Pseudoscalar Interaction in Nuclear Beta Decay Chander Perkash Bhalla University of Tennessee - Knoxville

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Masters Theses Graduate School

6-1960 Pseudoscalar Interaction in Nuclear Beta Decay Chander Perkash Bhalla University of Tennessee - Knoxville

Recommended Citation Bhalla, Chander Perkash, "Pseudoscalar Interaction in Nuclear Beta Decay. " Master's Thesis, University of Tennessee, 1960. https://trace.tennessee.edu/utk_gradthes/2950

This Thesis is brought to you for free and open access by the Graduate School at Trace: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of Trace: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a thesis written by Chander Perkash Bhalla entitled "Pseudoscalar Interaction in Nuclear Beta Decay." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, with a major in Physics. Dr. M.E. Rose, Major Professor We have read this thesis and recommend its acceptance: J.A. Cooley, Edward Harris, G.E. Albert Accepted for the Council: Dixie L. Thompson Vice Provost and Dean of the Graduate School (Original signatures are on file with official student records.) May 26, 1960

To the Graduate Council:

I submitting herewith a thesis written by Chander Perkas Bhalla entitled am"Pseudoscalar Interaction in Nuclear Beta Deea;y." I recommendh . that it be accepted in partial. tultillment or the requirements for the .; degree .Doctor or Philosophy, with a ajor i� Physics.

Major Professor We have read this thesis recommend its acceptance:and

·II /JJf�uJ-

Accepted for the Council: PSEUDOSCAIAR INTERACTIOM IN NUCLEAR BETA DECAY

A Dissertation

Presented to

the Graduate Council of

The University of Tennessee

In Partial Fulfillment

of the Requirements for the Degree

Doctor of Philosophy

Chander Perkash Bhalla

June 196<> ACKNOWlEDGEMENTS

E. The author is deep� indebted to Dr. M. Rose, who suggested this problem and underwhose guidance i� was accomplished. The many I stimulating discussions with him throughouI t this work contributed much to its success. The author acknowledges the kind considerations of the staff of the Physics department of the University of Tennessee, where he held a graduate assistantship for four years. In particular the author is grateful to Professor A. Ho Nielsen and Prof'essor J. w. White for arranging his teaching load and schedule, so that it was possible to spend considerable time throughout each week at the Oak Ridge National Laboratory. The author is also grateful to the administration of the Oak Ridge National Laboratory, who made their facilities available in the caupletion of this work. Considerable thanks are also due to Mrs. · Lee Pogue and Mrs. Beverly Cunningham, who typed this dissertation.

Finally, the author is grateful to Dr. T. A. Pond for making available a copy of the dissertation of Dr. w. A. Wo Mehlhop and to 66 1 • Dr. R. L. Graham for private c ommunications on Ho

457883 TABLE OF CONTENTS

CHAPTER PAGE I. INTRODUCTION 1 Experiments Indicating the Vector and the Axial Vector Interactions 1 Allowed and Forbidden Transitions 4 Experiments for Investigating the Pseudoscalar Interaction 14 Statement of the Problem 15

. . 17 Historical !ackground . . . I..

II. BETA DECAY INTERACTIONS IN NON-RElATIVISTIC FORM AND N

Notation and Representation • • • 38 Representation of Dirac Equation 39 Irreducible Tensors and Wigner-Eckart Theorem 42 III. FORMUlATION OF THE PROBLEM 45 First Order Time-Dependent Perturbation Theory 45 Beta Longitudinal Polarization in Nuclear Beta Decay 50 Polarization Operator for Electrons 51

Beta Longitudinal Polarization in 0 � 0 (yes) Transitions 54 iv

CHAP� PAGE The Two-Component Theory of Neutrin·o ...... 85 Formulasfor the Longitudinal Polarization f3 and the f3 Spectrum in 0 ..-,o {yes) Transitions Using the Validity of the Two Component Theory

of .Neutrino·· o •• o o o o • o o • o • 89 EXPERIMENTAL ON (YES) TRANSrriONS IV. DATA 0 -? 0 ITS ANALYSIS Wri'H THE DEVELOPED THEORY AND THE RESULTS 92 Experimental Data on 0-)0 {yes) Transitions 92 l Pr 44 Ndl44 ( i 93 � f3 166 166 o- Ho E r 1 ) • 0 • 96 � \P 97 97 98 Method of Computation 98 Finite Nuclear Size Effects 99 Finite DeBroglie Wavelength Effects 100

Longitudinal Polarization and Shape Factor

� 166 - Formulas for 144 ( 0 and Ho ( 0 0 +) Pr � 0 �· 1 � with Numerical Coefficients 101 Method of �lysis of and Results 103 De.ta

Nuclear Matrix Elements • o • • o o • 106 144 - Analysis of Data on Pr ( + ) o o o 0 -.. 0 108 v CHAPTER PAGE

tfLongitudinal Polarization 108 Pure Axial Vector Interaction . . . 108

Axial Vector and Pseudoscalar Interactions . .. . 108 - t3 Shape Fact or . . � . . . ., . . . . . 111

Pure A.· Interaction 0 .. 115 A and P Interactions 116 - Analysis of Data on ( 0 Ho166 � o+) 117

f3-Longitudinal Polarization • 0 0 • • 0 • • • • • • 117

Pure A Interaction 117 A and P Interactions 117 Results 121

V. SUMMARY AND CONCLUSIONS . . 122

. . 122 Main Points of the Procedure 122 Conclusions 125 Discussion 126

BIBLIOGRAPHY 128

APPENDIX A.. Clebsch-Gordon COefficients, Symmetry Relations of Racah Coefficients, and X-Coefficients 1 5 3

APPENDIX B. Neutrino Wave:function in Negative Energy State, General Nuclear Matrix Elements for A and P Interactions

and Matrix Elements of H _ for 0�0 (yes} Transitions 9 t3 13 vi

CHAPTERA PPENDIX Evaluation of Time Reversal Invariance PAGE C. 4, in Strong Interactions Time Reversal Invariance and in Weak Interactions o • • • o • o o o • • o • • o 170

APPENDIX D. Analytical Expressions for the Electronic Functions in the Polarization Formula, Up To an Order

of Nuclear Radius oo••••• ••••••• •••••••••• 175 LIST OF TABLES TABLE PAGE

I.. Nuclear Operators for Allowed (n = 0) and First Forbidden (n Transitions = 1) 12 IIo Angular Momentum and Parity Selection Rules for Allowed and First Forbidden Beta Transitions 13 III.. Experimental Data on Beta Longitudinal 144 Polarization of Pr + •••... • • • • • • • 0 (o-� o ) 95

144 "" Pr . ( """ Numerical Coefficients for Beta IV. 0 � 0 J,

Longitudinal Polarization and Shape Factor Formulas .. • o • o 104 l66 Numerical Coefficients for ;seta Vo Ho (0--+ o-f),

Longitudinal Polarization and Shape Factor rormulas o .. • • o 105 LIST OF FIGURES FIGURE PAGE l 4 1. Decay Scheme of Pr lt- of Porter and Day, Pbys• Revo 114,

1286 (1959) • • • ' 0 • 0 • 0 0 0 • 0 • • • • • • • • • • • • • 94 1 4 - 2 o Pr 4 (o � O+). Calculated �'Longitudinal Polarization in Units of ( - v/c ) versus Beta Momentum for Pure Axial

Vector Interaction. . o for Various Values of the Ratio

of the Nuclear Matrix Elements ( l.) • • o o • • • • • • • • • • 109 144 - C 3 o Pr ( 0 0 +) o Permissible Values of the Parameters p . A i J�r_, and for the - Longitudinal Polarization and the� Shape j/.� tl Factor Data of Mehlhop, and Porter et al. , Respectively • 110 144 - 4. Pr (o _, o+) o Calculated �- Longitudinal Polarization C in Units of v c) Versus�- Momentum for p 0.05 (� / iiJA =

for = -65, -150, -190, 90 and: 100 • • • • • • • • . • • • • 112 ·· 144 l - - . 5. Pr (o 0 +) °. Calculated� Longitudinal Polarization in Units� of v c) Versus Momentum for -0.05 ( _. f (f � = for = 150, 175, 190,- -110, -150 and -190 o A o • 113 6. Pr144 l(o -_, o+) o Calculated�- Longitudinal Polarization -35 in Units of (- v/c) Versus�- Momentum for • and Cp o.oo2, oooo , oooo6, o.ooa, -o.oo2, -o.ool. , o.oo6 - = 4 4 -

M:!Aand -o.ooa 0 0 0 0 0 0 • oooe ooeo•o •••••••• 11 4 ix FIGURE PAGE l66 7 o Decay Scheme of Ho of Cork et a1., Pbys·. Rev • .!!Q., 526 (1958) as modified by Strominger et a1., Revs.

Mod. Phys. �� 743 (1958) •••••• o •••• o o o ••118 l66 - 8. Ho (0 �o+). Calculated ��Longitudinal Polarization in Units of (- v/c) Versus �-Momentum for Pure Axial

Vector Interaction for 10, 5 and ••119 A. = 30, 130, -30, - 0 -130 6 C 16 - p · 9 Ho o Permissable Values of ·the Parameters . (0 � ct) . 1 J 7 5 _ �A and · for the f3 Longitudinal Polarization Datum jilo f ,. of Buhring, o Physik 155, o o o o • • o o o o • • Z , 566 (1959) 120 CHAPTER I

IN'l.'RODUCTION

The determination of the nature of beta interaction has been the subject of investigation for several years0 The experimental con- firmation of parity breakdown1 in nuclear beta dec� opened a new field of experimentation,a.nd a very clear understanding of the main interactions has emerged from the "post91 parity experiments. The experiments1 to be briefly described below (Section I), lead uniquely to the vector and the axial vector interactions; but have no bearing on the pseudo scalar interaction. In Section II, it is discussed why these experiments do not have aJJ:i' bearing on the pseudoscalar interaction; then the experiments, which can best determ.iiAe the existence, and hence, the contribution, of the pseudoscalar int(!r�S.·�tion are described (Section III) • It is the o this dissertation discuss the pseudoscalar interaction by purpose f to 2 formulating the theoretical expressions for these experiments and by' comparillg them vith the existing experimental data.

1 s. c. Wu, E. Ambler,. R. Hayward, D. D. Hoppes and R. P. Hudson, Phys . Rev. 105, 1413 (195 ). The hypothesis of nonconservation of .. parity � 7 T. N in decay. was origiD&lly suggested by D. Lee and c. . Yang, Phys. Rev. lo4, 254 (1956). 2we follow the formulation of the pseudoscalar interaction given by Mo Eo Rose and Ro K.:Osborn, Phys. Rev. �' 1315 (1954). 2

I . EXPERIMENTS niDICATING VECTOR AND AXIAL VECTOR INTERACTIONS (a) One consequence of the parity breakdown is that the 13 particles are longitudinally polarized in the nuclear beta decay . The longitudinal polarization of 13 particles has been measured in many cases of "allowed" 3 + v beta transitions, and the results for 13- and 13 are -v - and - respectively c c within an experimental error of 10�. Here is the ratio of the �c particle velocity and the vacuum velocity of light.

(b) The particle of spin -! and mass zero accompanying 13 emission is called an antineutrino, and for 13+ emission, it is called a neutrino . To explain the experimental polarization data, the vector and the axial vector interactions require the neutrino to be "left-handed"; whereas the scalar and the tensor interactions demand the neutrino to be a "rig�t- handed" particle . The left-handed and right-handed particles have negative and positive helicity respectively. The experimental obser- vation of the neutrino hellcity was made by Goldhaber, Grodzins and Sunyar.4 This experiment involves Eu152 {o-), which by K-capture goes

3c. s. Wu, Proceedings Rehovoth Conference on Nuclear Structure {North-Holland Publishing Company, Amsterdam 1958) -:;-P. 359; and J. Heintze, 150, 134 (1958). Zeits. fur Physik For a recent summary of t3 polarization measurements, see A. I. Galonsky, A. R. Brosi, B. Ketelle and H. B. Willard (to be submitted for publication in Nuclear Physics) . 4 M. 109, 1015 Goldhaber, L Gro.dzins, and A. w. Sunyar, Phys. Rev . (1958). This has been confirmed by I. Ma.rklund and L. A Page, Npclear 88 (1958). Physics 2, 1 2 to the excited state of Sm 5 (1-); which in turn decays by a dipole . + gamma transition to the ground state of 1� ( • By ob serving the sm 0 ) resonance scattering of these gamma rays in �03, only those 7-rays, which go in opposite direction to that of the neutrino, were considered. The 7-ray helicity is the same as the neutrino helicity. The 7-r� helicity was found to be negative and therefore the neutrino helicity is negative. (c ) Thus the experimental data on � longitudinal polarization in allowed transitions and the helicity of the neutrino lead to the vector and the axial vector interactions.. The relative sign and strength of the vector and axial vector interactions are fixed the nuclear � tran- by sitions, where these interactions interfereo The most informative and carefully analyzed case is that of a polarized neutron transforming into a proton with the emission of an electron and an antineutrino. Burgy et al5 measured the anisotropy of the electron the antineutrino with and respect to the spin direction of the neutrcin,.. 'fhe result of this experiment is that the relative sign of the ·vector the axial vector and coupling constant is negative.. Comparison of "ft-values" (cCllltparative half lives) of a neutron and 14 gives (1.21 as the ratio of the o ! 0.03) ab solute itudes of the axial vector and the vector coupling constants . magn The � interaction in the form of V- 1.2A law is consistent with· the other

5 T. Burgy, .. Krohn, T. B. Novey, R. Ringo and M. V E. G. V. L. Telegdi 1 Phys. Rev. 11 , 1214 ( 1 ) and Phys. Rev. Letters 1 8 • 0 958 .!_, 324 ( 95 ) 4 experiments on �allowed� beta transitions.* To understand why the ex-

periments on 8'a.llowedv' tran.si tions do not have an:y bearing on the pseudoscalar interaction, we give below the classification of the allowed and the forbidden beta transitions which is commonly used.

II. ALLOWED AND F'ORBIDDEN TRANSITIONS

The most general parity nonconserving interaction hamiltonian density for the nuc�ear � decay** is

� *A number of recent review &!"ticles appear in the literature. See references 7, 8, 9, 10, 11, 12. 7 Invited papers at the Conference �n Weak Interaction, Gatlinburg, Rev. MOd. Phys. 31, 782 (1959). 8 E. M. Rose, HandbC)ok of Physics (�Graw ...Hill Book Co., New York, 1958} 9-90o Po 9 5 n. L. Pursey, Proco Royal Soc� of London, Series A, 246, 444 (19 8). 10 o k 5 o Eo J Konopins i, A:rtm.ual R�v. Nuclear Science 2,, 14 (1959 ) 11M. Deutsch and Ko oed ...Hansen, erimental :Nuclear Physics f Exp ·· Oo New Y k Vol. III ( J©>hn Wiley and Sons Inc. 9 or , 1959) p. 427. (2), 12-r.. Smorodinskii9 Soviet Physics Uspekhi §1. No. l., 1 (1959) .. ** M .. E. Rose9 ref��n.ce 8o 5

The summation over x implies the five possible interactions. The first ter.m represents �- emission, and the hermitian conjugate of this gives the + ' � emissiono C and C are generally called the parity conserving and parity X X . nonconserving* coupling constants. � is a 4 x 4 matrix and in terms of . X the 7-matrices which obey the following commutation rules:

+ = 2 ( ,AA..and = 1, 2, 3, 4) 1M- 'Yv � 1v �..M-Jv }I

Let = Then 0 has the following forms for the respective inter- J'Lx - 7 4 0 . X X actions Scalar 1 (one component) (four camponents) Vector 1,M- I Tensor 7 ; (six components) � ., .-". "* �. Axial Vector 7 (four components) � 5 Pseudoscalar (one component 75 ) ** � Making use of 7 = -� (k = 1, 2, 3); 7 = -� ' we write {lola} as k ak 4

H= H + hermitian conjusateo (l.lb ) i= l X . where X

* It is the occurrence of the cross terms between. ex and ex. that give the parity noncanserving effectso

this representation: ; **rn � = (� -�) a �� ) 1 =(l . \o 6 H ) S = (\f�4'N ( lJ':f3 LCs+C�75) � )

c 0 Rv = <'f;'fN) clf: [ v+c;7 5 J � > - <'1-';ctLYN> (�a r cv+c�75 1 � > * * .... ' \..V* ... * � ' \JJ � o = N) ( f3 C'l'+C'f7 ) .+ ) 0 ( � CT+CT7 � ( 1 � .,.- \f lf'e 6" [ 5 J 'If, ( , � �N \fe (. 51 � ) * o * ' * * ' HA = (\f' e;t'fN) (Lf' ..... (CA+CA75) ) ... ( P75 N)( 7 lCA+CA75J p e � � \.f' \f 'fe 5 � ) ' ( * 7 N) *f3 7 C + Ji, � '¥� 5 'f' ( 4'e 5 l p �7 5) Y'v )

In equations ( 1 o 1), lf;and \f:represent the creation operators for a proton and electron respectively, whereas and are the de"" ·· an lpN � struction operators for a neutron and a neutri�� in the negative energy state o On.e can consider a neutron. and a proton as the two states of a nucleon, and we define an operator Q which transforms a neutron into a proton. Thus we can write the equation (lola) as

= Q o [C C l.lc � �4 ( :J\ X+ �75J � ) + ho Co ( ) X \fJ x

Strictly speaking, the setting up of the interaction hamiltonian density for th� four fermions . is a field· theory problem and one requires second quantization of the field amplitudes to insure the Pauli exclusion principle and to describe properly the creation and the destruction o� the particles o The usual field theoretic ..:� approach is to set up the first order perturbation theory fcrmu+a for the transition probability between the initial and the final nuclear stateso Then in this formul.a, \.f'e 'and � are treated as the proper Dirac wave tunctionso Using the relativistic units � = c = m8 = 1) the transition probability for �- emission between T the initial and the final nuclear states, represented as lfJi and �f' is given by the following:-

(1 ..2a ) is the density of final states and. is the stmliDation over f the S all un observed observables ..

In ( 1. 2b) beta interaction operator is written in the space of the kth

nucleon and the lepton covariant * ( ' uJ is be evaluated (lpe � Cx+Cx75J 'Til ) to at the position. � of' the k ... th nucleon.. The integration is over all nuclear co-ordinates and it is to be summed over all nucleons. The selection rules for the allowed and the first forbidden beta

transitions depend on the rotational properties of the interaction operators The in the nucleon space.. Coulomb effects on the � particle do not in- fluence the rotational_properties and as such, in the following discussion, I the electron and neutrino are represented-as Dirac plane waves. -- = e ip•r lf.e ue

(1.3b)

In (1.3), p and q are the (physical) momenta of' the electron and the anti...

� � � neutrino respectively. Letting P = p + q; substituting (1.3) in (1.2 ) 8

(1.2c)

rk is the position of the decaying nucleon and rk � nuclear radius (r ).

In the above we have introduced the conventional notation�

..... � A ( -iP•rk -iP•-; d d * k ( * e Q ')' ••• e Q( ) s :r-nx � 1 '("A If:r �(k) lf/1 Y' 4'1 k=1 J J

P is at the most as lar e a.s 10 in units , r = 0.02 for = 210; g ( me ) A �me

Prrv Pr therefore 0. 2. In general < < 1· u3ing the Raleigh expansion =£�.-r = < c e � �1 1� (1.4a.) e �-+ -iP·r <; )t = 4 ( -i e 1t L l -iP•r e = 4 (1.4b) ..,� 1t �M

In arriving at (1.4b), we have used

and �� : / ��> We write (1.4b) as 9

-iP•r _ M e < a � 5 b M � ( rJ (1. a) -L t. , ,e,M t. where

(1.5b)

5 Substituting (1. a) in equation (1.2c)

(1.2d)

Now is a tensor of a certain rank, depending upon the inter- -!lx action. For example, .J"tx = �� 1, .,5, �.,5 a.re zero rank tensors, because

·these are scalar undelj three dimensional rotation. �t us denote them by

T ) • Similarly when.Jl. == �;, G- , is a tensor of rank (� d, (3(1, ...fl. · 0 X X X and let us �epresent it by ). the nuclear matrix element one T1 {Jl.x In of equation (1.2d), we have

...{)_ ( ) M = 0 or = l. x = - ; L L "<( ; �- (Jlx) "cJ: (�)

= b T (...fl.. •. L ll l x' r) (l. 6) 2l b * where L l l. is a constant, which for our discussion is irrelevant: ·�Sub-

* b = C(L l.;,«.- 1 M, is a Clebsh-Qordon Coefficient. L t l. e M ,u..) 10

stituting (1.6)-in (1.2d) .we have

(1.2e ) ..fA. are the co onents of tensor rank where, in steps of unity, Tl. � l. (1.7) . Thus L, f. and l. must form a triangle : ·,,�� and we represent it as 6. (Lll.). This notation will be used wherever needed. To conserve angular momentum

in the nuclear matrix element of 1.2e , we must have J l.J • J Eq. ( ) 6. ( f i ) f and J are the spins of the final and the initial nucl ar states. · If we i � . represent the parity of as then = the Tl (�,�) 1C(Tl), n:fui n:(Tl). In Raleigh eXpansion, 1.5a , we get the leading when is zero. The I( . ) term� e higher the value ·of l , the smaller is the term in .the expansio�o Also from 1.1 we observe that the even operators in the Dirac sense Eq. ( ' ) ( 1 like �) appear on the left side and odd operators (in the Dirac sense, like a) appear on the right-hand side. As pointed out earlier, these 4 4 operators in the ·nuclear. space are x matrices. In the nuclear matrix elements the even operators connect the large component vith the large component and tl_le small component with the small components of the nuclear wave functions involved. On the other hand, the odd operators connect

. v the large and small components. The small component is of the order -c of the large component. As such the even operators with the leading term 11 of the Raleigh expansion (for t = 0) give maximum contributions to the transition probability. Such transitions, for whic� the selections are given by 6 (J�i) and • ui = u('f�), and which have the largest tran sition probability, are caflled allowed transitiona. Now we also note that for the pseudoscalar interaction, only an odd operator �75 (a zero rank tensor) is involved. Therefore, the max imum contribution of this interaction arises from the selection rules 1. u = - f !!!:.!. i i 6. ( Jf OJ i) and Now, i there no other interact ons n A 1) nuclear 13 decay, then this transition ( = 0; = - would be 6 J 6U called an allowed transition. But we know that there are other interactions (the vector and the axial vector) which have much larger contributions; therefore the transitions with these selection rules are called first forbidden. In the first forbidden beta transitions the contributions to the transition probability come from (1) The even operators with the term in the Raleigh expansion for

e = 1. (2) The odd operators with the term in for the Raleigh expansion

1, 2 The selection rules for the first forbidden are l AJ I = 0, 1. and = - Au In Table I we list the nuclear operators for the allowed (n = 0) and the first forbidden (n 1) transitions. In Table II, we explicitly = 12 I TABLE FORBIDD EN OPERATORS ( n = 0) FIRST (n = 1) NUCLEAR FOR ALLOWED TRANSITIONSAND

T T n .Jlx Interaction 7C(./LX ) l l 7C( :\.) Rank Even Operators � Scalar � 0 0 ... + t3r + 1 1 1 Vector + 1 0 -tr 1 + 10 6- fr'- t3 Tensor + t3 1 + � . � 0 6 ·r 0 ·1 �� x? 1 *� .1 ( 3 zr z-"' ...• r ... ) 2 1 t;-.. � Axial Vector � 1 - . + e-•r 0 + 10 ... - e; � 1 1 . * � .. (3�. z r - G"' ·r) 2 1 z Odd Operators � � Vector 1 1 (3(1a Tensor a� 1 1 75 Axial Vector 75 0 1 �75 Pseudoscalar � �15 0 "" 1

* One component of the tensor has been shown. 13 II TABLE ANGULAR PARITY SELECTION M>MENTUMF ANDT TRANSITIONSRULES FOR ALLOWED AND IRS FORBIDDEN

Interaction T A(JiJfA) A A7C Allowed: Scalar � 1 .O(JiJfO) Vector 1 1 (JiJfO) Tensor �� 1 6A (JiJr) Axial Vector .. 1 (JiJri) � A First Forbidden 1 Sc �; -1 A(Ji J f )· Vectoralar �r -1 A(JiJfl) ..... -1 �(JiJfl) Tensor � e-·��a r -1 A.(JiJfO) �� x1 -1 6(JiJfl)

- � • .. -1 ( ) * (3 cs-zr z <5' r ) 6 JiJf2 Axial Vector � 6-·r -1 b.(JiJfO) o-xr... � -1 (JiJfl) 4 � .. (J ) *. < 3 erz r z - C5"' •r ) iJr2 -1 6 ) Pseudo scalar �75 A(JiJfO

. *Only one component of the tensor is indicated. 14

give the selection rules for the allowed and the forbidden � transitions.

In general, the nth forbidden transition has the follow·ing selection rules

A J = n, n+l for n ) 1

III. EXPERIMENTS FUR INVESTIGATING THE PSEUOOSCALAR INTERACTION

Due to the parity selection rule, the experiments on the allowed beta transitions do not have any bearing on the existence of the pseudo scalar interaction. The operator �15 (in the nucleon space) is a zero rank tensor and bas odd parity. Therefore, the pseudoscalar interaction

= = - The pseudoacalar and axial contributes when ,6 J 0 and L::J.. 1t 1. vector interactions contribute to nuclear transitions with Ji = Jf = 0 and 1ti1tf = - 1. Generally this type of transition is written as 0 _., 0 {yes). Since the nuclear matrix elements ·are very bard to evaluate they are treated as parameters, which are adjusted to fit the experimental

, ' data. Though, in principle, the contribution of the pseudoscalar inter-

action can be determined also from transitions J = J 1t 1 - 1 ( i f � 0; i f � ) where the pseudosc alar1 axial. vector and vector interactions contribute, there are more unknown (nuclear matrix elements) parameters, thus making the analysis harder . The best cases far investigating the existence of the pseudoscalar interaction are 0 --) 0 (yes ) transitions because the vector interaction does not contribute at all. With the known negative hell-

city of the neutrino, the pseudoscalar and the axial vector interactions, 15 taken separately, give opposite longitudinal polarization of t3particles. The relevant experimental data on 0� 0 (yes) transitions are (1) The t3 spectrum (2) The longitudinal polarization of t3 particles. 166 At present, 0 -?0 (yes) transitions occur in the decay 13 of 144 , Ho , 1 2 206 Pr Ce144 , Eu 5 , and possible Tl \ ..

STA'l$MENT PROBLEM IV. OF THE The problem, considered in this dissertation, is to investigate the existence of the pseudoscalar interaetion in the interaction hamiltonian density for the processes of nuclear beta decay, by: (1) Formulation of the theoretical expressiqns for t3 longitudinal polar ization and the t3 spectrum in 0 0 (yes) transitions with the correct � for.m of the pseudoscalar interaction2 and the axial vector interaction. (2) Making an extensive numerical analysis of �he presently available �x- perimental data, using the derived formulas, with the calculated electronic

functions, which include accurately the nuclear finite size* and the effect

due to finite deBroglie wavelength.**

13 n. Str,ominger,. J. Hollander G. T. Seaborg, Rev. Pbys. 30, 585 (1958). M. and Mod.

* Rose, Phys. Rev. 82, 389 (1951); E. Rose and D. K Holmes, RidgeM. E.National Laboratory Report �o. 1022 M.(Unpublished). Oak ** M. E. Rose and P�rry, Phys. Rev. �' 479 (1953) c. L. ·16

The remainder of this chapter summarizes the history and the present status of the pseudoscalar interaction in nuclear � decay. In Chapter II find an· expression £or the interaction hamiltonian density, we by removing the odd operators with the Foldy-Wouthuysen transformation. llso the representation and notation used is discussed. Section of In I· Chapter III, time-dependent perturbation theory is outlined and the asymptotic wave function of the beta particle is given. a After brief discussion of the polarization operator i� Section IIA, the main problem of longitudinal polarization and � spectrum is set up for 0 � 6 (yes) transitions. After assuming time reversal invariance in weak and strong interactions and the two-component theory of neutrino as valid, we give the resulting formulasfo longitudinal polarization and spectrum r� f3 in (3.35) and (3.37) on pages 89 and 90 respectively. The relevant ex perimental data on 0 0 {yes) transitions are s ized in Section ummar --? II of Chapter Section III of Chapter IV, after a brief discussion IV. In of finite nuclear size and finite deBroglie wavelength corrections, the methods of analysis are described. and results are given. Chapter V con tains the conclusions of this investigation and a discussion of these conclusions. Appendix A, for clarity, the symmetry relations of In Clebsch-Gordon coefficients, Racah coefficients and X-coefficients are listed along with some "relations" which already exist in the literature.

Appendix B contains the neutrino wave function �n the negative energy state in (Section 1)1 and the general expression of � matrix elements for the 17 axial vector and the pseudoscalar interactions is worked out in detail ; in Section II In Section III. Appendix B the results of Section 2 are of specialized to 0 � 0 (yes} transitions. Appendix C shows the details of a certain Raca.h recoupling in Section 11 and a short discussion of time reversal invariance in strong and weak interactions in Sections 2 and 3 respectively. Appendix lists the expressions of certain functions D (introduced in polarization expression of Chapter III) up to order R (the nuclear radius).

V. HIS'IDRICAL BACKGROUND 19341 Fermi14 formulated a field theory of beta decay in close analogyIn wi th the field theory of electromagnetic radiation. He- considered only the vector interaction by taking the interac. tion hamiltonian density

* * * . as a scalarpr oduct of 4-vectors _ (lf'p74� �N) and (tye 741,M- � ). Soon it vas realized thattwo o ne could make other possible combinations15 in the interaction hamiltonian density, and they are called the ( i) scalar1

• In ( 2) tensor 1 ( 3) axial vector1 and ( 4) pseudoscalar interactions the

14:E. Fermi, Zeits. Physik 88, 161 (1934). fur * A 4-vector transorms under Lorentz transformation as do x, y 1 z 1 ict. l5H. A. Bethe and R. F. Bacher, Rev. M:>d. Phys. �� 189 (1936). 18

sett� up of the hamiltonian density, the following were assumed:

( 1 ) The interaction ha.miltoni an density is hermitian so as to

treat e - and e+ on the same footing.

(2 ) The beta interaction is direct, i.e., no derivatives of the

field amplitudes exist.

(3) The beta interaction is local, i.e., the field amplitudes

are taken at the same space-time point. (4 ) The c�assical beta hamiltonian was, a priori, considered a 1 scalar. The experiment· proved it otherwise.

In eq�tion (1.1) setting b' equal to zero gives the classical X beta interaction hamiltonian density.

In 1941 the theory of forbidden beta transitions was given by 6 1 ,l Konopinski and Uhlenbeck1 and extended l&ter by others. 7 B The ex-

perimental data on � spectrum, half-life and electron-neutrino corre-

lation were compared with the theory to determine the nature of �

interaction. The energy dependence of the � spectrum in allowed tran-

sitions indicated that there is little or no interference between the

vector and the scalar interactioRs nor between the axial vector and the tensor interactions. · The ·above statement is generally eJq>ressed as

16 E. J. Konopinski and G. E. Uhlenbeck, Phys. Rev. 60, 3o8 (1941). 1 5 42 7E. Greuling, Phys. Rev. 61, 68 (19 ). 1 42 , 51). 8n. L. Pursey, Phil. Mag. 1193 (19 19 19 20 that the Fierz interference terms are al.Jr¥:)st absent. As late as·

September 1957, the scalar and the tensor intera.ctions were considered 21 as the main interactions mostly due to erroneous results of Rustad and 22 Ruby. . 2 23 24 25 From 1952 through 1957, several authors ' ' ' expressed doUbts 16 18 about the correctness of the conventional treatment ' of the pseudo-

scalar interaction. In the conventional treatment the parameters

describing the lepton field were considered as independent of those

describing the nucleon. In the "new" formulation of the pseudoscalar

interaction a gradient operator appears, which operates on the lepton co- 2 variant.* As an illustration Rose and Osborn applied the formula for the �

spectrum in 0 0 {yes) transitions. using the pseudoscalar and the tensor �

19 M. Fierz, Zeits. fur Physik, ·lo4, 553 (1937). 20 J. B. Gerhart, Phys. Rev. 109 897 (1958): gives b = 0.00 ! 0.12. , F R. Sherr and R.+H. Miller, Phys. Rev. 93, 1076 ( 1954) : gives

b = - 0.01 - 0.02. G.T. 21 :. J. Konopinski, Proceedings Rehovoth Conference on Nuclear Structure (North-Holland Publishing Company, Amsterdam 1958)p. 318 2 �. M. Rustad and. S. L. Ruby, Phys. Rev. §2, 880 ( 1952). 23 T. Ahrens, E. Feenberg, and H. Primakoff, Phys. Rev. 87, 663 (1952). 24 T. Ahrens, Phys. Rev. 2Q1 974 (1953). 25 G. Alaga and B. Jaksic, Glansik Mlt-Fiz. i Astr. Tom. 12, No 1-2 (1957).

*R eference 2 contains an excellent discussion. See also Chapter II of this dissertation. 20 interactions to but t turned out later tha.t in (Bi210 the RaE$ 1 RaE ) 6 beta trans!tion was 1- Laubitz 2 and Zyrianova27 made a detailed � o+. 44 analysis of {yes ) in Pr1 using the Rose and Osborn formula for 0 � 0 the spectrum. Al.aga and aksic25 applied essentially the Rose and � J Osborn formulation with some extra parameters describing the nuclear 166 forces effect, to the analysis of spectrum of 0 0 (yes) in o • � 4 H Alaga, Sips and Tadic28 · also consider a tensor and the pseudoscalar 144 interaction for the lysis of spec of • At present, our � � trum Pr knowledge of nuclear ham11tonian; is not adequate to calculate the nuclear matri elements, and in the usual treatment of decay, these nuclear x � matrix elements are considered as parameters. Alaga and Jaksic25· intro- duce more parameters depending upon the nuclear forces. Now. if one did know how to calculate the nuclear matrix elements , with some confidence, then it might be interesting to see how many other parameters (depending on the nuclear forces) are required to fit the experimental data. . . But with our present knowledge of nuclear forces, it is neither practical nor desirable to complicate the theoretical calculations with such parameters. . 2 6 J. Laubitz, Proc. Pbys. Soco (London) A69, 789 (1956). M. . 27 L. N. Zyrianova, Bull. Acad. U.S.S.R.--Pbysical Series 20, 1280 ( 1956). (Translated by Columbia Technical Translation, New York). k II, 28 . Alaga, L. Sips, D. Tadic, Glansi Ma.t-Fiz. 1 Astr. Ser. 12, 207 (1G957). 21 After the experimental verifieation1 of the breakdown of parity symmetry law in beta decay, a number of experiments on the longitudinal polarization of particles, the anisotropy of particles from oriented � � nuclei, and circularly polarized correlation were done. the �-7 ( ) . In meantime Lee and Yang ,29 �0 and also Landau31 gave independently what is termed as the two-component theory of neutrino. �so �alloWing different approaches, Sudarshan and Marshak 32 Feynman and Gell-Mann33 , 4 and also Sakurai3 proposed the vector and the vector theory of axial � decay. As pointed out earlier in this chapter, the experiments on the � longitudinal polarization in allowed transiti ona , the experimental de termination of the neutrino-helici and the anisotropy of e- and from ty v

the polarized neutron, uniquely indicate the vector and the axial vector interactions. These interactions-are consistent with the electron-neutrino

29T. D. Lee and C. N. Yang , Phys. Rev. 105, 1671 (1957). 30 A. Salam, Nuovo Cimento _2, 299 · (1957). 3 lx,. Landau, Nuclear Physics 21 127 (1957). 3 . Phys. Rev. � E. Marshak and E. c. G. Sudarshan, 109, 1860 (1958). 33 R. P. Feynman and l Pbys. Rev. 1 9, 193 (1958). M. Ge 1-Mmu, 0 34 J. J. Sakurai, Nuovo Cimento 1, 649 ( 1958) • 22 ,3 , 19 correlation experiments35 6 37 on He6, Ne and A35. ce of the above development le�ding to t�e of As a conse��e� V-AA law beta the .pi-eVious estimates of the pseudoscalar interaction based . . ,, decay ' on the of a mixture of the .tensor and the pseudoscalar interaction� � -. analy�isI • 'I ' � 144 are no longer· correct. �ecently the· ·l>eta of Pr ( - has spectrum 0 � .o+) been studied experimentally.38,39,40 Gr� et a138 set up an upper limit 2 for �he pseudoscalar interaction using the Rose and. Osborn formula with the axial vector and the pseudoscalar mixture . To investigate the existence of the pseudoscalar interaction one must consider all the experimental data in particular � transition, and the best transition as pointed out earlier, any ' . is 0 4o {yes) . Apart from_the � spectrum, we have additional information 42 ab out the � longitudinal polarization. A number of treatments. 4 1 ' ' 43 of

35 B. B , P. Stab.e in, J. Allen and W. Herma.nnsfeldt, R. L. urman l s. T. H. Bird, Phys. Rev. Letters!, 61 (1958) and J. S. Allen, Rev·. Mod. Phys. 31, 791 (1959). Also see F. Pleasanton, C. H. Johnson and A. H.

l. Phy . Soc. 78 (19 9). Snell,:Bul Am. s �� 5 36 J. B. Gerhart, Phys . Rev. !Q2, 897 (1958). 3 7 W. B. Her sfeldt, J. S. Allen and P. Stahlein, Phys• Rev. mann 121, 6�r· < 1957) • B 3 R. L. Graham, J. S. Geiger, and T. Eastwood, Can . J. Ph.ys. 36, la34 (1958) . E. 39F. T. Porter and P. P. Day, Phys. Rev. 114, 1286 (1959).

4�. J. Freeman, Proc. Phys . Soc. 1l1 600 (1959) . 41c;. E. Lee-Whiting, Can. J. Phys. 36, 1199 (1958) . ani and Ross, Prog . Theor . Phys . 20, 643 (1958). 42T. Kot M. B. B. Ioffe, A. P. Rudik, and A. Ter 43v. Berestetsky, L._ K. Martirosyan, Nucle� Phys. 21 464 (1958). 23 the � longitudinal polarization in first forbidden.transitions appear in

the literature. In all of these treatments, the "conventional" form of the pseudoscalar interaction has been used instead of the correct formu- 44 lation of the pseudoscalar interaction. Geshkenbein gives the longi- ( tudinal polarization in 0 � 0 yes) transition, still not using the 4 . correct form of the pseudoscalar interaction. T.adic 5 has analyzed the

earlier less accurate (2�) measurement of the � longitudinal polari 44 zation in 0 0 (yes ) of Pr1 • His treatment, though it introduces 46 � parameters depending on the nuclear forces, is not rigorous and his analysis is inadequate because of the approximations used therein. Re 4 cently BUhring 7 measured the longitudinal polarization of � particles in 166 the 0-� 0 {yes) transition of Ho • His analysis of the longitudinal polarization measurement is not correct because he uses the for� of 48 Lee-Whiting. Cuperman has analyzed his measurement of the longitudinal polar - 207 ization of � particles in the (!+ � ! ) transition of Tl. . Apart from

44 • . 4 4 ( B. v. Geshkenbein, Zhur Eksptl' i Teoret Fiz. 3 , 13 9 1958). 4 5 ( M. n. T.adic Private communication to Dr. E. Rose ). 46 J. S . Geiger, G. T. R. L.· Gr and R. Mackenzie, Phys. Rev. 112, 1684 (1958). Ewan, aham D. 47 BUh ( w. ring, z. Phys . 155, 566 1959)·. 48 s. Cupernan (to be published in Phys. Rev. ) 24 the fact that there are more parameters occurring, it appears that the correct formulation of the pseudoscalar interaction has not been used. Recently Mehlhop49 reported the measurement on the longitudinal polari-

- + 144 zation in the 0 �ransition of Pr • In his analysis, very crude -> 0 approximation_s were used, along with the incorrect formulas of Lee Whiting.41 The effects of the finite deBroglie wavelength50 and the l, 2 finite size of the nucleus5 5 are important in 0 �0 (yes) beta transitions, and have not been properly considered. Thus, until now, no consistent treatment far the search of the pseudoscalar interaction existed in which the correct formulation of the pseudoscalar interaction was used. This dissertation presents such a treatment in which all the relevant experimental data are analyzed with . large scale computing programs using the accurate electronic functions.

49w. Mehlhop, �asurement of the Longitudinal Polarization of Prl44 BetaA. Partiw. cles" (unpub"A lished Ph.D. dissertation. Washington University, Saint Louis 1959).

5°M. Rose and C. Perry, Phys . Rev. 479 (19 3). E. L. � 5 51M. E. Rose, Phys. Rev . 82, 389 (1951) . 52 M. E. Rose and D. K. Holmes, Oak Ridge National Laboratory Report No. 1022 (unpublished) . CHAPTER II

BETA DECAY INTERACTION IN NONRELATIVISTIC FORM AND NOTATION USED

In this chapter, we start with a brief discussion of the diffi- culties which arise in obtaining the nonrelativistic limit of the nuclear matrix elements involving odd operators. The prescriptions of removing

the odd operators in the hamiltonian by the Foldy-Woutbuy�en;l,(can'dtii!cal) transformation for a free Dirac particle and for nuclear beta decay are given in Section I and II respectively. Section III contains the appli- cation of the results of Section II to the axial vector, vector and pseudoscalar interactions and it is explained, why the conventional treatment of the pseudoscalar interaction is not correct. For clarity, the notation and the representation, used in later chapters, is explained in Section IV .

the relativistic dynamdcs of a nucleon in a nucleus are not presently known and a.s such we are ignorant ab out the details of relativistic nucleon wavefunctions . In nuclear beta decay, the transition probability between the initial and final nuclear states depends upon the matrix (4 1 elements of operators x 4 matrices) in nucleon' s space . Due to the ab ove reason, these nuclear matrix elements are very �d to evaluate. . . However, there are some\ nucleap rodels like? the shell model, the Wigner model, the optical model and the unified model, which have some success in explaining some qualitative properties of nuclear stru9ture. These 26

models are nonrelativistic in nature and it appears that the relativistic corrections are not very important . Thus , at least there is a possibility of evaluating the nuclear matrix elements, provided the nonrelativistic limits of these matrix elements are known . We represent the nuclear wave function

where v and u have two components. In the nonrelativistic limit:

then v and u are called the small and the large components of 'f' . In the following the subscripts i and f refer to the initial and final nuclear states. Consider a matrix element of an even operator in nucleon space,

• ) e.g. , � L( a- Q (in the axial vector interaction)

L(�) ::. ('f:� ( CA+C�75] � )

L( }Q u et}u + v ) v 2.la )LJ'; � • iT yJ1 = S ; 0!- ·L( 1 S ; .5' ·L( � 1 ( ) ....lJo In the above equation, u' s and v' s are two component functions and es- is a Pauli matrix (in the nucleon space) on the right-hand side of (2.la) . Also the first and second term in ( 2 .la) involve only the large and small components "of the nuclear . wavefunctions . To obtain the nonrelativistic limit, we can neglect the second term as compared to the first one 27

(2 .lb) In the pseudoscalar interaction, we are interested in the matrix elements an � �7 �1 . of odd operator (which anticommutes with ) 5L( 5 )

L( ) L ) ( )v ( ) l'lr5 I'Ir5 lf'1 = v (l'l 5 u1 u L l'lr5 1 2.2a Y'; ; r -J ; whereS we have used the representatiS on 1 1 � = o ; 75 = o (2.2b ) ( 0 -1 ) (1 0 ) �d * lepton covariant L(�7 ) �1 ( Cp+Cp7 ] � ) (2.2a), both 5 ::. (\fe 5 ' 5 \l) • thethe terms involve the large component and the small componentsIn of the nuclear wavefunctions, as such these terms are of the same order of mag- nitude. There is no simple prescription, as in the case of even operators, to obtain the nonrelativistic limit. There aretw o methods, which, in principle, canbe utilized for obtaining the nonrelativistic limit of the matrix elements of an odd operator in the nuc.leon space: (1) By eliminating the small components by making use of the re- lation between the small and the large components of the nuclearwave function. As an illustration: if we take the wave equation for stationary states of a nucleon ( in units 11 = me = c = 1) as

v (W-V+M -1 ...... 1 � .... = ) M pu (2.2c ) : ( ) ; then v - •pu � - 2 <5"" • � u 6'" 28 - M 2M if we take W - V <� Substituting (2.2c) in (2.2a) gives

(2.2d)

In (2.2d) 1 operates only on L(�15 ) and if L(�15) is considered as a o, constant (as done in the conventional theory); then (� ) = · and 6 ·P L 15 hence there is no contributionfram tre pseudoscalar interaction. The above procedure may not be quite correct when the f'�elds are present 8.nd is very cumbersome in many body problems. ( 2) By applying a Foldy-Wouthuysen (canonical) transformation to the total hamiltonian of the system compri&ed of the decaying nucleon, the lepton ( e- V ) field and the lepton, so as to remove the odd operators . The odd operators can be eliminated to any order in 1 This proceci,:ure up . M is theoretically more sound and we shall follow this prescription.

1 2 I. FOLDY-WOUTHUYSEN TRANSFORMATION (NOFIELDS ) 1

The equation of motion of a Dirac particle of mass M, with no

�. L. Foldy and s. A. Wouthuysen, Phys . Rev. 78, 29 (1950) . 2 E. Rose, Relativistic Electron Theory (to be published by M. 8 John Wiley and Sons, New York) Sections 1 and 22. 29 fields present is

I \ '() H =i � (2.3) � t 0 3 where in the standard notation {using� = c = 1)

�� H = - - a p ( . �M • 2 4) In the equation ( 2 . 4) d is an odd operator and we want to eliminate, for

_., � 1 the time being, a•p terms up to an order • Consider a unitary trans- aS M formation generated by S and � = 0 s �· = e p (2.5a) then the equation of motion (2 .3) can be written as

H' �· (2.5b) C:>t where the new hamiltonian H' is given by

H' = e S H e -S

3 t. I. Schiff, Quantum Mechanics (McGraw-Hill Book Company, New York 1955) Sec . 43 . 30 2 s s2 1 • • 1 H' = + S + + . H - S + ! - . . . 2! ] [ 2 J 1 (S, ••• H' = [H + H) + 2(s, (S, H)) + (2.6a) where the commutator of S and H is written as (S, H): SH - HS We choose

-- .... 2 . • where o1 - is the odd part in the hamiltonian ( 4) We evaluate � a ·P H' in equation (2.6a) up to order � (2.6b) Using

(2.7a)

(2. 7b) Substituting (2.7) in (2.6b)

... 2 H ' 13 13 .. 2 • • • = 0 p H - 1 - M + 2M P + 31

-"> � - 1 �2 H� = - f3M - �·P Cl•p - p + .... 2M f3 + • • • - 13M - 13 �2p (2. 6c H' • • • = 2M + } We get the correct nonrelativistic hamiltonian by sUbstituting - 1 2 13 � 2 in ( 2.6c as M. reroove odd operators in to order } � + To H' up ( �) , this transformation is applied taking S : dd �t in By - �M (o H' }. successive applidation of the Foldy-Wouthuysen transformation, odd operators in the hamiltonian can be removed to any order in For 2 ( �). large M {say for a nucleon), the terms containing 1 or of higher orders ( M ) are very small . A similar prescription can be applied when the fields are present by considering

S = - �M ( o1 + odd operator involving fields J In the following, this procedure is applied to remove the odd operators in nucleon space for a system of nucleon "'source" coupled to the lepton (e- JJ}fi eld .

4 5 II . FOLDY- WOUTHUYSEN TRANSFORMATION FOR NUCLEAR BETA DECAY '

The total. hamiltonain is then composed of three the parts ( 1) nuclear hamiltonian (2) f3 interaction hamiltonian (3) the (�), (\), and lepton hamiltoni�.

4 M. E. Rose and R. K. Osborn, Phys . Rev. 93 , 1315 (1954). 5a. Alaga and B. Jaksic, Glasnik Mat-Fiz. i Astr . Tom 12; No . 1-2, (1957). 32 H=�+ Ht (2.8a) . \+. In the space of the decaying nucleon, considering the nucleon obeying the Dirac equation, we have M -.... a• -- V = - f3 n '?' p � [ ('+ + \ _ J + . . . - f3M - ...a• ..... - � M M V = p ( n - p ) tyz (2 .8b) HN + In the above Mn and are the neutron mass and proton mass. The ration M M M P n + alized relativistic units = c m = are used. , (1i = e 1ec tron 1) 2 M� P and V represents the nuclear potential. In the following, f3- emission is considered.

�- = g N 2.J\cX ( }Q•L{flx}

(2.8d)

.fL in the nucleon space, for clarity is writtenas .11.. (N). f3+ emission X X is obtained by hermitian conjugating �- in (2.8c ) . We write (2.8b ) and (2.8c} as

� = B:N{even) + o1 (2.9a) {even Hf3- = \- ) + o2 (2.gb) where the odd part of - ...a•p ... Byq: o1 a. and the odd partof is o • �- 2 {even) and -{even in {2 .9) represent the even parts of HN and HN \ } Hf3-· 33 Taking S - ( ) (2.1 a) = �M 01+02 0 •p o s = - �2M (- a + 2 ) (2.10b) , the calculation of H', the transformed hamiltonian, can be easily done by using :

H' = H + (S, H) + �(S, (S, H)) + . . . . (2.lla) H' H ' (2.llb ) = HN+ � + To remove odd operators i� (2.11) up to order in a consistent. er, � mann the following are used: (1) The terms containing. � or higher orders are neglected. (2) The terms of second order in the coupling constants are ignored. The term )� '( does not contribute (3 ) - -l(Mn -M p z to a:. J:S

I (4) There is no contribution to � arising from Ht .

( 5 ) The odd part of the nuclear force operator does not contribute . 6_ , 7· �king use of the above :

- 1 (- �.....·p � � - �M - - �p ) (s, HN) = 2M + 02' a· + ve

� .. � .... 2 � -'l ...... , -t ( ) = a•p - 02 - 2 ( 02a·p a·p0 (2.12a) s, HN 2M P + + 2 ) M 1 In the commutator ( S, ( S, . HN) we take up to order M

6 Chraplyvy Rev. z. V. , Phys . 91, 388 (1953 ) . Cbrap yvy, Phys . Rev. 92 1 1 (1953 ). 7z. v. l , 3 0 � ..:t ... ..,.,. ( -a•p ....., - o �n - -(� o a·p + a·pO ) (2 .12b) s, 2 ) - �M M 2 2

(S, �- (even)) is an odd operator because S is odd operator and it is of order The only terms from this commutator are of the � . contribut ing order and so are neglected . (1M) 2

1 .. � ) = ( �·p ) (s, 02 - 2M - + �02' 02

) = ( , ) ( o , ) (s, 02 �M �·p 02 - �M � 2 02

(a.12c)

Substituting (2 .12 ) in (2 . 11 ) ,

+ (o .p ) + t erms or higher �M 2, (i + f"�) 2 H = �M + v - P + + (2 .llc) ' L- e �M J [\ �M 2 + J Ht.. ...,2 ' M � ( 2 .13a ) - � + e - 2M HN : V P

(even (o , (2 .13b ) H� : � � + �M 2 �·p) + (2 .13c ) HL =:. HL ...... � _., ..... The anticommutator ( o , a•p ) o a.·p + a.po and o : odd part of the 2 ...... : 2 2 2 + Using (2 .13b) the odd operators in the vector, axial vector, and H�-· the pseudoscalar interactions will now be transformed into even operators . 35 III. BETA-DECAY OPERA'IDRS IN THE NONRELATIVISTIC FORM

For �- emission, the interaction hamiltonian for the vector, axial

vector and pseudoscalar interactions is �- = HV + HA + Hp lQ•L l - �·L(a) (2.14a ) lly = ( ) H Q•L( ; ) - 1 Q·L(1 ) (2.14b ) A = � 5 5 (2. 14c )

In the

following, (as usually is done ) , we suppress the operator Q which con verts a neutron into a proton. (2.13b) gives

' � ...:;. � H - = H - even (o , · p ) � � ( ) + 2M 2 a +

For the vector interaction 02 = - a·L(o)

< - L , a.p ) = - 2L( • -p . L i L( > c;. + L a) I> + + �. p x d J

(- 1 1(7 ) , = - 2L(1 )"cr •pL(1 ) 5 5 a.p)+ l 5 •p + 6 5 J

( - 1 L( 1 ), - 2L( 1 ) ·pL( 1 ) (2 .15b) �M 5 5 a•p)+ = �M [ 5 6 •p + 5 5 ]

For the pseudoscalar interaction: 02 = �15L(�75 ) ( .15c) (�75L(�75),a•p ) = - L(�75) 2 �M + �M a-.p d 5 Up to order ( �) 1 .� is from (2.14) an (2 .1 )

I I H I I �- = H,j + A + lip where

(2 .16a)