A MICROSCOPIC DESCRIPTION of HYPERNUCLEUS PRODUCTION using FAST KAONS • by. Robert J . Esch B. Sc. (Hon.), University of Guelph, 1970 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the department of PHYSICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA APRIL, 1972 In p r e s e n t i n g t h i s t h e s i s 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 U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may by h i s r e p r e s e n t a t i v e s . be granted by permission. Department of PHYSICS The U n i v e r s i t y o f B r i t i s h Vancouver 8, Canada Date 6 A p r i l , 1972 Department or I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n written the Head of my Columbia s h a l l not be allowed without my To my wife, Rosemary Preface and Acknowledgements In the first year of my studies, Professor Vogt and I worked on such varied problems as low energy nuclear reactions, nuclear astrophysics and medium energy pion physics. Needless to say I received exposure to many varied problems in physics. Because Professor Vogt was to go on Sabbatical the following year, he personally made arrangements with the National Research Council of Canada, The Department of Physics, U.B.C. and the Department of Theoretical Physics, Oxford, so that I could go to England with him. Shortly after arriving in Oxford, Dr. N. Tanner of the Depart- ment of Nuclear Physics, Oxford, informed us of the need for the calculations out of which this work arose. I would personally like to thank several friends for their help and encouragement: To my friend and teacher, Professor Erich Vogt, for his help, patience, and encouragement throughout the course of this work; Dr. N. Tanner, who pointed out the need for these calculations and for his constant help and encouragement and to Dr. R. Rook, who calculated the lambda wave function and for his Interest in the problem, I would also like to thank Professor R. E. Peierls and Professor E. Vogt for making i t possible to come to Oxford; the National Research Council of Canada for support through a I967 Science Scholarship and the Oxford Computing Center where most of the calculations were performed.,. Finally, and above a l l , I wish to express my gratitude to my wife, Rosemary, for carefully typing the thesis and without whose constant patience, love and encouragement this work would not have been possible. t * TABLE OF CONTENTS Abstract Chapter 1: Introduction Chapter 2 i Reaction Mechanism Chapter 3s Nuclear Wave Functions Chapter 4: Many Body Transition Operator Chapter 5: Applications 1 : (a) Helium (b) Carbon (c) Chapter 6: Oxygen Conclusion and Discussion References Appendix 1: Multiple Scattering Theory Appendix 2: Why the f a c t o r of^A/ i n ~ff; ? Tables and Graphs ABSTRACT The differential cross sections for the production of definite lambda hypernuclear states, within the single scattering, impulse approximation, are calculated from the reaction r\(K.~ Tf)^ on nuclear targets of helium, } carbon and oxygen at various K-meson Incident momentum. It is shown that these predictions are very sensitive to the three momentum transfer and to the wave function of the bound lambda in the hypernucleus. From the calculations, i t is shown that i t is possible to observe their production by studying the missing mass spectrum of the emitted pion. Introduction Chapter 1 The production and study of hypernuclei has been and continues to be a very fruitful and exciting overlap between nuclear and particle physics. Since the early fifties, hypernuclear physics has provided basic information concerning the lambda-hyperon-nucleon interactions and hypernuclear studies have, also supplied the nuclear physicist with information basic to understanding nucleon-nucleon forces in matter (Davis, D.H. and Sacton, J., 1967 and references therein). Unfortunately, however, their production has largely been limited to manufacture from slow negative K-mesons, which are captured in Coulomb orbit3 about a nucleus. atomic (is) orbital. the nucleus. The captured K-meson cascades down toward its As i t does, i t has a larger and larger overlap with But because the X-meson reacts strongly with nucleons, the cascading kaon is absorbed'by the nucleus. The energy released in the reaction n(K~,Tr~)A may leave the produced lambda-hyperon In the nucleus, 0 in some "bound" state or more often both the lambda and pion will escape, leaving an excited nucleus. The nucleus may even "explode", perhaps pro- ducing a hypernucleus fragment. The life and death of hypernuclei are usually recorded by the tracks left (before and after decay) in emulsion photographs. It is the study and analysis of these tracks that have yielded binding energies, angular distributions, and branching ratios of decay channels. This data has then been used to study, phenbmenologically, the lambda-nucleon force and nucleon-nucleon forces in nuclei. From this theory, i t is possible to make models of the hypernucleus states and to check nuclear models. Capture from Coulomb bound orbits is the easiest way to make hyper- 2. nuclei, but the process has an unfortunate drawback. In the capture of a K-meson at rest, the momentum transfer to the lambda is typically 250 MeV/c. As a result, the production rate from captured K-mesons is limited to less 12 than 2fo for a l l stopped kaons. For example, in C, only one in approx- imately 350 K-mesons that are captured in Coulomb orbits produce lambda hypernuclei (Davis, D.H. and Sacton, J., I967). This is quite easy to understand when one considers what the production rate will depend upon. If one can consider the reaction C (K^TT)'?^. occurring with only a single U nucleon, the production rate will be roughly proportional to the Fourier v transofrm of the product of the struck neutron and the final bound lambda wave functions. But because the lambda is bound (for p-shell hypernuclei) by only ~ lOMeV, in its s-state, the high momentum components in the form factor will be suppressed. As a result, the production rate will be re^ duced. This method makes i t almost impossible to study excited hypernuclear states i f they exist. In order to study hypernuclei and their excited states, this problem of high momentum transfer must be overcome. Dalitz (Dalitz, R.H., I969) 1 proposed that i t may bojpossible to study hypernuclear levels, by using "fast"kaon beams of approximately 600 MeV/c. The reaction (1) 1 K" + n •—^ A° + TT at 600 MeV/c has one very encouraging feature. When the incident kaon has roughly 550 MeV/c, the emitted pion comes off with 550 MeV/c momentum at zero degrees and hence the produced A° comes at rest in the laboratory. Thus, at these incident momenta, i t is possible to "deposit" a lambda into the nuclear system. This peculiar behaviour in the kinematics is basically because the reaction (1) is strongly exothermic with a Q*value of 178.19 MeV. This method for producing hypernuclei has become known as "producing hypernuclei with walking lambdas", (Bonazaola, G.C. et al, 1970). By detecting the pions emitted at forward angles in the reaction K~ -r nucleus —* per k\u.oUu.s -+- TV the determination of hypernuclear levels can then be made from a direct kinematlcal analysis for the missing mass or energy loss. The missing mass then gives, the binding energies of the hypernuclear levels produced. For excited states, i f production rates are appreciable, more complex decay schemes of the hypernucleus will be available for study, by using If -rays in coincidence with the emitted pions (Bonazzola, G.C. et al, 1970). One can estimate that the total differential cross section for reaction will be given by where /d£l)£ree is the differential cross section for reaction (1) at the same incident K-meson momentum and K £^(A j<r. jQTj.) the effective neutron e number (Kolbig, K.S., Margolis, B., 1968). This estimate contains a sum over a l l possible final states and also contains approximate absorptive and multiple scattering effects in N£j(A,<ri ,<TV)> <T~, is the total k^-rV cross e section, T t theTT-fs/ total cross section. Calculations for a uniform sphere model for O ^ reduce the effective number of p-shell neutrons from 1 6 to 1.85 (^2). (cl<r/jLXi.)free at 600 MeV/c 0°, is 4mb/st (lab). From this model one gets that the (<*<r-/cLfZ)hyper, at 0° is roughly 8 mb/st. Neglecting multiple scattering and absorptive effects (d<r/dL£X) hyper, at 0° is *-2^ mb/st. The true answer will probably f a l l between these two estimates. With only these rough estimates available and the experiments scheduled to staxt i n the spring of t h i s year, i t became i n t e r e s t i n g to make more accurate estimates f o r the d i f f e r e n t i a l cross sections and to study not the closure approximation to G^/otQ^hyper., but the contribution from various hypernuclear states (Tanner, N, 1971). I t i s t o these questions that the present work Is addressed. The d i f f e r e n t i a l cross sections f o r hypernucleus production on ^He, and 0 have been calculated i n the single scattering, impulse approx- imation a t incident momenta 500, 600, 700 and 800 MeV/c to d e f i n i t e hyper- nucleus states. The predictions made here are found to be very s e n s i t i v e to the momentum transferod to the bound lambda and to the wave function of the lambda. The treatment presented here consists of e s s e n t i a l l y four steps: (1) phenomological description of reaction (1) at incident K-meson l a b . momenta 500, (2) 600, 700 and .800 MeV/c; a proper treatment of the nucleax wave function, including a n t i - symmetry ; (3) some appropriate description of the A -hyperon wave function and (4) a proper treatment of kinematics i n the i n i t i a l and f i n a l states. When these steps are completed, they can be put together under the single scattering impulse approximation (Appendix 1) t o y i e l d t h e o r e t i c a l estimates f o r the production of varrbus hypernuclear energy l e v e l s at forward angles. 5. Chapter 2 Reaction n (K~> IT" ) A° In the d e s c r i p t i o n of hypernucleus production, as In a l l descriptions f o r the i n t e r a c t i o n of a p a r t i c l e with a many p a r t i c l e system, i t i s necessary to r e l a t e i n some l o g i c a l way, the elementary two body interactions to the t o t a l t a r g e t - p r o j e c t i l e i n t e r a c t i o n . In the formal theory of mul- t i p l e s c a t t e r i n g (Goldberger, M.L. and Watson, K.M., 1964), the single scattering impulse approximation t e l l s one that the i n e l a s t i c or c o l l i s i o n cross section w i l l be e s s e n t i a l l y t&e product of three f a c t o r s : ( l ) the number of e f f e c t i v e scatterers, (2) the i n e l a s t i c form f a c t o r squared and (3) the elementary free two body d i f f e r e n t i a l cross section. The s t r u c t - ure of the target has no dynamical e f f e c t on the process except i n a t r i v i a l kinematical way (see Appendix 1) and (Figure l ) . In t h i s section, we present the information concerning the free d i f f e r e n t i a l cross section f o r the reaction which i s necessary f o r our t h e o r e t i c a l p r e d i c t i o n of the hypernucleus production cross section. At low energies (less than 300 MeV/c. kaon momentum) the reaction rv(K.",TI')A i s predominantly s-wave and has been extensively (Kallen, G., 1964). In the region between 300 MeV/c. to 600 discussed MeV/c. the experiments have been scarce and not a great deal i s known, but p-wave e f f e c t s are noticeable. From 600 HeV/c. to 800 MeV/c, the experimental data i s not very good but f i t s to the d i f f e r e n t i a l cross sections have been made by expanding the d i f f e r e n t i a l cross section i n terms of Lege'ndre polynomials of the cosine of the scattering angle i n the center of mass (3) with k, the momentum i n the center of mass (^ = C- i) . The c o e f f i c i e n t s are 6. incident momentum dependent and the Legendre polynomial coefficients versus incident K-meson momentum are shown in Fig. 2 (Armenteros, R. et al, I 9 6 8 ) . In Fig. 3 the experimental differential cross section is shown for reaction (1) at ?77 MeV/c. incident kaon momentum. From these graphs i t is clear that the reaction in the momentum range 500 - 800 MeV/c. is predominately s-wave and p-wave with l i t t l e interference between them. Briefly, i f we expand the scattering amplitude -jOk^S") in terms of Legendre polynomials and keep only terms up toX= 1 , one has The differential cross section is I^Ck.eM**. To this order From the Fig. 2, A| for reaction (l) is approximately zero and in this work, i t was taken as identically zero. This is not necessary but i t simplifies the work without introducing a significant error. Now from relativistic scattering theory, the differential cross section in the center of mass is given by (6) Ljh) "IT'S I *• ' where s is the square of the total energy in the center of momentum, ^ , K are final and initial momentum and (7) XtXjUj,^ = x * + y * + a - 2-^l -2.** From equations (5) and (6), the square of the absolute value of the free two body transition amplitude is (8) 1*1*= 21 A.Ck^ P^Cccs &) In the calculations, that have been completed, we used the exper- 7. imental f i t to the free two body t r a n s i t i o n amplitude. are momentum dependent. The c o e f f i c i e n t s But because l i t t l e more i s known about ± , i t i s not possible to carry out more than the simple single scattering, impulse approximation to the hypernucleus production rates. 800 MeV/c,, the sum In the region 500 - was truncated atn.=*A because of the d i f f i c u l t y i n de- f i n i n g the other c o e f f i c i e n t s . I f the cross sections where measured more c a r e f u l l y at many angles then i t would be possible, i n the framework of the model presented here, to put a l l the experimental information into the calculation. However t h i s has not been done and would not be a s i g n i f i c a n t Improvement considering many other e f f e c t s are more important and w i l l also be neglected l i k e nuclear d i s t o r t i o n , spin-orbit s p l i t t i n g and multiple scattering e f f e c t s . 8. Chapter 3 Nuclear Wave Function For the purpose of this work, only doubly magic (N = Z) nuclei will be considered and the choice for a model wave function of the nucleus will be based on the physical process we have under consideration. For example, suppose that a deep one hole state is created in the final hypernucleus, say 0si^y'^ ""C , then the system will be in a very highly excited state and will have many fast decay channels with lifetimes much shorter than the lambda's lifetime in the nucleus. Processes, like the aforementioned, are interesting in their own right but reactions where the final hypernucleus has a small percentage of the available energy are more likely to form "stable" hypernuclei. Therefore, i t is reasonable to take the interaction between the incident kaon and the nucleus to be one with only those nucleons on the top of the nuclear fermi sea. For example, in the case of ^ 0 i t is assumed that we have an inert helium core on which we build an anti-symmetric state of 6 protons and 6 neutrons. To describe the nucleons near the top of the fermi sea, independent particle wave functions, with spin and isospin, are used in an L-S coupling scheme, neglecting the spin - orbit interactions. The average nuclear potential is taken to be given by a harmonic oscillator with an experimentally determined oscillator strength (Harmonic oscillator wave functions are used because of the facility for expressing the cross sections in a closed form). The single particle wave functions are denoted ^(£,4., , w^, w c ) simply 4>(«0 » where n is the radial total quantum number, 1, s, t, are the n orbital, spin and isospin quantum numbers and m-,, m , m. , are the z-compos "t> nents of 1, s, t respectively. Since i t has been assumed that only two body interactions are important or for the K-Nucleus interaction, i t is useful to factorize the target vrave function into one for a single nucleon times a wave function for the remaining nucleons, summed over possible single particle states. This factor- ization is the well known fractional parentage expansion (Elliott, J.P. and Lane, A.M., 1957). > To conclude this section the method of fractional parentage expansions is sketched in order to establish notation. (McCarthy* I.E., 1968) If the anti-symmetric state of A nucleons in the configuration CO is denoted by A ^ - (J-A,SA>TA , rv\ , ^ M r * ) * the anti-symmetric where an< u state of A-l nucleons in the configuration UjJ) t>y % ~\ X~'> ^ _,Vthen i t A A A is possible to write = L <x {_\ (9) where {nf-'u'-WAttM]** tf-'O A is the single particle state of the Ath nucleon and <. £.{ > §>*L<fC) is the one-particle fractional parentage coefficient. The summation extends over a l l possible states ^>,6>0 allowed to the Ath nucleon. A Finally £ denotes the vector coupling of the Ath particle to the state of the A-l nucleons to give the quantum numbers of the total wave function of the A nucleons. In the case when spin-orbit interactions are neglected one has; ~ WW*" ( l - A -A 2 L * (T,., ± (10) A , S "V, « J L , W J I TA m )(L ( V , * -"V, ^ l 5 A W s J A m^., TA A 5A w 3 a iT A m T A ) The fractional parentage coefficients are determined by requiring < 0 > to be anti-symmetric and normalized to unity. C L _i Si W\ A L yv\jL \ ^ A ^ L A ^ ) a r e The coefficients such as c l e t s c h - G o r ( i a n coefficients. 10. For conciseness Eq. (9) will be abbreviated with and the summation extending over .» wi^wi^and . In a similiar way the wave function of the final hypernucleus is ex- pressible as .the lambda wave function coupled to the A-l nucleon system to give a final state with the total quantum numbers of final hypernucleus: The fractional parentage coefficient contained in . is unltv since the hypernucleus must only be anti-symmetric with respect to the A-l nucleons, not the total A-CA-1) nuclear system. 11. Chapter 4 Differential Cross section The initial state of the K~-Nucleus system will be given by the product of the meson wave function and the nuclear state in the overall center of mass; (13) = i ^ V " ^ 1 * ^ ^ where t is the momentum of the kaon in the overall center of mass(O.C.M.). K The wave function of the nucleus is the product of the internal wave function 1 % i A , A* ) > A and the motion of the center of mass of the nucleus, A described by the plane wave • ^ andR.are the coordinates of the kaon and nucleus in the O.C.M. the plane wave states are given by (14) and whereco andco are the relativistic energies of the kaon and nucleus K respectively. A The normalization used here is the same as that used in the parametrization of the transition amplitude for n ( KT, TT ~ ) A given, in Chapter 2 (Kallen, G., 1964; Ma|rtin, A.D., Spearman, T.D. , 1970) in defining :£ . Similarly the final state will be the product of an outgoing pion state and the hypernjicleus wave function; (15) \ \%> - l&X Kw>aJ ^''C^- ^. A)> , / 1 j with "fo^ the'.O.C.M. momentum of the outgoing pion. Because the cross sections we are describing involve the Initial struck nucleon and the final lambda both in bound states, i t is necessary to know their wave functions in momentum space relative to the residual A-l 12. nucleons. If h. is the position of the nucleon relative to the center of mass of the A nucleon system, the wave function in momentum space is given by (16) = ^ ^ p U ^ . ^ 4a*-) where £ is the momentum of the nucleon relative to the center of mass of the nucleus. A similar expression holds for the bound lambda's wave function. In order to find the momentum of the neutron relative to the A-l nucleons, consider Fig. *K If is the coordinate of the nucleon relative to the O.C.M., R the center of mass of the nucleus nucleon system, then the momentum conjugate to and in the O.C.M., K - V* + \ R 9 r -k and~r =, r K f f t of the A-l - r is p . From these results the momenta of the nucleon and the residual nucleus In the O.C.M. are given by; fc« = 'kr - m& k K (18) L> = ~ fcr ~ ^ £ tec Because single particle wave functions will be used to describe both the nucleon and the lambda, i t is necessary to relate the wave functions which are defined with respect to the center of mass of the nucleus to wave functions expressed in terms of the separation between the nucleon (lambda) and'the.CM. of the A-l nucleon system. The separation between, the nucleon and the nucleus center of mass is given by (19) • . £ - J^-r rY» A 13. It is TL which is used in the description of the single particle wave function, The initial state of the nucleus is given by (20) <n R l 1L*> & A e,p(-ik,.R) 2 ^ 1 , tt-^ ^ " ' ^ A H , ) . Using Eq. (16) one finds (22) <aglTL*>-4*3? Z i C ^ - , j ^ / J ~ ? However i t is simple to show that (23) k'-* - with V £ . 5 + tP'^P . ^ / Then the initial wave function becomes (24) < * r e l * > A 2X / A _, i ^v*^' ^ 1 e i t ^ ^'^pC-fep^ C Similar expressions hold for the final state lambda-pion system. As remarked earlier, the choice of plane wave normalization was related to the definition for the transition amplitude of the process v\L%> ,TT ) A . Consider Eq. (24), i t is clear that one can interpret the exponentials as plane waves for the nucleon and the residual nucleus i f normalization factors are introduced. The initial nuclear wave function is then given by t . 14. S i m i l i a r l y the f i n a l hypernucleus wave function i s (26) Cv^l^,^^^^!^ ^> J^^ a Using these expressions the t r a n s i t i o n operator f o r the r e a c t i o n nuc leuT> T~t (27) K + Kypev v\v*.tl€u.s i s given by * j ^ K m ^ * ^ < T ^ t » <ktrk l *\ A V fc > n , where N i s the number of i d e n t i c a l scatterers and F i n a l l y i f we assume the i n i t i a l and f i n a l r e s i d u a l nuclei are unaffected by the reaction, so that ') 0t*~ ^ , =: » then the t o t a l t r a n s i t i o n amplitude i s ( 2 8 ) * n ^ ^ m / ^ ^ t W K f e r ^ U I V ^ > By making t h i s assumption, we assume that there i s no nuclear deformation i n the r e a c t i o n and hence, the nucleus only plays a kinematical r o l e and acts as a source of scatterers. Fortunately, i f we are only interested i n doubly magic n u c l e i l i k e helium, carbon and oxygen, then the f r a c t i o n a l one p a r t i c l e parentage c o e f f i c i e n t s are equal, hence K because they are normalized to unity, and L l "> \ — V^N ' » 1 5 - . The transition amplitude is then given by ( 3 0 ) V ^ ^ ^ ^ i H . C t ' , ^ x CS ^ p A ^ m / l p <t ,^Ulfefe >' ! 1 l S wi^) ( S A. mj -m^loo) + P p I / * where we have used the fact that the initial nucleus has = 5^ =~Tx. = O . In Appendix 2 the factorJIT times the amplitude is explained in terms of Slater determinants. The integral in Eq. (30) is rather complicated and so i t is desirable to replace i t be some convenient approximation. Because of the use of harmonic oscillator wave functions, the integral will be dominated by small momenta and will have vanishing contributions for large "k' .. Similarly <.TTf\\-klvcvn> will not be a strong function of ^ for small Ik/,. Hence as a first approximation to.the integral in Eq. ( 3 0 ) , < I^J £.\ is replaced by its value when k'=o . The total transition amplitude may henceA. forth be written as: y (3D T^. = with <irAlA.IKr\> (^V^lY <TAlM K w y, %S%) to be evaluated in the K-nucleus center of mass. This is now a high energy approximation to the total transition amplitude, that neglects most of the nuclear effects and puts the major structural effects 16. into the inelastic form factor ^jf^- , where 3^/-^) has been defined as (32) Here is the three momentum transfer. When m A tw i t reduces to the A momentum transfer for elastic scattering. The differential cross section is then given by (Kallen, G., 1964), (33) S is the total energy squared. 17. Chapter 5 Applications (a) Helium In the reaction *H«_(fc.~)1T~) Hei the production of the lambda-hyperon + A occurs with a neutron i n a (Is) state i n the nucleus. The hypernucleus He has only one state i n which the binding energy i s p o s i t i v e and t h i s state has the s h e l l model configuration ( l s ) ~ ^ ( l s ) . / , - both the nucleon and the bound lambda are i n states of zero angular momentum r e l a t i v e to the center 3 of mass of the hypernucleus. The parent or r e s i d u a l nuclear system has the quantum numbers Lp=0i ^p ^' ^ p ^ 1 = hyperon has the quantum numbers 1 =0, a n i ^ t m = + 2 ' • He, ^ e bound lambda s =j and t =0. Because the i n t e r - a c t i o n contains no s p i n - f l i p terms, the bound lambda must have the same s p i quantum numbers as the struck neutron. That i s , the t o t a l spin (spin i s a good quantum number i n our model; of the hypernucleus He^ must be ident- i c a l l y zero. Three dimensional harmonic o s c i l l a t o r wave functions are used to describe both the struck neutron and the bound lambda. The r a d i a l wave functions are given by (34) with the o s c i l l a t o r strength ively. and a A f o r the nucleon and lambda respect- The value of Q i s well known from electron s c a t t e r i n g and has the n value 1.38fm. The value of u. f or t h i s form of the lambda wave function i s not well defined. A This i s b a s i c a l l y because the Gaussian form f o r the wave function i s not a very good d e s c r i p t i o n of the lambda hyperon s-state i n h, He.. . In order to assign a value to Q. f o r our c a l c u l a t i o n s , the non- 18. r e l a t i v i s t i c Schrodinger equation was Saxon p o t e n t i a l of range 2.0 lambda i n ^He fm. integrated numerically f o r a Woods- and diffuseness 0.6 has a binding energy of 2.25 MeV. fm., given that the The r e s u l t i n g wave function A Is shown i n figure 5. well radius. The wave function i s seen to peak just inside the In order (to f i t t h i s curve with a Gaussian form, one must take an o s c i l l a t o r strength <x =1.90 / fm. approximately. This was the value used A i n a l l the c a l c u l a t i o n s that are reported here. The I n e l a s t i c form f a c t o r was calculated using the above wave functions, and i s given by k^n^^- -"^ (35) 0 with -L- = — + K ( ~ WW*. The d i f f e r e n t i a l cross section f o r production of ^He 1 ^ J C . M . 64TT £ I U V' W > I W A is J At forward angles, where the momentum transfer i n e l a s t i c form factor w i l l be maximum. v A Figures (6,7) T w i l l be small, the show ( d . T ' / d l - f l ) f t K . as a function of the incident kaon laboratory momentum and as a function of the s c a t t e r i n g angle i n the center of mass. The r a p i d drop i n the c a l - culated cross sections as' a function of angle i l l u s t r a t e the large momentum transfer encountered as one goes to larger s c a t t e r i n g angles. the curves i s a measure of the value of a , A z the value of <x taken i n our c a l c u l a t i o n s . A The slope of which i s a d i r e c t measure of 19. (b) Carbon The production of 1?C is the production of a "p-shell hypernucleus", A (p-shell refers to the nuclear shell of the struck nucleon, not the A wave function). These have been discussed in some detail (Gal, A., Soper, J.M., Dalitz, R.H., 1971). In the calculations we have performed, i t is the p-shell nucleons which are responsible for the production of the hypernucleus. The s-shell nucleons remain inert throught the reaction. The residual nucleus which couples to the lambda-hyperon wave function can then be considered to be (Lp)^ hole state, with rv\ Tf = L = \ S ~-^ , T? = p } P and . In p-shell hypernuclei there is also the possibility of states other than the lambda in a s-state which may be bound. In our calculations, we have taken both the s-state and the p-state of the A-hyperon as "bound" by 10.0 MeV. and;.5 MeV. respectively. One word concerning the p-states of the lambda which we have considered bound. Experimentally and theoretically, i t is not clear whether some, a l l or none of these states will be bound. Hence our calculations concerning the p-states of the lambda actually mean that i f a l l the p-states CI pYy^ ( t p ^ C (ipYj/^ , C'pYy^CipY^ , ( i are degenerate and bound by ~^ C ip")y^ , 0.5 MeV., then one should expect p-state cross sections of the approximate size and shape predicted by our calculations. Clearly, our results should therefore be taken with a grain of salt and considered very speculative. Again because there is no spin flip involved in reaction (1) in our model, the final spin of the hypernucleus is taken as zero. The final states of the hypernucleus are then nucleon is - X 1,0 ,$4*0,7.$ = JL . } The radial wave function of the struck 20. with a , the measured value fm. from electron scattering data. For the n final state of the lambda in a s-state, the radial wave function is given by (38) R*0O- (^'"Pt-''-.') • For p-shell hypernuclei i t is found^that 0.^= 1.74 fra. gives a reasonably good f i t to the calculated wave function (Gal, A., Soper, J.M., and Dalitz, R.H., 1971). In this case, the final state produced is CtpY^(iV) A and the inelastic form factor becomes (39) 3V> • i- l ^ f ( ^ ] o ! % ^ - ^ / ^ where and The differential cross section for the production of the state is then given by dLa t4nV UJV. W r v U ^ ; t «.o (40) The differential cross sections are plotted in Figures (8,9|10,11)(curve a) as functions of the cosine of the scattering angle in the kaon-nucleus center of mass. At forward angles the cross section vanishes because Q*~-? O but i t rises to an appreciable value for cos G ~»95* At these Tc.m, v angles, the production of the state Cip} ClS^ is seen to be an appreciable K A part of any reasonable estimates for the total differential cross section for A production summed over a l l possible final states. \ • • 21. For the production of the lambda-hyperon in a bound p-state the lambda wave function was integrated for a Wood-Saxon potential of range 2.6 fm., diffuseness 0.5 fm. taking the A. to be bound by only 0.5 MeV. The wave function is plotted in figure (12). It peaks at 2.50 and for a 3- dimensional harmonic oscillator f i t , this implies that the approximate oscillator strength is 1.76 fm. This value is in very close agreement with that used by Gal et al for the lambda in s-states (for models of hypernuclei, see Iwao, S., 1971; Shakin, C.H., et al, state will then be £lp)"^ C J p ) i,£ =2,1,0. A I967). The hypernucleus which will have the possibility of The sum over these states, gives an inelastic form factor squared, (41) when the radial wave function of the lambda is given by . R,;OO (42) = (dhi)~ • Thus the differential cross section for the production of the hypernucleus state CvpV^OpV is given by (43) dST ^ * .lisl-f ^"^"CvW ^X m \<TTA1 * Un>\ v Asc£-*o, this differential cross section has a maximum. This corresponds roughly to changing the struck neutron into a lambda without any momentum transfer and without changing the spatial distribution of the system by 22. , and Q^-^> o that this differential a great deal. Notice that as cross section would tend to /W x(d<^/cl£)5free. This corresponds to the case in which the struck neutron and hound lambda have exactly the same spatial distribution and inelastic form factor becomes a quasi-elastic form factor which is unity for zero momentum transfer. The differential cross sections are plotted in Figures 8 , 9 , 1 0 , 1 1 , (curve b) for the production of the (lp)/\ states, at incident kaon momenta of 5 0 0 , 6 0 0 , 7 0 0 , and 800 NeV/c. The apparent dip in the cross section comes from the state (Lp* IvClp^A coupled to zero angular momentum. When <x<£- (° , this term becomes zero, in l analogy with elastic form factor from electron scattering. contribution It is the from lj=Z. which f i l l s in the gap in this region (see momentum transfer graphs 1 8 , 1 9 ) . (c) Oxygen The calculations for oxygen follow exactly the same pattern as for carbon. In order to find a value for the oscillator strength for the .p-state 16 in 0 A , the Schrodinger equation was integrated for Woods-Saxon well of radius 2 . 9 0 fm., diffuseness 0 . 6 fm. and with binding energy of 1.0 MeV. The wave function is plotted in Fig. 1 3 . An appropriate oscillator f i t gives an oscillator strength of u A = 1 . 9 1 fm. This is the value used in our calculations. The only other trivial changes from carbon to oxygen are, the effective number of scatters and the kinematical changes. The oscillator strength for the nucleons was taken as I . 5 6 fm. In Figs. (14,15,16,17) the differential cross sections for both s and p state production are presented for various incident kaon momenta. These results have the same structure as the carbon results. 23. Chapter 6 (a) Conclusion and Discussion General Comments The f i r s t and most obvious comment i s that the d i f f e r e n t i a l cross sections are strongly momentum dependent. This occurs from tvro sources; the i n e l a s t i c form factors <^6j_x) and from the behaviour of (f^T / cL£±) free i n the lab. At high incident kaon momenta, the momentum t r a n s f e r i s large everywhere except i n a very narrow cone around the zero scattering angle. As a r e s u l t of the r a p i d increase i n 1^.1 with angle, a l l the d i f f e r e n t i a l cross sections are strongly peaked toward forward scattering angles. One of the most important parameters i n the model we have presented is a A , the parameter which determines the s p a t i a l wave function of the lambda hyperon. In a l l the model calculations we have presented, these o s c i l l a t o r parameters where choosen with some care. Either the e x i s t i n g values In the l i t e r a t u r e where used or i n cases of uncertainty, the wave equation was Integrated numerical and f i t t e d by eye to an appropriate o s c i l l a t o r value. The f i n a l r e s u l t s are very sensitive to Its choice and can vary by as much as a f a c t o r of 2. As an example, i n the production of the hypernucleus state ^ P V K C I P \ ^ 0 * , the square of i n e l a s t i c form factor i s proportional to i A Q r » / ( a K X ^ to the f i f t h power. A When a =Q,,, t h i s A A 16 f a c t o r i s unity but i n the actual case of i s .73 . 0 t h i s f a c t o r to f i f t h power As 0L i s made larger or smaller than O^, t h i s f a c t o r decreases K from i t s maximum value of 1. & A i a l cross section f o r the'state also determines the peak i n t h e . d i f f e r e n t through oj . Changes i n 3* , x move the peak outward vrhen d >a^ and inward . when n d < a . ^ . Thus the A predicted cross sections are Indeed sensitive to <2 . A F i n a l l y , throughout the c a l c u l a t i o n s , the kinematics where treated 24. relatlvistically. Only the motion of the nucleons and bound A treated n o n - r e l a t i v i s t i c a l l y . Also, when the dependence on k 1 where in was ignored, t h i s assumption neglected the f a c t that the i n t e r a c t i o n a c t u a l l y occurs i n a nuclear p o t e n t i a l . This means that the reaction proceeds at a s l i g h t l y higher energy than was assumed i n the calculations by an amount equal to the depth of the average potential well f o r the lambda. These o f f - s h e l l a f f e c t s , should not be s i g n i f i c a n t however f o r the hltfh momentum caaoa studied i n t h i s work. (b) Experimental Consequences The r e s u l t s we have predicted mean l i t t l e u n t i l we ask what are the experimental consequences, i f any of our r e s u l t s . The model we have used i s rather simple, lacking i n many f i n e d e t a i l s but the e s s e n t i a l features of the calculations are s i g n i f i c a n t . The c a l - culations concerning the p - s h e l l hypernuclei are the most i n t e r e s t i n g , because i t i s here that the p o s s i b i l i t y of excited hypernuclear states e x i s t and the p o s s i b i l i t y of being able to study them i n t r i g u i n g to the experimentalists. ^ In t h i s section, we s h a l l confine ourselves to discussions related to *^0 . What i s s a i d i s equally true f o r * C 2 A 0^ the A A . was taken as bound by 10.0 HeV. i n i t ' s r e l a t i v e Is state and by 1.0 MeV. i n i t ' s r e l a t i v e Ip state. But i n point of f a c t t h i s i s an extreme s i m p l i f i c a t i o n there are two Is states, CVpYv Cvs^ A In our calculations of A ( Ip)34 C l s } A and separated by roughly 6 Mev. (Ajzenberg-Selove, F., 1970). These, states are again s p l i t by the spin-spin i n t e r a c t i o n into 4 states, each separated by approximately 1 MeV. the p-state e x i s t s . No evidence f o r the existence of I f i t did however, i t would i n point of f a c t be the I I I / following states C\p}"^CI 25. p^ A , C VpY*,^ <-\?h'/ tr >C i p ^ ( I pY^ OpT^Clp^ a ^ c \ a total of 4 different configurations, with total of 8 different states, when spin-spin forces are taken into account. Many of these states will be unbound and some will probably be just slightly bound. No one knows how many will be bound, i f any, and the hope is that experiments may be able to see these states i f background is not high. The experimental missing mass plot, assuming our s-states and p-states are separately degenerate, would look something like figure 2 0 (a). The peak at 1 0 MeV. represents production of the (Is)^ state. MeV. represents the production of the (lp)^ state. The peak at 1 . 0 However i f one considers the situation somewhat more carefully the results would look more like figure 2 0 (b). The strength to the (is) state is now spread into 4 states while the (lp) state strength is almost completely washed out by the large number of states. It is clear from these simple considerations that i t may be very difficult in point of fact to see the excited states "p-states" when experimentally one must fight both the It" background associated with K-beams and the problems of good resolution, so necessary for meaningful interpretation of the data. Furthermore, one is faced with absorptive effects associated with the finite size of the nucleus and the other reactions which could remove kaons from undor^oln/"/ reaction (l). In our calculations, no account was taken of those absorptivo effects, but estimates (Chapter l) of those effects aro something like a correction factor of 2 or 3 down from our calculated results. These estimates are not excessive or outrageous. to the total K-N cross section (Kolbig, K.S., Margolis, B., They are related I968). When a l l these effects are put together i t appears that the production of the lowest s-state will be less than 0 . 5 mb/st at 18° in the lab. For "p-states" of 2 6 -. the lambda one is in much greater doubt about their observablity in these reactions, especially when one considers their low binding, background effects and finite resolution problems. But however difficult i t may be to see these states, i f they exist, i f some selection rules operate then a careful experiment of this kind probably has as much chance seeing these states as any other. We eagerly await the experimental results. 27. APPENDIX 1 Scattering by a Many Body System For completeness i n t h i s section, a summary of the formal theory of scattering of a p a r t i c l e by a general many body system i s described, i n order to show the r e l a t i o n of t h i s scattering compared to the scattering from the separate constituents and to show the l o g i c a l connection to the single scattering, impulse approximation. (Rodberg, S.L., and Thaler, R.M., 1967). A l l approximations to the many body problem seek to reduce the problem to a s e r i e s of two-body i n t e r a c t i o n s . The multiple-scattering equations can be expressed i n terms of two-body scattering amplitudes appropriate to the target. Consider the scattering of a p r o j e c t i l e by a complex target composed of N p a r t i c l e s which may each i n t e r a c t with the p r o j e c t i l e . I f the target has a f i n i t e s i z e , the incident p r o j e c t i l e and the f i n a l outgoing p a r t i c l e w i l l be free before and a f t e r the i n t e r a c t i o n respectively. The i n i t i a l and f i n a l states are described by the Hamiltonian H = Ko + H (1.1) where K 0 D T i s the k i n e t i c energy operator f o r the p r o j e c t i l e and H T i s the Hamiltonian f o r the target, including whatever interactions bind i t s constituents Let <^ be eigenstates of U . together. many body operators a from two-body operators, upper and lower case symbols are used r e s p e c t i v e l y . The p r o j e c t i l e - t a r g e t i n t e r a c t i o n Is the sum of two- body interactions (1.2) To d i s t i n g u i s h the V = Z ^ , 28. where v\ i s the i n t e r a c t i o n between the p r o j e c t i l e and p a r t i c l e n target. In i n e l a s t i c processes, v destroys p a r t i c l e s . of the may be an operator which creates and n For example i n the reaction K-viaA+TT" , v-^ a n n i h i l a t e s the kaon and neutron and produces a lambda and pion i n the f i n a l state, The f i n a l outgoing state i s given by the i n t e g r a l equation - <k with & - — - — vf*> the i n i t i a l free p a r t i c l e state ( S c h i f f , L., 1970). The tran- s i t i o n amplitude f o r e l a s t i c or i n e l a s t i c scattering i s I f the potenial V V . i s s u f f i c i e n t l y weak, T f i may be expanded i n powers of But a more general r e s u l t , separates two-body e f f e c t s from the multiple-scattering e f f e c t s . I t i s possible to completely describe the scattering by a single p a r t i c l e and w i l l generate a series showing a succession of scattering by d i f f e r e n t target p a r t i c l e s . Equations (1.3) (1.5) (i.6) a n ( i ^ ^ (1«^0 (1.8) n <k- , 4. - T - - (1.?) and ca ^ = 1^ D e rewritten as — - + 1 — i -t- ^ \ t — 2U^„ — 2L —S * . | + TL^ A > . This a d d i t i o n a l complexity i s j u s t i f i e d by the f a c t that these equations I provide a description of the scattering process i n terms of a multiple/ 29. scattering sequence. Substituting Eq. (1.6) and (1.5) © expand"T^ w in powers .of the t r a n s i t i o n operator - t ^ . ^ 2L ( 1 , 9 ) — — 1 *™—*—*^+--- 14x> Each term i n t h i s s e r i e s i s a multiple-scattering sequence i n which the p r o j e c t i l e scatters successively from d i f f e r e n t p a r t i c l e s i n the medium. In the f i r s t term the p r o j e c t i l e enters the target, scatters from p a r t i c l e in. , and emerges. In the double scattering term the p r o j e c t i l e scatters from wv, propagates to p a r t i c l e v\. , where i t scatters again and then i I emerges. / I Single Scattering | I f the target i s s u f f i c i e n t l y small, then only one scattering i s l i k e l y to occur, then~T^. may be approximated by the f i r s t term i n Eq. (1.9). It w i l l be a v a l i d assumption I f the target thickness i s small compared to the mean free path of the p r o j e c t i l e . With t h i s assumption This expression f o r the t r a n s i t i o n operator i s s t i l l rather complicated, because i t requires knowledge of ;fc i n the target, but i t may be evaluated n i f the impulse approximation invoked. The impulse approximation b a s i c a l l y replaces.the two body t r a n s i t i o n operator ib by the free two-body t r a n s i t i o n n fx* 6 6 amplitude ;fc , . f o r the elementary process on one of the free p a r t i c l e s n i n the target (l.ii)" T f i - £ «M** "l<k> r N 30. where "SjL-C^is the Fourier Transform of the product of the initial target wave function and the final residua^, system's wave function. The last equation above is the high energy approximation, where one assumes that the ^t^ r e e is not a strong function of the momentum of the struck particle. In the case under consideration, the struck neutron has a much smaller momentum than the incident kaon and hence this approximation would probably be rather good. If a l l the target particles are identical (1.12) Ta = N <3\±. "U> ^ C c p 4r The differential cross section in the center of momentum, is given by with Jfa^ are the final and incident momenta of the particles in the center of momentum of the target-projectile system,fc.,^are the initial and final momenta in the equivalent system for the elementary process under discussion and S, s are the total energy squared of the target-projectile and the nucleon-projectile systems respectively. The assumption of replacing the two-body scattering amplitude ^ by the n free K-N amplitude implies that the structure of the target nucleus has no dynamical effect on the elementary process under consideration. The corrections to the impulse approximation involve the nuclear structure corrections and these are closely related to the multiple scattering corrections. Furthermore the impulse approximation assumes that one knows the free T matrix off the energy shell. In practice one must extrapolates the off- 31. s h e l l value from the on-shell T matrix f o r s i m i l i a r kinematics. The struct- ure of the target system then only enters the impulse approximation i n a kinematical way and higher order corrections i n the multiple scattering theory may be calculated using the extrapolated o f f - s h e l l T matrix. 32. APPENDIX 2 Why the f a c t o r In t h i s appendix, we show how the f a c t o r of^NJ in~T^ a r i s e s naturally, when one considers nuclear wave functions as S l a t e r determinants, instead of the abstract f r a c t i o n a l parentage c o e f f i c i e n t s used i n Chapter 3. To describe a nucleus with an independent p a r t i c l e model wave function, we s t a r t with a nuclear Hamiltonionian f o r N i d e n t i c a l fermions: (2.1) with The single p a r t i c l e p o t e n t i a l , harmonic o s c i l l a t o r p o t e n t i a l . may be a Woods-Saxon, square well or The nuclear wave function i s then a pure Slater determinant (2.2) C£ (l,*»'-,.N) = N where the single p a r t i c l e wave functions <£<A^ are the solutions of the Schrbdinger equation (2.3) £Ti vV(^]4> CJu^= ^ ^ U i V K The normalization i s (2.4) and cf »4cv>r<^ = 1 / A- - Consider the reaction K~-»-nucleus — ^ nucleus^ + the two-body matrix elements of r e a c t i o n K +n t ~ „ A -V Tf In order to extract , we can write the Slater determinant i n terms of the co-factors of one of the columns as (2-5) *„<••• N> - - £ <h£} 1 t U , » , - - - , N ^ M 33. where the expansion has been choosen so that /iX^U.-.N^lW^.--=- \ (2.6) Since a l l the p a r t i c l e s are i d e n t i c a l , i t i s only necessary to consider the i n t e r a c t i o n of the incident K-meson with p a r t i c l e 1. The i n i t i a l state w i l l be product of nuclear wave function and the r e l a t i v e motion between the center of mass of the nucleus and the incident K-meson (2.7) & ~ o ? U V ^ - ! > ) ) % N I f we neglect multiple scattering e f f e c t s and changes i n the N-l nucleus, the f i n a l state w i l l be (2.8) % - C -it,-Cv w - S} XV-. £AA >?A^A) where 7] 6i)is the lambda wave function. The matrix element, neglecting A of mass motion, Glebsch-Gordan c o e f f i c i e n t s and other complications The f a c t o r center w i l l be instead of N comes b a s i c a l l y from the reaction being i n - I • e l a s t i c and the lack ofjanti-symmetry of the lambda-nuclear system. In e l a s t i c s c a t t e r i n g the i n i t i a l and f i n a l system would be anti-symmetric with respect to N p a r t i c l e s and one would have a double sum over i n i t i a l and f i n a l states r e s u l t i n g i n a f a c t o r of N times the free t r a n s i t i o n amplitude. \ \ 34. Figure Captions Figure 1: Diagramatic Interpretation of the K~-Nucleus Interaction. Figure 2 : The c o e f f i c i e n t s of the Legendre polynomial expansion Eq. (3) versus the incident K~ lab. momentum. Each of the curves i s the r e s u l t of a 9th order polynomial f i t i n momentum, to the c o e f f i c i e n t s obtained by averaging the K"-proton and K~-neutron I results. Figure (yjxlapted from Bonazzola, G.C., et a l , 1970) The experimental d i f f e r e n t i a l cross section f o r + A.-t TT at incident lab. momentum 77? MeV/c. measured i n the center of \ Figure 4: mass. \ This diagram i l l u s t r a t e s the p o s i t i o n of the nucleon r e l a t i v e to the (A-l) nuclear system and the o v e r a l l center of mass. The notation i s the same as i s used i n the text. Figure 5: A plot of cutO = v Rlr}, where RCrHs the r a d i a l wave function of 4 the bound lambda i n H e A . The peak i n the wave.function i s reproduced by an appropriate Gaussian f i t toRXrYwhen a o s c i l l ator parameter of o. = 1.90 A fm. i s used. The depth of the well, which reproduced the known binding energy of 2.25 Figure . 6: 29.34 MeV. MeV/c., was 4 D i f f e r e n t i a l cross section f o r the production of He^ at incident kaon momentum of 800 MeV/c. 4 Figure 7: D i f f e r e n t i a l cross sections f o r H e production at incident kaon A momenta (a) 500 MeV/c, (b) 600 MeV/c., and (c) ?00 MeV/c. Figure 8: D i f f e r e n t i a l cross sections f o r the production of MeV/c. at 500 Curve (a) corresponds to a sum over the states of the 35. configuration C l f ^ n C l s ) A , Curve (b) corresponds to a sum over the states of the configuration Op^ ^ C l p ) . A A at 600 A at 700 A at 800 Figure 9". Differential cross sections for the production of *^C MeV/c. 1? Figure 10: Differential cross sectiqjas for the production of C MeV/c. 12 Figure 11: Differential cross sections for the production of C MeV/c. Figure 12: A plot of ulr) - RXr) , where R6r"Hs e radial wave function of t n 1? the bound lambda in, C in p-state. The peak in the Wave A function is reproduced by a 3-3-imensional harmonic oscillator wave function with an oscillator parameter a = 1.76 fm. The A depth of the well, which reproduced the assumed binding of 0.5 MeV. was 37.06.MeV. Figure 13: A plot of u.Cr"> =• v, where &CT) is the radial wave function of 16 the bound lambda in 0 in a p-state. The peak in the wave A function is reproduced by a 3 iiraensional harmonic oscillator _( wave function with an oscillator parameter a =1.91 fm. The A depth of the well, which reproduced the assumed binding of l-.O MeV. was 30.61.MeV. Figure 14: Differential cross sections for the production of ^ 0 A at 500 0 A at 600 0 A at 700 A at 800 MeV/c. Figure 15: Differential cross sections for the production of MeV/c. Figure 16: Differential cross sections for the production of MeV/c. Figure 17: Differential cross sections for the production of MeV/c. 1 f, 16 0 I /• i 36. Figures 18 and 19: Tije momentum transfer l^j as a function of the cosine of the scattering angle G c m at incident kaon momenta 500, 600, '•\ 700 and 800 MeV/c. for ^He^ and 0 1 6 A respectively. The dashed line corresponds to the momentum transfer resulting from a stopped K~-meson. Figure 20: ( . In this figure we try to interpret our results experimentially for ^ 0 . A (a) corresponds to the ideal case in which there is no background, the (lp)^Cli) states are degenerate and A similarly the (lp)^ ^'P)A states. The missing mass corresponds to the energy of any bound states that exist. The Op) '(instate rt has a binding of 10.0 MeVi and the (tpT,llip) state a binding of A 1.0 MeV. In (b) we have tried to put some real physics into the picture. The ^p^CtS^ states are split by roughly 6 MeV. and further by spin-spin interactions. The p-states are almost washed out by energy splitting and background. REFERENCES A.jzenberg-Selove, F. , Armenteros, R., Nuclear Physics et a l , Bonazzola, G.C., et a l , A15_2, 1, 1970. Nuclear Physics B8, 183, 1968. CERN Internal Document, PH I I I ? 0 / 3 9 . D a l i t z , R.H., Proc. 1962 Ann. Int. Conf. High Energy.Physics, CERN, 1962. D a l i t z , R.H., Proceedings of International Conference on Hypernuclear Physics, High Energy Physics D i v i s i o n , Argonne National Lab. Argonne, I l l i n o i s , Davis, D.H., 1969. and Sacton, J . , Hypernuclear Physics i n "High Energy Physics", Edited by E.H.S. Burhop, E l l i o t t , J.P.,-and Lane, A.M., Gal, A., Soper, J.M., Goldberger, M.L., New Iwao, S., and Watson, K.M., Annals of Physics p_3_, 5 3 , N.Y, 1957. 1971. C o l l i s i o n Theory, John Wiley and Sons, 1964. Progress of Theoretical Physics, 4 6 , 1 4 0 ? , 19?1. Kallen, G., Elementary P a r t i c l e Physics, 1964, Kolbig, K.S., Lock, L.O., Academic Press, Encyclopaedia of Physics, V o l . XXXIX, and D a l i t z , R.H., York, Vol.11, 196?, Addison-Wesley, London. and Margolis, B., Nuclear Physics B6, and Measday,- D.F., 85, I968. Intermediate Energy Nuclear Physics, 1970, Methuen and Co., London. Martin, A.D., and Spearman, T.D., Elementary P a r t i c l e Physics, 19?0, North-Holland Publishing, Amsterdam. McCarthy, I.E., Introduction to Nuclear Theory, New I968, J . Wiley and Sons, York. Middleton, E., M. Sc. Thesis,, University of Liverpool, 19?1. Rodberg, S.L., and Thaler, R.M., Scattering, Introduction to the Quantum Theory of I967, Academic Press. Schiff, L., Quantum Mechanics - 2nd Edition, McGraw-Hill, 1970. Shakin, G.M., Wihmare, Y.R., Hull, M.H., Tanner, N., Private Communication, 1971. Physical Review 161, 1001, I96? TABLE I J PARTICLE PARTICLE MASS (MeV) 139.578 A 1115.60 %ie 3727.32 12c III74.67 a 14894.82 0 n a < > ls A a (IP) A K 493.82 n 939.55 3910.44 12 C A 16 MASS (MeV) lfi &*T^*\ 11357.44 (ipJ^Cipl 11366.44 (1,0^(13)^ 15073.53 . 15002.53 ^He 12c 16 1.38 fm. 1.56 fm. 1.56 fm. 1.90 fm. 1.74 fm. 1.90 fm. 1.74 fm. 1.90 fm. 0 FIG- I J K" laboratory momentum (GeV/c) FIG. 2 n co O U E UJ FIG-4 IO j.O -95 -90 -35: -80 -75 COS(O) -70 . -65 / - 6 0 ' FIG-6 IO rnb. St. l 2 C(K,¥pC A K.= 5 0 0 MeV/c (a) 10 l-O •95 •90 •35 •80 COS(G) 75 70 •65 •60 •• FIG- 8 COS(G) • FIG-9 IO i-O -95. \ .90 -35 ' -80 '-75 COS(0) 70 . -65 -60 FIG-IO £2 6 0(K"TTf 0 ,6 A K.= 500 MeV/c •95 -90 •85 •80 ' -75 •70 •65 •60 FIR. 14 IO j.O -95 -90 -85 -80 75 r.OS(O) 70 -65 -60 FIG • 15 rncjcM FIG. 17 3 10 •I r'l J-.O i i -95. -90 • j-J—i | .85 . I i i' -SO 75 COS(G) i 70 i . -65 __i -60 FIG-19 (a) MOD EL RESULTS -i -1 (lp)nOs) <'p> H A n • MISSING MASS IO lO-O (b) MORE PROBABLE MeV. RESULTS MISSING MASS IOO ip\ FIG - 2 0 M e V
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Microscopic description of hypernucleus production using fast kaons Esch, Robert J. 1972
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Title | Microscopic description of hypernucleus production using fast kaons |
Creator |
Esch, Robert J. |
Publisher | University of British Columbia |
Date Issued | 1972 |
Description | The differential cross sections for the production of definite lambda hypernuclear states, within the single scattering, impulse approximation, are calculated from the reaction n(K⁻,π⁻)Λ° on nuclear targets of helium, carbon and oxygen at various K-meson incident momentum. It is shown that these predictions are very sensitive to the three momentum transfer and to the wave function of the bound lambda in the hypernucleus. From the calculations, it is shown that it is possible to observe their production by studying the missing mass spectrum of the emitted pion. |
Subject |
Particles (Nuclear physics) Mesons |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-04-21 |
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. |
DOI | 10.14288/1.0084827 |
URI | http://hdl.handle.net/2429/33940 |
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 |
Aggregated Source Repository | DSpace |
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