Intro to Nuclear and Particle Physics (5110)

Transcription

1 Intro to Nuclar and Particl Physics (5110) March 09, 009 Frmi s Thory of Bta Dcay (continud) Parity Violation, Nutrino Mass 3/9/009 1

2 Final Stat Phas Spac (Rviw) Th Final Stat lctron and nutrino wav functions hav th usual fr-particl form, normalizd within th volum V: ϕ 1 r ( r ) xp( ip r / h) V 1 r ( r ) xp( iq r / h) V If th lctron is confind to a box of volum V thn th numbr of final lctron stats dn, for momnta in th rang p to p+dp, and th numbr of final nutrino stats dn in th rang q to q+dq ar, rspctivly 4πVp dn 3 h dp 3/9/009 ϕ ν dn ν 4πVq 3 h On intrprtation of this is that ach stat occupis phas-spac (ral spac x momntum spac 6D) volum of 3 r 3 p ( x y z) ( p X p Y p Z ) h 3 (Uncrtainty Principl?) and th numbr of final stats which hav simultanously an lctron and a nutrino, with th momnta p and q, is: d (4π ) N dndnν V p 6 h dpq dq dq

3 Trating th Ovrlap Intgral as a Constant For an lctron with 1 MV kintic nrgy, p 1.4 MV/c and p / h fm -1 Thus ovr th nuclar volum, pr / h <<1 and w can xpand th xponntials, kping only th first trm: ip r ip r xp( ) 1+ h h iq r iq r xp( ) 1+ h h This approximation is known as th allowd approximation. W thn hav for th ovrlap intgral V ' g V fi g f ν X i f X i V whr M * * * [ ψ ϕ ϕ ] O ψ dv [ ] [ ] * * * * ψ ϕ ϕ O ψ dv ψ O ψ dv ( M ) fi ψ * O ψ dv is calld th nuclar matrix lmnt f X i 3/9/009 3 ' fi g f ν g V X i fi

4 With lctron s momntum bing within p p + dp and nutrino s momntum bing within q q + dq, th partial dcay rat undr th allowd approximation is: dλ π g h V π g h π V h M M ' fi fi fi dn de f (4π ) V 6 h (4π ) π g h V p p dpq de dpq 6 h f M fi dq de dq f dn de f thrfor π h dpq h dλ g (4π ) M fi 6 p dq de (*) f Th final nrgy E f is just E f E + Eν E + qc and so dq/de f 1/c at fixd E. 3/9/009 4

5 As far as th shap of th lctron spctrum is concrnd, all of th factors in th quation (*) that do not involv th momnta p and q can b combind into a constant C, and th rsulting distribution givs th numbr of lctrons with momntum btwn p and p + dp is: N( p) dp Cp q dp N(p)dp ~ dλ N(p) Cp q is th total numbr of lctrons with a momntum p. Q If is th dcay nrgy, thn ignoring th ngligibl nuclar rcoil nrgy (NOT th rcoil momntum), q Q T c Q ( ) 4 p c + m c m c c Whr T is th kintic nrgy of th daughtr lctron and th spctrum shap is givn by C N( p) p ( Q T ) c C 4 p [ Q p c + m c m c ] + c 3/9/009 5

6 C N( p) c c 4 p [ Q p c + m c m ] + (1) p 0 N(0) 0 () Q T N(p max ) 0 at th ndpoint Expctd lctron momntum distribution with Q.5 MV 3/9/009 6

7 It is mor common to plot th nrgy spctrum, for lctrons of kintic nrgy T T + dt. With c pdp (T + m c ) dt, w hav C 1/ N( T ) ( T T m c ) ( Q T ) ( T m c c ) (1) T 0 N(T ) 0 () T Q N(T Q) 0 at th ndpoint Expctd nrgy distribution with Q.5 MV 3/9/009 7

8 How Wll Dos It Work? Cu35 30 Zn34 β (38%) Cu35 8 Ni36 + β (19%) Th rmaining 43%: lctron captur. 1. In this figur th gnral shap of spctra is vidnt, but thr ar systmatic diffrncs btwn thory and xprimnt. Ths diffrncs originat with th Coulomb intraction btwn th β particl and th daughtr nuclus.. From th mor corrct stand point of quantum mchanic, w should instad rfr to th chang in th lctron plan wav brought about by th Coulomb potntial insid th nuclus. 3/9/009 8

9 Th ffct of th nuclar Coulomb fild can b accountd for by introducing an additional factor, th Frmi function F(Z,p) or F(Z, T ) which modifis th β -dcay spctrum whr Z is th atomic numbr of th daughtr nuclus. F( Z', T ) x 1 x x m παz' c / v for β ± - dcay and α is th fin structur constant. In trms of momntum: In trms of nrgy: N( p) C p ( Q T ) F( Z', p) c C 1/ N ( T ) ( T + mc T ) ( Q T ) ( T + m c c ) F( Z', T 5 ) 3/9/009 9

10 How Wll Dos it work now? Liftd from HyprPhysics ( Thory fits xprimnt rathr wll: But did you notic th diffrnc in th scal of th horizontal axis? Prviously th ndpoints in th momntum spctrum wr about ½ th valus sn hr. Rason for this is that in ths plots th momntum is plottd as a ratio of p/(m c), and not th absnc of unit labl. 3/9/009 10

11 Th original plots Th plots of th spctra on th prvious pag, which I liftd from HyprPhysics, ( wr originally takn from: John R. Ritz, Th Effct of Scrning on Bta-Ray Spctra and Intrnal Convrsion, Phys. Rv. 77, (1950) Ritz did th calculation of th Coulomb factors. Th data wr from: C. Sharp Cook and Lawrnc M. Langr, Th Bta-Spctra of Cu64 as a Tst of th Frmi Thory, Phys. Rv. 73, (1948), and from C. S. Wu and R. D. Albrt, Th Bta-Ray Spctra of 64 Cu, Phys. Rv. 75, (1949) 3/9/009 11

12 Th Kuri Plot A convnint way to plot th bta dcay spctrum is th Kuri plot W had: N( p) C p ( Q T ) F( Z', p) c N( p) ( Q T ) p F( Z', p) If w plot N( p) / p F( Z', p) against T thr should b a straight lin which intrcpt th x axis at th dcay nrgy Q. Such a plot is calld a Kuri plot. ndpoint nrgy Kuri plot givs us a convnint way to dtrmin th dcay ndpoint nrgy. 3/9/009 1

13 Masurmnt of Nutrino Mass in Bta Dcays [ ] 1 / 4 4 Q p c + m c m c N ( p) p + Nar th ndpoint of th β spctrum, th nutrino nrgy approachs zro. If th nutrino has rst mass thn E ν ~ m ν c and our prvious calculation of th statistical factor for th spctrum shap is incorrct. Th nutrino kintic nrgy can b tratd nonrlativistically, so that q m ν T ν and or N( T ) ( T + T m c ) 1/ ( Q T 1/ ( T m c ) ) + dn dp dn dp 0 if m ν 0 if m ν > 0 W can thrfor study th limit on nutrino mass by looking at th slop at th ndpoint of th spctrum as indicatd in th figur. N 0 as p (or T ) 0, poor statistics!! 3/9/009 13

14 Exprimntal Dtrmination of th nutrino mass from th β dcay of tritium 3/9/009 14

15 Parity Violation of Wak Intraction Th parity transformation P rflcting all 3 coordinats of a systm: r r P( ) r If th parity transformation givs us a physical systm or st of quations that obys th sam laws as h original systm, w conclud that th systm is invariant with rspct to parity. In that cas, th original and rflctd systms would both rprsnt possibl stats of natur, and in fact w could not distinguish in any fundamntal way th original systm from its rflction. In fact, thr ar thr diffrnt typs of rflctions. 1. Parity Opration (P).. Charg Conjugation Opration (C). 3. Tim Rvrsal Opration (T). th spatial opration r -r. rplacing particls with antiparticls rvrs th tim dirction t -t. 3/9/009 15

16 In this figur thr procss undr th P, C, and T oprations ar shown. Vctors that chang sign undr P ar calld tru or polar vctors. position, vlocity, forc, lctric fild. Vctors that do not chang sign undr P ar calld psudo- or axial vctors. angular momntum, magntic fild, torqu. In this figur, ach rflctd imag rprsnts a ral physical situation that w could achiv in th laboratory, and w bliv that gravity and lctromagntism ar invariant with rspct to C,P, and T 3/9/009 16

17 On way of tsting th invarianc of th nuclar intraction to P, C, T would b to prform th sris of xprimnts dscribd in this figur. In ach cas w could compar th probability of th rvrsd raction with that of th original, and if th probabilitis provd to b idntical, w could conclud that P, C, T wr invariant oprations for th nuclar intraction. 3/9/009 17

18 W may us th mthod ncircld to tst parity consrvation in a laboratory. In such kind of arrangmnt th spin of th dcaying particl A dos not chang dirction undr P. Th original xprimnt shows particl B mittd in th sam dirction as th spin of A, whil th rflctd xprimnt shows B mittd opposit to th spin of A. W may simply align th spins of som dcaying nucli and look to s if th dcay products ar mittd qually in both dirctions or prfrntially in on dirction. 3/9/009 18

19 Th parity consrvation was found to b corrct in many xprimnts and was considrd a fundamntal principl of natur. It was not until 1950 s th validity of parity consrvation was qustiond du to th θ-τ puzzl. At that tim thr wr two particls, calld θ andτ, which appard to hav idntical spins, masss, and liftims but dcay into stats of diffrnt paritis! θ π + π τ π + π + π Sinc th dcays wr govrnd by a procss similar to nuclar β -dcay, L and Yang, in 1956, suggstd that θ andτ w r th sam particl (today calld a K mson) which could dcay into final stats of diffrnt paritis if th P opration wr not an invariant procss for β-dcay. 3/9/009 19

20 Shortly aftr th proposal of L and Yang, C.S. Wu and hr co-workrs compltd a dlicat xprimnt dmonstrating that parity was not consrvd in th β -dcay of 60 Co. Thy alignd th 60 Co spins by a magntic fild at vry low tmpratur, T ~ 0.01 K. Rvrsing th magntic fild dirction rvrsd th spins and in ffct accomplishd th P opration. Th obsrvd fact was that at last 70% of th β particls wr mittd opposit to th nuclar spin. 3/9/009 0

21 Improvd rsults of C.S, Wu and co-workrs on th parity violation in th 60 Co β -dcay. 3/9/009 1