Dube, G P

Jha, S N

English

IACS Institutional Repository

ON THE THEORY OF EMISSION OF ALPHA-PARTICLES FROM RADIOACTIVE NUCLEI*B y  G. P. DUBEANDS. N. JHAft)} puhlu [ituni. N(iv .'“/,/0/^iIABSTRACT. Tlif urLiii^ ^Mihn potuilial linlr of Cniuloii-riuriuy-IL'mifav iiiuflcl Ii.en 1)0lmiipplaiad !)>'an ( Npoiiciitial I)nt(‘iiti-il funcliAii V - “ V„p  ^^ " and an expression ha*- heeii u'orkc d out jor the deiay t'onslant A. J1 is luinul llial Ifu'exaet form and magnitude f>f the poll nlial fimiTion inside (he nuelens plavs a \er\ insignificant role in tin* prohleni In llie two limiting cases V,, very small and V,, verv large, the dreay constants bothiiideix ndeiit of V,, and ratio Aa* is i»f Hu-ordi-i s only 'FIk* iimdear radius r,, lias been laK'ulated using the e\[u riineiital data foi the decas constant a and the cneigv h) and all the ladii except th<'M of T!it\ Ka( an fuund lu lu'hr!weeii y.b and o 7 io ' cm. aIik'Ii is oI the right onU-i of magiiituile and agrees in a satlslac^)l  ^ niamiei with those, uhtaimal hy the IMe NlnllS WeJlkciS Wit]) tlu iMlc hod) piclUK* ol tllC mK'lcllS.A satisfai'tory explanation of tlic emissiiui of alpha-particles froiif certain ladioaclive siil)Stau(T\s was fust given by Condon and (inrney (h):2o) and iiidepcii' deiitly, by (hainow (lo^d), who considered the i)roI)ieii) as a transmission of alplia particles lliroiigli the nnclcai ])otential barrier. Tor case in calculation the nucleus was represented l)y a rectangular potential liole of a certain constant de])th and a width r(, where /q is defined as the nuclear radius. In the range2/c-Pi -  / the potential function was taken to be the Coiiloiiil) one V - " —1wlierc /  is the atomic niiml)cr of the product nudeus and .*c is the charge of the al])ha-i)articlc. Sudi a picture of the nucleus is very crude. As a matter of tact, c\])oriiiiental cx'idcnces go to [irove definitely that llie nucleus consists of protons and neutrons held together l\v strong and short-range exchange forces, d'hns the nuclear iMoldems imnst be regarded as nlany-body proiilems and not as one body problems. The extreme easey with which energy is interchanged between densely j)acked particles inside the iiudeus, can be easily understood with this model, 'i'lie emission of an alpha particle into free space after crossing the potential barrier which surrounds the nucleus is understood in this way, viz , the interaction between the several constituents of the nucleus leaves an alpha- pailicle with a certain kinetic energy; possessed of this kinetic energy the alpha- paitide penetrates through the barrier, 'flie ideal calculation would be to find out the probability of the formation of alpha-particles from neutrons and piotoiis'• ClunimmiriiU'd l-)^ ' Prof M N Sulia, P.RS.46Theory oj Emission oj Alpha-parlidcs J 45aud their being left with a eertaiii kinetic energy and the next task wonKl be to find out the penetrability of tile alplia-pai tides thiuugh the midciu iioleiitial barrier. This is how the problem should be tackled if the nucleus is considered as a niauy4K)dy ])rol)leui. Uufoituiialuly Hk* uiatlicmalical a>nii»]cxilics arc so great that the niaiiy-l^ody piuldciii may remain unsolved io\ a considcniMx  ^long lime.The exact faw of force between an alplia-parlieie and othei micleai j^artieles is at present unknown. It may be ex[)ceted that the exact foim of the i)otenlial function inside the nucleus will not dnislically modify the lesiills olilaimd in the rectangular ]>otential hole model but a detailed mathematical aiiaKsi.^ of this exj)ecled result has not liceii attempted. Hetlie ( i m e n t i o n s  that the exact value of the potential depth is of no gieat inii>oi taiiee. It w as tlierefoie thought necessary to examine this inoblein in detail.In  the  p resen t [)a[)er, the  rectangu la i tioUiitial hole of Coiuhant luniey- ('■amow m odel has been rei^laced by an expun.eiitial fuiietion given IwgV = :- V o C " ' ... (0aud an expression has been worked out foi the dteay coiishuil A, It has l)ceii found that the decay cousl.inl A docs not sensibly deiiciid on tbc magnitudeof V() and in the twe^  liiiiiling cases, Vn \ciy small and V,, vei\ large it is in fact iiide])eiident of V<^  and the lalio is of thk- ordei 5 only. It is ciitiielya new’ result and [>rovides a deal and detailed analysis ami in su])i)Oi l (^ 1 a Inllierto expected result, losing the ex]jerimeiiial data foi the decay consUmt, the nuclear radii have been calculated and the values olitamed arc i)uite satisfactory and agree well with those given by previous w^ r^Keis with the oiie-hody picture of the nucleus. In view of the mure correct manyd)ody i)ii'lme of the nucleus, the results presented in tliis paper are purely of theoretical inleresl.The exponential potential function is shown in iMg. 1 as a function of 1 the distance from the centre of the nucleus. If vve consider alpha-tiarticlos of a given energy ly, vve may divide the whole space into three legions as indicated in Fig. I.346 G . P. Dube and S. N . Jha(/) 'I'lic intfriol of tlie nucleus o- ^0, vvlicTC tliu [potential ciierj^y- V (U■ r; 0(//J The ic>;ion of llic ]>otcntial barrier \vliere u is the classical(lislan'L of clnsesl api^roadi of an alpha-particlc of eiierf.’y K falling on the nuclcns from oulsidc. In this icgi(m the potential energy i.j greater than the energy of the alpha-[>article.(;//) d'lie ouivside region i >  g in which the potential energy is less than H-We consider the case when tlu- wave function of the alplia-]>article is sjdierically syminelricah This coiresponds to an ali)ha-parlicle of orbital momtuiUnii / ■ c oi what is knt>wai as the .s-state of the particle. The wave functic»n of an alpha-particle can then be written in the form, \//where ^ satisfies tlie equation(IH . eJ\J ^ ^ I , (h.“  V ) v - u.yr" ii-where M is the mass of an ali)lia-f)article and h is Planck’s constant divided 2n. We take the values (birge, U),] t1M -  o .bpi X io '“ "  ^ gm.h —-1.05 ] X 10“ ' erg-secIn the three regions marked in the liginc i , the equations areRegion I, Region 11,, r - V• ' t  + ( u - v ) ^ ^ ,{{t  ^ 11 ** ... G)' f  'f + (E -V ) i^ -n-22Region III,In equation (;p, Vfagain - “  but K —V ts positive.fb'irst of all we give the solntiou of eqiiatiou (2). Putting_  ~ ' i i (Iv =  c(.1)and p; —V is negative wlici'cas in equation (4) V iswe have di_V_ dip and d H  _ ldr'^d''^^  _l 1 dip IJUi dy^ y dy \Theory oj Emission aj Alpha-particles Hence the equation (j) transforms itself to, 1  , 2SI . yj ( . .  ^ n \ , _We define1\ I /  ^ \ '■2 T' __ - I ^  _ ’‘2 2n  i ’  ’ '>/ i - ~ -  ,h- V„'riierefore the eiinatioii (5) takes the form,d" i l /  I d\p Idy- \' il Vy (tM 7(5)'0)(7)(S)Tins is :i standard ty|>e of cMiualion and is known as UesstTs ecinalion and lias been well-sludit^d by tnalhematirians, Hei(‘ m is no1 an integer, llierefou the general solnlion of this equation is,ifi =  C.] (/>y) -i-lkjor-I 1> JI ':!hi (o)}}l ' — INwhere C and D are constants to be deteimined by the boundary conditions. The first tenn corresponds to a ^vave travelling towards the right and the second one to a wave travelling towards the left after being reflected at the potential barrier.We next consider the equation in wliieh case k --V is negative. IajI ns define' lo)The solution i/^ ii in this region is of an exponential (diaracter rather than of a wave ty[)e. As has been shown by Jeffreys and later on, in(Upen(lentI\by Kramers (1026', the two fundamental solutions of approximate vciy closely to( \ rn ■j 0 '^) ‘ oxp.-^  I 0(01 /0(r) , . exi3,| - J 0(0 (11)I 'It is to be noted that decreases from the nucleus outwards and incieases. At the boundary of the nucleus, function V'l <'(>) " ’>11 i»e imich Kreatcr than i/'2(ro) because \l>i coutaius au exponential with a larj^e iiositive exponent while \1'2 contains one witli a large negative exponent.Lastly we consider the equation hi) in which ca.se b-~ V is positive. At large distances from the nucleus, V is very small and therefore it can bem ( j .  P. Dube and S . N . Jh ancf^lcclcd. 'J'lic cquulion thfii becomesd-V 2M+ T*7  V .^  —  nof wliicli a general solution is,2 „hZ■>f=Ac^'" -IwhtTt' 1/'^1111(1 A and H arc sonic coiistaiils. By multiplying it with the time factorexp .f it can be seen tlinl the first term represents an outgoing wave whilethe second term gives an incoming wave. In the physical problem considered here, the nlpha-])aitide leaves tin* m.u'ieiis and none comes towards it from\ ontside, henee we ])nt B =  <»."rherefoj e, ^ -  A d  . *. (I :,dTo ol>lain the wavi* I'liiiction for smaller values of 7, it is again convenient to use tlie Wcir/el- Krauieis- Brillouiii (1932) ap])ioximation in the form given by Kiamers (uj.nd. According to this metJiod, the two fairly good approximate solution^ of the ettuation (,)), whieh lit in smoothly at ? =  /, with the two solutions given in ( 11) rcs[)ectively are,JI I I 'P' M I (/ ; r  - /•r  r. M1 \ \) y I cos \ J </>i?) fi j)5IJ -1( 0 ( 7) [  ' COS I  f  \ f l ^ h ) ] ' ' ^ d r -  ^  {r  o 4 iThe most general solnlion of the eciuation ( j) is (lien,In order that (l,]) may be of the same form as (i;d, we must chooseB 0 -/ B 1Since, for large t, V can be neglected therefore,4'\u — ... (15)Ccmipariiig (12) and (15) we have,A —Bi/v"*! ... (16)At the boundary of the nucleus i =  the wave function of the region I will be equal to that of region II and so will their first derivatives be. As has already been pointed, Vq^n,) will be much greater than hence we may neglectV-jlro) and pul,d^ ('^dr '0# 1dr 0^Theory nf Emission oj Alpha-particles 349or1 “^  '0 )a»d, + ......y’< (■ ' /« \whcie J  J^,pc j.  ^ iKolcctcd n term ;  mcomparison with 0 { ; as it is miiiierically very small. We define,pe~- “  V]n I (a ) Cl J^ .(A -)-a '!'()h, I< / t  =  P(17)'riicixfore,This ftives ns CC» I D/i-n,{v-!io)}'<C"' +1)/-)'' — lij ;i; ( i^(i,)) —(>.I)ororB]==L. ,-cThe decay constant A is given by,/N --A N d /1 ( I N(ir-;)A =N (II.where ' is the ratio of all the particles coining out iter second from theN dtnucleus to the total number incident. Tims we have,\  =  /\7Tl‘^  .1 . I 'b I®= I A=  4iLl' . I 111 “ k '• V i  is exp .l-i) and.sljoiilcl not be ((jnfnstd with the ck-aronic charge, for which the greek symbol e tepsilon) lias been n.sed.7-  ni5ST’- V I350 G. P . D ube and S . N . Jh aNow =  ^  ; n i d = 1 ?^ .(Vi -  H) whore V i ~ ami definingA' JaI n- ),i( k j )irhwe have, A j C | ■ 'i '“  T •r  ... (.o)It Ii;is been [irovccl on i‘a,L;e c|3, 'Dieory of Bessel Functions, Watson (1022)* that if V be not an iiitc‘c,cr,2 Sm V7Tkcplan'iij^ i' 1)y ni aiul llic are,iiinenl , by \, this becomes identical willi y defined in dliLTcjoie ue lia\e,2 sin {inrr) , ,, ay-‘- ”  and \r »-T,V \ ~I'sin^ the expression foi in and Mob ’^^ c havea'v-^^' (V - ■ h;)1 1“ -- c "Also* Vl'^X =^-(2-0)“Hence, A -^ — 3. sinh-»in- c2V.: . irM Gfi ( V i - 1 ' , ) “ ' ' d ! 1^ '.I ,2I - PI C !" can is unity. Tlieicfore we have,(:m )(.;.o(:^ 3)v\ a VoPutting : “  J’i' ■ rhu\Xdz then -8-;olC|®. J  =  ’To perforin this integration, we inahe use of the relation (7), page 147, Watson (1022), viz.,T H J =,s^ 0, + l)Putting v = , ni( - 1 ) ' ' .( 2 .S-) !;• 2S-'^V=r.O !)‘^ F(?h/ +5  + i).r(—<hf ^. \Theory oj Emission of Alpha~paitides Therefore we have, i f  J  < n  ( s ^ J — ;  w  ( c )  1 ___ __________ IJ 7  ‘ '■ 'T T ; 'T 7 7 ,T i '(351- -  )oK I- l ' O/ I /'.{s 1 '(ill -I ,s +  l) l'( -  III 4 y -I- I )„ „ i _____ /.-7 i - r 72r(r l-h!)l’(l + //,) , ,H-,)l^ 'or sninll values of y? and hvju'c of Vo, we can needed all the leiins v'ontai and lii^ di^ *!’ powers and we have,nmia.*,cr^- . P i n  //PP(T-/a) -  ' . - -  --4^^) sinlu/TT .'|?oSinh ?otIf /j and hence Vn be not nei;li^ihk', we ha\'e In lelain three or ioni lei ms nl tliu SCIies containing 'J'hc tlie ranee of integration being small, a snlliciently accurate value is given byJ - " - '  M j . , 4  ' 1 . lov' />n , . J ^ -i /-j-I /'OI . / M iJ _iV 1 —a! wA I l> {2b)Next we turn tc» the evaluation of the terms containing ft and fi\ hoi all X'alues of the Ikssel function is defined by,ws n d‘ t JS‘l) . (i-'l .s! -P('' 1 6 I)Hence, / ' -  T I \ ' , _  (i-v)'-'h'or hinull values nf .v, wc liave,/- (i-v)-J IV  ■ \ | 2 .... ""^,2 =  ' ( I - .'- ) ! - (27)r ( i  — in)When order and argument l>oth are large and comparable, a very good appioxi- inatiou to the Bessel function J (2) is Watson (1922), page 220,J  fc)-M ^,coswlicrcV " ;r v i“ - v faud Q =  _ ”J  + vsh i- 'I  ^ ^111 the prolileiii considered here v — in, hence352  (j.  Dube and S . N . ]hda _ r  cosh (^>i ;^’e)or m  =n V -x;'^  + 11^  sinh nnand also 4r.silllw/77Tliercfore wc have ilie following results,2|ci/!> .Iff) siiih^ //rrlor small V<' =  lor .V and n large and comparable.:/M :i / () siiih^ 'ar^  2,c ?rA;(A H- /^ ) for A large and »  n.8). sinh^bjff iMaking use of tlie relation.I'„(c)-- 1 J„-(28)(2<j)\vc have 1^' _  1 _ J-U'«| l)('lSince we have the relationJv( 0Ji»—*j(0   ^ 1 “ .KIheiefore, for small values of \ ,J,.(-v)'■ 1 i’(v+ ij , 1 A \  ^  ^ A*  ^ (I )^i’(v+i) II \orJe _ |(a)  ^ ''A' ?/'/_ 2ui  ^ KA A* A "  •!^  v;,For large n and for large values of a , we IiaveJ < 91 -1 n J “ i 91 (A).Therefore— T 4- 4. •- =  I + — — j +X '■‘’V v -V 0t'shig these lelatious, we get,I «' ./ VoU v H " ' ' -\- h'j ,)  “ i! ~  ' "^V] -  F ~  g  ioi" small values of V,).(.10)(31)(32)Theory oj Emission oj Alpha-particles i 5Jfifor Vo comparable with Iv, audI  ^ , ] ^ »'■ ' < V j -"2 V V ' i  -  I'J ... ^Vtr(V i-lvfor Vo lar^c.By the help of the relations (7), (::o) and (31), for very small values of V\vc therefore have,An — '  ■ r!' (V ,-] ';)^■'ll(S'()I'or very larj;e values of V,), we have, using llie relations (6), ( i7), (23), (20) and (33),III V 2Ml.ks)h'or values of Vo coniparahle with li, we have, using (..’3), (au) and (33),, / ,  , V v o  i rlij-' _A- •' V  (\ -' / \ ... (301Vo + e V ,-2 V e V „(V ,-K )It is to be iioLicecI that Ao arid A.^ , are both iiidependeiit of V'o and that (3()) reduces to (3,1) and (35) in tlie limitinjL* case of V(, very small and V,, very larp,e respectively. Also we Jia\x'^ 0 ’ Iv _  . 1;■ ' ' '  '  v ,(37)If we choose for i„ the nuclear radius the value o x m"'®ciii. derived with <uie body model from experiinenl, Ikthe (1037), tor Z the average nuclear charge of radioactive elements, about bo, V i comes out to he 27 MV ; and for h, we choose an average energy 0 MV. Therefore the ratio A,,/A„ comes out to be .17 only. 'I'his leads us to conclude that the decay constant A does not dejiend sensibly on the exact form and magnitude of the potential function inside the nucleus.The most important factor is the exponential term and for tlie sake of complete­ness w e discuss its nature. As dehned in (17),\ip=y..2M .. / - l i  U r .0^KtSubstituting cos'-*^  ^=  we havcip -  V »M ,4 Z.“ 2. . Ik V R  i  ^ V EG. F.  D ube am i S . N . fh aU'lltrlLJ K:rm'me1 / Jv^^0- a /'  V 1, ,, V' :^M /  K /ll]Lrd(U’L J — , . ”,c(^ s A/ —[I. V v ,  I v ,V ,r -ILwlicrc Iv is ill iiiillKui volts uirI I Iif (0j~- ci )s * " -O'^ii—0)''Tlic valiicb of j^ iO) ;is a fiiudinn of 0 are ^ivuii below :I “ H, M )V| / ((.vs)'rAiii.K I0 !pf)|11) ! 1;-;«n0‘ 1 jS I ■ dSs i ,2.]Cl .SGtji . , J G 7-^7S -CS 1 G I -r *1*■■0 , c/.vi : .y.172It, 1 Si hi !1 1 -'‘7 5SJ21(111 ,GhS3 .:lSJ V177--’ , 6,>|(yS. V' 53'?Sl‘\)r c()ini)arisf)i\ with llie experimental data, it is loiiiid coiiveiiicnl to use the L'Xperiniuiilal data for llie decay constant and the ejiergv of tlic alpha-parielc and to cc)ni])uU  ^ llie ineleat jadius from these data \\ith the help of tlie derived theo- letieal formula. If the formula is correct, the radius/(, must come out to he nearly the same for all radioactive nuclei. It has alreatly been shown tliat the exact shape and mai;iiitude of the potential function inside the nucleus plays only a very insigndicanl role in t)ie i>roblein and llie ratio A ^ / A is of the order 5 (eg. A7)> hence we adopt as the final formula,A— ;:.5Ai, :Tliis can be wriUt;ii in tlie furtnV M? 2(•10)A =  A (Ii,Z . >o)c- jl-or log 1 (|A = log K* A -  'tibSoi'w h ereTheory uf Emission oj Alpha-particlcs'og,oA =  .';j.3Sj,>+]oj;,„ ■ ^log , „(V i K )355K, V i being in million volts. In valnilaling .A, uo haw tabon the value of V’ l COT 1 es{)oii(Hni^  to m  ^ vm, u c 1j:ivc\ ii \'  ^ he- in million \’o!ls,V,Usiii^ llic ivlation*' ilogi()A-log|„Ai[ o H.m / (.(T.Tin's unables ns to calctilalo Irom tlio i xpcrimcailal data of the decay coiistaiil A and the energy I , in mil]it>n volts. The lesnlls t;iven in Talkie 1 ha\a' hu n iisecl to caienlale and then the nneleai vadins is aiveii \)y the lelalioti,loe j 0 / ,i 1 s - 150 I  ^ lo;.', | , ,A Iol*, | ^ ei ' 'Tlie ealeidaled vaiues of the iuielt.:n radius aie ^^ p^ 'cn in 'i'ahle II and all radii e‘xcept tliose (_)f J a^C' lie between 7.-' and o 7 x ‘ eins 'These au* id theright order of niagnilnde and compare in a salisfaetoiy manner with tliose oIk tained liy (umiovv ( loml ,  '107^71 and by Hetlie (1077)'T\iiU-. IIiiit«m niiiip Pld'lluMl'il(ie\ \i ! K'( ;i\ Ml Uiml, Nm leaf i ;Xiu leir i\I\ 111 ‘\i( 1 1 1Wi in in ' ;ili lil;iti <1 fiTil AFl'l,! s: 4 SI 1 2 / J' I ih ■ V 7;Kd'll i ! 'HiX MS 1 . . , i . 1 ^ y l( • h' ■'>■7 1Tl iX Tliii ,Sf» 1 M pi ;m A 1' * h : ''Till, 'riiA S 1 1 f) 1 t r m 2 1 S inTh \ 'rm: S.' i <-0'' ' i Hi i'I'liC 'HiC'' s, i n 2m 7 y j 1 - 7 "7' IhC' 'i FT) I Mm t 1in 1 XI ' I TS n 1 I'i i '1lo K m , SS l -''7 ' 1 If ' 1 . : f 1. ,Km Kii ! so ■1 ; ' r / 1m 11 ! S,0SKii Ka \ S 0^ i 1 M y I M a ' >i 00K m A KmH Mj e 1 1 : 3 M \ ) ; s isK;a RaF s, 01 1 ■, ; !> f M > r.^iKmC KmI) 1 Sm 7 1 IM' j '1 "7KaF KmP C ;  ^ 1 ' y 1 ( 1 1 1 ,1T*a \r So 3. j (. /, 0 y IM, 1'; 7 omRrlAe AeX ss 0 U) ' '1 TS K I I  1- 7 7  ^iAeX An Se 1 " S"' V '■ y H i 7 S ' ( ,An \eA S4 ‘-bS , 7 7 X l( t i S MlAc A AcH s.- /  ^f 2i S y JM" S A',AcC \e(' Si 0.7-1 y J r  J 2 O.Sf,ArP' A el ) Si' 7 es '' —Xo h xperinit ntal v<ihic s rivnil;il)le356 a .  P. Dube and 5. N. Jha»)ur thanks arc due to Principal Kanita Prosad, for his keen interestin this work.I*n\sieve JjAHok\T()K\ ,Suexnv 0)Uj r:».,PatnaR H I' P, R H N C li SHillic, IP \ , K;37, Niirlr.'u- I’livsic'!., TP Nnrli'fir Dvnamics  ^ Theriretical, Rci\ Mod Vliys,,9, jof)R T , , ig p ,  \  new Tal>](.'of value '»f tliv j e^iieral i)hysieal (^aistants, Rev. Mod. i /'//ys' , 13, j,y;.I P T / , Nntions de ne'eaninue fiinlnlaloiie, Us iiuHliode <P approxiinalifjns,] ( t imlil ii's St iniiifiqm t‘l ] ud iislurllc ,^ f lerniami, T’ai is, ij,C oridoii, p. (P and (diriiey, R \VP, IJiiaiitiim !\revJiaiiies and Radioaeliva I)isiiite|:i;ra’ liun, l'hy\ Rt’i' , 33, i j/;.(lann>v\, <P, (JnaiitcidliLsn ie des Mniukerns, /e/7,s /. 1‘liy.s , 4i,Prminw, (P and IiDUlvrman ,^ IP, Zur niianleiitliL-Dtit' dvs radioakliwn Kerns,Zeds / Rhys , 42, [(jO,Pamnw, ( P and lIuiiliTMiaiis, 1^ ', i()37, \loiuii Niirlci and Radioai livif \>, CainP. Pniv TVcssj Sec nS,[efiieys, n  . j9.; ] , ( )n certain a[)pnjxiiuate solution ol lineai didercntial equations of the- second order, Proc. f^ tnid. Math Soc , 23, ,|jS. »Kiainers, 11 A., WPdleiinieclianik iind liid)tzal)li£»e (hmntisiernn^^ /riL\. f I*liys.t 39, S2NWatson, tP N , u)2d, Thcoiy of Rcsscl PdincliO)!, Canih. nniv„ Press.Wcn/el,(P, Idne Veralljjjcmcinernii,t  ^der f_)nantenl)e<linguiigen fnr die Zucckc derWelleiinu’clianik, Zr/Ps ( /’kys., 38, si8.\

On the Theory of Emission of Alpha-Particles from Radioactive Nuclei

http://arxiv.iacs.res.in:8080/jspui/bitstream/10821/1335/1/On%20the%20theory%20of%20emission%20of%20alpha-particles_Dube%20G%20P.pdf

On the Theory of Emission of Alpha-Particles from Radioactive Nuclei

Abstract

Similar works

Full text

Available Versions

IACS Institutional Repository