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 in the department of PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA APRIL, 1972 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 for 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 available for reference and study. I further agree that permission fo r extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It 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 PHYSICS The University of B r i t i s h Columbia Vancouver 8, Canada Date 6 April , 1972 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 Sabbat-ical 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 calculat-ions 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 all , 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) Oxygen Chapter 6: Conclusion and Discussion References Appendix 1: Multiple Scattering Theory Appendix 2: Why the factor 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 cal-culations, it is shown that it 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 under-standing 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. The captured K-meson cascades down toward its atomic (is) orbital. As it does, i t has a larger and larger overlap with the nucleus. 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~)A0 may leave the produced lambda-hyperon In the nucleus, 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 yield-ed 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, it 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 all 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 UC (K^ TT)'?^ . occurring with only a single 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 it almost impossible to study excited hyper-nuclear 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 it 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, it is possible to "deposit" a lambda into the nuclear system. This peculiar behaviour in the kinematics is basic-ally 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 kine-matlcal 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 coin-cidence 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 Ke£^(A j<r. jQTj.) the effective neutron number (Kolbig, K.S., Margolis, B., 1968). This estimate contains a sum over all possible final states and also contains approximate absorptive and multiple scattering effects in Ne£j(A,<ri ,<TV)> <T~, is the total k^-rV cross section, Tt theTT-fs/ total cross section. Calculations for a uniform sphere model for O1^ reduce the effective number of p-shell neutrons from 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 fall between these two estimates. With only these rough estimates available and the experiments scheduled to staxt i n the spring of this year, i t became interesting to make more accurate estimates for the diff 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). It i s to these questions that the present work Is addressed. The d i f f e r e n t i a l cross sections for hypernucleus production on ^ He, and 0 have been calculated i n the single scattering, impulse approx-imation at incident momenta 500, 600, 700 and 800 MeV/c to definite hyper-nucleus states. The predictions made here are found to be very sensitive to the momen-tum transferod to the bound lambda and to the wave function of the lambda. The treatment presented here consists of essentially four steps: (1) phenomological description of reaction (1) at incident K-meson lab. momenta 500, 600, 700 and .800 MeV/c; (2) a proper treatment of the nucleax wave function, including anti-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) to yie l d theoretical estimates for the production of varrbus hypernuclear energy levels at forward angles. 5. Chapter 2 Reaction n (K~> IT" ) A° In the description of hypernucleus production, as In a l l descriptions for the interaction of a particle with a many particle system, i t i s nec-essary to relate i n some logical way, the elementary two body interactions to the total target-projectile interaction. In the formal theory of mul-tip l e scattering (Goldberger, M.L. and Watson, K.M., 1964), the single scattering impulse approximation t e l l s one that the inelastic or c o l l i s i o n cross section w i l l be essentially t&e product of three factors: (l) the number of effective scatterers, (2) the inelastic form factor squared and (3) the elementary free two body di f f e r e n t i a l cross section. The struct-ure of the target has no dynamical effect on the process except in a t r i v -i a l kinematical way (see Appendix 1) and (Figure l ) . In this section, we -present the information concerning the free d i f f e r e n t i a l cross section for the reaction which i s necessary for our theoretical prediction 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 discussed (Kallen, G., 1964). In the region between 300 MeV/c. to 600 MeV/c. the experiments have been scarce and not a great deal i s known, but p-wave effects 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 dif f e r e n t i a l cross sections have been made by expanding the di 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 in the center of mass (^ = C- i) . The coefficients are 6. incident momentum dependent and the Legendre polynomial coefficients versus incident K-meson momentum are shown in Fig. 2 (Armenteros, R. et al, I968). In Fig. 3 the experimental differential cross section is shown for reaction (1) at ?77 MeV/c. incident kaon momentum. From these graphs it is clear that the reaction in the momentum range 500 - 800 MeV/c. is predominately s-wave and p-wave with little 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, it was taken as identically zero. This is not necessary but it simplifies the work without introducing a significant error. Now from relativistic scattering theory, the differential cross sect-ion 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 l- 2-^- -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 transition amplitude. The coefficients are momentum dependent. 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. In the region 500 -800 MeV/c,, the sum was truncated atn.=*A because of the d i f f i c u l t y in de-fining the other coefficients. If the cross sections where measured more carefully 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 this has not been done and would not be a significant Improvement considering many other effects are more important and w i l l also be neglected lik e nuclear distortion, spin-orbit s p l i t t i n g and multiple scattering effects. 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 it 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^, wc ) or simply 4>n(«0 » where n is the radial total quantum number, 1, s, t, are the orbital, spin and isospin quantum numbers and m-,, m , m. , are the z-compo-s "t> nents of 1, s, t respectively. Since i t has been assumed that only two body interactions are important for the K-Nucleus interaction, it is useful to factorize the target vrave function into one for a single nucleon times a wave function for the remain-ing 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 COA is denoted by where ^ - (J-A,SA>TA , rv\u , ^Mr*) a n <* the anti-symmetric state of A-l nucleons in the configuration UjJ) t>y % A~\ XA~'> ^ A_,Vthen it is possible to write (9) = L <x A{_\ tf-'O {nf-'u'-WAttM]** where §>*L<fC) is the single particle state of the Ath nucleon and <. £.{ > is the one-particle fractional parentage coefficient. The summation extends over all possible states >^,A6>0 allowed to the Ath nucleon. 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*" ~ 2 ( l - A - A "V, « J L , W J ( V , * -"V, ^ l 5 A W s J L A , S A (10) * (T,., ± m^., I TA mTA) ( L A 5A w 3 a i T A m T A ) The fractional parentage coefficients are determined by requiring < 0 > to be anti-symmetric and normalized to unity. The coefficients such as C LA_i Si W\L yv\jL \ ^ A ^ L A ^ ) a r e 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 K is the momentum of the kaon in the overall center of mass(O.C.M.). The wave function of the nucleus is the product of the internal wave function 1 %Ai AA , A* ) > and the motion of the center of mass of the nucleus, 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 wherecoKandcoAare the relativistic energies of the kaon and nucleus respectively. 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 K w > a J ^ ' ' C ^ - , / ^ . 1 j A ) > 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 and~rf of the A-l nucleon system, then the momentum conjugate to =, r f t - r p is and in the O.C.M., K R - V* + \ 9 r - k K . 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, it 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 Rl 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 it is simple to show that (23) k'-* - V £ 5. + tP'^P with . ^ / Then the initial wave function becomes (24) < * r e l * A > 2 X / A _ , i ^ v * ^ ' 1 ^ e i t ^ C ^ ' ^ p C - f e p ^ 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 . t . 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 14. Similiarly the f i n a l hypernucleus wave function i s (26) ^>aJ^^ C v ^ l ^ , ^ ^ ^ ^ ! ^ Using these expressions the transition operator for the reaction K + nuc leuT> T ~ t Kypev v\v*.tl€u.s i s given by (27) * j ^ K m ^ * ^ < T ^ t » < k t r k A l * \ V fcn> , where N i s the number of identical scatterers and Finally i f we assume the i n i t i a l and f i n a l residual nuclei are unaffected by the reaction, so that ') 0t*~ ,^ = : » then the total transition amplitude i s ( 2 8 ) * n ^ ^ m / ^ ^ t W K f e r ^ U I V ^ > By making this assumption, we assume that there i s no nuclear deformation in the reaction and hence, the nucleus only plays a kinematical role and acts as a source of scatterers. Fortunately, i f we are only interested in doubly magic nuclei li k e helium, carbon and oxygen, then the fractional one particle parentage coefficients are equal, hence K Ll "> \ — V^N ' » because they are normalized to unity, and 1 5 - . The transition amplitude is then given by ( 3 0 ) V ^ ^ ^ ^ i H . C t ' , ^ < t ! , ^ U l f e f e 1 > ' x CS p^ A ^ m / l p l S+wi^) (S P A. mjp-m^loo) 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 it 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. (30 ) , < I^J £.\ is replaced by its value when k'=o . The total transition amplitude may hence-A. forth be written as: y (3D T^. = ( ^ V ^ l Y < T A l M K w y , %S%) with <irAlA.IKr\> 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 mA twA it reduces to the 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~)+He i the production of the lambda-hyperon A occurs with a neutron in a (Is) state in the nucleus. The hypernucleus He has only one state in which the binding energy i s positive and this state has the shell model configuration (ls)~^(ls) /. , - both the nucleon and the bound lambda are in states of zero angular momentum relative to the center 3 of mass of the hypernucleus. The parent or residual nuclear system He, has the quantum numbers Lp=0i ^p 1^' ^p =^ a n i^ m t = + 2 ' • ^ e bound lambda hyperon has the quantum numbers 1 =0, s =j and t =0. Because the inter-action contains no spin-flip terms, the bound lambda must have the same spi quantum numbers as the struck neutron. That i s , the total spin (spin i s a good quantum number in our model; of the hypernucleus He^ must be ident-i c a l l y zero. Three dimensional harmonic oscillator wave functions are used to describe both the struck neutron and the bound lambda. The radial wave functions are given by (34) with the oscillator strength and a A for the nucleon and lambda respect-ively. The value of Q n i s well known from electron scattering and has the value 1 .38fm. The value of u.A f or this form of the lambda wave function i s not well defined. This i s basically because the Gaussian form for the wave function i s not a very good description of the lambda hyperon s-state i n h, He.. . In order to assign a value to Q. for our calculations, the non-18. r e l a t i v i s t i c Schrodinger equation was integrated numerically for a Woods-Saxon potential of range 2.0 fm. and diffuseness 0.6 fm., given that the lambda in ^He has a binding energy of 2.25 MeV. The resulting wave function A Is shown in figure 5. The wave function i s seen to peak just inside the well radius. In order (to f i t this curve with a Gaussian form, one must take an oscillator strength <x =1.90 fm. approximately. This was the value used / A i n a l l the calculations that are reported here. The Inelastic form factor was calculated using the above wave functions, and i s given by (35) k^n^^- 0-"^ with -L- = — + ( K WW*. ~ The d i f f e r e n t i a l cross section for production of ^HeAv i s 1 ^ J C . M . 64TT £ I U V ' W > I W A J T At forward angles, where the momentum transfer w i l l be small, the inelastic form factor w i l l be maximum. Figures (6,7) 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 scattering angle in the center of mass. The rapid drop in 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 scattering angles. The slope of the curves i s a measure of the value of a A z , which i s a direct measure of the value of <xA taken in our calculations. 19. (b) Carbon 1? The production of C A is the production of a "p-shell hypernucleus", (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, it is the p-shell nucleons which are responsible for the production of the hyper-nucleus. 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 L p= \ } S P ~-^ , T ? = and rv\Tf = . 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 cal-culations, 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, all or none of these states will be bound. Hence our calculations concerning the p-states of the lambda actually mean that i f al l the p-states CI pYy^ ( t p ^ , ( i ~^ C ip")y^ , C (ipYj/^ , C'pYy^CipY^ are degenerate and bound by 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 - X} 1,0 ,$4*0,7.$ = JL . The radial wave function of the struck nucleon is 20. with an, the measured value fm. from electron scattering data. For the final state of the lambda in a s-state, the radial wave function is given by (38) R * 0 O - ( ^ ' " P t - ' ' - . ' ) • For p-shell hypernuclei it 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 C t p Y ^ ( i V ) 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 d L a t 4 n V U J V . 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 it rises to an appreciable value for cos G ~»95* At these T- c.m, v angles, the production of the state Cip} KClS^ A is seen to be an appreciable part of any reasonable estimates for the total differential cross section for A production summed over all 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 hyper-nuclei, see Iwao, S., 1971; Shakin, C.H., et al, I967). The hypernucleus state will then be £lp)"^ C J p ) A which will have the possibility of i,£ =2,1,0. The sum over these states, gives an inelastic form factor squared, (41) when the radial wave function of the lambda is given by (42) . R,;OO = (dhi)~ • Thus the differential cross section for the production of the hypernucleus state CvpV^OpV is given by (43) dST ^ * .lisl-f ^ "^"CvW m^X \<TTA1 * U n > \ 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. a great deal. Notice that as , and Q^-^> o that this differential 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 <xl<£- (° , this term becomes zero, in analogy with elastic form factor from electron scattering. It is the contribution from lj=Z. which f i l l s in the gap in this region (see momentum transfer graphs 18 ,19). (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 fit 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 Conclusion and Discussion (a) General Comments The f i r s t and most obvious comment i s that the di f f e r e n t i a l cross sections are strongly momentum dependent. This occurs from tvro sources; the inelastic form factors <^ 6j_x) and from the behaviour of (f^T / cL£±) free in the lab. At high incident kaon momenta, the momentum transfer i s large everywhere except in a very narrow cone around the zero scattering angle. As a result of the rapid increase i n 1^ .1 with angle, a l l the diff e r e n t i a l cross sections are strongly peaked toward forward scattering angles. One of the most important parameters in the model we have presented i s a A , the parameter which determines the spatial wave function of the lambda hyperon. In a l l the model calculations we have presented, these oscillator parameters where choosen with some care. Either the existing values In the literature where used or in cases of uncertainty, the wave equation was Integrated numerical and f i t t e d by eye to an appropriate oscillator value. The f i n a l results are very sensitive to Its choice and can vary by as much as a factor of 2. As an example, in the production of the hypernucleus state ^ P V K C I P \ ^ 0 * , the square of inelastic form factor i s proportional to i A A Q r » / ( a A K X ^ to the f i f t h power. When aA=Q,,, this 16 factor i s unity but in the actual case of 0 this factor to f i f t h power i s .73 . As 0LK i s made larger or smaller than O^ , this factor decreases from i t s maximum value of 1. & A also determines the peak i n the.different-i a l cross section for the'state through oj x. Changes in 3* , move the peak outward vrhen dn >a^ and inward . when d A<a.^ . Thus the predicted cross sections are Indeed sensitive to <2A . Finally, throughout the calculations, the kinematics where treated 24. r e l a t l v i s t i c a l l y . Only the motion of the nucleons and bound A where treated non-relativistically. Also, when the dependence on k1 in was ignored, this assumption neglected the fact that the interaction actually occurs in a nuclear potential. This means that the reaction proceeds at a slightly higher energy than was assumed in the calculations by an amount equal to the depth of the average potential well for the lambda. These off-shell affects, should not be significant however for the hltfh momentum caaoa studied in this work. (b) Experimental Consequences The results 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 results. The model we have used is rather simple, lacking in many fine details but the essential features of the calculations are significant. The c a l -culations concerning the p-shell hypernuclei are the most interesting, because i t i s here that the possibility of excited hypernuclear states exist and the possi b i l i t y of being able to study them intriguing to the experimentalists. ^ In this section, we shall confine ourselves to discussions related to *^0 A . What i s said i s equally true for * 2 C A . In our calculations of 0 ^ the A was taken as bound by 10.0 HeV. in i t ' s relative Is state and by 1.0 MeV. i n i t ' s relative Ip state. But i n point of fact this i s an extreme simplification there are two Is states, ( Ip)34 C l s } A and CVpYvA Cvs^A separated by roughly 6 Mev. (Ajzenberg-Selove, F., 1970). These, states are again s p l i t by the spin-spin interaction into 4 states, each separated by approximately 1 MeV. No evidence for the existence of the p-state exists. If i t did however, i t would i n point of fact be the I I I / 2 5 . following states C \ p } " ^ C I p^ A , C VpY*,^ <-\?h'/tr > C i p ^ ( I pY^ a ^ c \ O p T ^ C l p ^ 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, if 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 20 (a). The peak at 1 0 MeV. represents production of the (Is)^ state. The peak at 1 .0 MeV. represents the production of the (lp)^ state. However i f one considers the situation somewhat more carefully the results would look more like figure 20 (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 it 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. They are related to the total K-N cross section (Kolbig, K.S., Margolis, B., I968). When all these effects are put together it 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 it may be to see these states, if 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 this section, a summary of the formal theory of scattering of a particle by a general many body system i s described, in order to show the relation of this scattering compared to the scattering from the separate constituents and to show the logical 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 series of two-body interactions. The multiple-scattering equations can be expressed i n terms of two-body scattering amplitudes appropriate to the target. Consider the scattering of a projectile by a complex target composed of N particles which may each interact with the projectile. If the target has a f i n i t e size, the incident projectile and the f i n a l outgoing particle w i l l be free before and after the interaction respectively. The i n i t i a l and f i n a l states are described by the Hamiltonian (1.1) H 0 = Ko + H T where KD i s the kinetic energy operator for the projectile and H T i s the Hamiltonian for the target, including whatever interactions bind i t s constituents together. Let <^ be eigenstates of Ua . To distinguish the many body operators from two-body operators, upper and lower case symbols are used respectively. The projectile-target interaction Is the sum of two-body interactions (1.2) V = Z ^ , 28. where v\ i s the interaction between the projectile and particle n of the target. In inelastic processes, v n may be an operator which creates and destroys particles. For example in the reaction K - v i a A + T T " , v-^ annihilates 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 integral equation - & - — - — vf*> with <k the i n i t i a l free particle state (Schiff, L., 1970). The tran-si t i o n amplitude for elastic or inelastic scattering i s If the potenial V i s sufficiently weak, T f i may be expanded i n powers of V . But a more general result, separates two-body effects from the multiple-scattering effects. It i s possible to completely describe the scattering by a single particle and w i l l generate a series showing a succession of scattering by different target particles. Equations (1.3) a n ( i (1«^ 0 ca-n D e rewritten as (1.5) ^ - <k- - — 1 — 2U^„ (i.6) ^ , 4. + — i — 2L (1.?) - 1^ -t- ^ — S T L ^ and (1.8) T ^ = \ t * . | + A > . This additional complexity i s j u s t i f i e d by the fact that these equations I provide a description of the scattering process in terms of a multiple-/ 29. scattering sequence. Substituting Eq. (1.6) and (1 .5) w© expand"T^ in powers .of the transition operator -t^. ( 1 , 9 ) ^ 2L — 1 — * ™ — * — * ^ + - - - 14x> Each term i n this series i s a multiple-scattering sequence i n which the projectile scatters successively from different particles i n the medium. In the f i r s t term the projectile enters the target, scatters from particle in. , and emerges. In the double scattering term the projectile scatters from wv, propagates to particle v\. , where i t scatters again and then i emerges. I / I Single Scattering | If the target i s sufficiently 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 valid assumption If the target thickness i s small compared to the mean free path of the projectile. With this assumption This expression for the transition operator i s s t i l l rather complicated, because i t requires knowledge of ;fcn i n the target, but i t may be evaluated i f the impulse approximation invoked. The impulse approximation basically replaces.the two body transition operator ibn by the free two-body transition fx* 6 6 amplitude ;fcn ,.for the elementary process on one of the free particles in the target ( l . i i ) " T f i - £ « M * * r " l < k > 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\±. 4 r"U> ^ C c p 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 ^ n by the 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. shell value from the on-shell T matrix for similiar 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 off-shell T matrix. 32. APPENDIX 2 Why the factor In this appendix, we show how the factor of^NJ in~T^ arises naturally, when one considers nuclear wave functions as Slater determinants, instead of the abstract fractional parentage coefficients used i n Chapter 3. To describe a nucleus with an independent particle model wave function, we start with a nuclear Hamiltonionian for N identical fermions: (2.1) with The single particle potential, may be a Woods-Saxon, square well or harmonic oscillator potential. The nuclear wave function i s then a pure Slater determinant (2.2) C£ N(l ,*»'- , .N) = where the single particle wave functions <£<A^ are the solutions of the Schrbdinger equation (2.3) £Ti vV(^]4>KCJu^= ^ ^ U i V The normalization i s (2.4) cf »4cv>r<^ = 1 a n d / A - - t Consider the reaction K~-»-nucleus —^ nucleus^ + ~ „ In order to extract the two-body matrix elements of reaction K + n A -V Tf , 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 MU ,»,- - - ,N^ 33. where the expansion has been choosen so that (2.6) / i X ^ U . - . N ^ l W ^ . - - = - \ Since a l l the particles are identical, i t is only necessary to consider the interaction of the incident K-meson with particle 1. The i n i t i a l state w i l l be product of nuclear wave function and the relative motion between the center of mass of the nucleus and the incident K-meson (2.7) & ~ o ? U V ^ - ! > ) ) % N If we neglect multiple scattering effects 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 - £AAS} XV-. >?A^ A) where 7] A6i)is the lambda wave function. The matrix element, neglecting center of mass motion, Glebsch-Gordan coefficients and other complications w i l l be The factor instead of N comes basically from the reaction being i n -I • elastic and the lack ofjanti-symmetry of the lambda-nuclear system. In elastic scattering the i n i t i a l and f i n a l system would be anti-symmetric with respect to N particles and one would have a double sum over i n i t i a l and f i n a l states resulting in a factor of N times the free transition amplitude. \ \ 34. Figure Captions Figure 1: Diagramatic Interpretation of the K~-Nucleus Interaction. Figure 2 : The coefficients of the Legendre polynomial expansion Eq. (3) versus the incident K~ lab. momentum. Each of the curves i s the result of a 9th order polynomial f i t in momentum, to the coefficients obtained by averaging the K"-proton and K~-neutron I results. (yjxlapted from Bonazzola, G.C., et a l , 1970) Figure The experimental di f f e r e n t i a l cross section for + A.-t TT at incident lab. momentum 77? MeV/c. measured i n the center of \ mass. Figure 4: \ This diagram ill u s t r a t e s the position of the nucleon relative to the (A-l) nuclear system and the overall center of mass. The notation i s the same as i s used in the text. Figure 5: A plot of cutO = v Rlr}, where RCrHs the radial wave function of 4 the bound lambda in He 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.A= 1.90 fm. i s used. The depth of the well, which reproduced the known binding energy of 2.25 MeV/c., was 29.34 MeV. . 4 Figure 6: Differential cross section for the production of He^ at incident kaon momentum of 800 MeV/c. 4 Figure 7: Differential cross sections for He A production at incident kaon momenta (a) 500 MeV/c, (b) 600 MeV/c., and (c) ?00 MeV/c. Figure 8: Differential cross sections for the production of at 500 MeV/c. 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 . Figure 9". Differential cross sections for the production of *^CA at 600 MeV/c. 1? Figure 10: Differential cross sectiqjas for the production of C A at 700 MeV/c. 12 Figure 11: Differential cross sections for the production of C A at 800 MeV/c. Figure 12: A plot of ulr) - RXr) , where R6r"Hs t ne radial wave function of 1? the bound lambda in, C A in p-state. The peak in the Wave function is reproduced by a 3-3-imensional harmonic oscillator wave function with an oscillator parameter a A = 1.76 fm. The 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 A in a p-state. The peak in the wave function is reproduced by a 3_(iiraensional harmonic oscillator wave function with an oscillator parameter a A =1.91 fm. The 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 MeV/c. 1 f, Figure 15: Differential cross sections for the production of 0 A at 600 MeV/c. Figure 16: Differential cross sections for the production of 0 A at 700 MeV/c. 16 Figure 17: Differential cross sections for the production of 0 A at 800 MeV/c. 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 1 6 0 A respectively. The dashed line corresponds to the momentum transfer resulting from a stop-ped 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)Astates are degenerate and similarly the (lp)^ ^ 'P)A states. The missing mass corresponds to the energy of any bound states that exist. The Op)rt'(instate has a binding of 10.0 MeVi and the (tpT,llip)A state a binding of 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. , Nuclear Physics A15_2, 1, 1970. Armenteros, R., et a l , Nuclear Physics B8, 183, 1968. Bonazzola, G.C., et a l , CERN Internal Document, PH III ?0/39. Dalitz, R.H., Proc. 1962 Ann. Int. Conf. High Energy.Physics, CERN, 1962. Dalitz, R.H., Proceedings of International Conference on Hypernuclear Physics, High Energy Physics Division, Argonne National Lab. Argonne, I l l i n o i s , 1969. Davis, D.H., and Sacton, J., Hypernuclear Physics i n "High Energy Physics", Edited by E.H.S. Burhop, Vol.11, 196?, Academic Press, N.Y, E l l i o t t , J.P.,-and Lane, A.M., Encyclopaedia of Physics, Vol. XXXIX, 1957. Gal, A., Soper, J.M., and Dalitz, R.H., Annals of Physics p_3_, 53, 1971. Goldberger, M.L., and Watson, K.M., Collision Theory, John Wiley and Sons, New York, 1964. Iwao, S., Progress of Theoretical Physics, 46, 140?, 19?1. Kallen, G., Elementary Particle Physics, 1964, Addison-Wesley, London. Kolbig, K.S., and Margolis, B., Nuclear Physics B6, 85, I968. Lock, L.O., and Measday,- D.F., Intermediate Energy Nuclear Physics, 1970, Methuen and Co., London. Martin, A.D., and Spearman, T.D., Elementary Particle Physics, 19?0, North-Holland Publishing, Amsterdam. McCarthy, I.E., Introduction to Nuclear Theory, I968, J. Wiley and Sons, New York. Middleton, E., M. Sc. Thesis,, University of Liverpool, 19?1. Rodberg, S.L., and Thaler, R.M., Introduction to the Quantum Theory of Scattering, I967, Academic Press. Schiff, L., Quantum Mechanics - 2nd Edition, McGraw-Hill, 1970. Shakin, G.M., Wihmare, Y.R., Hull, M.H., Physical Review 161, 1001, I96? Tanner, N., Private Communication, 1971. T A B L E I J PARTICLE MASS (MeV) PARTICLE MASS (MeV) 139.578 K 493.82 A 1115.60 n 939.55 %ie 3727.32 3910.44 12c III74.67 12C &*T^*\ 11357.44 A ( i p J ^ C i p l 11366.44 1 60 14894.82 l f i (1,0^(13)^ 15073.53 . 15002.53 ^He 12c 16 0 a n 1.38 fm. 1.56 fm. 1.56 fm. a <ls> A 1.90 fm. 1.74 fm. 1.90 fm. a (IP) A 1.74 fm. 1.90 fm. 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 -70 . -65 / - 6 0 ' COS(O) FIG-6 IO rnb. St. l 2 C ( K , ¥ p C A K.= 5 0 0 MeV/c (a) 10 l-O •95 •90 •35 •80 75 C O S ( G ) 7 0 •65 •60 •• FIG- 8 COS(G) • FIG-9 IO i-O -95. \ .90 -35 ' -80 ' -75 7 0 . -65 -60 COS(0) FIG-IO £ 2 6 ,60(K"TTf 0A K.= 500 MeV/c •95 -90 •85 •80 ' -75 •70 •65 •60 F I R . 1 4 IO j.O -95 - 9 0 -85 -80 7 5 7 0 -65 -60 r.OS(O) FIG • 15 r n c j c M FIG. 17 3 10 • I r'l i i j-J—i i i ' i i __i J-.O -95. -90 | .85 . -SO 75 7 0 . -65 -60 • I COS(G) FIG-19 (a) MOD EL RESULTS -1 -i <'p>nH (lp)nOs)A • MISSING MASS IO lO-O MeV. (b) MORE PROBABLE RESULTS MISSING MASS i p \ IOO M e V FIG - 2 0
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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 |
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UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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