Contemporary, atomic, nuclear, and particle physics

Transcription

1 Contmporary, atomic, nuclar, and particl physics 1 Blackbody radiation as a thrmal quilibrium condition (in vacuum this is th only hat loss) Exampl-1 black plan surfac at a constant high tmpratur T h is paralll to anothr black plan surfac at a constant lowr tmpratur T l. Btwn th plats is vacuum. In ordr to rduc th hat flow du to radiation, a hat shild consisting of two thin black plats, thrmally isolatd from ach othr, is placd btwn th warm and th cold surfacs and paralll to ths. ftr som tim stationary conditions ar obtaind. By what factor is th stationary hat flow rducd du to th prsnc of th hat shild? Nglct nd ffcts du to th finit siz of th surfacs. (1.5 points) Solution: Undr stationary conditions th nt hat flow is th sam vrywhr: J ( T T ) h J ( T T ) J ( T 1 1 TL dding ths thr quations w gt 3J ( Th TL ) J0, whr J 0 is th hat flow in th absnc of th hat shild. Thus J J 0 taks th valu 1 3. ) ns.

2 Exampl Tak Earth as a blackbody, stimat its surfac tmpratur (givn Sun s surfac tmpratur T s, radius R s, sun-arth distanc d, Earth radius R ). Solution: Earth taks all its nrgy from th sun, so first w nd to know th amount of nrgy, in trms of radiation from th sun, that rachs Earth. 4 Th total amount of nrgy mittd by th sun is 4 R s Ts. Whn this amount of nrgy rachs a distanc d from th sun its light intnsity I is 4 4R s T s 4d I. Th amount of nrgy Earth rcivs is thn, considring only half th Earth is facing th sun and Earth s projction, E R I. In thrmal quilibrium, this must qual to th amount of nrgy mittd by Earth, again in th form of blackbody radiation. 4 E 4R T. Combining th abov quations, w hav T 1/ 4 Ts ( Rs / 4d ). ns. W s that th rsult is indpndnt of th Earth diamtr. In fact, objcts of any siz at th sam distanc to th sun would hav th sam tmpratur, if all rgardd as blackbody. If 1/4 th objct has a wavlngth indpndnt rflctivity r, thnt Ts((1 r) Rs / 4 d ). If r is wavlngth dpndnt, which is usually th cas, thn w nd to intgrat th blackbody I I r d, radiation to find th total amount of nrgy rcivd pr unit ara ( ) 1 ( ) whr I() is givn by th blackbody radiation formula I Nuclar physics hc 1 ). 5 k T hc B 1 ( / 0 Introduction nuclus is mad of particls calld protons and nutrons. proton carris positiv charg + whil nutrons carry no nt charg. Th protons and nutrons in a nuclus ar bond togthr by strong intraction forc which is strongr than th Coulomb forc in th short distanc but drops xponntially ovr larg distanc. Protons and nutrons togthr ar calld nuclons. nuclus can hav a total numbr of nuclons, of which Z ar protons and N=-Z ar nutrons. Th numbr Z also dtrmins th numbr of lctrons th atom has, and its chmical lmnt. Elmnts with sam Z but diffrnt s (and thrfor N s) ar calld isotops.

3 B Binding nrgy stabl nuclus should hav a total mass m that is lss than th combind mass of th total numbr of protons and nutrons it contains. Th diffrnc is calld th binding nrgy B, which is simply dfind as B/ c ( Z) m n Zm p m. (1) m m n, p ar th masss of fr nutron and proton, rspctivly. Thr ar many othr particls lik lctrons, nutrinos (lik photons, thy hav narly zro mass so always movs at spd of light), pion ( ), muon ( ), W-particls, Z 0 -particl, K (kappa),,, tc. But quarks ar nvr in fr particl form. Thy ar always glud togthr by gluons. Th liquid drop modl Th intraction btwn nuclons is short-rang, similar to intraction btwn molculs in a liquid. nuclus is thn modld as a drop of liquid, and blow ar th nrgis associatd with th liquid drop. Volum trm B a Surfac trm B V S V a S /3 Z( Z 1) Coulomb trm BC ac 1/ 3 ( Z) Symmtry trm BSym asym ap Pairing trm BP 3/ 4

4 B(, Z) a ( 1) a ( Z) a S acz Z sym p av 1/3 4/3 7 / 4 () Total mass of a nuclus with (, Z) is m ( Z) mn Zmp B/ c (3). m m n, p ar th masss of nutron and proton, rspctivly. For atomic mass, add th mass of Z lctrons Zm. a V =15.5 MV, a S =16.8 MV, a C =0.7 MV, a Sym =3. MV, 34MV,(N odd, Z odd) a P 0, (N Z odd) (4) - 34 MV, (N vn, Z vn) C -dcay nuclus (mothr) with (,Z) mits an lctron and a nutrino and bcoms a daughtr (,Z+1) nuclus. Th total kintic nrgy shard by th thr particls is Likwis, for K = (m m m d - m )c =B(,Z+1)-B(,Z)+(m n m p )c - m c (5) -dcay, a positron (anti-lctron) is cratd so K = (m m m d - m )c =B(,Z-1)-B(,Z) - (m n m p )c - m c (6) Obviously, if K is ngativ thn th procss will not occur. Othr nuclar procsss (fission of U 35 into two fragmnts, -dcay, tc.) can b calculatd in th sam way. tomic unit mass (u) = MV/c. Nutron mass m n = u. Proton mass m p = u.

5 Exrcis: Vrify th rsults shown in th figurs blow. D Gnral dcay procss Within a unit tim intrval th probability of a nuclus dcay is. If thr ar N(t) idntical nucli at tim t thn aftr tim intrval dt thr ar dn=ndt nucli that hav dcayd within th tim intrval dt. Th diffrntial quation is thn dn Ndt (7) Th solution is t N( t) N(0) (8). If thr ar svral simultanous dcay procsss with probability 1,, 3..., thn dn N( 1 3..) dt (9) Th solution is thn ( 1..) t N( t) N(0) (10).

6 Sinc nuclus is much havir than lctron (and nutrino) it shars vry littl of th total kintic nrgy (rcall E k = p /m), th xcssiv nrgy K is shard by th lctron and th nutrino. K is typically about 0.5 MV, which is comparabl to th lctron rst mass (0.511 MV), so th lctron is quit rlativistic. Not that th momntum of th lctron plus that of th nutrino is not zro bcaus th daughtr nuclus can hav comparabl momntum. So th lctron and th nutrino can just fly in any uncorrlatd dirctions. 3 Enrgy lvls In th microscopic world, vrything is govrnd by quantum mchanics, and th systm (lik atoms which is mad of lctrons surrounding th nuclus, or a nuclus which is mad of a bunch of protons and nutrons) is dscribd by stats with associatd nrgy lvls (and othr quantitis lik angular momntum, tc). For isolatd atoms and nucli th nrgy lvls ar discrt. For solids and liquids whr atoms form molculs and molculs intract with on anothr th nrgy lvls ar continuous and ar thrfor calld nrgy bands. Excitd- Excitd-1 Ground Howvr, vn th discrt lvls in atoms (and nucli) ar not without width. nd an atom at a highr nrgy lvl will not stay thr forvr. ftr a whil it will transit back to th lowst lvl (ground stat) and mit a photon (spontanous mission). On can dfin a liftim = 1/ and th transition procss is mathmatically th sam as th dcay procss. Furthrmor, th nrgy lvl width is rlatd to by 1 1 h (11). whr h is th Planck constant. Typical nrgy of photons mittd/absorbd by atoms involving th chang of nrgy lvls of an lctron in th outrmost shll is of th ordr of V and in th visibl to nar ultraviolt wavlngth. s a convnint formula nrgy (V) = 140/wavlngth-in-nanomtr. Th nrgy lvls in th innr shlls ar of th ordr of KV, and th photons mittd/absorbd by transitions to/from ths lvls to th outrmost ons ar X-rays. toms in a molcul (lik H O or CO) can vibrat around thir quilibrium positions and th nrgy is typically 10 mv (=, whr (~ 10 m). k m ) and th photons ar in th mid-infrard Transition nrgy in a nuclus (a proton or nutron dropping from a highr lvl to ground stat) is around MV and ths ar calld -ray photons. Pauli xclusiv principl Each stat can hold only on frmion (lctron, proton, nutron, tc.). Bosons (photons, tc.) can occupy a singl stat.

7 4 Lasr cooling of atoms photon can b absorbd by an atom and th absorbd nrgy is usd by th atom to transit from on nrgy lvl (usually th lowst, or ground stat) to a highr (xcitd) lvl if th photon nrgy qual to th nrgy diffrnc. Th momntum of th photon is also transfrrd to th atom. ftr spnding crtain tim on th highr lvl, th atom will spontanously drop back to th ground lvl but th photon mittd in th procss can go in any dirction. Th absorption-thn-mission procss can tak plac many tims, and on avrag, th absorbd photons all hav th sam momntum, but th mittd photons hav momntum in all dirctions, thrfor on avrag thir nt contribution to th momntum of th atom is zro. Th vlocity of th atom is thrfor lowrd if its initial dirction is opposit of that of th incoming photons. This is th gnral concpt of lasr cooling. 5 D- Brogli wavlngth atom Excitd- Excitd-1 Ground photon ny microscopic particls in motion can b dscribd by a plan wav with wavlngth diffraction and intrfrnc p k (1) whr p is th momntum of th particl. Elctron (and othr particl) bam can thrfor produc intrfrnc and diffraction pattrns in xactly th sam way as an EM plan wav with th sam wavlngth. Th wavlngth of a bam of lctrons with ~ 100 KV kintic nrgy is around 0.5 nm so it can b usd, as X-ray diffraction, to xplor th crystal structur of solids. This is th working principl of lctron microscops. 6 Hisnbrg uncrtainty principl p x E t (13) Estimation of ground stat nrgy in a confind systm <x>=a, so p p, and a p 1 E ( ) m m a