CARBONATED WATER. 5 2, 4 during the surfacing. The answer should be an. k 3. Neglect a change in the density of CO 2

Transcription

1 CARBONATED WATER Suppos you hav a rgular 1-litr fatory sald bottl of arbonatd watr You hav turnd th bottl ap slightly to unsrw it (so a hissing sound was hard) and srw th ap tightly again Now you would obsrv bubbls of arbon dioxid (CO ) rising upwards larg, at first, and thn smallr ons Lt us study th pross of bubbl surfaing on On an asily s that th shap of a small bubbl is muh losr to sphrial than that of a biggr 1) Estimat th siz of an immobil bubbl suh that th bubbl shap approximats sphr with an auray of 1% or bttr Watr dnsity is 1 g/m, th surfa tnsion, 7 N/m, and th fr fall alration is takn to b g 9, 8 m/s Th numrial answr should b in mm Considr a bubbl so small that it an b rgardd as almost sphrial For instan, lt th bubbl initial diamtr nar th bottl bottom b d, mm ) Figur out th bubbl alration right aftr it has dtahd from th bottom Th CO dnsity insid th bubbl at th givn watr tmpratur is, v g/m Th numrial answr should b in m/s Drag for xrtd on a bubbl, whn it is moving in watr, is linarly proportional to its rossstional ara, watr dnsity, and th bubbl vloity squard W assum th proportionality fator for this problm to b, ) Suppos that th bubbl volum rmains onstant What is th trminal vloity th bubbl an rah? Writ down th quation and valuat th numrial valu (in m/s) 4) Estimat th tim it taks th bubbl to rah th trminal vloity aftr dtahmnt, i whn its alration boms muh lss than g Writ down th quation and valuat th numrial valu (in sonds) Atually, th dnsity of CO moluls dissolvd in a liquid is muh highr than that in a gas bubbl Thrfor, th dissolvd gas diffuss into th bubbl and its radius grows It is rasonabl to assum that th growth rat of th bubbl volum du to th diffusion is proportional to th bubbl surfa ara and to an xss of th dissolvd gas dnsity and invrsly proportional to th thiknss of th liquid layr through whih th gas diffuss (alld th «dpltion layr») Th fastr bubbl is moving th thinnr is th fftiv dpltion layr (du to irulation of watr surrounding th bubbl) A kinti thory alulation givs for th dpltion layr thiknss: bubbl vloity and d is its diamtr d onst, whr v is th v 5) Suppos it taks tim T for a bubbl to surfa Figur out th bubbl diamtr as a funtion of tim t, providing it has inrasd by a fator k 4/ 5, 4 during th surfaing Th answr should b an quation for d (t) in trms of T, t, d, and dissolvd in watr during th bubbl surfaing 6) Dtrmin tim dpndn (t) 4 / 5 k Nglt a hang in th dnsity of CO v of th bubbl vloity (th answr should inlud a formula for v (t) xprssd in trms of th paramtrs listd abov and th trminal vloity v )

2 7) Dtrmin th law of motion of a bubbl, i tim dpndn of its lvation h abov th bottom (th answr should inlud a formula for h (t) in trms of T, t, and v ; tak th sam valu 4/5 k ) 8) Suppos th hight of watr olumn, whih th bubbl travrss on its way upward, quals H m Evaluat th tim of bubbl surfaing (th answr should b th xpliit formula inluding th paramtrs givn in th problm and alulat th numrial valu in sonds) During bubbl surfaing som hat is bing rlasd (th drag for dos a work by inrasing th kinti nrgy of turbulnt flow whih vntually dissipats as hat) and at th sam tim som hat is bing absorbd (du to CO vaporation from watr into a bubbl) 9) Assum that all th hat rlasd during th surfaing is onvrtd into hating th «olumn» of watr whih ross-stion quals th avrag ross-stion of a rising bubbl Using this assumption stimat (by th ordr of magnitud) th tmpratur inrmnt of watr in th «olumn» Th spifi hat apaity of watr is 4 J/kg K Th answr should b givn in Klvin w 1) Evaporation into bubbls of 1 mol of CO dissolvd in watr rquirs approximatly kj of nrgy Estimat th ooling fft by th ordr of magnitud and ompar to th hating fft (s 9)) Th answr should b givn in Klvin What will th nt rsult b? Th answr should b ithr «+» (th tmpratur inrass) or (th tmpratur drass) Assum that th prssur and tmpratur in th bottl hang slightly and rmain los to p 1 kpa and T 9 K

3 PROPOSED SOLUTION AND ANSWERS 1) Bubbl «oblatnss» is ausd by th diffrn of hydrostati prssur abov and blow th bubbl, whil surfa tnsion tris to kp th bubbl shap sphrial Thrfor, on obtains a rasonabl stimat by assuming that th bubbl shap rmains sphrial at th givn auray if th for of surfa tnsion xds th nt for of hydrostati prssur (i buoyany for for th bubbl at rst) by on ordr of magnitud (i approximatly 1 tims): d mm, so max 5g d mm d d 1g 6 Hn ) Thr is no drag for xrtd on th bubbl at zro vloity Only th gravity and buoyany fors at on it Howvr, using a «ommon» xprssion for th buoyany for givs th absurd rsult: th alration of gas bubbl must b about 5 g! Atually, at a non-zro alration th nt for of hydrostati prssur diffrs from th «stati» valu of buoyany for Th simplst way to undrstand this is to tak into aount that bubbl rising is quivalnt to going down of th displad amount of watr du to gravity for Obviously, th orrsponding alration annot xd g vn if thr is no drag At th start, th bubbl alration is los to th maximum, i a g Not: an aurat valuation of th alration of a body of dnsity v, whih is surfaing in a liquid of dnsity, yilds: v a g This indd tnds to g whn v v ) At th stationary rgim, whn th bubbl vloity quals to its trminal valu v, th drag balans buoyany for (whih at zro alration quals its «stati» valu), and th gravity for d d gd xrtd on th bubbl an b ngltd Thrfor, g v and v, 1 m/s 6 4 4) Th bubbl alration drass from g to zro whil its spd grows from zro to v Thrfor, it is rasonabl to stimat that th bubbl rahs th trminal vloity in a tim v d g / g, s Noti that th hight of watr in th bottl signifiantly xds th distan travld during this tim (about 1 mm), so pratially during th whol of surfaing th bubbl is going up at th trminal vloity 5) Aording to th prvious stimat th bubbl is going at th trminal vloity almost all th tim during th surfaing, i its vloity and th diamtr ar rlatd as g y v ( y) (whr y is th bubbl diamtr at a partiular tim) Aording to th problm statmnt, th rat of th bubbl d y n volum hang is A y (whr n is th diffrn btwn CO moluls dt 6 dnsity in watr and insid th bubbl and A is a numrial onstant) Diffrntiating this quation dy An on obtains: Now, using th quation givn in 5) for th thiknss of dpltion layr and dt

4 ralling that approximatly n d ( t) onst, on obtains: t dy dt B v y C y 1/ 4 (whr B and C ar nw 1/ 4 onstants) Thn y dy C dt Ct, whih givs aftr intgration 4 [( 5 d 4 / 5 5/ 4 5/ 4 5 / 4 5C d ( t)) d ] Ct d( t) d t Sin d( T) D k d onstant: finally: d ( t) d 4 / 5 5 / 4 5 / 4 5 / 4 t 5 / 4 [ D d ] d 1[ k 1] 4 / 5 ( ) t d t d 1 T T 4 t T on an dtrmin th 4 / 5 Substituting 6) Using th rlation btwn vloity and diamtr on obtains: / 5 5 / 4 t 1[ k 1] v 1 / 5 4/5 k, on gts gd( t) v ( t) or t v ( t) v T T 7) To dtrmin th law of motion on should intgrat th vloity: th lvation abov th bottom is t h( t) v( t) dt Hn, 7 / 5 5v T t h ( t) T 8) Now w substitut t T in th drivd quation and obtain 5( 7/5 1) vt H Thrfor T H, s Th xprimntal valu is a bit gratr baus th bubbl at th nd 7 / 5 5( 1) gd of surfaing is biggr and thrfor mor oblat, whih lads to inrasing drag for 9) Th hat rlasd is qual (up to th sign) to th work don by drag for, so it an b stimatd as Q S v H (th subsript «av» stands for th avrag valu of a physial quantity) Th av av mass of th watr «olumn» dsribd in th problm is of th «olumn» is stimatd as T v av w gd ~ w av ~ 1 m S 6 К av H Thn th tmpratur inrmnt 1) Th absorbd hat is proportional to th amount of vaporatd gas : Q r, whr 4 r 1 J/mol Sin p 1 RT 4 8 ( k 1) d 7 mol, T r m w w r ~ 1 S H Thrfor, th nt fft is, obviously, ooling Noti that surfaing of a singl bubbl has almost no fft on th watr tmpratur in th bottl av K TABLE OF ANSWERS Answr Maximum sor 1 d mm An answr dviating from this max valu by no mor than 1mm is aptd 1

5 A numrial answr must agr with th fr fall alration by ordr of magnitud = a g 9,8 m/s gd v,1 m/s 1(quation) + 1 (numbr) = 4 v d, g / g s 1(quation) + 1 (numbr) = 5 4 / 5 ( ) t t d 1 d T 6 / 5 ( ) t v t v 1 T, whr gd v / 5 5v T t h ( t) T 7 6 T H, s 7 / 5 5( 1) gd (quation) + 1 (numbr) = 4 9 gd av 6 T ~ ~ 1 K w 1 T ~ 1 K; + 1 = (th nt fft is ooling) Total 5

6 Problm : RADIOGRAPHY Industrial radiography is on of th basi modrn mthods of matrials sin A studid objt is plad in a ohrnt monohromati bam of X-ray photons of high intnsity whih sattring pattrn is thn analyzd Somtims mthods of sptromtry ar usd, i variation of intnsity of th bam passing through a matrial is masurd as a funtion of radiation wavlngth Howvr, th most ommon mthods of study of atomi strutur ar diffration mthods Thy ar basd on analysis of th diffration pattrn rsulting from lasti sattring of X-rays by atoms of th sampl Noti that radiation wavlngth rmains onstant in lasti sattring A ommon sour of ohrnt X-rays of high intnsity with a wid wavlngth sptrum is synhrotron a larg ring storag of hargd partils travling at a spd los to th spd of light Suh partils ar alld rlativisti baus thir motion is no longr dsribd by Nwtonian laws of lassial mhanis, instad on must us th spial thory of rlativity (STR) Th oprating prinipl of synhrotron is basd on th fat that a hargd partil mits ltromagnti radiation at trajtory turns Dirtion of partil vloity is hangd by spial bnding magnts A partil (g ltron) path in th synhrotron ring onsists of straight sgmnts, whr ltrons riv kinti nrgy, and sgmnts of almost onstant urvatur in a strong magnti fild of bnding magnts In th urvd sgmnts ltron motion is highly alratd, so thy mit ltromagnti radiation in th X-ray rang A powrful synhrotron is oprating at th Kurhatov Institut in Mosow Rlativisti quation of motion of a hargd partil in magnti fild is whr th partil momntum ), th quantity is th partil invariant mass and mass Th nrgy of rlativisti partil is masurd in ltronvolts: 1 V (1, C) (1 V) Th fator «Lorntz fator» If th partil spd is los to с, [ ], Hr с is th spd of light in vauum ( is th partil rlativisti Usually th nrgy of a miropartil is alld th 1) Evaluat th Lorntz fator for an ltron of nrgy E =,5 GV (th invariant mass is kg, th rst nrgy is MV) By how many prnt is th spd of suh an ltron lss than с? Th ltron harg is C Th answr should inlud formula and numrial valus ) Dtrmin th urvatur radius of ltron trajtory in th fild of bnding magnt if ltron nrgy in th synhrotron storag ring is maintaind at E =,5 GV and th indution of th fild of bnding magnt is B = 1,7 T Eltrons ar travling in a plan prpndiular to th magnti fild lins Th answr should inlud th formula and th numrial valu Any alrating hargd partil mits ltromagnti radiation Th important fatur of synhrotron radiation (i radiation of rlativisti partils with travling along a urvd path) is its «sarhlight» natur: almost all th nrgy is radiatd «forward» along th partil vloity in a narrow on with half an aprtur of (s Fig1)

7 Fig 1 ) Suppos an «obsrvr» О rsids at th irular orbit plan and an b rgardd as a point In this as sh dtts radiation flashs orrsponding to th short priods whn sh is insid th «sarhlight» on of orbiting partil Dtrmin th lngth of th ar travrsd by ltron on th orbit whn its radiation is dttd by th obsrvr Th ltron nrgy and th magnti indution ar givn in ) Th answr should b th quation 4) Dtrmin duration of th radiation «flash» dttd by th obsrvr It must b takn into aount that du to ltron rlativisti motion it taks diffrnt tim for th «initial» and «final» portions of th «flash» to rah th obsrvr Th answr should inlud th quation and th numrial valu Thus, radiation of rlativisti partil travling along a irular path is obsrvd as a bright short flash Th flash sptrum (frqunis and wavlngths) turns out to b vry wid: th width of th frquny rang orrsponds to «haratristi» (or «synhrotron») frquny A wid rang of wavlngths provids a lot of opportunitis for using synhrotron radiation in radiography Th haratristi wavlngth of a partiular sour is an important quantity for pratial appliations 5) Dtrmin th haratristi wavlngth of th sour dsribd in ) Th answr must b th quation and th numrial valu Th main mthod of diphring th strutur of a rystal matrial is X-ray diffration Th radiation is bing diffratd (lastially sattrd) by atoms of a sampl Th rystal srvs as a diffration grid for X-ray bam baus its wavlngth is of th sam ordr of magnitud as a spaing btwn atomi plans Whn th radiation is inidnt on th rystal at som angl, th rfltd radiation is dttd not only in th dirtion dtrmind b th laws of gomtrial optis but also at th angls for whih th wavs rfltd by adjant plans hav optial path diffrn qual to an intgr of radiation wavlngth Ths rfltd wavs mutually amplify at a rmot dttor rsulting in a signifiant ris of intnsity in th orrsponding dirtion (diffration maximum) 6) Using th ondition of diffration maximum driv th xpliit formula for th dirtion at a diffration maximum of X-rays rfltd by a rystal whih latti onsists of a singl st of paralll plans Th bam is inidnt at th angl to th plans, th intrplan spaing quals d (s Fig)

8 θ d Fig In a ral rystal strutur it possibl to introdu diffrnt sts of quidistant paralll plans Suh a st an b dfind by a vtor prpndiular to th plans whil dirtions at th diffration maxima dfind by th ondition drivd in 6) an b spifid by diffration angl (th angl btwn th inidnt and diffration bams) In what follows th obsrvd maximum of a givn ordr for a spifi st of paralll plans is alld «rflx» Any atomi plan nssarily passs through th nods of rystal latti, so oordinats of th vtor prpndiular to a partiular st of paralll plans an b givn by intgrs providing th oordinat axs ar alignd with th latti prinipal axs (dgs) and a distan is masurd in latti onstants Thus, a rflx an b dtrmind by a st of thr intgrs Suprondutor is a vry intrsting objt for th modrn matrials sin On of th most ommon and widly usd low tmpratur suprondutor is triniobium-tin Nb Sn For instan, this suprondutor is usd in ltrial iruits of th Larg Hadron Collidr A unit ll of its latti strutur (i a ll whih rptitiv translation along prinipal axs rprodus th whol rystal) is a ub of sid L = 5,9 Å (1 Å = 1-8 m) 7) Find th diffration angl (th angl btwn th inidnt and diffratd bams) for rflx (11) at th first ordr of diffration using th valu of haratristi wavlngth alulatd abov Th answr should b th formula and th numrial valu На рисунке надо заменить d на L Fig In th Cartsian fram, whih oordinat axs ar alignd with th dgs of rystal latti, th oordinats of Sn atoms (in th units of d) ar: (; ; ), (,5;,5;,5), and th oordinats of Nb atoms ar: (,5; ;,5), (,75; ;,5), (,5;,5; ), (,5;,75; ), (;,5;,5), (;,5;,75), (s Fig) Atoms in th unit ll ar numbrd from 1 to 8 in th ordr thy ar listd in th

9 txt A pattrn of X-ray sattring is dtrmind by th distribution of ltrons in a rystal latti (ltrons ar so muh lightr than atomi nuli, so thy rspond muh strongr to th ltromagnti fild of inidnt wav), i by distribution of atoms of various lmnts Th ability of an isolatd atom of a rtain lmnt А to sattr radiation is spifid by a quantity f(a) alld atomi sattring fator This quantity spifis th diffrn of wav sattring by ltroni shll of a givn atom ompard to that on by fr ltrons Atomi sattring fator is a omplx quantity (i, whr is imaginary unit) and its absolut valu squard dtrmins intnsity of sattrd radiation of an isolatd atom Th intnsity of an obsrvd diffration pak for a rystal latti is alulatd as th squard absolut valu of th rflx strutur fator, whih in turn is valuatd as: whr sum is ovr all atomi positions in th unit ll Hr (x, y, z) ar oordinats of atom at th n-th position; is th atomi sattring fator of lmnt А whih atom rsids at th n-th position, and is oupation of th position by th lmnt Oupation of a position is th avrag (ovr th whol latti) numbr of atoms of a rtain lmnt at th position In idal rystal (i whn all atoms of rystal latti rsid at thir nods) ah position is oupid xatly by a singl atom of a givn lmnt and thr ar no «xtra» atoms, i an oupation is ithr 1 or For xampl, in th unit ll of Nb Sn at th 1-st position: and Howvr, a ral rystal strutur has dfts distorting idal latti, so oupations an b diffrnt On of th most ommon dft of this sort is th so-alld antinod disordring whn atoms swith thir positions For instan, if in som lls atoms of Sn at position 1 swith to position, and atoms Nb in ths lls swith from to 1, oupation boms lss than 1 and boms non-zro Nvrthlss, th gross oupation of any position rmains qual to 1, i any atom laving its position swiths to position of anothr atom and vi vrsa 8) Suppos thr is an antinod disordring in th onsidrd strutur, so th oupation of Nb positions by atoms of Sn boms qual to δ, i Dtrmin othr oupations in th strutur Oupations of atoms of any lmnt at positions n = 1 and n = ar th sam, as it is th as for positions n =,, 8 Exprss th oupations via δ 9) Assuming and to b known valuat strutur fator of rflx (11), using th oupations alulatd in 8) Th answr should b givn as quation Hint: aording to Eulr s formula omplx xponntial is valuatd as 1) Dtrmin th ondition of qunhing th rflx (11) (whn its intnsity vanishs) for th sam strutur Th answr should b givn as a numrial valu of δ PROPOSED SOLUTION AND ANSWERS 1) Aording to th givn quations, i Using th dfinition of th Lorntz fator on finds:, whn Thrfor, th spd of ltrons in this mahin is lss than th spd of light by approximatly,% ) In th fild of bnding magnt ltrons ar subjtd to th Lorntz for whih is prpndiular to partil vloity and dos no work Thrfor, th absolut valu of ltron vloity (and th momntum) rmains onstant Th Lorntz fator rmains onstant as wll and th quations of motion oinid with th orrsponding quations of Nwtonian mhanis in whih mass is rplad with rlativisti mass Hn, th urvatur radius of ltron path (alld «Larmor radius») an b found from th quation Sin th ltron

10 spd is pratially th sam as th spd of light on obtains:, or [ ] [ ] [ ] [ ] [ ] [ ] m ) If th «obsrvr» of X-ray bam rsids at th orbital plan, th dttd radiation is produd by ltrons travling along th ar subtnding th angl qual to th bam aprtur (th on aprtur), s th figur Th orrsponding ar lngth Equivalnt answrs ar and ltron orbit R φ φ O φ 4) Th tim rquird for an ltron to ovr a distan l is, howvr, th duration of th flash is lss du to th finitnss of th spd of light To undrstand what is going on in dtail, lt us introdu th dttor rfrn fram and onsidr two vnts: 1) mission of light by th ltron at th bginning and ) at th nd of a flash Lt in th dttor fram Lt th axis x b dirtd from th dttor to th ltron (sin φ is small w an think of ltron going along this axis during all th flash) Lt ltron oordinat (in ltron fram) at b Th vnt in ltron fram thn happns at at th point (th ltron approahs th dttor) Now onsidr vnts: ) dttion of th bginning and 4) th nd of flash by th dttor Obviously, th orrsponding tims ar и ( ) Thus, th flash duration rgistrd by th dttor is ( ) s 5) Aording to th abov analysis th orrsponding haratristi frquny is and th orrsponding wavlngth is ( ) 6) Lt us alulat th optial path for th wavs rfltd from adjant plans (s th figur): θ A B D d C

11 Construtiv intrfrn (amplifiation) is obsrvd whn th optial path diffrn for ths two wavs quals an intgr of wavlngths Hn, th ondition of diffration maximum is:, whr n is an intgr (Bragg s law) Th absolut valu of n is alld th ordr of diffration maximum 7) Th diagram blow shows th projtion of a unit ll on th XY-plan Th rd arrow is th vtor prpndiular to th plans of st (11) Atomi plans prpndiular to vtor (11) Using th diagram and th fat that th ll is a ub on valuats th intrplanar spaing d x for th st of plans (11): Thn th diffration angl for th maximum of first ordr (for rflx (11)) follows from Bragg s law: Th angl btwn th inidnt and diffratd bams quals, i 8) Sin th nt oupation of th niobium positions (-8) must rmain qual to 1 th oupation of th positions by Nb atoms is If N is th total amount of Sn atoms (and th nt numbr of positions 1 and as wll) in th strutur, th numbr of Nb atoms (and positions -8) quals N (aording to th hmial formula) For a givn δ th numbr of atoms Sn swithing to Nb positions quals δ N; th sam amount of Nb atoms mov to Sn positions Thus, th oupation of positions 1 and by Nb atoms quals Aordingly, 9) Sin th sum in th formula for th strutur fator is quit umbrsom, it is asir to omput it by sparating ontributions of six Nb positions (-8) and two positions (1-) of Sn Aording to quation givn in 9) th ontribution to th rflx strutur fator F of an arbitrary rflx du to positions -8, whih oupations a by atoms of a singl lmnt with atomi fator f ar th sam, is [ ( ) ( ) ( ) ( ) ( ) ( ) ] [ ( ) ( ) ( )] (hr oordinats of vry position ar usd) Th ontribution of positions 1 and is For th rflx [ (11) this givs: ( ) ] [ ] [ ( ) ( ) ] [ ], and [ ] Now lt us omput F using oupations alulatd in 8) by summing th ontributions of all ight positions:

12 [ ] 1) Th rflx intnsity quals [ ], it vanishs whn Noti, that it is for this partiular valu of δ all oupations of atoms of a givn lmnt ar qual: for any position oupations and No wondr, this orrsponds to th hmial omposition of th matrial (5% of tin and 75% of niobium) baus avrag oupations must satisfy this proprty for any δ TABLE OF ANSWERS Answr Maximum sor 1 1+1= m (quation) +1 (numbr)=4 (any variant) 1 4 (quation) +1 (numbr)=4 5 1(quation) +1 (numbr)= ( ) 6, whr n is an intgr 5 7 (quation) +1 (numbr)= ,5+1,5=4 9 1,5+1,5+=6 [ ] 1 Всего 5

13 Problm : RADIOGRAPHY PROPOSED SOLUTION AND ANSWERS 1) Aording to th givn quations, i Using th dfinition of th Lorntz fator on finds:, whn Thrfor, th spd of ltrons in this mahin is lss than th spd of light by approximatly,% ) In th fild of bnding magnt ltrons ar subjtd to th Lorntz for whih is prpndiular to partil vloity and dos no work Thrfor, th absolut valu of ltron vloity (and th momntum) rmains onstant Th Lorntz fator rmains onstant as wll and th quations of motion oinid with th orrsponding quations of Nwtonian mhanis in whih mass is rplad with rlativisti mass Hn, th urvatur radius of ltron path (alld «Larmor radius») an b found from th quation Sin th ltron spd is pratially th sam as th spd of light on obtains:, or [ ] [ ] [ ] [ ] [ ] [ ] m ) If th «obsrvr» of X-ray bam rsids at th orbital plan, th dttd radiation is produd by ltrons travling along th ar subtnding th angl qual to th bam aprtur (th on aprtur), s th figur Th orrsponding ar lngth Equivalnt answrs ar and ltron orbit R φ φ O φ 4) Th tim rquird for an ltron to ovr a distan l is, howvr, th duration of th flash is lss du to th finitnss of th spd of light To undrstand what is going on in dtail, lt us introdu th dttor rfrn fram and onsidr two vnts: 1) mission of light by th ltron at th bginning and ) at th nd of a flash Lt in th dttor fram Lt th axis x b dirtd from th dttor to th ltron (sin φ is small w an think of ltron going along this axis during all th flash) Lt ltron oordinat (in ltron fram) at b Th vnt in ltron fram thn happns at at th point (th ltron approahs th dttor) Now onsidr vnts: ) dttion of th bginning and 4) th nd of flash by th dttor Obviously, th orrsponding tims ar,

14 Hn, ( ) ( ) Thus, th flash duration rgistrd by th dttor is s 5) Aording to th abov analysis th orrsponding haratristi frquny is and th orrsponding wavlngth is ( ) 6) Lt us alulat th optial path for th wavs rfltd from adjant plans (s th figur): θ A B D d C Construtiv intrfrn (amplifiation) is obsrvd whn th optial path diffrn for ths two wavs quals an intgr of wavlngths Hn, th ondition of diffration maximum is:, whr n is an intgr (Bragg s law) Th absolut valu of n is alld th ordr of diffration maximum 7) Th diagram blow shows th projtion of a unit ll on th XY-plan Th rd arrow is th vtor prpndiular to th plans of st (11) Atomi plans prpndiular to vtor (11) Using th diagram and th fat that th ll is a ub on valuats th intrplanar spaing d x for th st of plans (11): Thn th diffration angl for th maximum of first ordr (for rflx (11)) follows from Bragg s law: Th angl btwn th inidnt and diffratd bams quals, i 8) Sin th nt oupation of th niobium positions (-8) must rmain qual to 1 th oupation of th positions by Nb atoms is If N is th total amount of Sn atoms (and th nt numbr of positions 1 and as wll) in th strutur, th numbr of Nb atoms (and positions -8) quals N (aording to th hmial formula) For a givn δ th numbr of atoms Sn swithing to Nb positions quals δ N; th sam amount of Nb atoms mov to Sn positions Thus, th oupation of positions 1 and by Nb atoms quals Aordingly, 9) Sin th sum in th formula for th strutur fator is quit umbrsom, it is asir to omput it by sparating ontributions of six Nb positions (-8) and two positions (1-) of Sn Aording to quation givn in 9) th ontribution to th rflx strutur fator F of an arbitrary rflx

15 du to positions -8, whih oupations a by atoms of a singl lmnt with atomi fator f ar th sam, is [ ( ) ( ) ( ) ( ) ( ) ( ) ] [ ( ) ( ) ( )] (hr oordinats of vry position ar usd) Th ontribution of positions 1 and is For th rflx [ (11) this givs: ( ) ] [ ] [ ( ) ( ) ] [ ], and [ ] Now lt us omput F using oupations alulatd in 8) by summing th ontributions of all ight positions: [ ] 1) Th rflx intnsity quals [ ], it vanishs whn Noti, that it is for this partiular valu of δ all oupations of atoms of a givn lmnt ar qual: for any position oupations and No wondr, this orrsponds to th hmial omposition of th matrial (5% of tin and 75% of niobium) baus avrag oupations must satisfy this proprty for any δ TABLE OF ANSWERS Answr Maximum sor 1,5+,5=1 m (quation) +,5 (numbr)=,5 (any variant) 1 4 (quation) +,5 (numbr)=,5 5 1(quation) +,5 (numbr)=1,5 ( ) 6, whr n is an intgr 7 1 (quation) +,5 (numbr)=1, = ,5=,5 [ ] 1 1,5 Всего

16

17 Nutrino Nutrino is on of th most puliar lmntary partils It has no ltri harg and dos not partiipat in th strong intrations (whih ar rsponsibl for stability of atomi nuli) Physiists us th word «flavor» to spify a nutrino typ Thr ar thr nutrino flavors known to dat: ltron nutrino, muon nutrino, and tau-nutrino A nutrino of ah flavor has its antipartil (antinutrino) Th symbols usd for th lattr ar th sam as for nutrinos but with an uppr bar:,, and Nutrino partiipats only in th wak intrations, th most famous pross mdiatd by th wak intration is -day In this pross a singl nutron in atomi nulus days into a proton, an ltron, and an ltron antinutrino: n p (howvr, it would b a mistak to think than nutron is omposd of ths partils, thr is also a pross p n!) A nutrino is always ratd togthr with its antinutrino or a hargd antilpton (positron ( ltron antipartil), antimuon, or antitau-lpton ) An antinutrino, in turn, is always ratd togthr with its nutrino or th orrsponding hargd lpton Nutrinos ar xtrmly lightwight, thir masss ar svral ordrs of magnitud lss than masss of othr mattr partils Th pris valus of nutrino masss ar still unknown Du to thir small masss all nutrinos partiipating in nular rations ar ultrarlativisti, i thir vloitis ar vry los to th spd of light in vauum Th nrgy of suh a nutrino of mass m and momntum p is almost indpndnt of its mass: 4 E m p p A nutrino, lik many othr lmntary partils, has spin, i th propr angular momntum, whih is non-zro vn in th nutrino rst fram A spifi fatur of all dttd nutrinos (antinutrinos) is th ngativ (positiv) sign of its spin omponnt projtd on th dirtion of nutrino (antinutrino) momntum Loosly spaking, a nutrino dos not hav a «mirror rfltion» Othr lmntary mattr partils an hav both signs of th spin omponnt Physiists xplain this fat by saying that nutrinos with othr spin omponnt ithr do not xist, or do not partiipat vn in th wak intrations (so thy annot b dttd) Physial onstants and data (an b usd in any part of th problm) 8 spd of light in vauum 1 m/s; 11 gravitational onstant G 6,7 1 m /(kg s ); Plank onstant 1 4 J s; proton radius r 1 15 m; p Avogadro onstant N A 6 1 mol 1 ; hydrogn molar mass g/mol; Solar mass Solar radius M 1 kg; C 8 r 7 1 m; С man radius of Earth s orbit a 1,5 1 m; ntriity of Earth s orbit, 17 ; 11 radius of «ativ» solar or whr nular fusion prods and nutrinos ar ratd r a 1, 1 m; 1 rang of ltron dnsity n insid th Sun from th ativ or to outr layrs: from 5,9 1 m 9 to 1 m ; 16 pars (p), an astronomial unit of lngth, 1 p, light yar 1 m 8

18 ltronvolt (V) is th unit of nrgy qual to th work don by ltrostati for moving a singl ltron aross potntial diffrn of 1 V Part I: nutrino masss and osillations Th Nobl Priz of 15 was awardd for th «disovry of nutrino osillations indiating that nutrinos ar massiv» Nutrino osillations is a pross of intronvrsion of nutrino flavors Aording to modrn thortial modls th possibility of nutrino osillations is indd losly rlatd to thir masss (masslss nutrinos annot osillat) It should b notd that nutrinos lik othr lmntary partils ar not som «immutabl» ntitis, rathr thy ar quanta of a nutrino fild (similarly to photons whih ar quanta of ltromagnti fild) Thrfor, in diffrnt physial situations thy an appar in th stats with diffrnt proprtis For instan, nutrino stat of a rtain flavor (a stat in whih nutrino is ratd or annihilatd in nular rations) dos not oinid with nutrino stat of a rtain mass To b spifi, onsidr osillation (i w nglt th third nutrino flavor) An intnsiv flow of nutrinos an b rgardd as «almost lassial» radiation of a givn wavlngth (hr th analogy with ltromagnti wav, an «almost lassial» flow of a larg numbr of photons, applis again) Th xistn of svral nutrino stats an b dsribd by introduing a «polarization»: u( t, r ) u1 os( 1t kr ) u os( t kr ) Th quotation mark indiats that this polarization is not a polarization in th «rgular» spa, this is polarization in th «spa of nutrino stats» although for our purposs this is almost insignifiant Th flux of nutrinos is proportional to u Noti that th frquny and wavvtor k of th wav ar rlatd to th nrgy and momntum of nutrinos by th ommon quantum formula: E and p k, whr is Plank onstant Th diffrn of frqunis is du to diffrn in masss: for th sam momntum 4 E m p Obviously, an orthogonal «polarization» u 1, orrsponds to th 1, 1, 1, nutrino stat of rtain mass m 1, Noti, that stats of dfinit flavor ( and anothr pair of orthogonal «polarizations» u u, whih do not oinid with 1, Th polarizations u 1, orrsponding to rtain masss and polarizations, rtain flavors ar rlatd as: u1 u os u sin, u u sin u os Angl is alld th «mixing angl» of u ) orrspond to orrsponding to and In this as and indd do not hav «rtain» masss and do not hav a rtain nrgy for a givn momntum For instan, a masurmnt of th nrgy of ltron nutrino would «on avrag» yild th valu E E1 E E E1 os E sin E os, whr E p is th man nutrino nrgy and E E E1 Suh an outom ould b intrprtd as bing du to intration of th stats and, whr th intration nrgy is V E sin os Th quantitis introdud abov an b xprssd in trms of nrgy E and paramtrs m 1 m m, m m m1, and It has bn alrady mntiond that aurat valus of ths paramtrs ar not known yt but to solv th problm it would suffi to adopt th following approximat valus: m 4, 1 V, m, 1 V, and 1 1) Evaluat «man» masss of ltron and muon nutrinos for th givn valus of th paramtrs Th answr an b givn ithr in kg or V/

19 Now onsidr nutrinos radiatd from som small rgion and propagating along x-axis Lt th nrgy of a nutrino ratd in this rgion b E MV, all ratd nutrinos ar ltron nutrinos and hav th sam rtain momntum (this mans that nutrinos ar ratd with diffrnt masss and, thrfor, nrgis) Clarly, th nutrino wav propagating along x-axis is a mixtur of nutrino wavs orrsponding to nutrinos of diffrnt masss (hn, a mixtur of wavs with rtain frqunis at a givn wavlngth) Th phas shift of th wavs varis with distan x Sin th phas shift varis th ontributions to th rsulting wav du to ltron and muon omponnt would vary as wll Thrfor, at any partiular position x on would dtt not only ltron nutrinos but muon nutrinos as wll Th intnsitis of th orrsponding nutrino fluxs vary priodially in spa This phnomnon is alld nutrino osillations ) Dtrmin th osillation lngth, i th priod of spatial variation of th tim avragd flux of muon nutrinos Th answr should b givn as th formula and th numrial valu Part II: nutrinos and th Sun Osillations dsribd in Part I our in vauum At first glan, it would b rasonabl to assum that mattr dos not altr th pitur signifiantly sin nutrinos vry wakly intrat with any mattr whih dnsity is muh lss than th dnsity of atomi nulus Thr is a powrful nutrino sour los to th Earth It is th Sun Nular rations prod in th ntral rgions of th Sun suppling it with nrgy and rating nutrinos and antinutrinos, mostly, ltron ons Howvr, th obsrvd flux of ltron nutrinos turnd out to b only a half of th flux prditd from th Solar luminosity Is it possibl to xplain this «dfiit» by partial onvrsion of ltron nutrinos to nutrinos of othr flavors on thir way from th Sun to th Earth (vauum osillations)? ) Try to giv a justifid answr by using th data and th rsults from Part I of th problm Do nssary alulations to support your judgmnt In partiular, it is rasonabl to assum that nrgis of nutrinos ratd in nular rations in th Sun ar not vry diffrnt from MV Tak into aount th fat that a nutrino dttor aumulats data for a long priod of tim, up to - months Th answr should b givn as «+» (ys) or (no) A dtaild analysis must tak into aount that th nutrinos travl a part of thir path insid th solar substan It turns out, absorption of th nutrinos by th substan dos not hang muh th stimat for th flux of ltron nutrinos but thr is som additional irumstan whih is quit ssntial Th solar substan ontains a lot of ltrons (s th problm data) and ltron nutrinos intrat with thm muh strongr than muon nutrinos do Du to this fat th «man» nrgy of ltron nutrinos inrass by E, th quantity whih is proportional to th ltron dnsity: 1 n 1,7 1 1 E V At th sam tim th «man» nrgy of muon nutrinos and th nrgy 1 m of intration btwn th stats and rmain pratially th sam 4) Evaluat «nw» valu of mixing angl ~ (by taking into aount th solar ltrons) Th answr should b th quation 5) By how many prnt an th solar ltrons hang th xptd «dfiit» of th ltron nutrino flux? Evaluat in prnt (%) th maximum inrmnt of th flux of muon nutrinos du to osillations (ompard to th flux in th absn of mattr) 6) Plot an approximat dpndn (i show only th main faturs) of th mass diffrn of and as a funtion of th distan r travld insid th Sun from th ntr outwards

20 m μ m,5 1 r/r C Part III: nutrino and suprnova xplosion Aftr an «ordinary» star has xhaustd its nular ful th star ools down and its intrnal prssur annot withstand gravitational omprssion anymor As a rsult, th star is ontrating until its substan undrgos transition to a nw phas in whih all ltrons ar «shard», all nuli «float» in th ltron «gas», and only th prssur of this gas halts furthr ollaps Th star an xist in this stat for a long tim; howvr, if th mass of its dns or gradually inrass and xds M r 1, 5M C th or boms unstabl and ollapss At th onst of ollaps suh a or usually has a radius about 1 km with approximatly qual numbrs of protons and nutrons Soon aftr th ontration starts th rat of ltron-nular ollisions boms high, whih rsults in nutronization of th star substan du to ration of «invrs -day» ( p n ) Eltron disapparan rdus th prssur of ltron gas alrating th nutronization vn mor Th whol pross is ssntially th trmndous xplosion laving in th aftrmath a nutron star and an «outr nvlop» flying outwards Astronomrs all suh an xplosion «suprnova xplosion» or simply «suprnova» A nutron star is indd omposd mostly of tightly pakd nutrons, so its dnsity is approximatly qual to th dnsity of atomi nuli 7) Estimat an ordr of magnitud of th nrgy rlasd du to omprssion of th stllar or from th initial radius to th nutron star Calulat th numrial answr in Jouls Th rlasd nrgy onvrts to kinti nrgy of th star rmnants (th flying outr layrs and rotation of th nutron star) and to th nrgy of ltromagnti radiation and nutrinos Suprnova xplosion is on of th most powrful sour of nutrinos (somtims it is alld «nutrino bombs»), alulations show that mor than half of th rlasd nrgy onvrts to th nrgy of radiatd nutrinos Nutrinos ar radiatd both at th nutronization stag and aftr formation of th nutron star whih is initially xtrmly hot and subsquntly ools down mostly by radiating nutrinos Th nutronization and ooling tak just svral sonds Noti that only ltron nutrinos ar ratd during th nutronization and nutrino-antinutrino pairs of various flavors ar ratd during th ooling 8) Estimat th numbr of nutrinos ratd by a suprnova xplosion assuming that th mass of th initial stllar or is approximatly 1,5M C (th stllar substan dos not «go away» with outr layrs), th nrgy of radiatd nutrinos is 8% of th rlasd nrgy, and th man nrgy of radiatd nutrinos and antinutrinos is approximatly 1 MV You ould assum that nutron mass and radius ar approximatly th sam as thos of proton Calulat th numrial valus 9) Suprnova SN1987А xplodd at th distan of R 5 kp from th Earth (in th Larg Magllani Cloud) What is th total numbr of nutrinos and antinutrinos passd through an Earth basd dttor of th ross-stional ara S 1 m? Estimat th xptd numbr of dttd nutrinos and antinutrinos assuming that th dttor on avrag rgistrs 14 1 % of nutrinos of any flavor in th orrsponding nrgy rang Calulat th numrial valus It is important that nutrino radiation is asymmtri with rspt to th star magnti axis du to «puliar» nutrino bhavior undr mirror rfltion: th powr di radiatd in th infinitsimal solid

21 di I angl d sin dd is [1 os ], whr 1, is th angl of nutrino mission to d 4 th axis, and is th angl of rotation around th axis 1) Estimat th spd gaind by th star du to nutronization and ooling Calulat th numrial answr (in km/s) PROPOSED SOLUTION AND ANSWERS Part I 1 Th partil mass in STR (rst mass) is dtrmind from th rlation btwn th partil nrgy and momntum For ultrarlativisti partil E E 1 os ( m1 os E sin p and th «man» mass of ltron nutrino in vauum E с p m 1 p m sin p ) m с p Thrfor, p p ( m) m m1 os m sin m m m os 4 mv 9 Th numrial valu m,64 4,69 1 kg m p ( m) Similarly, for muon nutrino, m m m m os and 4 mv 9 m 5,4 9,66 1 kg An ltron nutrino ratd at t = and x = is a suprposition of stats of rtain masss: u u 1 os u sin Eah stat propagats at a distan x as a wav of rtain frquny and wavvtor, i u( t, x) u1 os os( 1t kx) u sin os( t kx) Substituting xprssions for u 1, in this formula on finds that at any point th wav ontains two «flavor polarizations»: u( t, r) u[os os( 1t kx) sin os( t kx)] u sin os [os( t kx) os( t kx)] Th wav of muon nutrinos (th sond trm) quals to th xprssion 1 1 sin sin t sin t kx, orrsponding to a propagating wav of th «man» frquny 1 and «amplitud» modulatd by th harmoni fator of th half-diffrn frquny Nutrinos travl a distan x from th point x 1 E E1 E of mission for a tim t and th half-diffrn frquny is It is important to not that small orrtions du to nutrino masss an b ngltd whn alulating th E1 E man nrgy ( E p ) and must b takn into aount whn alulating th nrgy m1, diffrn: E1, m1, p m1, E E, hn E 4 4 ( m m1 ) m m 1 E E E1 6 1 V Obviously, E E, so th «amplitud» E E 1

22 fator an b rgardd as almost onstant during th osillation priod orrsponding to th «man» frquny Thrfor, th flux of muon nutrinos as a funtion of x obtaind by avraging u ovr th m m priod orrsponding to th «man» frquny is: I( x) I sin sin x Hr I is E th initial flux of nutrinos Thrfor, th osillation lngth orrsponding to th spatial priod of th E 5 sond fator in I quals L 1 м = km m m mv 9 ANSWERS-1: th «man» mass of quals m,64 4,69 1 kg, th «man» mv 9 mass of quals m 5,4 9,66 1 kg, and th osillation lngth in vauum is E 5 L 1 m m m Not: th ontstrs ould stimat th «man» masss somwhat diffrntly, g by following this lin of argumnt: «Th partil mass is proportional to its nrgy at zro momntum, thrfor, th m «man» mass of ltron nutrino in vauum is m m1 os m sin m os Thn mv 9 m,59 4,611 kg Similarly, for muon nutrino m mv 9 m m1 sin m os m os, and m 5,41 9,6 1 kg» This approah is not logial from th STR prsptiv sin ths «masss» ar not invariant undr Lorntz transformations, th orrt mass must b dfind via th invariant E p Howvr, th numrial valus turn out to b quit rasonabl That is why suh an approah should b givn a partial rdit Part II To stimat th impat of vauum osillations on th flux of ltron nutrinos dttd on th Earth, noti, that nutrinos ntring th dttor travld various distans: thy hav bn ratd at diffrnt points of ativ solar or of siz r A L Bsids, during th data aumulation th Earth hangs its position with rspt to th Sun (th distan btwn th Sun and th Earth varis by 9 a,6 1 m L ) Thrfor, th flux must b avragd ovr variations of x whih signifiantly 1 xd th osillation lngth Sin sin x, th flux of muon nutrinos orrsponding to th L 1 drmnt of th flux of ltron nutrinos I ( x) I sin, 58 I whih is lss than 6 % of th initial valu! Thus, vauum osillations annot xplain th signifiant dras of th flux 4 Proding to dsription of th osillations in mattr, noti, that although th nrgy of nutrino intration with solar ltrons is vry small ompard to th nutrino nrgy (vn for th 1 1 maximum ltron dnsity in th or E 7,5 1 V), it is of th sam ordr as E 6 1 V! Thrfor, this intration annot hav a signifiant impat on th nutrino propagation and/or th total flux but it an notiably afft th osillations Thus, taking into aount th intration of nutrinos with mattr, th man nrgy of ltron ~ E ~ E ~ nutrino boms E E os E E os, whr th «tild» ovr a quantity orrsponds to that on in mattr Th man nrgy of muon nutrinos and th nrgy of intration of ~ E ~ E ~ ltron and muon nutrinos rmain unaltrd: E E os E os and

23 ~ E E ~ V sin sin Using th first two quations on finds that ~ E E E and ~ ~ E os E os E Combing this and th third quation yilds th rmaining paramtrs: ~ ( ) ( ) ~ tg E E E E E os and tg (th minus sign in th 1 E /( E os ) xprssion for E ~ orrsponds to E E os ) 5 First of all, lt us look at th nw valu of th mixing angl: if E Erz E os, th angl turns out to b rz / 4 45! This situation is alld «maximal mixing» baus th frations of muon and ltron nutrinos in th flow hang priodially in antiphas btwn и 1% Th phnomnon of sharp amplifiation of th osillations in mattr is alld «Mikhyv Smirnov Wolfnstin fft» or MSW-fft Th ondition of MSW-fft for nutrino with our 1 n 1 1 paramtrs rads 1, os, i n 1 n rz 4,44 1 m Noti, that th 1 m mixing angl away from th rsonan rmains small (th osillations in ths rgions do not signifiantly afft th flux) On an also s that thr is a layr outsid th ativ solar or whr ltron dnsity taks th «rsonan» valu Th width of th rgion in whih th dnsity diffrs from 7 th rsonant on by lss than 1% is RC r,1 nrz 1 () ( ) m, whih xds th n n RC osillation lngth by mor than ordr of magnitud Thrfor, a nutrino passing through this layr undrgos many osillation yls in whih th fration of muon nutrinos hangs harmonially from % to 1% Th initial phas of th wavs from diffrnt points in th solar or varis in a wid rang Whn valuating th flux of at th xit of rsonan rgion on should avrag ovr ths variations as wll Hn, th flux of muon nutrinos an b as larg as 5% of th total flux (ompar to 6% in vauum) Thus, th solar mattr inrass th flux of 85 tims, or approximatly by 75%! 6 Anothr intrsting obsrvation onrns nutrino masss It follows from th quations for E ~ and ~ that th diffrn of nrgis of muon and ltron nutrinos with th sam momntum ~ ~ E E E os boms ngativ for E Eos (i whn th ltron dnsity is highr than th rsonant valu)! Thus, in th solar or blow th rsonant layr th mass of th mass of Outsid th rsonant layr is lss than E Eos and th mass of xds th mass of Obviously, th mass diffrn tnds to its «vauum» valu whn th dnsity is small Thrfor, th plot looks approximatly as: m μ m r rz,5 1 r/r C 1)) Not: An aurat alulation yilds th sam rsult Using th xprssion for man nrgy (s 4 E1 E ( m1 m ) E p, on obtains for paramtrs m, m (and similarly for m ~, m ~ ): 4E

24 4 [4m ( m) ] E p, 8 ~ 4 4 ~ E E ( 4 ( ) ~ 4 ) 4 1 ~ [4 ~ E m m m m M E m ( m ) ] E E p 8E (th quantity M 1 is introdud to simplify th alulation) Similarly, using th xprssion for E E E 1 (s 1)) and th formula for E ~ drivd in 4), on obtains: 4 mm E E ~ ~ 4 M ~ ~ m m m ( m) sin [ mmos E E / ] 4 ~ mm E E (hr M is anothr shorthand notation) Solving th st of ths two quations allows on to dtrmin m ~ and m ~ Mass is positiv, thrfor, m ~ m~ and ~ m M1 M1 M, ~ 4 4 m M1 M1 M Th quantity M rahs its minimum at th rsonant dnsity (th sond trm undr th radial in th xprssion for M vanishs) whil M 1 hangs monotonially (slowly drass with dnsity sin E is proportional to th ltron dnsity and othr trms in th xprssion for M 1 rmain onstant) On an s that m ~ ~ drass whn approahing th rsonant dnsity Bsids, os at th rsonan Thrfor, th diffrn of masss of and vanishs It is obvious that os ~ swiths sign whn rossing th rsonant layr Whn th dnsity xds th rsonant valu th sign is ngativ, i ltron nutrino «on avrag» is havir than th muon nutrino Whn th dnsity is blow th rsonant valu, is lightr than and th masss tnd to thir vauum valus whn th dnsity drass Clarly, th aurat analysis lads to th sam onlusion as th proposd solution although th information obtaind and th amount of alulations hav signifiantly grown Sin th diffrn of man masss of and in mattr is xprssd via m ~, m ~, and ~ on ould writ down th xpliit formula for th mass diffrn as a funtion of ltron dnsity (it turns out to b quit umbrsom) ANSWERS-: ~ tg tg, solar mattr nhans th flux of approximatly 1 E /( E os ) at small ltron dnsity orrsponds to th vauum valu, by 75%, th mass diffrn of and drass whn approahing th rsonant dnsity, gos through zro and hangs sign whn passing through th rsonan layr, and thn inrass as th dnsity grows Part III 7 Firstly, it is nssary to stimat th nrgy sal of th phnomnon, i to dtrmin th nrgy rlasd during th ollaps Th «binding nrgy» of a gravitating sphr of mass M and GM radius r quals W g To stimat th radius of nutron star lt us quat its dnsity to th 5 r dnsity of nular mattr qual to th ratio of proton mass to its volum Th mass of N A (1 mol) of protons approximatly quals on half of molar mass of molular hydrogn ( = г/моль), hn 1/ M N AM r N r and r rp For a nutron star of mass 1,5M C on obtains r 1 km Th A p initial radius of th stllar or is gratr by thr ordrs of magnitud, so th rlasd nrgy is () GM 46 W 1 J 5 r

25 Atually, this numbr is an ovrstimation baus it has bn assumd that th or mass dos not hang during th ollaps Howvr, aording to STR a dras of nrgy is quivalnt to th G( M W / ) dras of mass, i on should atually writ W Sin 5 r 47 () M,7 1 J 9W, this fft is not too small Solving th quadrati quation yilds: W W () W, whr M () 46 W,85 W,475 1 J In any as, () 1 9 So, th bttr stimat of th rlasd nrgy givs 46 W ~ 1 J by th ordr of magnitud 8 Aording to th problm statmnt th total nrgy arrid away by nutrinos and 46 antinutrinos E,8W 1 J Sin th avrag nrgy of mittd nutrinos and antinutrinos quals E 1 1 MV, thir total numbr is E 58 N 1,5 1 Th numbr of proton-ltron pairs E in th initial stllar or is N p M m p N AM 1,9 1 58, whil in th final stat almost only nutrons ar lft Thrfor, th numbr of nutrinos mittd during nutronization is,9 1, and th rst ar mittd during th ooling Thus, th total numbr of mittd antinutrinos and nutrinos 1,5, ,5, is: N 1 5,8 1 and N,9 1 6,7 1 9 Th numbr of nutrinos and antinutrinos from SN1987A passing through th dttor on Earth is proportional to th ratio of th dttor ross-stion to th ara of a sphr of radius R 5 (dt) S 4 (dt) 16 (dt) 16 kp: N N,5 1 N Rsptivly, N 1 and N,5 1 Sin only 4R 14 th fration 1 % is dttd, th xptd numbr of dttd antinutrinos and nutrinos is ( ) (dt) stimatd to b N rg ( ) (dt) N 6 and N rg N 7 1 Sin all mittd nutrinos and antinutrinos ar ultrarlativisti, th momntum arrid by di thm pr small solid angl is Thrfor, th omponnt of th momntum arrid by th radiation projtd on th magnti axis z of th star is: Idt E and os d sin os p z 1 dt di d 4 d, on obtains os d E p z 4 Idt os 58 d Sin Th roil vloity is pz opposit to th magnti axis and its valu V ~, whr M ~ is th mass of th ratd nutron star M ~ W E Taking into aount th mass dft M M, on finally obtains V 8 km/s ( M W ) This spd is alrady notiabl against th bakground motion of stars in galaxis (a typial vloity of this motion is svral hundrd km/s) ANSWERS-: th total numbr of nutrinos and antinutrinos mittd by suprnova is N 6,7 1 and 5,8 1 ; th numbr of nutrinos and antinutrinos passing through th dttor is N (dt) ( rg) 16 N,51 and (dt) N ( rg) 1 16 ; and th xptd numbr of dttd nutrinos and antinutrinos is N 7 and N 6 Th roil spd of nutron star is V 8 km/s

26 TABLE OF ANSWERS Answr Maximum sor 1 mv mv +=6,6 m,7 or (if insid th «doubld» 6 6 4,6 1 kg m 4,8 1 kg intrval: +=4) mv mv 5, m 5,5 or 6 6 9,551 kg m 9,751 kg Expliit xprssion for u ( t, x) via «flavor polarization» or th + ( for th quation + 1 for amplitud of muon nutrino wav is writtn down th valu) = 5 E 5 L 1 m m m (no) full rdit is givn if thr is any rasonabl justifiation supportd by alulations 4 Th orrt st of quations for E ~, E ~, and ~ is writtn down +=5 ~ tg tg 1 E /( E os ) 5 Th fft of «rsonan» is disovrd 1+=4 Mattr inrass th flux of by approximatly 75% 6 m μ m (if it is not indiatd that th r orrsponds to r r :,5 points) rz r rz,5 1 r/r C 7 (any plot with a finit ngativ valu for positiv valu for Equation 46 W ~ 1 J r rrz ) r rrz and a finit GM is usd (th fator /5 an b omittd) 5 r W g 8 Nutrinos mittd at th nutronization stag ar takn into 58 aount: about, N 6,7 1, 57 N 5,8 1 9 (dt) 16,51 + = 4 (fator /5 is absnt or mass dft is not takn into aount: -,5 points for ah) + = 4 (mass dft is ngltd:, th rror gratr than 5% and lss than 6%: ) (dt) 16 N = (th rror gratr N and than % and lss than 5%: ( rg) ( rg) N 7 и N 6 1) 1 Intgral xprssion for th momntum omponnt is writtn down = 5 (mass dft is ke ngltd: 4, th rror is Momntum arrid away, p z gratr than 5% and lss than 5%: ) E V 8 km/s ( M W ) Total 4

27