Brief Review of the R-Matrix Theory

L. C. Leal

Introduction

Resonance theory deals with the description of the nucleon-nucleus interaction and aim s at the prediction of the experim ental structure of cross sec t i ons. Re sonance theory is basically an interaction m odel which treats the nucleus as a bl a ck box, whereas nuclear m odels are concerned with t h e description of the nuclear properties based on m odels of the nuclear forces (nuclear potential). Any theoretical m ethod of calculating t h e ne utron-nucleus interactions or nuclear properties cannot fully describe the nuclear effects inside the nucleus because of the com p lexity of the nucleus and because the nuc l e a r forces, acting within the nucleus, are not known in detail. Quantities related to internal properties of the nucleus are taken, in this theory, as param eters which can be determ ined by exam ining the experim ental results.

The ge ne r al R-m atrix theory, introduced by W i gner and Eisenbud in 1947, is a powerful nuclear interaction m odel. Despite the generality of the theory, it does not require inform ation about the internal structure of the nucleus; instead, th e unknown internal properties, appearing as elem ents in the R-m atrix, are treated as param eters and can be determ ined by exam ining the m easured cross sections.

A brief review of the R-m atrix theory will be given here and the interaction m odels which are specializations of the general R-m atrix will be described. The practical aspects of the general R- m atrix theory, as well as the relationship between the collision m atrix U and the level m atrix A with the R-m atrix, will be presented.

Overview of the R-Matrix Theory

The general R-m a trix theory has been extensively described by Lane and Thom as. An overview is presented here as introduction for t h e r e sonance form alism s which will be described later.

To understand the basic points of the general R-m a trix theory, we will consider a sim p le case of neutron collision in which the spin dependence of the constituents of the interactions is neglected. Although the m athem atics involved in this special case is over-sim plified, it nevertheless contains the essential elem ents of the general theory.

As m entioned before, the nuclear potential insi de the nucleus is not known; t h erefore, the behavior of the wave function in the internal regi on of the nucleus cannot be calculated directly from the Schrödinger equation. In the R-m atrix ana l ys i s t h e inner wave function of the angular m o m e ntum l i s expanded in a linear com b ination of the eigenfunctions of the energy levels in the com pound nucleus. Mathem atically speaking, if is the inner wave function at any energy E and is the eigenfunction at the energy eigenvalue E 8 , the relation becom es

1

Courtesy of Luiz Leal, Oak Ridge National Laboratory. Used with permission.

(1)

Both and are solutions of the r a dial Schrödinger equations in the internal region given by

(2)

and

(3)

Since all term s in this expression m u st be f i n ite at , both functions vanish at that point. In addition, t h e l oga r i thm i c derivative of the ei genfunction at the nuclear surface, say at , is taken to be constant so that

(4)

where is an arbitrary boundary constant.

Since we are dealing wi t h e i genfunctions of a real Ham iltonian, are orthogonal.

Assum i ng that are also norm a lized, we have

(5)

From Eq. (xx-1) and the orthogonality condition, we find the coefficients ,

(6)

To proceed to the construction of the R- m a trix, Eq. (xx-2) is m u ltiplied by and Eq. (xx-3) is m u ltiplied by . Subtracting and integrating the result over the range to (as in Eq. (xx-6)) produces the expression for the coefficients :

(7)

Inserting into Eq. (xx-1) for r=a at the surf ace of the nucleus and using Eq. (xx-4), gives the following expression for the wave function:

(8)

Equation (xx-8) relates the value of the inner wave function to its derivative at the surface of the nucleus. The R m a trix is defined as

(9)

or

(10)

where , the reduced width am plitude for the level 8 and angular m o m e ntum l , is defined

as

(11)

The reduced width am plitude depends on the valu e of the inner wave function at the nuclear surface. Both and are the unknown param e ters of the R m a trix which can be evaluated by exam ining the m easured cross sections.

The generalization of Eq. (xx-10) is obt a i ne d by including the neutron-nucleus spin dependence and several possibilities in which the reac tion process can occur. The concept of channel i s introduced to designate a possible pair of nucle us and particle and the spin of the pa i r . The channel containing the initial state is called the en trance channel (channel c), whereas, the channel

containing the final state is the exit channel (channel c’). The elem ents of the R m a trix in the general case are given by

(12)

where the reduced width am plitude becom e s

(13)

The next objective is to relate the R-m a trix to the cross-section f o rm alism so that cross sections can be com puted once the elem ents of the R-m a trix are known.

Relation betw een the R-matrix and the Collision Matrix U

The general expressions f o r the neutron-nucleus cross sections are based on the collision m a trix, also known as U-m a trix, whose elem ents can be expressed in term s of the elem ents of the R-m a trix. From basic quantum m echa n ics theory the cross sections for the neutron-nucleus interaction can be given as a function of the m a trix U as follows:

(1) Elastic Cross Section

(14)

(2) Reaction Cross Section which includes everything which is not elastic scattering (i.e., reaction=fission, capture, inelastic, ...)

(15)

3) Total Cross Section

where is the neutron reduced wavelength given by

(16)

(17)

W e first derive the relationship between the U and R m a trices, for a sim p le case of spinless neutral particles. The total wave f unction in the region outside the nuclear potential can be expressed as a linear com b ination of the incom i ng a nd outgoing wave functions. If and are the incom i ng and outgoing wave functions for a free particle, respectively, the solut i on of the radial Schrödinger equation can be written as

where is a norm a lization constant.

for r $ a , (18)

The presence of the U-m a trix in Eq. (xx-18) (in this case a m a trix of one elem ent) indicates that the am plitudes of the incom i ng and outgoing wave functions are, in general, different. The case of corresponds to pure elastic scattering which m eans that no reaction has occurred.

The Schrödinger equation for a n d is the sam e as Eq. (xx-2) with since the potential outside the nucleus is zero. The solution is a com b ination of the spherical Bessel ( ) and Neum ann ( ) functions

and

where .

(19)

(20)

The relation between the U and the R-m a trices is obtained by first noting that Eq. (xx-8) can be written as

(21)

where is given in Eq. (xx-9).

Equation (xx-21), when com b ined with Eq. ( xx-18), provides the relation between R and U- m a trices as

W e define the logarithm i c derivative as

Since from Eqs. (xx-19) and (xx-20), and are com p lex conjugates,

Equation (xx-22) becom e s

(22)

(23)

(24)

(25)

Equation (xx-25) represents the desired relations hip between the collision m a trix U and the m a trix R.

The representation of the neutron cross secti ons w ill depend on the reduced width am plitudes and which are unknown param e ters of Eq. ( xx-25). Those param e ters are obtained by fitting the experim e ntal cross section.

The general relation between the m a trices U a nd R is sim ilar to Eq. (xx-25) with each term converted to m a trix form :

(26)

All m a trices i n Eq. ( xx- 26) are diagonal except the R m a trix. The m a trix elem ents of are given by .

It should be noted that no approxima t i on was used in deriving Eq. (xx-26). That equation represents an exact expression relating U and R , and leads to the determ ination of the cross section according to Eqs. (xx-14, xx-15, and xx-16).

To avoid dealing with m a trices of large dim e nsions, several approxim a tions of the R-m a trix theory have been introduced. W e will discuss various of these cross-section f o rm alism s in the pages to com e ; we begin by introducing the level m a trix A.

Relation betw een U, R, and A

Another presentation of Eq. (xx-26) m a y be obtained by introducing the f o llow i ng def i nitions

(27)

(28)

and

(29)

wher e S and P are real m a trices which contains t h e shi f t and the penetration factors, respectively and .

From Eqs. (xx-20, xx-23, and xx-27), the penetration factors can be wri t t en as

, and Eq. (xx-26) becom e s

(30)

with .

It should be realized that the R-m a trix is a channel m a trix; i.e. it depends on the entrance and out goi ng c h annels c and c’. The level m a trix con cept introduced by W i gner attem p ts to relate the U m a trix to a m a trix in which the indices are the energy levels of the com pound nucleus, the level

m a trix of elem ents . In relating the channel m a trix to the level m a trix we recall that the R m a trix is defined as

where indicates the direct product between two vectors.

The expression can be written as

(31)

(32)

where we have defined , and is a sym m e tric m a trix. The form of Eq. (xx-32) suggests the f o llowing relation

(33)

where the indices and refer to energy levels in the com pound nucleus and A is determ ined as f o llows:

Multiplying Eqs. (xx-32) and (xx-33) and using the identity , we obtain the following expression,

(34)

Factoring the term in the above equation, we find that the level m a trix satisfies the equation

(35)

The evaluation of the m a trix which appears in Eq. (xx-30) is obt ained by com b ining Eqs. (xx-31) and (xx-33) which gives

Using Eq. (xx-35) as gives

(36)

(37)

Hence, the collision m a trix is related to the level m a trix as

(38)

The elem ents of the collision m a trix f o r entr ance and exit channels c and c’, respectively, are given as

(39)

where

is the level width, and from Eq. (xx-35) the level m a trix is

(40)

(41)

It should be rem e m b ered that no approxim a tion has been introduced in the form al derivation of the collision m a trix up to this point.

Simplified Models Derived from the General R-Matrix Theory

In this session we will present the approxim a ti ons introduced to the R-m a trix and, likewise, to the level m a trix A which leads to various sim p lif ied resonance formalism s . The cross section f o rm alism s f r equently used are the single-leve l Breit-W i gner (SLBW ) , the Multilevel Breit-W i gner (MLBW ) , the Adler-Adler (AA), and the Reich-M oore (RM) form alism (also known as the reduced R-m a trix form alism ) . A new m e thodology, called m u lti pole representation of the cross section, was developed at Argonne Na t i ona l La bor atory by R. N. Hwang; in this approach the cross section representation is done in the m o m e ntum space ( ) . W e will address the appr oxim a tions needed to obtain these sim p lif ied R-m a trix m odels.

The starting points in deriving these f o rm alism s will be the level m a trix A and its relation to the collision m a trix U.

The collision m a trix is given by

(42)

The level m a trix is represented as

(43)

1. Multilevel Breit-Wigner (MLBW) Formalism

In the MLBW approxim a tion the level m a trix is assum e d to be diagonal, which m eans that the off-diagonal elem ents of the second term in th e m a trix given in Eq. (xx-43) are neglected, i.e.,

(44)

Hence Eq. (xx-43) becom e s

(45)

From Eqs. (xx-27) and (xx-40) we have and , which leads to

(46)

where (energy shift factor for the MLBW ) and . Redefining , the level m a trix becom e s

(47)

The collision m a trix given by Eq. (xx-42) becom e s

(48)

From this point, we proceed to the derivati on of the cross section form alism in the MLBW representation. For a reaction in which (fission, capture, or inelastic scattering channels) the collision m a trix and the reaction cross section are given respectively by

(49)

and

(50)

where we have used the identity in Eq. (xx-49) . Inserting Eq. (xx-49) into Eq. (xx-50) gives

(51)

where we have m a de and . This e xpression can be further m odified by using the f o llowing identity

which gives

(52)

(53)

where . The second term in Eq. (xx-53) is the com p lex conjugate of the first term , hence

(54)

The term in the sum m a tion on can be expanded to give

(55)

where

(56)

and the line shapes and are defined as

(57)

and

(58)

Equation (xx-55) is the MLBW cross section fo rm for the reaction cross section. A sim ilar procedure can be followed to derive the elastic cross section.

2. Single Level Breit-Wigner (SLBW) Formalism

The SLBW cross section form alism is a partic ular case of Eq. (xx-55) when the second term in Eq. (xx-56) is zero, that is, .

3. Adler-Adler (AA) Formalism

The AA approxim a tion consists of applying an orthogonal com p lex transform a tion which diagonalizes the level m a trix as given in Eq. ( xx-43). W e are looking for a transform a tion such that

(59)

or

(60)

where . Here is a orthogonal c o mpl e x ma t r ix and is a diagonal m a trix of com p lex elem ents. The elem ents of the m a trix in Eq. (xx-60) are given as

The collision m a trix of Eq. (xx-42) then becom e s

(61)

(62)

where and . The elem ents of th e m a trix are determ ined f r om

(63)

where Eq. (xx-43) has been used.

B ecause of the energy dependence of through the penetration f actor , the elem ents will, in general, be energy-dependent. In the AA approach, the energy dependence of is neglected. This assum p tion works very well for fissile isotopes where the resonance region is predom inantly described by s-wave resonances (angular m o m e ntum corresponding to ) f o r which the penetration factor is energy independe nt. However, the assum p tion breaks down when p- wave ( ) or other neut r o n partial wave functions with angular m o m e ntum greater than 1 are present.

The reaction cross sect i on i n t h e AA form alis m can be obtained in a sim ilar way to that developed for the MLBW . The result is

where the following definitions were m a de

(64)

(65)

and

(66)

4. Reich-Moore Formalism

The approach proposed by Reich and Moore for treating the neutron-nucleus cross sections consists of elim inating the off-diagonal contribu tion of the photon channels. The rationale for this assum p tion is this: system atic m easurem ents of the resonance widths, m a inly in the case of the neutron and fission widths, show strong fluctuati ons am ong resonances of the s a me t o t a l angular m o m e ntum and parity. It should be expected, from Eq. (xx-40), that these fluctuations are connected either to the reduced widths or to the penetr ation factors . However, it is im probable that such

fluctuations are due to the penetration factors becau se they are either constant or a sm ooth function of the energy. Hence, the fluctuations m u st be related to the reduc e d wi dt hs. Porter and Thom as noted that the reduced widths of Eq. (xx- 13) a r e functions of the channel functions whi c h, i n turn, are projections of the eigenf unctions of the com pound nucleus onto the nucle a r surface and exhibit random si ze variations. Cons equently, the large num ber of gam m a channels im plies that is very sm all for . The sec ond term of the level m a trix in Eq. (xx-43) is divided in two parts as

and in the RM approxim a tion

The level m a trix becom e s

(67)

(68)

(69)

where, sim ilarly to the MLBW , the following definitions were m a de:

(Energy shif t f actor), and . Note that these quantities are different from that in the MLBW form alism . Again, redefining we have

(70)

From this point we are going to derive a re lation between the collision and the level m a trix in the RM representation. Multiplying Eq. (xx-70) by and sum m i ng over gives

(71)

Multiplying Eq. (xx-71) on the lef t by and on the right by and summing over and gives

(72)

If we define

(73)

and

(74)

then Eq. (xx-72) becom e s

(75)

Note that this R m a trix is an approxim a tion, not t o be c onfused with the exact R-m a trix defined earlier.

Rearranging Eq. (xx-75) gives

(76)

Hence, f r om Eq. (xx-42) the collision m a trix in the RM approxim a tion becom e s

(77)

Equation (xx-77) relates the collision m a trix to the Reich-Moore R-m a trix in a f o rm sim ilar to that in the case of the general R-m a trix theory. In the general R-m a trix, the elem ents are

(78)

whereas in the RM approxim a tion they are

(79)

Equation (xx-79) is frequently referred to as the reduced R-m a trix theory.

W e now proceed to obtain a form for the cr oss section in the RM approxim a tion, by writing Eq. (xx-77) as

(80)

where

(81)

It is usef ul to write the reduced R-m a trix as

(82)

in which the elem ents of K are given by

(83)

The explicit f o rm of is

(84)

Therefore becom e s

(85)

Recalling that and m a king , the expression for becom e s

(86)

The m a trix form of Eq. (xx-86) is

(87)

Equation (xx-87) can be f u rther reduced by using the identity . Letting , , and we have

(88)

If we then add and subtract the expression becom e s,

(89)

f o r which the elem ents are, explicitly,

The collision m a trix of Eq. (xx-80) then takes the f o rm

(90)

(91)

where the elem ents of are given as

(92)

The RM cross sections are written in term s of the transm ission probability, def i ned as

(93)

f o r which the collision m a trix can be written as

(94)

The cross sections can then be obtained by using Eqs. (xx-14), (xx-15), and (xx-16) as,

and

(95)

(96)

(97)

(98)

5. Conversion of RM parameters into AA parameters

A procedure to convert RM par a m e ters into an equivalent set of AA param e ters was developed by DeSaussure and Pere z . Their appr oach consisted of writing the RM transm ission pr obabilities and as the ratio of polynom ials in energy; these polynom ials can the n be expressed in term s of partial fraction expansions by m a tching the AA cross sections as:

and

where

(99)

(100)

(101)

(102)

(104)

(103)

and .

Equations (xx-99) and (xx-100) have pol es which are roots of the equation

(105)

and are identifiable as the param e ters of the Ad ler-Adler form alism . In deriving this m e thodology DeSaussure and Perez neglected the energy dependence of the neutron widths, i.e., . This assum p tion lim its the application of this m e thods to s-wave cross section. Hwang has extended the application of the DeSaussure and Perez approach to the calculation of cross sections for any angular m o m e ntum . In his approach, instead of using energy space, Hwang noted that the dependence of on suggests that an expansion in te r m s of woul d lead to a rigorous representation of the cross section. Since m o m e ntum is proportional to , Hwang calls his m e thodology a rigorous pole representation in the m o m e ntum space or, for short, a m u ltipole representation of the cross sections (MP). The transform a tion of the RM param e ters into the MP param e ters is obtained as

(106)

and

(107)

where

(108)

and is the num ber of resonance param e ters in the RM representation. The factor of Eq. (xx- 104) becom e s

(109)

where

and

(110)

(111)

Doppler Broadening and Effective Cross Sections

(112)

The Doppler broadening of cross sections is a well-known effect which is caused by the m o tion of the atom s of the target nuclei. Since the target nuclei are not at rest in the laboratory system , the neutron-nucleus cr oss section will depend on the relative speed of the neutron and the nucleus. The effective cross secti o n for m ono-energetic neutrons of m a ss m and energy E (laboratory velocity v ) is given by the number of neutr ons per unit volum e, m u ltiplied by the num ber of target nuclei per unit volum e, tim es th e probability that a reaction will occur per unit tim e at an energy equivalent to the relative velocity | v – W |, integrated over all values of W, the velocity of the nucleus . The relation between the cross section m easured in the laboratory and the effective cross section is

(113)

where is the effective or Doppler-broadened cross section for incident particles with speed v [laboratory energy mv 2 /2] . The distribution of velocities of the target nuclei is described by . A m a jor issue is the choice of the appropr iate velocity distribution function of the target nuclei. Let us now assum e that the target nuclei have the sam e velocity distribution as the atom s of an ideal gas; i.e. the Maxwell-Boltzm a nn distribution,

(114)

where M is the nuclear m a ss and kT the gas tem p erature in energy units. Com b ining Eqs. (xx- 113) and (xx-114) gives

(115)

Note that, from the above definitions, a 1/v cross section rem a ins unchanged.

Changing the integration variable from and choosing spherical coordinates sim p lif ies the integral to the f o llowing:

(116)

This equation, known as the Solbrig’s kernel, m a y be m o re fam iliar when written as the sum of two integrals,