Nuclear and Particle Physics

Transcription

1 Nucla and Paticl Physics Intoduction What th lmntay paticls a: a bit o histoy Th ida about th lmntay paticls has changd in th cous o histoy, in accodanc with th human s comphnsion and lat obsvation o natu. Ancint Gks blivd that th wold is mad o ou basic lmnts: ai, i, wat and ath. Dmokitos, 4th cntuy B.C.: th wold is composd o th smallst indivisabl pats atoms. 1/03/014 B. Golob 1

2 A bit o histoy: D. Mndljjv, 1869: piodic systm o lmnts E. Ruthod: All scinc is ith physics o stamp collcting. JJ. Thompson, 1897: discovy o lcton - studnt Enst Ruthod, 1911: xplains th stuctu o an atom with th atomic nuclus studnts Gig, Masdn: thy do th xpimntal wok Enst Ruthod, 1911: discovs that all nucli consist o th Hydogn nuclus, which is considd as th discovy o th poton p poton: gk wod o ist, πρῶτον. In this laboatoy J. Chadwich in 193 discovd nutons n 1/03/014 B. Golob

3 A bit o histoy: M. Gll-Mann, G. Zwig in 1964 suggst that n and p a composd o quaks quak: J. Joyc, Finngans Wak Gll-Mann civs th Nobl piz in physics in 1969, o th classiication o th lmntay paticls; Zwig dos not gt th Nobl piz J.I. Fidman, H.W. Kndall in R. Taylo pat th Ruthod xpimnt s p. 3 in Thy xpimntally conim th xistnc o quaks and liv to gt th Nobl piz in /03/014 B. Golob 3

4 Composition o th wold as sn today: siz in m siz in m atom nuclus poton quak lcton 1/03/014 B. Golob 4

5 Pat 1, Nucla Physics 1/03/014 B. Golob 5

6 1.1 Basic poptis o nucli Mass 1/03/014 B. Golob 6

7 binding ngy / nuclon [MV] Smi-mpiical mass omula Wizsäck omula: mpiical : wittn with th intntion o dscibing th xpimntal data smi : th a phnomnological asons o inclusion o individual tms in th quation mass o a nucli with Z potons and A-Z=N nutons p, n = nuclons: m A, Z Zm A Z m W / c p W: binding ngy o nuclons in th nucli, W < 0; ngativ binding ngy is a consqunc o th stong nucla oc binding th nuclons insid th nucli; altnativly, th binding ngy is a consqunc o a attactiv potntial among th nuclons 1 th avag binding ngy p nuclon is appoximatly constant: n 1/03/014 B. Golob A 7

8 Smi-mpiical mass omula Wizsäck omula: dviations om th avag valu a sn at low A; this is a consqunc o th act that o low A th a lativly mo nuclons at th suac o th nucli and a hnc lss bound bcaus o a low numb o nighbouing nuclons; th ct is a positiv tm in th binding ngy, popotional to th suac, i.. a tm popotional to 3 dviations a sn also at high Z valus; this is a consqunc o th Coulomb pulsion among positivly chagd potons; th ct is a positiv tm in th binding ngy, popotional to th lctostatic potntial ngy o Z potons, i.. popotional to Z / E W lc 4 Z 0 ZU ; U U Z 4 0 Ed Z U 1/03/014 B. Golob 8

9 Smi-mpiical mass omula Wizsäck omula: 4 it is ngtically avouabl o nucli to hav th sam numb o n and p; i w consid ngy lvls o nuclons in th nucli similaly at th ngy lvls o lctons in th atom: n p p and n a mions spin 1/ paticls; in accodanc with th Pauli xclusion pincipl ach ngy lvl is occupid with two idntical mions diing in th 3d componnt o th spin i in th abov pictu on n is placd by a p, it must occupy a high ngy lvl which mans it is lss bound; th ct is a positiv tm p nuclon in th binding ngy popotional to Z/A - 1/ a quadatic ct sinc it is lss avouabl o a nucli to hav a lag numb o p o n; o th total binding ngy this is A Z/A - 1/ Z - A /A 5 it is xpimntally obsvd that nucli with an vn numb o p and n vn-vn nucli a stong bound than th nucli with an odd numb o ith p o n and vn stong bound than th nucli with an odd numb o both p and n; this is a consqunc o th Pauli xclusion pincipl, similaly as mntiond und 4; 1/03/014 B. Golob 9

10 Smi-mpiical mass omula Wizsäck omula: th ct is a tm in th binding ngy popotional to A, Z vn vn vn odd odd odd Th a tms in th binding ngy dpnding on th adius o a nucli; xpimntally as dscibd on p. 4 w can s that th adius o nucli is givn appoximatly by = 0 A 1/3, with 0 1. m. Summing all th mntiond tms th binding ngy is W A, Z w Z A Z A A /3 3/ 4 0A w1 A w w3 w4 A A, Z 1/3 Constants w 0-4 a dtmind to dscib data, typical valus ound a w 0 =15.6 MV, w 1 =17. MV, w =0.7 MV, w 3 =3. MV, w 4 =1 MV Such a omula dscibs wll xpimntally obsvd valus: 1/03/014 B. Golob 10

11 Smi-mpiical mass omula Wizsäck omula: Homwok 1: calculat masss o som o th nucli in th chat xpimnt omula Homwok : stimat which nuclus o a givn mass numb is th most stabl; compa to th chat blow N A stabl nucli N=Z 1/03/014 B. Golob 11 Z

12 1.1 Basic poptis o nucli 1.1. Chag distibution 1/03/014 B. Golob 1

13 EM scatting Lt s consid an xpimnt wh a chagd pojctil is scattd o th nucli du to th Coulomb oc lctomagntic potntial; an xpimnt simila to th on which Ruthod did to discov th atomic nucli v -, p, a,... q v i pojctil, nucli, Z w wit th quantum mchanical but classical - as opposd to lativistic xpssion o pobability o scatting th pojctil though a scatting angl q intval o solid angl dw = sinq dq 1/03/014 B. Golob 13

14 EM scatting Fmi goldn ul: W i V i E W i : pobability o a systm in ou cas pojctil tansition om an initial stat i to a inal stat p intval o tim units s -1 V i : matix lmnt o th i tansition E i : dnsity o inal stats at th ngy o th initial stat E i initial stat i: pojctil with th vlocity, dscibd by th wav unction y i inal stat : pojctil with th vlocity, dscibd by th wav unction y wav unction: y is th pobability dnsity, intgatd ov a volum givs pobability o inding th stat in this volum; th simpls appoximation is to dscib th pojctil a away om th scatting cnt as a plan wav: y, i 1 V n 1/03/014 B. Golob 14 ik i, i all spac y 3 i, d 1

15 EM scatting V n is th nomalization volum, th wav unction is nomalizd as 1 paticl p V n ; V n is abitay and has to cancl in any inal xpssion dscibing any obsvabl k is th wav vcto: l: d Bogli wavlngth numb o inal stats in a volum lmnt o spac and momntum: : dnsity o inal stats 1/03/ B. Golob p k p h k ; ; l l ; h p d V N d h p d d N d n ; / ; / ; i n i n n n v m V E d d mp V d d m dp p de m p E de h dp p V d d de h d dp p V de N d d W W W W

16 EM scatting V i : matix lmnt; xpctation valu o th potntial causing th i tansition V i y * V y d i 3 Fom th Fmi goldn ul w constuct a nw obsvabl, mo dictly latd to th masumnt - th dintial coss-sction: ds dw dw it psnts th pobability o th tansition p unit o tim and intval o th solid angl W, nomalizd to th incoming i i lux o pojctils i v i ; sinc w intoducd th nomalization volum V n, th dnsity o incoming paticls is just 1/V n i v / dw units o ds/dw: 1/ 3 1/ m s m / s m 1/03/014 B. Golob 16

17 EM scatting th dintial coss-sction can b masud in th ollowing way: q tagt with nucli incoming paticls dintial o th tagt aa ds tagt thicknss l numb o scattd pojctils in th intval o th solid angl dw: dn ds dnt ; dw dw ds dnt 1 dnt dv l ds dn Z ds lt dw M dw dn dv t dnt t dm Z t M by masuing th numb o scattd pojctils into th solid angl intval dw and knowing th poptis o th tagt t, Z, M mass o nuclus, l, on can dtmin ds/dw 1/03/014 B. Golob 17

18 EM scatting to calculat th matix lmnt w ist inst th plan wav appoximation o th wav unctions: with q=k i -k Thn w mak us o th Gn s omula: wh w tak Th potntial ngy o th pojctil and th nuclus is and th potntial U is latd to th lctic chag distibution in th nuclon by is nomalizd so that 1/03/ B. Golob d V V i i 3 * y y d V V d V V V iq n k k i n i i S u d v v u d u v v u V S 3, V v u q i 4 0 U Z V 0 U Z d 3

19 EM scatting a away om th nuclus w can assum V, V 0 and hnc th ight hand sid o th Gn s omula quals 0. Th lt hand sid, taking into account iq V, q yilds V i 0 V 0 iq n d q 3 iq q d 3 V d V Th matix lmnt V i is thus latd to th chag distibution in th nuclus, mo pcisly to its Foui tansom, which w dnot by Fq and call th om acto. Now w inst all th ingdints to th xpssion o th dintial coss-sction: iq 0 n 3 q 0 0 F q iq ds dw dw i v i / dw i V mp n F q 3 0 Vnq vi 1/03/014 B. Golob 19

20 EM scatting at aangmnt w can wit a mo compact om in cas o lastic scatting E i =E, q quals and so 1/03/014 0 B. Golob s q F q m d d W sin 4 ; cos p q p k k k k k k k k q i i i i 4 0 sin 1 8 q F p m d d s W

21 EM scatting in cas o a point-lik nuclon, an assumption valid i wh R is th siz o th nuclus, chag distibution is just and th om acto is Z 0 F F q iq 3 iq0 q Z 0 d Z Z l R ; p h/ R th dintial coss-sction is ds dw Z m sin p 4 and dscibs what is calld th Ruthod scatting most o pojctils at small scatting angls, som also at lag. 1/03/014 B. Golob 1

22 EM scatting W can us a Taylo xpansion in th om-acto xpssion: F q 4 0 q Z 1 6 iq d 3 sin q q d 0 d iqcosa q 3! 4 By masuing ds/dw w can dtmin th avag squa adius o th chag distibution within th nuclus. This is how on can xpimntally viy th pviously givn lation = 0 A 1/3 p. 1. On can o cous do mo, and ty to dtmin th shap o mo pcisly. sin ada q 5! th intgation gos to ; howv, i is null o som >> R valus, and qr<<1, w can do th Taylo xpansion 4 d 1/03/014 B. Golob

23 EM scatting o an assumd in th om o a stp unction Fq homwok 3: calculat this Fq th calculatd Fq is: 4 [m] q [MV/c] q Th masumnt o 450 MV - scatting on 58 8Ni nucli givs Fq on can ty vaious ansatzs and th on which its th masud data bst is calculatd om = 0 A 1/3 o A=58 q [MV/c] [m] 1/03/014 B. Golob 3

24 1.1 Basic poptis o nucli Spin 1/03/014 B. Golob 4

25 Pincipl o Schöding quation solving shll modl o nucli 1 assum an avag potntial lt by nuclons solv th Schöding quation 3 nuclons ill th calculatd ngy lvls in accodanc with th Pauli xclusion pincipl 4 chck th magic numbs Z and N o nucli xhibiting lag than th avag binding ngy 5 in cas o discpancy with th xpimntal data coct th potntial 1 Som possibl potntial shaps: V R V R V R -V 0 -V 0 -V 0 impovmnt potntial wll impovmnt ham. oscillato 1 V V 0 Saxon-Woods potntial V0 V R / a 1 1/03/014 B. Golob 5

26 Pincipl o Schöding quation solving Schöding quation : ducd mass o a nuclon and th st o th nuclus, = m n m N /m n +m N m n solutions o th quation a E s - singl nuclon ngy lvls solutions a typically sachd o by th ansatz with Y lm dnoting th sphical hamonics with dnoting th opato o th squa o th angula momntum Th angula momntum tm acts as an additional potntial Th Schöding quation simpliis to Homwok 4: viy that with q. ducs to this om! 1/03/014 6 B. Golob y y y y E V E H ˆ u R Y R lm /, q y ˆ 1 ˆ 1, 1, ˆ Eu u m l l V u m Y l l Y n n lm lm q q, q y lm Y u y Ey H ˆ

27 Pincipl o Schöding quation solving solutions o th potntial wll: solutions o E and y dpnd on th main quantum numb n du to th bounday conditions and obital angula momntum quantum numb l wav unctions ngy lvls V 0 potntial l l 1 m n individual solutions a makd by nl, with l=0 makd as s, l=1 as p, l= as d,... o high l th nuclons a pushd to high adii 1/03/014 B. Golob 7

28 Pincipl o Schöding quation solving solutions o th hamonic oscillato - V 0 +1/m N : E n,l =n+ l -1/h o th hamonic oscillato th ist th magic numbs s p. 33, lt column a in accodanc with th masumnts,8,0; spaatly o n and p, th high a not. Using a init potntial changs th ngy lvls to som xtnt although th situation with th magic numbs mains th sam th magic numbs a thos, wh th a lag ngy gaps btwn goups o ngy lvls; s p. 33, middl column. To btt xplain th ngy lvls and hnc th magic numbs o nucli on nds to consid th spin-obit intaction. This is an additional tm in th potntial aising om th intaction btwn th total spin o two nuclons and thi lativ obital angula momntum. It taks th om wh l dnots th obital angula momntum and s th spin. E ls s 1/03/014 B. Golob 8

29 Pincipl o Schöding quation solving Inclusion o th spin-obit tm in th potntial changs th quantum numbs by which th individual solutions can b lablld; instad o good quantum numbs n, l, l z, s and s z good quantum numbs dtmin th xpctation valus o th cosponding opatos, in th abov cas o th opatos o ngy, magnitud and 3d componnt o th obital angula momntum,, and th magnitud and th 3d componnt o th spin,, z s, s z ; ths opatos commut with th Hamiltonian opato, and hnc th xpctations valus o thos opatos a consvd on has good quantum numbs n, j, j z, l, and s du to th act that th Hamiltonian now includs th tm, it dos not commut any mo with th opatos z s s, z ; it dos, howv commut, with th magnitud o th total angula momntum magnitud and its 3d componnt j,. In od to undstand th solutions o th Schöding q. with th inclusion o th spin-obit tm in th potntial, ist on nds th lation btwn th total angula momntum j and : s j s j s s s not: s, : opatos l, s : quantum numbs 1 s j j 1 l l 1 s s 1 1/03/014 B. Golob 9 j z

30 Pincipl o Schöding quation solving th last lin in th quation was divd using th act that y l l 1 y, s y s s 1 y i on dals with mions lik nuclons, s=1/, and hnc th total angula momntum o a nuclon can b only j = l ± ½: s 1/ l 1/ l 1 Evy ngy lvl with a givn l is now splits into two lvls, on with j=l+1/ and on with j=l-1/ apat om th lvl with l=0. With this inclusion o th spin-obit intaction th calculatd magic numbs using th init potntial ag with th masud ons s p. 33, ight column. j l j 1/ l 1/ 1/03/014 B. Golob 30

31 Schöding quation solutions o vaious potntials notation: n l notation: n l j magic numbs l +1 nuclons/lvl j +1 nuclons/lvl notation: l =0 s l =1 p l = d l =3 l =4 g : : l z = -1,0,+1, s z =±1/, 6 nuclons s z =±1/, nuclons V 1 j z =-3/,-1/,+1/,+3/ 4 nuclons 0 R/ a j z =±1/, nuclons s 1/03/014 B. Golob 31

32 Spin o nucli within th shll modl Th total angula momntum o th nuclus also calld th nuclus spin is a vcto sum o th angula momnta o individual nuclons. Individual obits ngy lvl dtmind by a givn st o valus n, l, j ully illd by nuclons hav th total angula momntum qual to zo. This is asy to s sinc at such ngy lvl all th sublvls dgnatd, i.. all having th sam ngy cosponding to th 3d componnt j z =-j, -j+1,..., j-1, j a qually populatd and hnc th vcto sum o thos nuclons is zo. Moov, i th nuclus has a singl nuclon mo than ndd to ully populat th low ngy obits, spin o th nuclus is dtmind by th total angula momntum o this additional nuclon. Simila is tu o nucli which hav on nuclon lss than th numb quid to ully populat ngy obits. Exampl: lt s tak th 17 8O nuclus on o oxygn isotops. It has 8 potons and 9 nutons. Looking at th ngy lvls shown at th ight column o th schm on p. 33, w s that 8 potons ully populat th 1s 1/, 1p 3/ and 1p 1/ lvls. Th sam is tu o 8 nutons. Th additional nuton populats th 1d 5/ lvl. Sinc th ist th ngy lvls a ully populatd, th nuclus spin is 5/, as a consqunc o th additional nuton on j=5/ lvl. Homwok 5: chck th ngy lvl population o som oth nucli,.g. 13 C, 39 K, 101 Sn, and ty to dtmin thi spin. 1/03/014 B. Golob 3

33 Intsting acts latd to th nucla shll modl Maia Goppt-May, Hans Jnsn: Nobl piz in physics in 1963 o th nucla shll modl shad with Eugn Wign Hans Jnsn: paticipatd in th dvlopmnt o cntiugs o Uanium spaation in Gmany duing th WWII Eugn Wign: on o th initiatos o th Manhattan pojct in th U.S. Wikipdia: hug amount o inomation,.g. on th Epsilon opation; a U.S. opation to siz th lading Gman physicists, including Wn Hisnbg, with th aim o gathing inomation on th Gman nucla pogam. Hisnbg lat clad Jnsn o accusations o coopation with th Nazis. Nobl piz H. Jnsn, 1963: "o his contibutions to th thoy o th atomic nuclus and th lmntay paticls, paticulaly though th discovy and application o undamntal symmty pincipls Nobl piz M. Kobayashi, T. Maskawa, 008: "o th discovy o th oigin o th bokn symmty which pdicts th xistnc o at last th amilis o quaks in natu" 1/03/014 B. Golob 33

34 1.1 Basic poptis o nucli Dipol magntic momnt 1/03/014 B. Golob 34

35 Dipol magntic momnt classic dipol magntic momnt o ciculating chag: dq v p ISn; I dt t0 m p n p m m m cosponding opatos; o th abov cas this mans that B : Boh s magnton ˆ m y m m Th cospondnc pincipl in quantum mchanics tlls us that th quantum mchanical dsciption o an obsvabl can b obtaind by xchanging classical quantitis by 1/03/014 B. Golob 35 ˆ l l ˆ 1 n y m B l l p l l 1 1 y

36 Dipol magntic momnt What i th chagd paticl cais also spin bsid th obital angula momntum? Accoding to th cospondnc pincipl on would assum ˆ ˆ Howv, this is not th cas. Th ason o this is that spin intinsic angula momntum o a paticl dos not hav a classical quivalnt, and hnc th cospondnc pincipl can not b applid. Th valuation o th dipol magntic momnt o a spin ½ paticls mions is actually a big succss o th quation namd at Paul Diac th Diac quation. Th quation s p.??? is a lativistic quivalnt o th Schöding quation which is non-lativistic o dsciption o mions. It can b shown s p.??? that it pdicts th dipol magntic momnt o such paticls to b ˆ wh g s is calld th spin gyomagntic atio and g s = o mions and not 1 as would b implid by th cospondnc pincipl. s g s m m sˆ s ˆ 1/03/014 B. Golob 36

37 Dipol magntic momnt A consqunc o is that o a chagd mion having a non-zo obital angula momntum th dipol magntic momnt dos not point in th diction o th total angula momntum: ˆ m g sˆ In th abov quation w intoducd g l, an quivalnt o th g s o th obital angula momntum, which quals 1 o chagd and 0 o nutal paticls. Nuclons p and n a mions and hnc on would xpct : ˆ ˆ s, p s, n g g s, p s, n sˆ; gs, p m sˆ 0; n is m l ˆ g s nutal chag d ˆ ˆ s ˆ j sˆ mion m m Supisingly, th masud dipol magntic momnts o p and n giv Th unxpctd dipol momnts o p and n a du to thi constitunts; p and n a not lmntay mions but paticls composd o quaks and can not b dscibd by th Diac quation. 1/03/014 B. Golob 37 g g s, p s, n

38 Dipol magntic momnt Hnc th spin dipol magntic momnt ignvalus o nuclons a s, p s, p g g s, p s, n N N s s 1 s s 1 with N m N calld th nucla magnton. As th dipol magntic momnt o a nuclus on usually quots th 3d componnt o in th diction o an xtnal magntic ild, at th maximal spin pojction. In tms o an xpssion o th xpctation valu this mans JJ ˆ z wh w usd th shot-hand notation JJ z > o th nuclus stat wav unction o total angula momntum J and its 3d componnt J z. < JJ z psnts th conjugat wav unction. Th abov xpssion is hnc a shot-hand notation o th xpctation valu ˆ z y * ; J, J z JJ J ˆ y ; J, J z z J d 3 1/03/014 B. Golob 38

39 Dipol magntic momnt Du to g l g s th dipol magntic momnt o a nuclus dos not point in th sam diction as th total angula momntum spin o th nuclus. On can nvthlss din an ctiv gyomagntic atio g so that and J a paalll: I th nuclus has a singl nuclon out o th othwis ully populatd obits th nuclus spin is dtmind by th total angula momntum o this unpaid nuclon s p. xx. W can wit ith o ˆ g JJ m N z J ˆ ˆ jj m JJ ˆ N z jj jj g g wh j dnots th total angula momntum o th unpaid nuclon. l m N g ˆ z N g JJ s j sˆ 1/03/014 B. Golob 39 z Jˆ z jj JJ g m N J g J N

40 Dipol magntic momnt Th ctiv gyomagntic atio g can b dtmind by valuating th xpssion jm ˆ ˆ j jm On on hand, using th dinition o g, this quals to jm ˆ ˆ j jm jm g m N ˆj jm g j N j 1 On th oth hand, using g l and g s, this is jm ˆ ˆ j jm To valuat th lat xpssion w nd to dtmin ˆ ˆ m jm j jm and jm s j ˆ N ˆ jm jm g l ˆ ˆ j g s sˆ ˆ j jm 1/03/014 B. Golob 40

41 Dipol magntic momnt Using a shotnd notation w can gt th ollowing xpssions: Th w hav valuatd alady bo s p. xx. Using this w gt 1/03/ B. Golob s s s j s jm j s jm s l l s s j jm j jm ˆ ˆ 1 ˆ ˆ ˆ ˆ ˆ ˆ 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ŝ ˆ ˆ ˆ ˆ ˆ s s l l j j j s s s l l j j j

42 Dipol magntic momnt Finally, w aiv at Equating this xpssion and th on obtaind using g p. xx, w gt to th ctiv gyomagntic atio with 1/03/014 4 B. Golob ˆ ˆ ˆ ˆ ˆ ˆ s s l l j j g s s l l j j g jm j s g j g jm m jm j jm s l N s l N j j s s l l j j g j j s s l l j j g g s l n p g n p g s l

43 Dipol magntic momnt Fo nuclus with a singl unpaid nuclon j=l ± ½, and N g j j 1 l g j 1/ g l So o nucli with odd Z on unpaid p th dipol magntic momnt is N whil o th nucli with odd N on unpaid n th dipol magntic momnt is / j 3/ g j.3 j j 3/.8 j j 1.9 N j 1 s s / j j 1 j j j j j j l 1/ l 1/ l 1/ l 1/ l 1/ l 1/ 1/03/014 B. Golob 43

44 Dipol magntic momnt Th xpssions o th dipol magntic momnt o nucli as a unction o th nucla spin J a known as th Schmidt lins. Th masud dipol magntic momnts o vaious nucli li btwn thos lins sinc not all o thm hav only a singl unpaid nuclon: / N odd Z / N =j+,3 / N odd N / N =1,9 j/j+1 / N = [jj+3/-,8 j]/j+1 / N =-1,9 Homwok 7: dtmin th dipol mag. momnt o 13 C and 39 K. J lins: xpctd o nucli with J = l± ½ Schmidt lins 1/03/014 B. Golob 44 J

45 Intsting acts: in basic scinc it happns otn that th discovis mad th ind thi way to vaious applications, although such an outcom can not b osn bo th puly scintiic sach is caid out. Such applications a calld spin-o. Exampl 1: www Wold Wid Wb was dvlopd at Cn du to communication nds o intnational scintiic collaboations ist sv Exampl : knowing what th dipol magntic momnts o nucli a, thi Robt Cailiau Tim Bns-L masumnts though th Nucla Magntic Rsonanc NMR tchniqu psnt a nowadays ssntial diagnostic tchniqu in th mdicin. NMR diagnostic appaatus B 1,5 T pat o today s computing cnt at Cn NMR pictus o vaious tissus 1/03/014 B. Golob 45

46 1. Nucla dcays 1..1 a dcays 1/03/014 B. Golob 46

47 a dcays In a dcays th initial nuclus ducs its ngy by mitting a H nuclus: Z,A Z-,A H H nuclus is also known as th a paticl; it s th sam paticl as usd by Ruthod, Gig and Masdn in th xpimnt xposing th atomic nucli, s p. xx yy Th ngy consvation in this pocss is: mz,a c = mz-, A-4 c + m a c + T a + T Z-, wh T a, Z- dnots th kintic ngis o th a paticl and inal stat nuclus. Consvation o momntum quis p Z- = p a and hnc T a = p a /m a = T Z- mz-,a-4/m a. Sinc usually m a << mz-,a-4 on can nglct T Z- in th ngy consvation quation. Th dcay is ngtically possibl i T a = - WZ,A + W a + WZ-,A-4 0 wh w xpssd th masss o nucli by th bounding ngy as givn by th smi-mpiical mass omula, p. xx. 1/03/014 B. Golob 47

48 a dcays W can now stimat om which mass numb A th a dcay is possibl by assuming A, Z >> 1, tating A and Z as continuous vaiabls and -witing th abov condition as a dintial: with Z= and A=4. Dintiating th smi-mpiical mass omula yilds In th abov condition w hav two indpndnt vaiabls, Z and A. In od to obtain a condition xpssd as a unction o A only w mak a uth appoximation, by including th lation btwn A and Z as valid o stabl nucli, i.. th xpssion obtaind om 1/03/ B. Golob A A W Z Z W W A Z W A Z W W T a a a 4,, 0, 3 / 1 4 / / 4 3 1/3 1/3 1 0 A Z A w A Z w A Z A Z w A w w W T a a / A A Z Z W

49 a dcays Homwok 8: chck th pcision o th ZA lation o vaious nucli by compaing th calculatd valus with th chat o known stabl nucli. A-Z This xpssion givs Z o nucli which o a givn A a most stabl. Claly this is not coct o th nucli which undgo th a dcay, but is suicint to obtain a ough stimat. By insting th abov ZA lation into th condition o a dcays ollowing om th ngy consvation on obtains a non-tivial xpssion that cannot b solvd o A analytically but can b solvd numically, o xampl by dawing th dpndnc as a unction o A taking into account th xpimntally dtmind valu W a =8.3 MV. T a [MV] chat o known stabl nucli Z A A W can s that om th ough stimat on xpcts nucli with A 155 to dcay though an mission o a paticls. 1/03/014 B. Golob 49

50 a dcays Th stimat is not conimd by xpimntal data. Th latt povs a dcays o nucli havi than 07 8Pb. This should not b supising givn th appoximations w mad. Moov on nds to includ quantum mchanical pictu in od to undstand at last qualitativly th lowst mass numb om which th a dcays a obsvd. I w imagin th initial nuclus composd o a H nuclus and th st o th nuclons, th potntial xpincd by th a paticl looks qualitativly as shown in th plot blow: V At small distancs btwn th a paticl and th st o th initial nucli th potntial is attactiv ngativ du to th stong nucla oc binding th nuclons. At lag distancs th pulsiv positiv Coulomb potntial btwn th two positivly chagd ntitis bcoms impotant. Hnc th a paticl with a positiv kintic ngy T a has to tunnllat though th positiv potntial bai in od to scap om th initial nuclus. T a 1/03/014 B. Golob 50

51 a dcays W can stimat th tunnling pobability by ist considing tunnling though a potntial bai: V pˆ V V y Ey Th ist stp is solving th Schöding quation o th two body poblm o a paticl and th st o th nucli with th ducd mass and momntum opato p as wittn; th solution can b ound using th ansatz involving th sphical unction Y l,m s p. xx. Th solution o th adial pat has th distinct intvals: wh T a 4mn A 4 mn A 4 4 m Amn A ˆ pˆ u y Y l, m, ik ik ui A B R y uii C B R R' ik u E R' n R R k T a, V ' T 3/1/014 B. Golob 51 a III

52 a dcays Th bounday conditions om which on dtmins th unknown coicints A,B,... a ui uii ui R uii R, R R uii uiii uii R' uiii R', Fom ths on obtains E A k calld also th bai tansmittancy. In th cas o a dcay th bai hight is not constant; w can appoximat this by dividing th Coulomb bai into thin slics: V R' R' 4k R' R Th tansmittancy can thn b wittn as L L 1 d Taking into account th Coulomb bai dpndnc w gt g T a L 1,L,... 3/1/014 B. Golob 5

53 a dcays Taking into account th Coulomb bai shap w gt Th abov xpssion dpnds on R ; w stimat R as th distanc at which T a bcoms lag than V, i.. VR = T a. Th sult o th intgation is Th pobability o tunnling 3/1/ B. Golob c Z Z V d T V g R R a a 4, 0 ' 1 actan a a a a a T R V R R V T R V T R V R T R V R g R V T ln, 1 R V R k T R RV k g w w g a homwok 9: typical T a o known dcays is 5 MV; compa this to th valu o Coulomb potntial at som typical nucla distanc.

54 a dcays Sinc V R 1 R w aiv at W can ad o two poptis o th sult: 1 ln w k3 k4 T - pobability o tunnling and thus o dcay incass with th mass numb A; w - pobability o a dcay invs o th nucli litim dpnds spciically on T a ; th latt dpndnc is also known as th Gig-Nuttal ul. Fo a dcays o nucli with A ~ 155 o which th dcay bcoms ngtically possibl th tunnling pobability is too low o dcays to b obsvd; it bcoms sizabl only o nucli with A 07. w a k k R 1/ 6 4 R k5a / Ta k3 3, / T a 3/1/014 B. Golob 54

55 a dcays Gig-Nuttall s law: Rd dots psnt xpimntal data on pobability o a dcays o vaious nucli. Lin dnotd Gig-Nuttal psnts a it with a staight lin to th data. Blu dots blu hoizontal scal dnots th sam data but in dint scal to distinguish thm om blu dots. Th staight lin though thos dots psnt th pdiction obtaind om th abov dscibd calculation tunnling though a potntial bai which psnts a good dsciption o xpimntal data. ln w 1 T a /1/014 B. Golob

56 1. Nucla dcays 1.. b dcays 1/03/014 B. Golob 56

57 b dcays In b dcays th initial nuclus ducs its ngy by mission o an lcton and lctonic anti-nutino: Z,A Z+1,A - n Nutinos a paticls discussd in mo dtails in th lmntay paticls sction. Fo now it should b suicint to know that ths paticls intact only via th wak nucla oc and hav mass almost qual to zo anti-nutino is an anti-paticl o a nutino, similaly as th positivly chagd lcton, calld positon, is an anti-paticl o an lcton. Th abov dcay is calld a b - dcay du to a psnc o th lcton in th inal stat. At th lvl o nuclons n p - n Sinc m n - m p c = 1.9 MV such a dcay o a nuton is actually possibl. Nutons a unstabl and dcays though a b - dcay with a litim o aound 880 s. 1/03/ B. Golob

58 b dcays Th b - dcay is ngtically possibl i m Z, A c m Z 1, A c Taking into account th abov nuton - poton st ngy dinc and th st ngy o an lcton, m c = 0.51 MV, th condition ads Anoth typ o b dcay is a b + dcay: Z,A Z-1,A + n with a positon in th inal stat. At th nuclon lvl this is p n + n Du to th mass dinc btwn th p and n such a dcay o a poton is not possibl. It can only tak plac o th p and n which a bound insid th nucli. Th nucla b + dcay is ngtically possibl i m c 1/03/ B. Golob 0 W Z 1, A W Z, A MV W Z 1, A W Z, A MV

59 Som intsting acts about poton dcay Poton is th lightst known bayon paticl composd o th quaks, s Pat II, p.??? Sinc th bayon numb s Pat II, p.??? is consvd in all so a known pocsss it ollows that th poton dos not dcay. Howv, in od o th Univs to volv om th Big Bang to th psnt stat th a stong agumnts that in th aly stags o th Univs th bayon numb consvation had to b violatd s Pat, p.???. Hnc many thois which a xpimntally not conimd pdict th dcay o a p with litim p yas. It is thus undstandabl that a sach o possibl p dcays should b xpimntally pomd. On o th most snsitiv xpimnts in this sach is th SupKamiokand xpimnt in Kamioka in Japan. It is placd in th cavn in th mountain to potct th xpimnt om vaious backgound soucs such as th cosmic ays. Kamioka Tokijo 1/03/ B. Golob

60 Som intsting acts about poton dcay Th xpimnt consists o a svoi with tons o ulta-puiid wat, suoundd by photomultiplis o dtction o singl photons. ~40 m ~40 m photomultiplis o photon dtction t ulta-puiid wat 1/03/ B. Golob

61 Som intsting acts about poton dcay On o potntial but unobsvd p dcays is p + 0, shown in th sktch blow: Th dcay can b sachd o by dtcting th photons o th Chnkov light poducd by th + in th wat, and by dtcting photons om th 0 dcays. So a no signiicant signal o such dcay was obsvd. Th cunt limit on th p litim is p > yas. O cous this dos not man that th xpimnt nds to b opatd o yas. Th p dcays - i xisting - would oby th xponntial dcay law, th numb o p dcaying in a tim intval [t, t+dt] would b dn/dt = N 0 1/ p -t/p, wh N 0 is th numb o all p bing obsvd. Hnc th lag p is compnsatd by th wast amount o p in t o wat. Homwok 9: stimat th numb o xpctd p dcays in t o wat in on ya i th xpctd p litim is yas. 1/03/ B. Golob

62 Som intsting acts about poton dcay In 001 th SupKamiokand dtcto was hit by an accidnt: du to th implosion o w multiplis th shock-wav popagating though th wat dstoyd 6600 o photomultiplis th pic o ach at th tim was 3000 US $. With hug ots th dtcto was paid in 006 SupKamiokand II. Th xpimnt is not ddicatd only to a sach o th p dcay but also to th study o anoth intsting phnomna - oscillations o nutinos. 1/03/014 6 B. Golob

63 Spctum o ± in b dcays In 1934 Fmi gav th ollowing dsciption o th nucla b dcay, stating om th his goldn ul p. 14: W i E Sinc Fmi did not know any dtails about th wak intaction causing th dcays h wot V i G F * * * y ny id 3 wh obviously no dtails o potntial a wittn and all th dtails o th intaction a dscibd by a constant G F, nowadays calld th Fmi constant. y, y i, and n a th wav unctions o th initial and inal nuclus, and o lcton and nutino, spctivly. As an initial appoximation on can us plan wavs to dscib th latt two, 1 ik n G F * ik 3, n, and th matix lmnt bcoms Vi i d V y y, V V with k=k +k n. Sinc typical ± ngis in b dcays a o th od o MV, and lcton st ngy is m c = 0.51 MV, on must us a lativistic lation btwn th momntum and ngy: p E mc k c. Th constant c 197 MVm is a usul convsion constant wothwhil to mmb. 1/03/ B. Golob i i

64 Spctum o ± in b dcays Th intgation in th matix lmnt is pomd ov ull spac, howv sinc th nuclus wav unctions a substantial only in th gion o th nucli, i.. in th gion with R~ w m, w can s 1MV k ~ 5 m ~ MVm. Hnc w can mak a Taylo sis xpansion o th xponntial in th matix lmnt: G F * k 3 Vi i 1 k d. Considing that th xponntial acto aiss V y y! du to th -n wav unction, and that th individual tms k l /l! hav th angula dpndnc o a sphical hamonic Y lm q,, which is th ignunctions o th obital angula momntum opato s p. 6, w can associat ach tm in th xpansion to a pobability that th and n in th inal stat cay th obital angula momntum l l 1. Th lagst tm in th matix lmnt is k 0, i this tm is non-zo. In th latt cas th lagst tm would b k 1, and so on. Accoding to th xponnt o th ist non-zo tm in th sis w call th dcays allowd k 0, onc obiddn k 1, twic obiddn k, tc. Moov, o th allowd dcays th is no dpndnc o th matix lmnt on th ngy o - n systm sinc thy involv k 0. Hnc th dpndnc o th dcay pobability on this ngy aiss only om th dnsity o inal stats s p. 15. Fo th obiddn dcays 1/03/ B. Golob

65 Spctum o ± in b dcays 1/03/ B. Golob th is som ngy dpndnc in th matix lmnt, howv it povs to b ath mild compad to th dpndnc aising om th dnsity o inal stats. Contay to th scatting discussd in 1.1.., in b dcays w hav th inal stat paticls. Du to th momntum and ngy consvation only two a indpndnt in tms o th momntum, and hnc th dnsity o inal stats can b wittn as wh in th last stp w intgatd ov th dictions o both and n. Sinc n s a diicult to dtct th obsvabl o intst is th ± ngy distibution. W can plac th indpndnt vaiabl p n by th total ngy o th lcton-nutino systm E: ,, n n n dp p dp p V N d de N d d p d p d V N d de E E dp p N d c de dp c E E p cp E c m p c E E E E c A Z M c A Z m E E i 4 ;,, ; 1,, n n n n n

66 Spctum o ± in b dcays Th lcton momntum distibution is dw i /dp d /dp d dp d de N dp d N dedp p E E I on plots th [dw i /dp / p ] 1/ as a unction o th lcton ngy i.. in obsving dcays o som b souc, a sampl o nucli undgoing th b dcay, w plot th numb o dtctd ± with a givn momntum, dividd by this momntum squad, as a unction o th ± ngy th dpndnc is lina. Such a plot is calld th Fmi-Kui plot. [Np /p ] 1/ E 1/03/ B. Golob E [MV]

67 Spctum o ± in b dcays Th lcton ngy distibution is obtaind by noting E de = p dp, and dw de NE i dw dp dw dp i i dp de p E E p E 4 E E E m c E E E b - m c b + E =T +m c ± ngy distibution in b dcay; th dviations om th calculatd shap a obsvd at low ngis du to th Coulomb intaction btwn th ± and th inal nuclus. E Coulomb intaction T [MV] 1/03/ B. Golob

68 Spctum o ± in b dcays Th chat o nucli with makd typ o dcays lading to th stabl nucli dnotd as black points is shown blow: Z a b b DZ=-,DN=- DZ=+1,DN=-1 DZ=-1,DN=+1 stabl nucli N 1/03/ B. Golob

69 Slction uls o b dcays Th consvation o th angula momntum quis J J J n i wh J is th total angula momntum spin o th inal nuclus, J n is th total angula momntum o th lcton-nutino systm, and J i is th th spin o th initial nuclus. J n is composd o th obital angula momntum lativ to th inal nuclus and th total spin o th lcton-nutino systm: J s n n Sinc both lcton and th nutino a mions o spin ½, s n = 0 o 1. In th om cas such dcays a calld Fmi dcays w will dnot thos with F, whil in th latt thy a calld Gamow-Tll dcays GT. Hnc in th cas o Fmi dcays J n =l, whil in th cas o Gamow-Tll dcays J n = l ± 1. Th dinc o spins btwn th initial and inal nuclus, DJ J i J, quals th total angula momntum caid by th lcton-nutino systm, D J J n. Paity o th wav unction dscibing a ctain systm dtmins th bhaviou o th unction und th lction o spac, -. 1/03/ B. Golob ˆ Py y

70 Slction uls o b dcays I w apply th paity opato on th wav unction onc mo, Hnc ˆ ˆ ˆ ˆ PPy P y Py y ˆ P y P y y P 1 wh th ignvalu o th paity opato Pˆ is dnotd by P. Th paity in b dcays is consvd. This is not a tivial act, which aiss om xpimntal obsvations. Whn discussing th wak intaction that causs b dcays w will s that som o th most intiguing poptis o th wak intaction ais du to th non-consvation o paity. Howv, at th ngis involvd in th nucla b dcays, th paity is consvd dspit th act that in gnal th wak intaction dos not consv it. Th paity o th inal stat thus quals th paity o th initial stat, Pi P n P As mntiond abov th lcton-nutino systm, caying an obital angula momntum l, is dscibd by a sphical hamonics Y lm. Th popty o sphical hamonics is P ˆ 1 1/03/ B. Golob Y lm l

71 Slction uls o b dcays Th paity o th lcton-nutino systm with th obital angula momntum l is thus -1 l. This psnts th chang in th paity btwn th initial and inal nuclus, DP = P i /P = -1 l. Fom th discussion abov w can dtmin a st o slction uls o th angula momntum and paity in b dcays. Fo xampl, i th spins o th initial and inal nuclus a qual, D J = 0, th dcay can b ith a Fmi dcay with l =0, o Gamow-Tll dcay with l = 1, which togth with s n = 1 can giv J n = 0. I uthmo th paity o th initial and inal nuclus a qual, this has to b a Fmi dcay sinc in this cas th chang o paity DP = -1 l = 1. Following simila agumnts w can build a tabl o slction uls shown in th nxt pag. 1/03/ B. Golob

72 Slction uls o b dcays D J =J n DP Typ 0 no F0 0 ys GT1 1 ys F1 1 no GT0 no F ys GT1 no GT : : : In th abov tabl F and GT dnots Fmi and Gamow-Tll dcays, spctivly. Th numb ollowing this notation psnts l, th obital angula momntum o th lcton-nutino systm. 1/03/014 B. Golob 7

73 1. Nucla dcays 1..3 g dcays 1/03/014 B. Golob 73

74 g dcays In g dcays a nuclus mits lcto-magntic adiation photon = g paticl and by this d-xcits om a high to a low, phaps a gound ngy lvl. Such dcays a causd by g E th lcto-magntic intaction. In th dcay th chag distibution as wll as th distibution o th magntic dipol momnts o nuclons is changd. Th lcto-magntic EM ild aound a systm o moving chags nuclons in th nucli can b dscibd by xpanding it into th multipol sis s p. xx. Fo xampl, an oscillating classical lctic dipol adiats a dipol ild, with an avag adiatd pow o 4 p0 P wh is th oscillation quncy and p 0 is th amplitud 3 3 0c o th lctic dipol, p 0 = 0. High multipols adiat a cosponding high multipol adiation. I th avag adiatd pow is intptd in tms o photons, th pobability o adiating a photon o ngy p unit o tim is 3 P k p0 w 3 0 1/03/014 B. Golob 74 E i

75 g dcays Multipol sis: any unction o angls and q in spical coodinats can b wittn as q, 0 m c Y,, q m m, wh Y l,m q, a sphical hamonics. Wign-Eckat thom: iducibl tnso opato o ank k is dind as any st o k+1 quantitis that tansom as th sphical hamonics Y k,q und otations; xampl: opato tnso o ank k=1 can b dind in tms o Y 1,q though Y 1,0 3 z ; Y1, 4 1 ˆ 3 x iy 4 Th Wign-Eckat thom stats that th xpctation valu o any componnt o an iducibl tnso opato T k q q=-k, -k+1,...,k btwn two angula momntum stats j,m > and j,m> can b actod as ˆ k jm ', ' ˆ k j, mtq j m Ckqj' m' j T j', wh 1/03/014 B. Golob 75

76 C g dcays k j Tˆ j' is th ducd matix lmnt and dos not dpnd on q, m and m. jm a th appopiat Clbsch-Godan coicints. Ths coicints qual zo unlss ' m' m = m +q and j -k j j +k! kqj Hnc also j, my, j', m' m 0, unlss j - l j j + l 1/03/014 B. Golob 76

77 g dcays Quantum-mchanically th xpssion is analogous with th dipol momnt dind as p * y y d i 3 A simila xpssion holds o th adiation o a magntic dipol, 0 m0 * w p y g g s y d m 3 k p 3 m N l s sct o dinitions o N, g l and g s. Expssions o high multipols a not asy to wit in a simpl vcto omat. It suics o th momnt to know that th cosponding lctic multipol opatos hav an angula dpndnc o Y lm q, and th magntic multipol opatos that o l Y lm q,. Why is this impotant? Th Wign-Eckat thom s p. xx tlls us that in od o th xpctd valu o an opato, * ˆ 3 O y Oy id 0, th angula momnta o th initial and inal stat J i, J, and th multipolnss o th opato l should oby th tiangula lation J +l J i J -l. Fom this on can div th slction uls o th angula momntum in g dcays. s i 3 1/03/014 B. Golob 77

78 Slction uls o g dcays Th matix lmnt o a gnal g dcay can thus b schmatically wittn as * m 3 V y Oˆ Oˆ y d i l l l Th supscipts m and dnot th natu o th opato - magntic and lctic, spctivly. Fo xampl, O 1 is just. Basd on th tiangula lation abov th dinc btwn th spins o th initial and inal nucli is latd to th multipolnss o th opato causing th tansition, J -J i = l. Th notation o vaious tansitions is basd on th l o th lowst multipol opato in th matix lmnt sulting in a non-zo V i, and on th natu o opato as ith lctic E o magntic M. Thus a tansition causd by th lctic dipol opato is dnotd as E1, th tansition causd by th magntic quadupol opato as M, tc. Fo a givn l th pobability o a magntic tansition is signiicantly low than th pobability o th lctic tansition. This can b asily stimatd o th dipol opatos : i 1/03/014 B. Golob 78

79 Slction uls o g dcays w w m k p m0 0 1 m0 3 3 k p0 c p0 As an appoximation o th magnitud o th lctic and magntic dipol w tak R with R bing th adius o a nucli and N th nucla magnton. W gt w w m Th atio is vn low o high multipol tansitions. p 1 N c 00 MVm 3 ~ ~ 10 R mnc R 900 MV 5 m c Elctomagntic intaction psvs th paity. Fo th lctic tansitions th lation P P Ylm = P i thus holds, wh th P Ylm is th paity o th Y lm sphical hamonic sinc this is th angula dpndnc o th lctic multipol opato, -1 l. Th slction ul gading th paity in lctic tansitions is thus P -1 l = P i. Fo th magntic opatos th angula dpndnc is Y lm q, and hnc th paity is -1 l+1 th divativs in also chang sign whn spac is lctd. Th slction ul o paity in th magntic tansitions is P -1 l+1 = P i. A summay tabl o slction uls o g dcays is givn on th nxt pag. 1/03/014 B. Golob 79

80 Slction uls o g dcays l J i J Paity chang Tansition typ ,1, 1,,3 3,3,4 1/03/014 B. Golob 80 No Ys No Ys No Ys No Ys : : : : 0 1 1,,3 0,1,,3,4 3 1,,3,4,5 No Ys No Ys No Ys No Ys : : : : : : : : : M1 E1 M1 E1 M1 E1 M1 E1 E M E M E M E M

81 Exampls o dcay schms b - dcay 13 5T 13 53I - n J P=± ngy o nucla stat lativ to th gound stat maximal ngy E E g Homwok 10: basd on th slction uls o g dcays dtmin which opatos a sponsibl o ths tansitions! dcay hal-tim t 1/ = ln 1/03/014 B. Golob 81

82 Exampls o dcay schms In som cass th g dcay om an initial to a inal ngy stat can only pocd though high multipl tansitions. In this cas th litim o such a stat is long and th stat can b tatd as quasi-stabl. Such stats a calld isoms. dcay hal-tim t 1/ =,5 months Homwok 11: dtmin which opatos a sponsibl o th tansition shown! 1/03/014 B. Golob 8

83 Tabl o Contnts: Intoduction Basic poptis o nucli Mass Chag distibution Spin Dipol magntic momnt Nucla dcays a dcays b dcays g dcays /03/014 B. Golob 83

84 Indx: a paticl 47 allowd dcay 64 b+, b- dcay 57, 58 bayon 59 Boh s magnton 35 convsion constant, hc 63 cospondnc pincipl 35 d Bogli wavlngth 15 dnsitiy o inal stats 14, 15 dintial coss-sction 16 Diac quation 36 ctiv gyomagntic atio 39, 4 ngy distibution in b dcay 67 xponntial dcay law 61 Fmi constant 63 Fmi dcay 69 Fmi goldn ul 14 Fmi-Kui plot 66 obiddn dcay 64 om acto 19, 3 Gamow-Tll dcay 69 Gig-Nuttall's law 54 Goppt-May M. 33 Gn s omula 18 Hisnbg W. 33 isoms 80 Jnsn H. 33 Kobayashi M. 33 magic numbs 5, 31 Maskawa T. 33 matix lmnt 14, 16 multipol sis 74 nutino 57 nuton litim 57 Nucla magntic Rsonanc NMR 45 nucla magnton 38 nucla adius paity 69 poton dcay 55 quantum numbs 9 Ruthod scatting 1 Schmidt lins 44 Schöding quation 5 slction uls b dcay 71, 7 slction uls g dcays 77, 80 Smi-mpiical mass omula Wizsäck omula 7 shll modl 5 spin gyomagntic atio 36 spin-o 45 spin-obit intaction 8, 30, 31 SupKamiokand 59 tiangula lation 77 tunnlling 50 wav unction 14 Wign E. 33 Wign-Eckat thom 77 Wold Wid Wb 45 1/03/014 B. Golob 84