Biswas, Sukumar

Patro, A P

English

IACS Institutional Repository

61 APPLICA nON OF GAMOW'S THEORY OF a-EMISSION TO (4n + 1) RADIOACTIVE SERIES .• By SUKUMAR BISWAS AJof,D A. P. PATRO (Reech'ed /0,. f>lIblicalioll, Sepf. 15, 1918) ABSTRACT. Gamow's theory of a-('mis'iifll1 is applied to neptuniulll (.1n+1) senes. 'rhe values of the a<;sulUed "nudear radius" "0 are fibnrl to \'ar~' irn'gl1larlyas in actinium (4n + 3) series. A large drop of Yo Ol'curs in 8Jni~13 slmilal' to {'-products of U. Tit and .l\e series A complete calculation of Yo values for all the memhers of {T, Th And Ae serics 's also included with tlwir exten',ioll to lran~urallic kgioll u<;ing 1.1t. 'it exptTinH'lltal values. It has been di~('t1sse,l that the exi<;ting tlu.:oric" of a-emi,,;()l1 ,dtll angular lllo11l{·ntu!l1 are inadequate in explaining the<;!:! irregnlnr vAriationS' of I u. ~peciallv in the odd radioactive series. It appell1's that the nnclear charge Z has something tt) 00 \lIth the irreguillrities of "0' INTROrnJCTION Gamow's th~ory of leakage of ex-particles through a potential barrier has been app1ied to three radioactive series known So long. The relation between thc disintegration constant'\ and decay encrgy E contaius a term To which is referred as "lIUCk,ll' radius" on the assll1l1ption of simplified potential field. This denotes the distance from the centre of the nucleus to the point where the inverse square law of repulsion suddenly changes to a force of attraction as assumed by Gamow. In reality, the fall of potential near the nncledr boundary cannot bc so abrupt; hut a calculation of To from the expcrimentally determined values of,\ and E are mude to sec whether these are consistent. The values of To are in general agreement with the liquid drop 1 model of a nucleus (y,,=R. A:f) for U, and '1'h scrics and less satisfactory for Ac series. But abnormally low values of To are obtaincd for all the C-products. Since G:1Inow's work To values have been calculated by Bethe (1937) ~nd by Preston (IQ46, 1947) for U, Th and Ac series. Recently the missing (411 + I) radioactive series has been identified by two groups of investigators (Hagemann ct aI, 1947 and EngHsh cI ai, 1Q47). Further data have been reported by Seaborg (1948). With these values of E and ,\ it is worthwhile to observe the consistency of 1'0 values for this series. The present work is undertaken with this end in view. Method of.Calculation The value of fo is calculated with experimental valt1es of decay constant A. and disintegration energy E. Various forms of the relations used by different * Communicated by Prof. M. N. Saha. 540 S. Biswas and A:r P. Patro investigators have been referred in a previous paper. transparency factor (Saha, 1944) and Laue's (1929) yield the relation Rigorous calculation of semi-classical arguments (I) where l' = velocity of ex-particle rolative to the product nuclens. T" = radius of the product nuclens. I6rrr2 (Z - 2) (. ) 1 u 0 - Sin U 0 cos U (I • 11' U o = cos-1 [ ~ J~ =cos-l[4CI;~~2~';)Jl Preston h940, r(47) uses more C'omplex expressions, deduced from complex cigeu-fl1nction, A = 2~' jl? tal~ c -2[( To II2 +tan'uo where 11 = (r - TJ/E,) k=21l'11t"'- . It These equations are very sensitive to the small variation in the exponential term, so the additional factor ~2-.!an ,,!i<' is not of much consequence. The II + tan- Un values of To calculated from relatioll (I) are given in Tables I and II. Preston (1946) remarks that the additional term in (2a) gives a refinement in the value of roo The method adopted for solution of (2a) and (2b) is, however, not referred. For the comparison of the values of, obtained from the two methods, the latter equations are also nsed in this work. Solutions of (2a) and (2b) for ro and J1 are done here graphically hy assuming a new variable :v = Illno• Two explicit relations of y and Un an: used to determine their values graphically. These come out as _ ± (kR) sin Uo cos Uo ,'. - [~~ ran u. (,: tan'-u~;'~] +r Y2 = [ .. - .... ------;KJ ... ~ X-~ tan UII (r + tan2 Uo) e - I From the value of '!lo, 1'0 is calculated, and from 1'0 and y, J1 is calculated which in tum gives the value of U. Application 0/ Gamow's Theory 0/ a-emission 541 TABLE I r ll values for (40 + 1) Neptunium radioactive series. Nuclei I I -----usA m241-':>!I3Np237 I " 93Np237~91Pa233 I 9'~Um_:>vo Th229 1 oo'1'h229-"fj8Ha225 : , , 89A('225-:>S7Fr221 ! 87 Fr22J-:>ssAt%17 i , 85At217~83Bi2J3 i i 8JBi23L-:>8111200 I ~+P0213~82Pb209 Nuclei ooCm240~ 90 Th232_~881\1 5Th 1m 9ORdTh!~q--+881hXZ24 88 ThX J24-+86T1l220 86Tn~2~-~8.ThAU6 84 ThA 21'~82 '1'1113212 81 ThCm~61 Th(,"208 .. ThC'212_-:..,ThD'08 I From relation From relation (I) (~) Ea. ill V x 10-· A ---(Mev) ('Ill I set· sec , -;:xw;f::; ffiX lOI:J R X 1,,13 I (,ID. CIlI. em cm. , S ,0 1661 '1.4" X 10 ~ll I 893 r.44 V·71l I 58 1 73 1.5·1° 9.78X Jo~15 V U9 148 9·79 1.5<) ~.S2S 1.556 1.37)( lOcI!3 906 Li8 9.65 1.58 4 85 1.561 3. 18)( 10""2 9 03 1.5 2 9.67 1.59 , 5 801 1.7oB 8 O(l x IJ"1 g.bS 1.~4 923 1.53 6.3 1 1.782 231Xro-* il 1)7 I 41 9·3~ 1.54 7 023 1.880 33 il 91 1.49 \}.60 1 61 586 1.718 3 IS )( 10-6 7 13 I..lu 743 1.25 I 8.336 2.<'49 1.5~ x 10-5 8AO I 1.42 910 r,53 I , ·rAHI.I~ 11 to values for Th, U and Ac -series. (4n) Th-serics. 1<:a. III (Mev) 3.92 5420 5.681 6.282 6·774 6.054 8.776 v x 10-9 cm/sec. 1·400 1.656 1,('90 1.778 1.847 1.746 2.10~ From relathn (1) From relation (2) cm. cm. cm. crn. Yo x 10131 R x 1013 TO x 1013 R x lOIS ~ _____ '.~ ____ ----- ----- - -----2.68)(]O-7 I·5Rxw-lb 990 r.b2 10.01 1.64 933 x 10-9 * 9 0 3 149 9·33 1.54 2.~XIO-6 8.87 ] 47 9 29 1. 54 1.27XlO-~ 8.9 1 149 9·28 1·55 439 8.69 1-46 9 12 1.53 1.75 x 10-5 • 7.0 7 1. 19 7·57 1.28 , 2.31 x 108 8.65 1.46 9. 15 1.54 ~, -------_. ----- ---- -- - -- ---.. PU230-+'2 U235 5.137 1.605 928XlO-J2 9·52 1.51 I 1013 1.64 I2U"'-+oo Uy231 4.36 1479 3.J9X 10-17 8m I.46 9·55 1.56 'lPaUl~8»Ac!27 5·01 1.586 5.59 x 10-13 * 8·38 l.J6 8.18 1·34 8ORdAc'-17_+88AcX223 6049 1·713 1.02 x It> , • 7·78 1 28 8·56 1·41 88AcX!2488An219 5·719 1./'95 2.86xxo-7 • 8.08 134 8.70 144 8sAn219-+84Ac A 216 6.824 1.854 0124 • 8.20 137 8·90 :·45 84AcAm~I2Acl:l~1J 7365 1925 3.79 II 102 8.65 115 899 1·51 asAcC"I-+81AcC"207 6.619 1.825 4.78XIO-3 * 7.65 1.:19 7·90 1·33 .,AcC'2l1-+82Ac DtO'I 7434 1.934 I.39 x10' 8.20 1·39 885 1.50 *. Values of AGo (partial decay constant for o-group p-particles) Application oj Gamow's Theory oj a.-emission 543 DISCUSSION A study of the To values for (4n + I) neptunium radio-active series reveals the following features: Firstly: The values of '0 do not vary in a regular way as required by the rule To = R.Ai. Such anomaly ill To value is predominant in actinium series. In fact the two odd series (4n + J) and (4n'" 3), behave in au irregular'way as regards the To values. On the other hand, ~o even series Th (4n) and U (4n + 2) show nearly regular variation of To t\ralucs with the exception of C-products. Calculation of To arc given hi Preston (1946, J947) for well-known members of V, Tll and Ac series. Rccntly these known series are extended to the transUlanic region and some of the experimental data have been revised. So a complete calculation of To values for aUlbe members of U, '1'h, and Ac series are also included with :the latest experimental data. These are given in Table II. The values for (In + I) series logdhcr with those of U, Th and Ac series arc given graphically in Fig, 1. Tile To 10'0 Np·S..,c.(l ) / ._ . .0-.-.-. ,~ Jr ••. t. .~,-.-fl.. • I  .. ~ ." , . ' .... " , , " . -, , • Th ·'Sent' I ''0-. ____ " A , ,'.", lIP's., " (I' , to -~ ~. -/ ,'''' '-..z, "0 t 8 It. IC ,.. 7'0 lot • I ! / I • : , 1;1 V " .. I • .,. • .'1, : ':'/ · . " .: " : I, " :' VV ; ,I . :' " 1 • I • , , " ! II lila l~ k.'!;..ri~ '. ,II '. J .•.. .....•. '" ' . au t&4 &a &01. ~ll6 ;140 Z44 1'10. I 544 S. Biswas and A. P. Pairo valut:s of Np serieb calculated according to both the relations (1) and (2) are plotted 011 curves. The!oc show that values obtained by relation (2) an! higher than the other values hy a nearly constant quantity. The plotted values of V, Th and Ac series are those calculated from relation (I). Ac series shows a sharp regular fall to RdAc227 and then a rise in To up to AcA. But in Np series the irregularities arc 110t so wide. Sl:condly: The value of To! for "3Bi211~~ITI2"" is abnurlnally low. DiU 3 is the correspol1dillg C·product of ner1hmiUIll series. It has been observed that in tT, Th, and Ac series there. occurs an abrupt fall in the value of To ill C-+l'/I (1i!o>integl ations. The value of 't) agaiu assullleS normal mugnitude ill C'~ D disintegrations. Similar Ilhellolllena occur also in the (411 + I) radioactive series. It is intcre!o>lillg 10 note that b.,RaE2111 which has been recently observed to he <x-active (Broda and Fc.ather, 1947) exhibits all ahnortually low vlJlue of III ill h.,RaFl11l ~ 8/fl20b disintegration. Tln1s all the <x-activt: isotO[ll'S of h ,Hi show abnormul value of "nuclear radius. " The drop in tIll! values of I" for tIle C-VlodllctS and the members of the Ac series have been attributed by Gamow (19.37) as due to emission of <x-particles with angular moment11m different frolll zero (1=/=0). The effective radius in such a case, as deduced by Gamow (11)37) is SUIlposed to follow the relation: h2 Tefl = T~- -.,- 2-(--) l(1 ... r) 4;r"11l C Z - 2 Allotments of i-values to diffe! ent <x-ray lines are rather arbitrarily made to fit the experimental data. No qnantitative treatment on the above linc bds been found to be satisfactOlY. Enlission of <x-particle with angular momentull1 1-=/=0 has been treated by Preston (1947)· Calculations are made by him with the complicated relations for a few <x-disintegrations having excited states. The method of calculation is very round about aud 1 values are chosen arbitranly to give a more or less consistent value of 1t1 for different excited states for 'l'hC - ThC" and a few others. On tbe whole, the problem of emission of <x-particle with {::po is at present far from satisfactory. As ill the Ac series, tht: irregularities in 1u for the Np series is probably due to emission of a·particles with angular momentum different from zero. • The experimental observations on (4n + I) series are rather pleliminary. Further investigations are sure to reveal complex a-spectra in many members of (411 + 1) radioactive serie~. A detailed experimental observations are required before any theoretical treatment 1S attempted. Since the irregular-.. ities in 10 are found with the nuclei witb odd mass number and in Bi which is the first member in the even series heWing odd atomic number it is plausible Application of Gamow's Theory of a-emission 545 that the even-odd property of a nucleus atTects the a-emission process to a large extenl. In this connection it may he mentioned that in Hand Th series 10 varies -~ . 1 more or less regularly. However, the valne of R = ':.11 IS secn to )C 110t a constant hut increases from lightest member to heaviest one in the series to the extent of about 15 %. '1'1Ie rule To = J?A \ can110t IJc cxpccted to hold accurately over the whole radioactive series hecause an increase in 7, increases the Coulomh repulsion which teuds to decrease the nuclear density, and increase the nudear radius. The simple rule would hold if the nuclear binding energy ('ontained the only term .E = a A. nnt E is given by the expressio11 Hence with increasing /, R the average distance hetweell the 1111cleons should drift to a higher value. This i!> actnally ohserved in the value of R calculatcd from 10' Present (T940) proposed the following formula for nuclear radius after employing the corrt:ctions for N=/=l, snrface tension aud Coulomb repulsion. [ 12 7,2 ] 10 = R.lli = R*A~ 1-10.8 -, -0.3A-1+o.01 -'-3 jJ- Il. 111 this relation R* in place of R should be COllstant for all members in the series. For two extreme mcmbers of TT serie"" R varies frol11 1.55 to 1.36 from PU21S _ + U n 214 to RaF21'.~ RaG20R • With above relation, R* comes out as 1.48 antl1.35. In case of 'I'h scties R for 'rh2"2~MsThJ22& is 1.62 and for 'l'hC'21~-+'l'hD2OR r .46. The correspolltling valucs of R* arc 1.55 ancI 1.40. Thus the proposed relation is far fro111 satisfactory. Although from the very definition of 1 fI, the relation hetwccll I" ano actual nuclcar rarlius is rather vague, the above relation cannot account for the variatIOn of R along the radioactive serics quantitatively. ACKNOWL1\Vti1\T1\NTS Thc authors exprcss their gratitude to Prof. M. N. Saba, F.R.S., for his keen intereht and guidance in the progress of the work. Our thanks arc also due to Dr. B. D. Nagchowdhury for helpful discussions. The senior author is grateful to C.S.l.R. for the award of a scholarship which enabled him to carry out the work. INS'tI'tUl'1t 0' NUCLEAR I'HYSH'S, C ... L('tTTT .... 546 S. Biswas and A. P. Palro REFERENCES Bethe, 1937, Rev. Mod. Phys., 9, 161. Broda and Feather, 1917, Proc. Roy. Soc, Lond., A, 190, 20. Engli~b, Cranshaw, DellllErS, Harvey, HillCks, Jelley and Ma~', J947, Phys. Rev, 72, 253, Galllow, 1937, Structure ot \tomic Nuclei and Nuclear Transformations, (Clarendon Press Oxford), p 104 Hal.!emanl1, Kat7il1, Studier, (jhiorso, and Sellborg, J947, ['/IYS. 1?cv., 72, 252. Laue, 1929, Zeit. t. Physili., 52, 7~b. Present, J940, Pltys. Rev., 60, 28. Preston, J946, Phys. T?cv., 69,535, Preston, J917, l'ltys. Rev., 71. 8f5S. Saha, A. K., 1946, 1'roc. NIlt. 111)t. Sf. 111<1., 10, 373 Seaborg, J948, Chell!. En,!!. News., 26, Tt)02. 

Application of Gamow's Theory of α-Emission to (4n + 1) Radioactive Series

http://arxiv.iacs.res.in:8080/jspui/bitstream/10821/2073/1/Application%20of%20Gamow%27s%20Theory_By%20Sukumar%20Biswas%20%26%20A.%20P.%20Patro.pdf

Application of Gamow's Theory of α-Emission to (4n + 1) Radioactive Series

Abstract

Similar works

Full text

Available Versions

IACS Institutional Repository