Theoretical halflives of ’forbidden’ beta-transitions

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SSSOmXOAX, BLUF-IOT8S m

^'FOHBiDDEiin wmk t m m m o m

Submitted to the F atuity o f . the Graduate Sohool in p a rtia l fu lfillm en t of ‘m e noquinesente fo r the &#g3?eet Doctor of Philosophy# in the D e ftftiw i of Physios# .In d ia n a

by

lugena 0remliiag s

April 1340

ProQuest Number: 10295108

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COKTSHTS CHAPTER I ,

THB FK’KI TBBOHT

J a sllffi

laa®

In

XntroduetioH *. * * * *

*.* .* * * * * * . * *

*

1

0#

System of tta lts * * * ■*

» *4 * * * * * * * * *

4

3#

Pertmrb&tlon theory* « * » * * « • * * *

* * *

4

* *

6

4# , Th© In v a ria n t Xntor&etions * • * * » * « # 3*

fli# Matrix Element© * « * * • * * * * * * « * *

8

6*

tt© S t a t i s t i c a l Factor© and th e C onservation o f Energy* * ** * « * . * * # * • * « * # * »

0

os&pmr ii*

m m n m im -B

«sx

1*

th© lo a n in g o f ^Forbidden** t r a n s it io n s * *

» *

IS

2m .

fh © ^av© f t o c b l o n g * *

« •

14

3*

The F ir s t Approximation * ^Allowed®f r a n a iiio n s

16

4*

Higher Approximation * "Forbidden* frtn a itio n ©

@0

5*

t h e Irre d u c ib le Tm®or M atrix B l m n t a * ♦ # «

@1

6*

S e le c tio n Rules* # # # * # # # « • * # * * * *

24

7*

0©ji©r&X F e r m i as f o r

*

06

8#

Magnitudes o f th© Unclear Matrix RXersiertis* * *

09

9*

P o sitro n M .s a lo n and O rb ita l E lectro n Capture

33

10* Campari ©or): with Marshak* a Method of Approximo^ tio n ^ t «

36

* ** * * » ,

,

* «

» * * • • * • * * *

OEAFflH III* 0OMFAHISOK WlfE M TBSW tm fAL MS?A

1*

Introduction * , »

i •« * * • • « # * « * «

2*

RaB*

3*

P52* »

4.

K40........................................................ ... .....................

t . * # ♦ . * , ♦ . *


e th e s t a r t i n g p o in t fo r n i l th a t fo llo w s .

*•

M s if f ii a sa a ttta a i I t t h i s p o in t i t i t w ell to c o n sid e r th® form o f th®

i n t e r a c ti o n proposed by Fermi# a

to® c l e a r l y re c o g n ise s t h a t

m is t h© an o p e ra to r capable o f producing t r a n s i t i o n s ho*

tween th® p ro to n and twmtron' s t a t e s o f a nucleon#

I f one

repp®cants th e two p o s s ib le Isofcopi® spin s ta te s o f a miofc cm as n eu tro n s ta t e s

%~(o)+ p ro to n s ta te s f ^ U ( J m m may In clu d e

ln S £ th e 1e c to p ic s p in o r i d e n t i ty o p e ra to rs Q said Q which r e s p e c tiv e ly & m m

and

tr a n s itio n s #

/- \ s«t® & Q and Q ■*a© th e m a tric e s II °t

°\

dI

F eral ra p ro -

and. {( °o oJ )) re sp e c tiv e ly * *

f t o s a n e u tro n s t a t e i s u n a ffe c te d by Q b u t Qm = p # Sim ilarly# Q F - 8 a n d Q f^ O *

F o rth sra o re # i t i s necessary# in o rd e r to

p re se rv e th e r e l a t i v i s t i e In v ar lane© o f

to in tro d u c e heavy

p a r t i c l e o p e ra to rs (denoted by th e su p e rs c rip t# Ht ) s m lo g m s to th e M rs® m a trix o p erato rs#

(See Appendix A}#

Ferm i’ s

o r ig in a l i n t e r a c ti o n H am iltonian had th© f o r a of a four** dim ensional polar- v e c to r In v a ria n t*

i l

= f r f [ R. + ( « * £ ) & + lR ? + (£ ','d f k * }

(8)

where \P , FJ) 1* a #0ur*$oHqpon&nt lig h t partial© operator

sim ilar to th e ©Xeetrm&afpaetl© vector p o ten tia l of a phot mi f ie ld *

$b© magoibta&e of the Feral constant,

which t ekes

th e place o f th e electron charge in the analegeus e l eetron~ photon to tera ctio n , give® the magnitude o f th e in ter action between a m oleen m& the eX®etron*m#utrino fie ld *

th e

Hamiltonian# ( 9 ) # i s mte the m l j p ossib le form o f invariant in te r a c tio n that one might too###*

th e re are f i v e poeeihh

In tera c tio n s, not involving d erivatives o f th# electron and neutrino wav# fu n ction s, which transform resp ectiv ely lik e foi>r-~&to©nsion&l ■scalar#, polar rector# tensor* axial vector or p«eude*«galar invariants*

the e x p lic it expressions for

each mmj he found t o s e v e ra l somroes*4*^ fha m atrix elem ents, H , # appearing to f t ) w y he w ritten In to# geaerallaed form? / - / ‘ = (r fc * li J*[(oH- 0 ) ^ +10* 0 ) * Q*J Rj

(9)

H

O l a th e abbreviated n o ta tio n fo r a cover isnfc s e a le r , v e c to r , or m tto a p T O tr io second ranis ten so r heavy p a r t ic le operator w h ile 0 represent® th e corresponding m n tr m m ttm k l i g h t par* t i d e operator*

th e f i v e in v a ria n t# th a t may bo formed by

th e c o n tr a c tio n o f auto op erators are denoted by (O •0)x toer© and P r e f e r s to to e fo llo w in g formal S oalart

Z5’^

(10S)

?*lap V e 9 ta v t( 0 ^ 0 ) v = > -

(10P)

Tensor t

(O^O)r - ( P S f ’I f c ) + {P2)*/P£)

(10T)

A xial fa c to r?

fO* 0)H - 5 T '2 “ "" ?r

F®eu&o~s0aX«r« ( = ^ w # Mm used In h a n g in g fm m th e in te g r a tio n mm* m m m tm i to th e energy in te g r a tio n In d ic a te d in (15} * 1m th e n ext oh apt or i t w ill b© shown t h a t t o f i r s t .approximation th e f o o te r s entitles e& in th e brack ets# [- - - j , « « 7 l i t t l e to comparison to f > W f ,

One » « ? show

very e a s il y t h a t ^w^JiA/\'%§. m erely th e s t a t i s t i c a l p r o b a b ility t h a t m e le c tro n i s em itted w ith m energy between W and W 4 &W along w ith an an tl-n eu tx d n o of energy# K =% -W# by macing th© simple assum ption th a t the em ission p r o b a b ility $ of th e two p a r t i c l e s are j u s t p ro p o rtio n a l t o th e ir re s p e c tiv e TOltaesie © Icm nts in ©omentum apm& su b je c t only to th e conserva­ ti o n o f energy condition# E quation (15) may bo used to r both e le c tro n and p o s itro n ©mission; f o r th e former I H j i s g iv e n by (IX) and; fo r th e l a t t e r by (12)«

The fo rm u latio n fo r th e ca p tu re process

le a d s to the ©mission of n e u trin o s w ith an energy given by th® co n se rv atio n of energy c o n d itio n j namely t Wo -

*

- 6

(16)

***12 m '

i s th© energy d iffe re n c e bet*©on th# i n i t i a l and f i n a l nucleus and £ l a th© energy of e bound o r b ita l electron# In ste a d of th© inte.gr a t Ion over e le c tr o n en erg ies m in US) one m a t mm over th© d is e ro te s ta te s of the atomic electro n s* ff^ r ^

fhns in (15)

&# re p l seed by 2 1 - , where

th e summation I s over the t o t a l up ante© numbers# etc*# o f th e bound sta te s*

1*2*3,

fh& energy s ta te s arc tts u a llj

re fe rr e d to as th e K# L» Sf etc** s h e ila o f th e atm *

th e

m m l i f e of th e cap tu re process i s given by m expression S j3 lU 8 r t 0

{ 1 5 ) * n 0 S K ,ly '

.

t'T

^ /r , -

x m >

lo r e i H j i s given by In sertin g in (12) the hound electron wave functions# fu n ctio n ft*

^ n e g a t i v e *

V

energy wav©

-13-

CHAPSSB I I , !*

THti "PORBIDCgBn TT.'ANSITIOSS

5 » . m m r n fig "fforbldaen" Th© designations*. "allowed" and ^forbidden'*1 tra n sitio n ® ,

are perhaps most e a s ily explained in term s o f th e so -c a lle d Sargent law*®

Sargent proved* by p lo ttin g th e decay c o n sta n ts

o f th e n a tu ra l ra d io a c tiv e o l aman ts a g a in st th e maximum en­ ergy o f the e le c tro n s em itted* th a t th e h a lf - liv e s of elem ents o f comparable sia e could be roughly separated in to d i s t in c t o rd ers of magnitude*

th o se elem ents having the sm allest

h a l f - li v e s are sa id to undergo "allowed" tra n s itio n s *

The

" f ir s i for'bidden’9 group of alemoxts have h a lf- l f r es approx­ im ately one ban&r©d tim es g re a te r than, the "allowed" group* Elements having h a l f - li v e s of s t i l l la rg e r o rd e r of magnitude arc said to undergo "second forbidden1* tra n s itio n s # or even— fo r such elem ents as Hb*^ and K ^ — *th ird * or "fo u rth fo r bidden ** tra n s itio n s * In a d d itio n to th® f a c t th a t the observed r a te s a t which v ario u s m c lo l decay seem to f a l l in to d i s t i n c t orders of mag­ nitude# the accum ulation of d ata concerning b e ta ra d io a c tiv ity has in d ic a te d th a t a "forbidden** tr a n s itio n sometimes g iv e s r i s e to a d ls b rlb u tio n in energy of th© charged p article® (e le c tro n s or p o sitro n s) em itted considerably d if f e r e n t from the "allowed" energy spectrum* The '-problem of accounting fo r th e assymmetrie energy d is tr ib u tio n of e le c tro n s from nuefei ap

such a® HaE and P '* was re c e n tly tre a te d th e o r e tic a lly by Konopinskl and Uhleribeek#^

The method of approxim ation in -

**14*

vented by these authors In order to calcu late th® energy s p e c tra r e s u l t to g from th e so*called Bforblddcntt tra n sitio n s s h a ll ho disc&'sscd la d e t a i l to t h i s chapter* aad the designs* tlo n s of th e order of ^forbiddennese* w ill be more p recisely d e fin e d * B i g W a v e E m anation.®

la order to ca lcu la te the ex p ressio n ,

Y L Ih J ,

yxw k a p p e a rin g .to (IS) i t w ill: be necessary to l i s t th e ap p ro p riate electron and neutrino wave fu n c tio n s involved to th e tre n s i* tlon* {See e la tio n ® (11) and ( I t ) ) * solution*

V

the fear component

.

o f the Dlrae aquations o f ea electron of energy

W> 1 in th e coulomb f i e l d of the nucleus are conveniently sop* arafced in to two types# type at

..*'=-f+'/a, -£>yo

-{j+ /je= .r*

-,- a * " £ L % + , f -e -1 / - i

y ' - [ t r r U + * » + 0 H -t~ » 0 !\

(Z -n n -0 ? T f x H * - * .) ? T u

fN m « b .

j-J t-'A ,

^-.l-Jgrw “" /a

'Y - \ y n (J-f

-

i ft-j-i

0

The spherical harmonics used are those given by Darwin?VJ namely t

- 15 *

Th© r a d ia l fu n c tio n s , f m d g f depend cm th© energy of th© e&setron* it© p o s itio n coordinate,. r 9 and th e n u clea r charge* In th e f i n a l f©ratal as they w ill appear as h i lin e a r combinations of f y |

* T heir d e f in itio n s f o r th e p resen t

f (v+t; and

ease (w> 1} a re i

^ +/

ftziQjsk ^ ^>s,«c] [p %

w h e re

w

r ~

~aee lTf ' - ^r ln r s^^ + QH) t

(SO)

{ t5a)

and C>0|

jgf[a, b, X^J -

“ T

th© complex confident hyper*

geom etric se rie s* e * £ * '«' a S * - l' 'fr

u/

(2 3 )

Th© eon s te n t, S>> » m d y depond on the fin e s tru c tu re c o n sta n t, c\-i/i37# and til© atomic noasher, &* 5

—[^C V t 0

]

^ - o ( 2 U //jp

= £ q s - t x) ;

{24)

For th e a n ti-n e u trin o , o n e'in tro d u ces th e negative m * orgy so lu tio n of the D irac em iations of a eh arg eless p a r tic le of seero r e s t mass*

h -k ) 3,

cp = B e

*

,

B

^

(th)

b:

(th) b; 8

' (25)

V h)

(th )

She amplitudes, B * depend only on the p o s itiv e energy and momentum, K and &, of the anti-neutrino.

The superscript, h=+',+2

— 16—

r e fe r s to the two o rien tetlon a of the anti-neutrino spin* to e + sign in d ica tes m anti-neutrino ware function in con­ t r a s t to a * sign fo r a neutrino o f p o sitiv e energy* ®* &

& m

a a m sa g tt

to e square of th e m atrix element* (11), may he evaluated in su ccessiv e approxim ation by expanding th e wave fu n o tio is (IS ) m d {20) i n powere of the p o s itio n coordinates* #,*■* *-

and x3 * S tr ic t ly speaking th e se fu n c tio n s should be ev al­ u ated a t th e p o s itio n of th e transform ing nucleon*

Since

th e wave length o f to e e le c tro n and anti-neutrino wave func­ tio n s are la r g e in comparison w ith r which h m the magnitude -