Kar, K C

Roy, R R

English

IACS Institutional Repository

aA SELF CONSISTENT METHOD OF DETERMINING THE MASS OF MESOTRONBy K. C. KARANDR. R. ROYfor piihlicatiou, \ABSTRACT A m u incllif)rl is di'vtTopcd for niidinjc: 11r‘ iiia^ s^ of inesuttoii in lieu-tinns «ii(l piotoiis. The method is si lf-tMusistenl. The neetiraey of tlie mass thus dcleiTiiiiied, w'fiieh is 1 1 ) electron unit, depends on the accimile value (T tlie ljiiidin^ >^  ener|Ty of deuleron aln'adv known from the expel iineiil of Cluidwick. The intf i* u'tion potential is taken in tfie fmaii Mi|;,eested hv Yukawa. 'I'lie mass delmnined hy the ]uassent method is the same as the. mass obtained l>v Kar Uo4-) fiom the them v of proton-proton si atteriii^.In tile i>resent paper we shall discuss a new method of delcniiiiiiug the mass of mesulroiK The method is iierfectly rigorous and sdf-cousistciit. The accuiacy willi whieli the mass is delermiiied depends on the knowledge^of the acemale value of tlie binding energy of deuteron.It is well known that the neutron-proton attractive force in a deuteron is of Yukawa type, derivable fiom a iiotential of the formV ' A 'u (■ r(r)Again, because there is a special aflniity between a neutron and a proton, the attractive force between them should lie of c.ulnijifyc lytn\ The particle wliicli takes part in the exchange is evidently the mesotron. The experiments done so far prove definitely that the charge of this particle is c the electronic charge ; but the mass determined by several experimenters till now varies between xoo*->4oo electron mass approximately. Our proposed self-consistent method gives the mass accurately inasmuch as the binding energy of deuteron is known with fairly great accuiacy.Now, it is already knovviF (Bethe ami Ikiclicr, 1936-37) lliat in an oc.particle the exchange force is comiiletely salinated both with respect to the spatial and spin coordinates, whereas in a deuteron it is satinated with respect to the spatial coordinates only. Consequently the interaction potential between a neutron and a iiroton should be as given by Majorana. Thus referring the motion to the centre of mass of the interacting neutron and proton and assuming the potential to be s[)herically symmetrical and of Yukawa type, we have foi theS elf Consistent Method of Determining the Mass of Mesotron 317 wave equation of the neutron or proton initially at and after cxchanseat n - 0  ^  ^+ 0)i - r t  ... Uwhere M is the mass of the jnoton or neutron and !■: the himling energy heinji negative. Now, because\'( - r )  -  R( »)P’F(cos(  ^- x(r; = R(,)p-;'(ros(,R) -t h ; . f , R ) = “ " ' (-R) (3)andwe at once get from (2)4»*MThere ar c reasons lo believe that no excited stale of tlie deuteron exists wliicli differs from the i r^onrirl slate with res[)ect to the orbital motion. 'iMiiis I -  o and so (3) becomes( - R )  I T v . ( < R ) = -  . ( rR )  .. .  ( . y r )Ivveii at tlie ground state the deuteron may have two varieties according as the si)ins of proton and neutron are paiallel and aiilij^Jiralleh In the former case the deuteron is in a tiiplet state having threefold degeneracy, while in the lallcr it is in a non-degenerate singlet state. However, if we assume Yukawa interao tion of i>urely Majorana tyi)C, the binding energies shoidd be same in both the cases.Now, because a is very great, the Yukawa potential is api>reciably dianged for a very slight variation of i due to the exponential factor the factor A/ r prac*tically remaining constant- Thus to solve the wave equation (3. r) for deuteron, we may take A / / to be constant to a lirst approximation and equal to A/?, where T is, strictly, the mean distance between neutron and proton. More rigorous solution of of (3.1) may, however, be obtained by using <Yfunclion. i^'lms i)utUng Vo =  A /r, we have from (3']) up to first approximationLet us again put 3 - i/i.s.sV-Adft ( .R )- l- ( J i  +  VoC “ ")(rR)=-o 4it2M dr-(3-a), (3.2) reduces to| . r   ^ ( r R ) i+  (P: + V ,.v) (»R) =  odx 1 d r   ^ “  li-  (3-3)Vo-v* (3-4)318 K. C. K ar and R . R . RoyHeuct (3., )^ reduces to; , { r R ) +  ; + =  od:. (.1.5)wliicli is tin- usual Hessd equation, when J-i is iieKalive. Its well known solntimi is) ... (,)w l i c u * _^4 (^ -  KM )" (4*Jmul C 11k* averaijjin^ farlnr. Hec’aiisc R slniuld l)c honiuled al u, wc have fioni (1)j ,I, j••• (.S)iMoin tal)lcs of Bessel luinciions J .^ (;:) one can easily iind the value of (say) for v\ lii('h llie funelion is zero for a ^iveii n. In the same way one finds easily that foi a given s, Ivq(,s) is satisfied if- B ,  ... (5.0ahwliere the numerical value of B , is obtained from tables of Bessel Fnnttions (Watson.)Now, from (3.^) wc have for the wavestatistical value of the binding energy of deuteronwhere is given in (4). Iboin tlic sulxstitntions already made {ride (3.4))we have^  __ tid / 2and thereforec/=J., '/‘' J -d, - •1 " (  • - dz I— n/c = -2and also ru _”  j 6-^']\JVo ■ ‘Thus we easily get from (6), denoting differentiation ith respect to the argument by dashK '4^HI“ T y .  (cj.." t - j , ' ) d c - ... (6.1)2 / 47r“ M 2 JSelf (Consistent Method of Delermininf; the Mass of Mesotron 319where 11, is given in (5.1). The uveraging factor C in(().i) is detennined ill the usual way from the condition(6.2)a ./o 0Again, from the wdlknown recurrence formulae of Bessel functions^ vi::,,we haveOn using the above relations, (6.i) may be easily transfonncil to,H.:)“f0 ndzj.i.S MV Of tliese( )u SLihstituling the cxi)ei iincnlal values of binding energy (J*d vi.,=  3,421 X io““ '^ erg, we get sets of values of *s’ and f* salislying ( |.i). only one set will obviously be eoireet. To find out these correct values we make use of the wavestalistical eq. (b.3) of binding energy* Now, lor a given value of s, B., is known from tables and so the integrals in ((>► 3) arc evaluated f)y the grapliical method. The averaging faetoi L may also be determined fiom the condition (6.j). Thus the binding encigy is calcnlatcd from U).3) for a given set of values of ,s' and satisfying (4.1). 'bhe binding energy so calculated IS not in general tJie correct one. We, llicrefoie, calculate a number of biuding energies from (0.3) for different values of and satisfying (4*1) and draw a gral)h giving 1\ for different values of .’i and taken, lids enables us to find the exact values of .v and a fui which t)ie binding energy is just what is obtained experimentally. Having got tlie correct values of .s and by the above method which is evidently selbconsistent, we get the mass of mesotron from the usual formulaa//2nc(7)The accuracy of the mass so determined depends on the accuracy with which the binding energy of deuleron is known. The values of x and a cakulatedby the present method are 1.59 and .2848 >« lo^ respectively. With the abo\e value of a in (7) the mass of mesotron is found to be n o  e.u. This is the valueobtained experimentally by Corson and ErodeIt should be noted that from the theory of proton-protcii .scattering already developed by one of us, (Kar, 1942) the value of - ha.s been obtained asJ 2 0 K . C. K ar and R . R , Royjk’ " :iik1 tliu c’orrc.spondhig mass calculated from (7) is iio ,S  e.ii. Il is ill very g^ *od agiceiiieiit with the mass calculated l>y the present method. This slronv.ly suggests that the origin of the short range attractive force must be same in both the cases. The attraction is due to meson field surrounding a neutron and a proton.If, however, it is assumed that the proton-]a'oton interaction is due to exchange and the attractive force is of Majorana tyjie, then, it is clear, the ))artic:le taking part in the exchange cannot he mesotron having chargee. It .slionld l)C ‘neulietto’ having the same mass Imt without the free charge c. As, lujwever, in the formation of nucleus, the ])roton is not found to exhibit any affinity foi proton, it ap[icars to us that there is no exchange although there is attraction beh\ecn the protons. Accordingly, the assmnptioii regarding the existence of a new kind of exchange particle, neiitrelto, is unnecessary.It is thus evident that the attraction between a neutron and a proton or I)etvveeii two protons is due to the meson field. In the former case there is exchange while in the latter ea.se there is none. Accordingly the interaction between protons cannot be of Majorana type. It should be that given by Wigiier* But for 1 — 0, the wave equation is same as (3.1). The next (jueslioii that is to Ire settled is whether the short range meson field ])ennaneutly exists in the I»rotoiis and neutrons. As, liowever, the short range forcu is always found to be altiactive and nevei ie[>ulsivc, it appears likely that the field is of electrical origin and is temixnarily developed during close interaction, due to poloiisation oi neutrons and ])iotous and consujuciit creation of mesotrons within the jiarticles.In conclusion \vc may note that as and IT. are already determined we readily c\Mluate V'o i.c., A j J  from (5.1). Tlie unknown iraramcler A of VukaWa potential cannot be determined by the present method. It is detcniiined from m utroii-proton scattering which we shall discuss on a different occasion. How­ever, we may just mention at present that the value of v"" A .so cleteriiiined is ^  6e which is just the value obtained frciii protornprolon scattering (Kar, 1942). <hi coin]>aring the values of A and Vo we find ? 3.1 jS x 10"'^ from which a roughestimate of the scattering cross-section may be made.Pn\MCM, 1. \ltOKA'l'()K\ .pRi'Sini'Ni.^  Cnujaa-,CArcirrTA.R !•; 1'^  Jv R K NlUtlu' and It.'u lier, R r v t i ^ M o d ,  P h y s . ,  8.Co’ SDii and lUndc, 193s, P h y s .  Rra., 5 3 , 7 ;y  Kill', K. C., h u i .  J o u r ,  P h y s . ^  16, 1S7.O. N., IJcNsel Finu'tiun.s, Cambridge.

A Self Consistent Method of Determining the Mass of Mesotron

http://arxiv.iacs.res.in:8080/jspui/bitstream/10821/1313/1/A%20self%20consistent%20method%20of%20deermining%20the%20mass_Kar%20K%20C.pdf

A Self Consistent Method of Determining the Mass of Mesotron

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