A CALCULATION OF THE TRITON BINDING ENERGY USING SOFT-CORE POTENTIALS by TAI PING NG A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the department of PHYSICS We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1966 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y aval]able f o r reference and study. I further agree that permission., for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of The University of B r i t i s h Columbia Vancouver 8 , Canada i i ABSTRACT In t h i s thesis we calculate the symmetric S-state contribution to the binding energy of the t r i t o n (H^) using the nucleon-nucleon potentials recently developed by Wong (1965). Towards t h i s end we employ a modification of the equivalent two-body method used by Peshbach-Rubinow (1955). i i i TABLE GP CONTENTS I Introduction 1.1 II Reduced Schrodinger's Equation 2.1 11.1 The V a r i a t i o n a l P r i n c i p l e 2.1 11.2 F i r s t attempt: The polynomial fa c t o r 2.4 11.3 Second attempt: The exponential fa c t o r 2.8 III Numerical Results 3.1 IV Summary and Conclusions 4.1 APPENDIX BIBLIOGRAPHY ACKNOWLEDGEMENTS I am indebted to Dr. J. M. McMillan for suggesting the problem and for generous assistance with i t . 1 .1 CHAPTER I INTRODUCTION In t h i s theses we s h a l l calculate the binding energy of the t r i t o n (H^) using recently developed nucleon-nucleon potentials. In focussing our attention on the t r i t o n , we are imme-d i a t e l y faced with t r y i n g to solve a three-body problem. Major steps i n t h i s d i r e c t i o n were taken by Derrick and B l a t t ( 1 9 5 8 , 1 9 6 0 a , 1960b) who have constructed a complete set of states l n terms of which the t r i t o n wave function may be expanded, and derived a set of sixteen p a r t i a l d i f f e r e n t i a l Schrodinger equation i n three variables f o r the expansion c o e f f i c i e n t s . That work, while providing a great s i m p l i f i c a t i o n of the nuclear problem, s t i l l c l e a r l y leaves one with an extremely d i f f i c u l t mathematical problem. Indeed r e l a t i v e l y l i t t l e has been done i n f i n d i n g i t s s o l -ution. Instead, the common procedure has been to compute t r i t o n wave functions and energies using a v a r i a t i o n a l parameter approach. That i s , a form of the wave function containing various parameters has been assumed, and the expectation value of the Hamiltonian of the system has been computed and minimized with respect to these parame-te r s . This approach has obvious disadvantages; there i s 1.2 no guarantee that the assumed function approximates the actual wave function, and there i s no d i r e c t correspon-dence between the wave function parameters and the param-eters appearing i n the nueleon-nucleon p o t e n t i a l . Some work on a non-variational parameter approach f o r fi n d i n g a t r i t o n wave function and energy has been done, i n p a r t i c u l a r , by Feshbach and Rubinow (1955) and by McMillan (1965). Feshbach and Rubinow assumed that the symmetric S-state component of the t r i t o n wave function (which component i s known from the work of B l a t t et a l (1962) to be strongly dominant) i s a function of the sum of internucleon distances only, and derived an ordinary d i f f e r e n t i a l equation based on t h i s assumption. McMillan (1965) discussed some properties of the Feshbach-Rubinow equation and solved i t numerically using potentials that f i t t e d low energy data. In t h i s thesis we are interested i n using the Feshbach-Rubinow method with the Wong (1965) "soft core" potentials which give a good f i t to the two-nucleon data to 250 MeV and (which have been used successfully i n nucl-ear matter c a l c u l a t i o n s . These pot e n t i a l s have a strong repulsive core but not a "hard core". We have found that when the Wong potentials are used d i r e c t l y i n the Feshbach-Rubinow equation,, no bound three-nucleon state Is found. 1.3 The reason f o r t h i s i s that the Feshbach-Rubinow approxi-mation makes no special concession to the form of the t r i t o n wave function f o r small i n t e r p a r t i c l e distances and, as a re s u l t , i s inadequate f o r potentials with strong rep-ulsi v e cores. Indeed, the repulsive core i n the two-nucleon pot e n t i a l requires that the t r i t o n wave function be small f o r small i n t e r p a r t i c l e distances, and we must b u i l d t h i s into our approximate form, i n the same way thay Feshbach and Rufelnow have required that f o r hard core potentials the t r i t o n wave function vanish inside the core. In the remainder of t h i s thesis, attempts w i l l be made to show the a p p l i c a b i l i t y of the Wong p o t e n t i a l to the t r i t o n problem. 2.1 CHAPTER II REDUCED SCHRODINGER'S EQUATION I I . 1 The V a r i a t i o n a l P r i n c i p l e Aa stated i n the Introduction, we s h a l l i n t h i s theses use the Feshbach-Rubinow (1955) method f o r f i n d i n g an approx-imate t r i t o n energy. The es s e n t i a l step i n t h i s method i s the restatement of the more customary Schrodinger d i f f e r e n -t i a l equations i n terms of a v a r i a t i o n a l p r i n c i p l e . In t h i s subsection we s h a l l outline the steps leading to the v a r i a -t i o n a l p r i n c i p l e used by Feshbach and Rubinow and by McMillan (1965). The set of coupled d i f f e r e n t i a l equations f o r the comp-onents of the J=l/ 2 , T=l/2 wave function of the bound three-nucleon system can be found from Derrick ( i 9 6 0 ) . When a l l but the symmetric S-state component f ^ i s neglected, the following equation r e s u l t s - t t ^ T T 1 2 3" 1 V H r j U = E f x (2.1) m 'fr-i r i 2 r i : , 2 r 2 r 3 ^ r 2 9 r 3 J c y c l i c where E i s the energy i n the centre-of-mass system, the r ^ denote the internucleon distances, and where i n terms of the usual nucleon-nucleon potentials V = ~ ^ v s i n g l e t - e v e n + V c e n t r a l - t r i p l e t - e v e n ^ ( 2- 2) 2.2 We use, with Derrick and B l a t t (1958), the internucleon distances as'generalised coordinates. Indeed, there are six generalised coordinates needed i n the centre-of-mass system; Derrick and B l a t t use the three r± which specify the size and the shape of the t r i a n g l e formed by the three p a r t i c l e s , and the three Euler angles which specify the orient a t i o n i n space of t h i s t r i a n g l e , r e l a t i v e to some standard orien t a t i o n . The function f 1 ( r 1 , r 2 , r ^ ) i s symmetrical under the i n t -erchange of any two nucleons; i t w i l l be taken to be normal-ised as folows C O dr. «6 dr. <6 dr-lr;-r2i rlr2r3 ^ r l ' r 2 , r 3 ^ ' 2) =1 ( 2 . 3 ) The e s s e n t i a l step now i s to replace ( 2 . 1 ) by the following v a r i a t i o n a l p r i n c i p l e 6 O drl\dr2\dr3 rlr2r3 2i *Fi 2 r5+r l-r?8f, 3f, ) , g j * 3 xl " 2 r 2 r 3 ^ c y c l i c 1 8 f i ^ 2 ^ = 0 ( 2 . 4 ) - V ^ f ^ j + E f * from which equation ( 2 . 1 ) can be obtained using the Euler-Lagrange equation. Feshbach and Rubinow (1955) and McMillan (1965) proceed at t h i s point to assume that f x = f x ( R ) ( 2 . 5 ) 2.3 where R = ^ r l + r 2 + r 3 ) which approximation may be regarded as a generalization of the t r i a l exponential function. When (2.5) i s used i n (2.4), some of the in t e g r a l s may be performed, and subsequent use of the Euler-Lagrange equation y i e l d s the Feshbach-Rubinow ordinary d i f f e r e n t i a l equation. McMillan (1965) has solved t h i s d i f f e r e n t i a l equation numerically using p o t e n t i a l s that f i t t e d low energy data. In t h i s t h e s i s we are interested i n using the Feshbach-Rubinow method with the Wong (1965) "soft core" potentials which give a good f i t to the two-nucleon data to 200MeV and which have been used successfully i n nucl-ear matter c a l c u l a t i o n s . These po t e n t i a l s have a strong re-pulsive core; we have found that when they are used d i r e c t l y i n the Feshbace-Rubinow equation, no bound three-nucleon state occurs. The reason f o r t h i s i s as follows: approximat-ion (2.5) makes no special concession to the form of the t r i -ton wave function f o r small i n t e r p a r t i c l e distances and, as a re s u l t , i s Inadequate f o r our purpose here. Indeed, the repulsive core i n the two-nucleon p o t e n t i a l requires that the t r i t o n wave function be small i n small i n t e r p a r t i c l e distances, and we must b u i l d t h i s into our approximate form, i n the same way that Feshbach and Rubinow have required that f o r hard core potentials the t r i t o n wave function vanish inside the core. 2.4 In the remainder of this Chapter, we describe two att-empts made to find a simple wave function which would mark an Improvement to the t r i a l wave function used by Peshbaefe and Rubinow (1955) and McMillan (1965) and which would be suitable with the Wong potentials. II.2 F i r s t attempt: The polynomial factor The f i r s t simple function proposed which i s symmetrical under the interchange of any two particles and tends to zero as any one of the r.'s tends to zero, i s (2.6) where R =?( rl + R2 + r3) and a i s a parameter to be determined. When assumption (2.6) i s used i n (Z4) we obtaine dR 4 ' DK L 2 + ^2 U e f f ( R ) C l ( a ) R 6 a + 5 + ^ C0(a)R6o+%2J= 0 (2.7) 2 .5 where U e f f = £ (dx dy [y(Rx) { ( 2-x-y)xyj 2 a + 1J (2 .8) and 0 ( a ) = | dx jdy | { (2 -x -y )xy] 2 a + 1j ^(a) = j d x J d y | ( 2 - x - y ) x y } 2 a ( - 8 + l 6 x - x 2 ) ( l 6 - 8 x ) y - 8 y 2 J C 2(a) = \dx | dy | ( 2 - x - y ) x y y a _ 1 ( - l 6 x + 4 G x 2 - 3 2 x 3 + 8 x 4 ) o H X +(-l6+64x-60x 2 +l6x 3 )y+(40-60x+24x 2 )y 2 ( -32+ l6x )y 3 +8y 4 ) + C 3(a) = (dx (dy [ | 2-x-y)xy} 2 Q ! ~ 1 ( - l 6 + 3 2 x - 8 x 2 - 8 x 3 + 2 x 4 ) o »~* +(32-48x+8x 2 +4x 3)y+(-8+8x+6x 2)y 2 +(-8+4x)y 3 +8y 4 } Application of the Euler-Lagrange Equation i n ^ y i e l d s m R D a + : : > dR dR m R dR m C^ ^ Rd' e l l / e i C where 8 = o~2- (2.10) L ' l Now defining u(R) by -(3a4 + fo ^(R) = R d d u(R) (2.11) 2.6 Then ( 2 . 9 ) may be written as h 2 H 2 „ / R x 8c n (a) aR 0^{a) where Veff< R> * i| 2|^ 2+(3a+5/2)(3a +3/2-p) +4aC 3/C 1^ ^ l ^ e f f ^ (2.13) After transformation (2.11), normalization i n t e g r a l (2.3) now becomes oO r dR u 2(R).R - 6 a ^ - 5 ~0 — " Equation (2.12) i s the reduced Schrodinger Wave Equation and u(R) i s uniquely determined once the two-body p o t e n t i a l i s given. We note that setting a=0 i n (2.13) f o r a range of a (a=0, 0.1, 0.5, 1.0, 5.0) f o r the potentials used by McMillan (1965) and f o r the Wong (1965) p o t e n t i a l s . In a l l cases the minimum values of V ^ ( R ) occurred when a = 0 (see^ f o r example, Figure 1 which shows V e f f,(R) f o r the PBB&2 p o t e n t i a l used by McMillan (1965); further, V e f f ( R ) was positiv e f o r a l l values of R and a f o r the Wong potentials, showing then that no bound state was possible. The reason 2.8 f o r our f a i l u r e i s t h a t while the polynomial f a c t o r i n (2.6) makes the three-nucleon f u n c t i o n small f o r small i n t e r p a r t -i c l e d i s t a n c e s , i t becomes la r g e f o r l a r g e I n t e r p a r t i c l e d i s t a n c e s . As a r e s u l t the k i n e t i c energy term become larg e and dominates the p o t e n t i a l i n t e g r a l to the extent that no bound s t a t e i s p o s s i b l e . Thus approximation (2.6) i s i n f e r -i o r t o the o r i g i n a l Feshbach-Rubinow approximation. I I . 3 Second attempt: The exponential f a c t o r In view of the inadequacy of the polynomial f a c t o r bu-i l t Into the t r i t o n wave f u n c t i o n (2.5), we repeated the procedure w i t h a more complicated f a c t o r which i s s t i l l com-p l e t e l y symmetrical, which goes to zero more q u i c k l y as the i n t e r p a r t i c l e d i s t a n c e vanishes and which goes to u n i t y f o r la r g e i n t e r p a r t i c l e d i s t a n c e s . More s p e c i f i c a l l y , the t r i -ton wave f u n c t i o n we propose i n t h i s s e c t i o n i s where a and n are parameters to be determined. For I l l u s t r a t i v e purposes, we have p l o t t e d I n Figure 2 the f u n c t i o n exp(- —• ) f o r a=0.5 -«*d .n=l, 2,3. r When equation (2,15) I s used i n (2.4) the E u l e r -2.10 Lagrange Equation i n (j) y i e l d s the following reduced Schro-dinger equation m dR^ e i r F 1(R) where y(R) i s defined by (2.16) R 5 / 2 ( P l ) 1 / 2 (2.17) The e f f e c t i v e p o t e n t i a l V e f f ( R ) i s found to be W * ) - I > ! ( 8 ^ -2 / F 0 16P, 'eff 4 P R « 2 « 2 m v p 1 P l P^ P, J R ^ 4(4-n)F„ an 15 - £ ) — + _ ( ( 3 - 2 n)P 4 Rn+2 4 R 2 24 r1 r1 — fdx /dyJv p0 J 1 2aP(x,y) ) (Rx)(2-x-y)xy exp(- n )V ( 2 . l 8 ) R o i - X where the P's are functions of R and are defined by f' C1 (- 2aF(x,y) > P Q(R) = dx dy (2-x~y)xy exp( — — - ) J 4) l-x R (2.19) 2.11 I I ( 2aP(x,y) ^ P 1 (R) = dx J d y J G ^ y ) exp( ) j (2.20) f' C r 2aP(x,y) ") P 2 (R) = J d x J d y | Q 2 ( x , y ) exp( — )J- (2.21) I f ( 2aP(x,y) >, P 3 (R) - jdx jdy - |G 3 (x ,y) exp( — )j (2.22) r1 f ' f 2aP(x,y) ) P 4 (R) = j d x j d y j p f o y ^ f o y j e x p f - - J J )j (2.23) "0 .1 r1 r f Cr \ 2 2aP(x,y) ") P 5 (R) = J d x J d y | [ p ( x , y ) J G 1 (x ,y)exp( )J (2.24) 0 i-x f C ( 2aP(x,y) ^ P 6 (R) = Jdx Jdy | P ( x , y ) G 2 ( x , y ) e x p ( )\ (2.25) o where the G ( x , y ) ' s and P(x,y) are defined as fo l lows 2 G 1 ( x , y ) = z(6x+6y-4) - z(x-y) ( 2 . 2 6 ) c y c l i c ^ ^ x x y xy <y x,y ,x l««cf; [WE]] ^ 2 + ^ i V («.*) 2.12 P(x,y) (2.29) cyclic'' x1 x,y,z n wherein z = 2-x-y. Normalization i n t e g r a l (2.3) now becomes oo (2.30) o F i n a l l y , we note that setting a=0 i n (2.16) y i e l d s the Feshbach-Rubinow (1955) equation. Some rather general remarks can be made about equati-ons (2.16) and (2 .18) . We note f i r s t that the functions F i(R) (1=0,1,2,3,4,5,6) approach non-zero constants as R—*00, and vanish as R — > 0 . Further, one can see that since the two-nucleon po t e n t i a l i s short ranged, the i n t e -gral i n (2.18) i s 0(R~3) as R->°°. (That a short-range two-nucleon po t e n t i a l leads to t h i s long-range behaviour re-sult s form the f a c t that large values of R do not preclude the occurence of small i n t e r p a r t i c l e distances as was poin-ted out by McMillan.) Thus one sees that as R-^ -op, V e f, f(R) i s dominated by the terra 15/4R , as i n the o r i g i n a l Feshbach-Rubinow case. The behaviour of Vef^.(R) at the o r i g i n i s more d i f f i c u l t to see a n a l y t i c a l l y (since a l l of 2 . 1 3 the vanish). Our numerical r e s u l t s indicate however that F^ vanishes the quickest, so the V e f f has a repulsive core near the o r i g i n which i s steeper than that occuring with the o r i g i n a l Feshbach-Rubinow equation. This w i l l be demonstrated numerically i n C h a p t e r l l l . 3.1 CHAPTER I I I NUMERICAL RESULTS The eigenvalue problem (2.16) has been solved numeri-c a l l y f o r v a r i o u s phenomenblogical Yukawa-like p o t e n t i a l which f i t the two nucleon data. The f i r s t two p o t e n t i a l s used were the re l e v a n t p a r t of the g=0 Peshbach-Pease (1952) p o t e n t i a l and the PBBK2potential used by McMillan ( 1 9 6 5 ) i . e . W e t = V c e n t r a l - t r l p l e t = - ^ . 4 8 2 S L^8M H e v ( 3 . 1 ) ( 3 . 2 ) The combination of p o t e n t i a l s (3.2) w i l l be l a b e l l e d PBBK2-potential f o l l o w i n g McMillan. P o t e n t i a l s (3.1) and (3.2) f i t the low energy two-nucleon data. For each p o t e n t i a l used, we c a l c u l a t e d the three body e f f e c t i v e p o t e n t i a l (2.18) f o r a=0, 0.25, 0.50. 0.75, 1 and n=0., 0.25, 0.50, 1.0, 2.0, 3.0, 4.0. I t was found f o r both PP g=0 p o t e n t i a l and PBBK2pote-n t i a l that the lowest value of V e f f ( R ) oceured when a=0.5 3.3 and n • 0 . 5 . The three body e f f e c t i v e potentials and the corresponding wave functions that s a t i s f y the reduced Schrodinger equation (2 .16) f o r t h i s case are plotted on Figure 3 . The eigenvalues are given i n Table 1. Potential Eigenvalue E of Feshbach-Rubinow Equation (a=n=0) (MeV) Eigenvalue E of (2.16) (a=n=0.5 ) MeV) Feshbach-Pease g=0 -2 .47 -3.55 PBBK2 -4.19 -5.39 Table 1: Comparison of r e s u l t s : the computed eigenvalues. The experimental value i s E=-8.492MeV. From Table 1, we see that the eigenvalue E of the reduced Schrodinger Equation (2.16) i s roughly 1 MeV lower than that of the Feshbach-Rubinow equation, thus inducating an Improvement of our t r i a l wave function over the Feshbach-Rubinow wave function. The calc u l a t i o n s were repeated with the more recent phenomenologlcal Wong (1965) potentials: V(r) - v^M-M) + Yz**ti-/v) . £^m^fM-N*)) (3-3) 3-5 where ft = mc/h Is the inverse range f o r the pion of average mass m . TT M = average nucleon mass. g£ = the OPEP coupling constant. v l > u l = parameters specifying a short range repulsive p o t e n t i a l . v 2 , u 2 = P a r a m e t e r s specifying a longer range a t t r a c t i v e p o t e n t i a l . k The masses that Wong used f o r f i t t i n g the softcore p o t e n t i a l are l i s t e d i n Table 2 (Table 4 of Wong (1965)). The r e s u l t i n g p o t e n t i a l parameters are shown i n Table 3 f o r d i f f e r e n t choice of the coupling constant g£ from 14 to 10 (Table 5 of Wong). For each p o t e n t i a l ( 3 . ^ - ) used, we again calculated V f f ( R ) f o r a=0, 0.25, 0.50, 0.75, 1.0, and n=0, 0.25, 0.50, 1.0, 2 .0, 3 . 0 , 4 . 0 . I t was i n t e r e s t i n g to f i n d that f o r a l l Wong potentials, the lowest value of V e f f ( R ) occured when a=0.5 and n=2 . 0 . The manner that e f f e c t i v e p o t e n t i a l (2.18) varies with a change i n n and a i s demonstrated on Figure 4 f o r Wong's Ynp2 p o t e n t i a l . As i s shown there, V e f^.(R) changes quite markedly when n i s varied. We note also that V e f£.(R) i s everywhere p o s i t i v e when a=0 (the Feshbach-Rubinow case) 3.6 which indicates no bound state in this case. We further note the steep repulsive core in the effective potentials, as noted in the previous section. m TT 2M m m 7T+ V 2 M p = 134.97 MeV = 139.58 MeV = 939.505 MeV = 938.211 MeV + Table 2: The masses used in the soft core potential (eq.(3.4)). Potential Yppl Ypp2 Ypp3 Ynpl Ynp2 Ynp3 4 14 12 10 14 12 10 ra^rneV) 134.97 138.04 / ^ ( f n f 1 ) 3 . 2 5 2.63 2.25 2.95 2.48 2 .20 v2(MeV-fm) -4622.8 -1751 -989.8 - 2 9 4 9 A -1435.3 -951.7 v3(MeV-fm) 14280 7930 5620 11100 7200 5650 v1(MeV-fm) -446.1 -447.9 -449.3 -460.1 -461.3 -462.2 Table 3: Parameters f o r Wong's ^SU s t a t e soft-core p o t e n t i a l s . 3.7 In Figures 5 and 6, the equivalent three-body poten-t i a l s (2.18) are plotted f o r Wong's nucleon-nucleon poten-t i a l s . And i n Figures 7 and 8, the wave functions y(R) corresponding to Wong's potentials are plotted f o r a=0,5, n=2.0. The corresponding energy eigenvalues are tabulated i n Table 4. Wong Potential Depth of V e f f ( R ) Energy Eigenvalue (TWo-body) (2.18) E ( MeV) (MeV) Yppl -115.8 (R=1.7F) -23.8 Ypp2 -101.2 (R=1.7F) -19.5 Ypp3 - 92.1 (R=1.7F) -16.6 Y f e l -111.2 (R=1.7F) -23.0 Y$p2 - 99.6 (R=1.7F) -19.7 ' YiSp3 - 90.9 (R=1.7F) -17.7 Table 4: Energy eigenvalue E f o r Wong potentials 4.1 CHAPTER VI SUMMARY AND CONCLUSIONS We have c a l c u l a t e d the symmetric S-state c o n t r i b u t i o n to the b i n d i n g energy of t r i t o n (H^) f o r the Wong (1965) p o t e n t i a l s u s i n g Feshbach-Rubinow (1955) method. The r e s u l -t s found are t a b u l a t e d i n Table 4. I n view of the f a c t we found that when the Wong poten-t i a l s were used d i r e c t l y i n the Feshbach-Rubinow equation, no bound three nucleon s t a t e occurs, two attempts were made to f i n d a simple symmetrical wave f u n c t i o n which would mark an inprcvement to the approximation used by Feshbach and Rubinow and which would be s u i t a b l e w i t h Wong's s o f t core p o t e n t i a l s . In the f i r s t attempt, the t r i a l wave f u n c t i o n w i t h a polynomial f a c t o r f a i l e d to support a bound s t a t e . The reason f o r the f a i l u r e I s that while the polynomial f a c t o r makes the three-nucleon f u n c t i o n small f o r small i n t e r -p a r t i c l e d i s t a n c e s , I t becomes l a r g e f o r l a r g e I n t e r p a r -t i c l e d i s t a n c e s . As a r e s u l t the k i n e t i c energy term becomes l a r g e and dominates the p o t e n t i a l i n t e g r a l to the extent that no bound s t a t e i s p o s s i b l e . In the second attempt we then repeated the procedure w i t h a more complicated f a c t o r which i s s t i l l completely 4.2 symmetrical, which goes to zero more q u i c k l y as the i n t e r -p a r t i c l e d i s t a n c e vanishes and which goes to u n i t y f o r large i n t e r p a r t i c l e d i s t a n c e s . With t h i s f u n c t i o n we found a s t r o n g l y bound s t a t e as i s c l e a r from Table 4. I n f a c t , the bi n d i n g energy may even be bigger than we found since we have not made an extensive parameter search. Indeed, the bi n d i n g energies we found are so la r g e that the r e s u l t s seem to i n d i c a t e the Wong p o t e n t i a l s are a l s o not completely sucessful nucleon-nucleon p o t e n t i a l s . V*Jj(R) M $ov R (F FIGURE 5 EFFECTIVE POTENTIALS FOR TRITON USING WONG 'S Y>P - NUCLEOLI • NUCLON POTENTIALS 80-44 60h 4oV toy -/20V 4-5 F / G . 7 : WAVE. FUNCTION CORRBSpONDlNQ TO WONG 'S Yf>p - POTENTIALS CF/G-5) J I I I I ' I I i / 2 3 4 5 6 7 8 RCF) FIG 8 WAVE FUNCTION CORRESPONDING TO WONG'S vnp- POTENTIALS cno.e) APPENDIX The numerical computation was ca r r i e d out on the IBM 7040 computer of the UBC compting centre. We calculated V e f f ( R ) f o r a=0, 0 .25, 0 .50, 0 .75, 1, n=0, 0 .25, 0.50, 1.0;, 2 .0 , 3 .0 , 4 . 0 . For a given a and n we compted V e f f ( R ) from R=0.05F to R=8.0F with 0 .05F i n t e r v a l . The time taken to compute each point i s 13 second. A l l the double integration were ca r r i e d out by the Gaussian double integration subroutine of the UBC computing centre l i b r a r y with minor modifications to enable i t carry-ing out r e p e t i t i v e integrations. Eight point Gaussian was found to give s u f f i c i e n t accuracy. Solving the reduced Schrodinger d i f f e r e n t i a l equation involves solution to the eigenvalue problem (2.16). Numer-i c a l integration of the d i f f e r e n t i a l equation was done by the Runge-Kutta method. The eigenvalue E was found as follows: 1) an arbitary value of E was chosen arid the d.e. was integrated forward from R=0.1F to 3.OF at 0.1F i n t e r v a l and the logarithmic derivative of R=3.0F was calculated. 2) the d.e. (2.16) was integrated backward from R=8.1F to R=3.0F at 0.1F i n t e r v a l with the same value of E as i n 1) and the logarithmic derivative at R=3.0P calculated again. 3) The procedure was repeated with d i f f e r e n t value of E with the logarithmic derivative calculated i n 1) and 2) d i f f e r e d by lCT^F" 1 (Successive values of E were chosen using a l i n e a r approximation.) The corresponding value of E i s the eigenvalue. The wave function obtained i s then scaled by the nor-malization i n t e g r a l (2.30). BIBLIRGRAPHY Biedenharn, L . C , B l a t t , J.M. and Kalos, M.H. 1958. Nucl. Phys. 6, 359 Derrick, G.H. and B l a t t , J.M. 1958. Nucl. Phys. 8, 310 Derrick, G.H. I960. Nucl. Phys. 16, 405 Feshbach, H. and Pease, R.L. 1952. Phys. Rev. 88, 945 Feshbach, H. and Rubinow, S.I. 1955. Phys. Rev. 98,188 McMillan, M. 1965. Can. J . of Phys. 43, 463 Preston, M.A. 1962. Physics of the nucleus (Addison-Wesley Co., Inc., Reading, Mass.) Wong, CW. 1965. Nucl. Phys. 71, 385
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A calculation of the triton binding energy using soft-core potentials Ng, Tai Ping 1966
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Title | A calculation of the triton binding energy using soft-core potentials |
Creator |
Ng, Tai Ping |
Publisher | University of British Columbia |
Date Issued | 1966 |
Description | In this thesis we calculate the symmetric S-state contribution to the binding energy of the triton (H³) using the nucleon-nucleon potentials recently developed by Wong (1965). Towards this end we employ a modification of the equivalent two-body method used by Peshbach-Rubinow (1955). |
Subject |
Particles (Nuclear physics) |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-08-06 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0103709 |
URI | http://hdl.handle.net/2429/36548 |
Degree |
Master of Science - MSc |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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