Theoretical Reinvestigation of the β-spectrum of RaE

Masami YAMADA The Sspectrum of RaE is reinvestigated taking into account the finite de Broglie wave length effect pointed out by Rose et al. 'and the correction to' the nuclear matrix elements pointed out by the author, and it is shown that the conclusion of Petschek and Marshak that only the assumption of tensor + pseudoscalar, spin chan/Si 0-0, parity change yes is able to explain the experimental ,results is incorrect. Taking into account the above two effects, many other assumptions can explain the experiments. The results are as follows: TPo-o (tensor+pseudoscdar and spin change o-o), STI-O and VAt-O fit the experiments only with the finite de Broglie wave length effect, Ao-o (including APO-O) and VTI-O requires both elects, but VT1-'0 seems too artificial. Other cases can not fit the experiments_ It has been well known long since that the p-spectrum of RaE deviates from the allowed shape, but there had been no definite theory to explain it. By the Mayer's nuclear shell model it is made clear that this transition is parity change yes, and last year Petschek and Marshak!) concluded that only the mixture of tensor and pseudoscalar interaction and spin change 0-0 (of course parity change yes) is able to explain the experiments. Some time after, however, Ahrens, Feenberg and Primakoff2) insisted using their method:!) to evaluate the ratios between nuclear matrix elements that the nuclear matrix element l1r;; in pseudoscalar type is very small and the large ratio of the coupling constantslGI'/GII ~133 is required to fit the Petschek's and Marshak's interpretation. However, this large ratio causes many awkward affairs, * and recently Ruderman6) has shown that if the nuclear force includes a large pro type one the relation of Ahrens, Feenberg and PrimakofPl becomes invalid and the magnitude of J9rs is of the order of other momentum type nuclear matrix elements (\ a, I pa and frs). According to his theory it is sufficient to take the coupling constant Gp to the same order as G7, to fit the result of Petschek and Marshak!) On the other hand, if the 'nuclear force is entirely due to the pseudovector coupling of 1r-mesons, the evaluation of Ahrens, Feenberg and Primakoff2) is valid, and a large pseudoscalar coupling constant is required. Therefore, if the foundation of the article of Petschek and Marshak were reliable and if it becomes clear that a - Introduction and summary Theoretical Reinvestigation i f the {1-spectrum i f RaE Calculation I Lo= (1 +s) /2-uZp/(2s+1). {(2s+3) W+s/ W} +';/ (2s+ 1)2. (_2p2_Sp2+7u.2Z2+8sU2Z2+ 8u.2Z 2Y+4su.2Zp2) , -aZp/ {(2s+ 1)2(s+ I)} . (-6p2_5sp2+ 3u.2Z 2+4su.2Z W2+ 8u.2Z 2p2) ,(2) Of course the mixture of both Ps and Pv couplings can be admitted. _ p21 {(2s+ 1)2 W} . (p2 + 2Sp2- 81j,2Z2_7su.2Z2_9u.2Z2p2_ 8s0.222p2 N o-=-saZ/(2pW) +U.2Z 2 + p/ {(2s+ 1)2W} . (aZp2+saZp~-3u.3Z3_4su.3Z3W2_4u.3Z3 p2). Gil ~pJ'12 {(1/3) K2LO+ (2/3)KNo+Mo+ 2L1 } + G~'[I ipa l 2Lo+ Iipa X rl2 {(1/6)K2Lo- (2/3) KNo+ Mo+ (1/2) L j } - {\fa*it9ax-t'+c.c.} {(1/3)KLo,..-No}] Ik(r)/rk- 1 = -=---,-,-~~:-:---tk' g_k(t')/rk - 1 Ik(P) I pH g -k (p) I pH I-k(r) Irk _ gk(r) Irk -m". l-k(p)1pk gk(P) I pk Theoretical ReimJestigation 0/ the {i-spectrum 0/ RaE The correction factor including C. N. M. E. is + ZL]I f (3'1'12) 12] + G~[Lo II (3at]) 12 + (1/6) K2Lo IJ(3cr X 'rl]) 12 - (z/3)KNol (3cr X '1"/1) *i (3cr X 'I'm]) + lVIoI i (3cr X 'l'm1) 12 - (Z/3) KLof (pall) *i (3cr X '1'0.) + ZNoi (3al1) *i (3cr X 'I'm]) ] + zGsGz[Ll i ((3'1"12) *. if ({icr X 1'12 ) M.,l (3'l"ml) *. i i (pcr X 'I"ml) + U/3)KNo{ J(31'1111) ~ ii (3cr X r'll ) - i (3'1'11) *.i! (3cr X r'Inl )} - (1/3) KLo I (3rl[) * .i i (3al]) - N O\(3f'1n1) *i i (fal])]. + U/Z)Lll i (3cr X r'/2) 12 Then expression (7) becomes + 3Noi (3al]) *i (pcr X rml) + (1/4 )K2Lol i(pcr X r'll) 12 We put - (I/Z) KNoi (3cr x'f'll) *I (3cr X 'I'm!) + (9/4) Mol f (pcr X "'ml ) 12]. (9) C~'X~-~ Kxy (p2+0.2Z2). as (13 _ _-9-c0.--3c_Z3 { ( Zs + 3) W + ~ }. Z(1 +S)3(ZS+ l)p , W Theoretical Reinvestigation 0/ the {i-spectrum rf RaE GAf(a x 'I'li) =2iGv~ ('1"1,), GA~ (a x 'I'm;) =2iG v ~ ('f'111i) ' and corresponding to (10) we put To cancel the large energy independent term we put then using ( 1) "" (~) the correction factor becomes X=2/(1 +s) (aZ/2p) +x, X' =2/ (1 +s) . (aZ/2p) +x', +~2[x'2_ 23 Kx'y' p~+(J.2Z2 X'+(~K2+ A p2 )y'2 W 6 18 4K 3(1 +s) (2s+ 1) (P2+(J.2Z 2)] +~[~xx'_2sK (x '+:1;"' )+slAp2 , _ B s W 3 W Y - Y 9 urtY (1 + s) 2 (2s + 1r where A was used by Davidsonl6l : Theoretical Reinvestigation of the f3-spectrum of RaE other as in the preceding three cases. Therefore, to get the desired correction factor we have to take an unnatural ratio of the nuclear matrix elements. Neglecting the scalar type and putting To remove the large energy independent term we put then the correction factor becomes X=-----+a Z.:I:, 2 l+s 2p 4 %y+ (1+s) (2s+1) p2+u.2Z 2 + 1 KO IV % "9 -y ----,-4--K----:- -.-/-}-2/+'--a-2'-Z-2--)' + 3 (1 + s)(2s + 1) W V(.;,-" + a-O~z'0-) . Putting However, if we take ~f9r;, a little smaller, there is a case in which the desired correction factor can be obtained without C. N. M. E. Putting Calculation II and discussion +G~ni1ar2*If?axfr+c.c.} { - (1/6)K 2N o+ (1/3)KMo- (1/2)N]} + G~{\19ar2* I{1a+ c.c.} (1/3) KNo+ GsGl'{ i\/3ar2* L8r+c,c.} {(1/3)KM;;+~} Theoretical Reinvesti{jation if the /1-spectrum if RaE + G~I Jr(/1ar) 12 {(1/2)K2J1fo+ (9/2)M;} + GH fr(/1a.r) *\/10"' x r+c.c,} {(1/6)K2N o- (2/3)KMo+-(3/2) N;} + G~{\r(/1ar) *i.Ba+c.c.} {- (2/3)KNo} + GSCT {iir (/1a '1') *i/1r +c.c.}{- (1/3-) K 2JVo-(2/3) KM,;-3N;} , in which C N. M. E. is' neglected. We put (8), (10), (12) and u=O(l). v=O(l). then with the same normalization as (13), (28) becomes K2v2 + Ap2U 2+ Ayv2 + Kux Corresponding to the omission ofC N. M. E*., we put C~ (1 + LJT.)LOX2- (1/3)K(1 + LI"O)LoXY+3 (1+ LI"'o) NoX + (I/4)K2(1 + LlLo) Loy2- (1/2)K(1 +LlNo)NoY+ (9/4) (1 + LIMO) 111;;. (32) where LILa etc. represent the deviations from the case of the pure Coulomb field, and to match the notation the suffices of Lo etc. are written as one smaller than those of Rose and Holmes.17) We put 1 +LI[o X=X', 1 + LlNo then the cQrrection factor can be written as C",,(1+LlNO)2[L X'2-~KLX'Y'+3N,X'+~K2L y'2-~1(N y' (1 + LlLo) 0 3 0 0 4 0 2 0 6.4 X lOR in this case. Theoretical Reinvestigation i f the (i-spectrum i f RaE References (...truncated)