by high energy scattering experients). The classical radius of the electron can be found iposing that the energy of its electrostatic field (assued to be that of a spherical capacitor) is equal to its rest ass: 1e 1 e e 15 E= 0 c ; E= = r el = =.810 C 0 r el c The radius of atos nuclei, containing alost the entire ass of the ato and a charge Ze, as found by Rutherford, was of the sae order of agnitude (10 15 ). Nuclei were shown later to have an inner structure ade by charged protons and (neutral) neutrons, both coposed by other fundaental particles called quarks (having fractional charges 1/3e, /3e, but they are always bonded to for integer charges). A siple calculation for condensed atter show that the average diension occupied by an ato in solid systes is quite bigger. For etallic Cu, atoic weight A= g/ole, density at roo teperature 8.94 g/c3 we find: Volue per ole V=A/ = c3. The average interatoic spacing results then to be: dint= (V/N)1/3 = (Ang.) Di Cicco UniCa Italy 009, rev

14 Atoic structure as seen by x ray diffraction Early after the discovery of the x rays, scientists were using the to investigate atter in various ways. C. G. Barkla (Nobel Prize 1917) and others showed the properties of transission and excitation of atter by using x rays, identified as electroagnetic radiation at very short wavelength (in the range around ). The experients in 191 of M. Von Laue, (Nobel 1914) Friedrich and Knipping and of W. L. and W. H. Bragg (Nobel 1917) showed unabiguously that crystalline atter was able to produce interference patterns on x ray beas. The basic experient showing the reflection of x ray radiation fro crystalline speciens provided: 1) the direct proof that crystals are built up of discrete entities, the atos ) the lattice constant of crystals, so a direct easure of the interatoic distances in a crystal, being in the range (subatoic particles are instead below 10 15, atter is thus pretty epty!) 3) the wavelength of x ray radiation The exact derivation of the interference conditions can be obtained using a tridiensional atoic arrangeent and the incoing and outcoing electroagnetic waves (Laue/Ewald construction). A siplified picture showing a Italy relationship with lattice planes is due to Bragg. Di Cicco UniCa 009, rev

15 Bragg reflections A typical set up for diffraction can be seen in the figure. When a single crystal is shined by a polychroatic x ray bea a certain nuber of spots, corresponding to well defined reflections of x rays of given wavelength is obtained. Experients of this kind were perfored for any crystals. X ray diffraction can be regarded as a reflection of x ray radiation by a specific set of lattice planes at well defined scattering angles. Each set of plane is distinguished by spacing, orientation and ato density on the planes. As shown in the figure each ato belonging to a plane is a source of an eleentary scattered wave, interfering with all the others generated by the other atos. The condition for constructive interference is siply given by iposing that the difference in the optical path ( ) ust be equal to a ultiple of the wavelength: = AB BC AE=AB AE = AD= d AD cos sin d d = 1 cos =d sin tan sin n =d sin Di Cicco UniCa Italy 009, rev

16 X ray diffraction ethods Besides the Laue ethod with polychroatic radiation (useful for single crystals), other ethods were devised using onochroatic radiation by Bragg for single crystals (interference conditions fulfilled rotating the crystal at well defined angles), and by Debye/Scherrer for powdered/polycrystalline saples. A typical Debye Scherrer set up showing any lattice planes of MgO is shown in the figure. Di Cicco UniCa Italy 009, rev