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Dispersion formulas in the theory of nuclear reactions
Nuclear Physics 21 (1960) 245--255; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
D I S P E R S I O N F O R M U L A S IN T H E T H E O R Y OF N U C L E A R REACTIONS V. I. S E R D O B O L S K Y
P. N. Lebedev Physical Institute, Moscow, U S S R Received 4 March 1960 A b s t r a c t : The p a p e r offers a t h e o r y of resonances in nuclear reactions, in which the concept of t h e R - m a t r i x is n o t used. U n d e r consideration is a certain p a r a m e t e r whose smallness m a k e s it possible to select, o u t of t h e total n u m b e r of formal levels obtained in the theory, the resonance levels a t t r i b u t a b l e to the f o r m a t i o n of the long-lived c o m p o u n d nucleus. F o r the levels which do n o t overlap the nuclear-reaction thresholds the c o m p o u n d - n u c l e u s resonances are singled o u t b y tile poles of a Green's function at c o m p l e x energies. The dispersion f o r m u l a s t h u s o b t a i n e d are linear w i t h respect to resonances and contain the complex reduced half-widths. These f o r m u l a s do n o t depend on auxiliary " h y p e r s u r f a c e radii" a n d take into a c c o u n t the tails of w a v e functions responsible for multi-particle processes occurring close to the surface of the nucleus.
1. I n t r o d u c t i o n
The invariably successful application of the Breit-Wigner resonance formulas for the analysis of a great variety of nuclear reactions has contributed to the evolution of a number of general mathematical theories justifying and specifying the simplest resonance formulas. In 1947 Wigner suggested the well-known theory 1) based on the application of the R-matrix. In a number of subsequent papers this theory was perfected and extended to embrace various particular cases. The success of the theory for the past ten years has been summarized in an extensive paper 2). Yet at present it has been clear that the Wigner theory is unsatisfactory in many respects. A large number of auxiliary parameters without physical meaning figure in the R-matrix formalism. These are the energy levels and halfwidths of formal "remote levels" and arbitrary "radii of nucleus hypersurface". It is not clear how the processes involving the emission of two or more real or virtual particles should be described. On the other hand, practical computation has shown that in many cases the apparatus of the Wigner theory proves cumbersome and unjustifiably complex. Accumulation of experimental data has shown that processes occurring without the formation of a compound nucleus bulk large in nuclear reactions. Papers concerned with the "unified theory of nuclear reactions" 3-6) have appeared. It was found that scattering on an optical potential as well as direct i:~teractions and processes involving the formation of a compound nucleus may conveniently be described b y a single mathematical apparatus. Dispersion 245
246
v.I.
SERDOBOLSKY
formulas in the framework of the unified theory of nuclear reactions have been studied b y Feshbach 5) as well as Sano, Yoshida and Terasawa 6). The formulas obtained in ref. 5) have a limited range of application. In ref. 6) the permeability coefficients in dispersion formulas are represented as infinite series which makes their practical application rather difficult. The present paper has two basic aims: to overcome the most essential theoretical drawbacks inherent to the current formal theory and to develop a procedure for evolving dispersion formulas with allowances for optical interaction and other processes weakly depending on energy. Sec. 2 introduces a parameter, defined b y eq. (3), whose smallness makes it possible to regard the formal quantities Ez and Yxe as compound-state resonance parameters. Sec. 3 investigates the analytic properties of the wave function and the Green's function ( E - - H ) -1 of the general problem of collision with rearrangement of particles. The expansion (8) for the Green's function near the poles is established. Sec. 4 discusses a dispersion expansion (11) independent of arbitrary radii and taking account of the possibility of multiparticle decay. Under consideration is the problem of the reality of the quantities ~'ac included in practical formulas. These quantities prove to be complex as a rule. Sec. 5 demonstrates that the compoundstate wave functions in the region of energies far from the threshold form a biorthogonal system. Hence follows eq.. (23) for parameters entering into the dispersion formulas. Sec. 6 discusses the problem of correlations in the signs of Yac and allowances for the contribution of the "remote levels".
2. Selection of Compound States As in the previous papers concerned with the unified theory of nuclear reactions, we shall proceed from the integral equations of the formal theory of rearrangement collisions. Let ¢ be the wave function of free motion of particles of the ~ type with the spin quantum numbers sv and the momentum hk in the centre-of-mass system. Let H be the total Hamiltonian of the problem, and H o the Hamiltorfian of free motion in the channel ~, H being equal to Ho+V. Two solutions ~vc+) and ~v~-) are obtained from eq. (4) for Im E --~ i 0 . (see below). Let us introduce an optical potential V ° and the wave functions ~0~+J and ~o ~-~ describing scattering on an optical potential in each of the channels. The indices and operators relating to some other channel with other quantmn numbers will be designated b y dashes. Let ~ be equal to V--V°; then the following identity (see refs. ~,e)) can be written:
+(~o~-~']¢"(E--H+ie)-~Yrl~o(+~).
(1)
The first term in (1) describes an optical interaction and vanishes for ~' ¢ ~. The second term represents the amplitude of direct processes in the distorted
DISPERSION
FORMULAS
247
wave approximation method. In ref. 6) the authors regarded the third term of the right-hand side of eq. (1) as a resonance one and applied the Kapur-Peiefls theory 7). As is well known, only some of an infinite number of formal levels correspond to those really observed. Let us determine a parameter whose smallness allows of regarding a given level as conditioned b y the formation and disintegration of a compound nucleus. We shall proceed from the time description of a nuclear reaction suggested in ref. 8) and isolate compound states b y their large lifetimes. Average reaction times are closely connected with the nature of the energy dependence of wave functions and amplitudes: according to the uncertainty principle A E ~ ?i/t, where A E is the interval over which the wave functions change essentially. The characteristic periods of potential scattering and direct processes are, in general, determined b y the time needed b y the particles for their flight across the nucleus R/v. For particles with the highest orbital momenta this time increases, due to slowing down near the centre, b y a factor (kR)/Po, where Pc is the permeability coefficient determined in ref. 2). The decay time of the compound state ~ with respect to the channel ~ s l J M (index c) is equal to ~/2Pe[y~e[ 2. The ratio tnight :
tdecay
[7~c12 ER '
~2 where E R - - - - - - - , 2mR 2
(2)
is at the same time the ratio of the width of the compound-nucleus level to that of the optical resonance width -Pc ER- The parameter describing a compound state m a y be determined more accurately with the help of wave functions. The ratio of the square of the wave function modulus of a decaying state averaged over the hypersurface of the nucleus to the internal volume average is open
(4~) - z E I(CJkr-/~.)R[2 c
open ~, 2 I N" J¢~cl
The sum over c includes only open channels. The quantity (3) describes a relative time which the system spends in one-particle states possible at the given energy. This time should be small for the compound nucleus. Treatment of experimental data performed in ref. 10) shows that even for light nuclei the parameter (3) rarely exceeds 0.1. 3. A n a l y t i c
Properties
Analytic dependence of the wave function of the stationary problem on complex energy calls in principle for special proof. With real E this proof can probably be performed with the aid of general theorems on the regular dependence of the solutions on the form of the equation and the boundary conditions.
248
V. I. S E R D O B O L S K Y
With Im E ~ 0 the proof of analyticity offers no difficulties. The spectral analysis 1) shows that the Green's function G -= ( E - - H ) -1 is analytical with respect to E at all points which do not coincide with the points of the H operator spectrum. The wave function ~v is connected with ( E - - H ) -1 through the Gell-Man-Goldberger equation ~v = ¢ + ( E - - H ) - I V ~ .
(4)
The continuous spectrum of the operator lies on the half axis of real energies E > El. For Im E -+ +0, E > E~, G passes into the Green's funtions G~*), and ~vinto the ~vc±) of the physical problem. The asymptotic form of the wave functions (by the radiation condition) contains wave numbers proportional to ~/~,, where E , is the energy of the nth threshold of tile reaction. Tile analytic continuations of ~oc*) ,therefore, are unique analytic functions only of the arguments ~ / ~ 1 . . . . ~ / E - - E . . . . . . The region of determinations of these functions constitutes a system of Riemann sheets with brarmh points at E~ . . . . E . . . . . . From the continuity equation it follows that in the principal Riemann sheet in which all Im v / ~ n > 0, the Green's function G may have poles only at real E. These poles describe the stationary states of the system. Apart from them, complex poles lying in other Riemann sheets are possible. If the pole approaches the half-axis E > E~ of the principal sheet on which a continuous spectrum lies, resonances can be observed ill the cross sections. Tile time reversibility of quantum mechanics leads to a certain symmetry in the analytic continuations G and ~. Let 0 be a time reversibility operator, such that H* = OHO +. (5) It is convenient to introduce the operation B which, in acting on the point E in the system of Riemann sheets, leads to reflection with respect to the real halfaxis oi the principal sheet E < E 1 . For example, tile result of the action of B on E is E*, B ~ V / E - - E , is equal to -- ( ~ / E - - E , ) * , etc. For real energies it is obvious that OG~±~O+ ---- BG(±~'. (6) The operation B* changes the signs of all roots in the arguments of analytic functions and consequently does not violate analyticity. The relation (6) therefore holds good for the complex E as well. It is clear that the poles of G are paired, a pole ~ at the energy B E a corresponding to each pole 2 at the energy E a. Physical meaning can evidently be attributed only to the first order poles. The function ERes VJEa satisfies the Schroedinger equation with complex energy E a and contains only divergent waves in all channels; therefore tile residue [Res(JM]v)]E ~ can be regarded, except for a proportionality factor, as the wave function ,q,~of the quasi-stationary state with given integrals of motion
DISPERSION
249
FORMULAS
J and M. The constant phase of ~ should naturally be chosen so that 1PX*= (-- )J-M O~a (--M) •
(7)
With the help of the relations (4), (6) and (7) the resonance part of the Green's function can be expanded near the poles over the quasi-stationary state wave functions. In Dirac's notations ~ is a ket I~) and ~pX*the bra <~tI. The expansion has the form
IX>O.I _]_Gsmootla.
(8)
It will be noted that if the compound-nucleus level overlaps the nuclearreaction threshold, the optical wave function changes essentially over energy intervals of the order o f / ' owing to the permeability coefficient. The isolation of compound states according to energy dpendence therefore becomes impossible. In refs. 11) and 4) the approximation / ' << ]E--EtkreBh I is actually used, where Emre8h is the energy of the threshold closest to E. In the same approximation, the second term ill (8) which remains after the subtraction of the polar part of G can be regarded as the Green's function of direct and optical processes.
4. Dispersion Expansion As has been established, the compound-nucleus resonances are contained in the last term of eq. (1). To obtain a dispersion formula it is sufficient to substitute (8) and pass to the representation with quantum numbers otslJM. Employing the formulas given in the appendix, we obtain
sL =
i
< l lcue> x
E--E~
J + (so,o),mooth.
(9)
With the help of relations like (5), (7) and (25) we can easily see that the resonance part of (9) is symmetrical with respect to the indices c' and c. For the sake of convenience we introduce the notation
The expansion (9) assumes the form
s L = (sL),moot,,+2i 2: .,o,u.o E~--E"
(")
An essential theoretical advantage of eq. (11) is that it was derived without the assumption of the orthogonality of the wave functions of the channels on the nucleus hypersurface. Thus, there arises no "problem of cutting off the tails of wave functions" characteristic of the current theories. It is clear
250
V. I. SERDOBOLSKY
that multi-particle processes occurring on the nucleus surface do not distort the shape of the dispersion formula and the influence of closed channels (in the energy region far from threshold) is inessential. The decay of the compound nucleus into three or more particles can be described with eqs. (9) and (11) through a proper choice of the function u c with allowance for a continuous spectrum of outgoing particles. Eq. (11) does not contain the radii of the channels, the choice of which is largely arbitrary. If, on the other hand, it is assumed that there exists a hypersurface on which the wave functions [c> are orthogonal, the usual results will be obtained. In the energy region far from threshold the values of Uc depend weakly on energy and it m a y be assumed that u c = ue(k~r). Let us replace ~e" in the brackets b y the difference H - - H o - - V ° and make use of Green's theorem. We find that uae = Oe Pe½7xe •
(12)
Here £2e and Pe are the same as in ref. 2). The amplitudes of the reduced halfwidths are as usual equal to
[~2 R \ ½
(li' R ~½
7ae = t-2-m--m)R = \-~--m/ <'~[c>a"
(13)
The quantities 7ac are complex, in general. It wall be noted that the 7ac in the Kapur-Peierls theory ~) are also complex. In the R-matrix theory, with its procedure of eliminating the infinite background of levels, the dispersion formula contains, not the Yac, but certain modified quantities Otaesmoothly depending on energy. In the simplest case of the equation with one level (see ref. 2)) •A =
(1--R°(L--B))-I~tA
•
(14)
If the remaining part of the S-matrix corresponds to scattering on an optical potential, R ° has only diagonal elements equal to (Le°--Be) -1. In the energy region far removed from optical resonances, [Le°l >> 1 (an infinitely high potential wall) and ~ae reduce to the real ?xe of Wigner's theory. Thus, the quantities ?ae in (12) are real only when there is one isolated level of the compound nucleus, while other processes are well described b y scattering on an infinitely high potential wall. 5. P h a s e s of the R e d u c e d H a l f w i d t h s
The complex character of Yae leads to doubling the indefinite parameters and at a first glance considerably depreciates the dispersion equation (11). It turns out, however, that the phases of these values can be determined from certain general considerations. To begin with, it will be noted that if the halfwidths of levels are considerably less than the distance to the nearest threshold of the nuclear reaction, the wave
DISPERSIONFORMULAS
251
functions of the compound states ~0a and VrXform a mutually orthogonal system. To show this, the equations (Ea--H)~px = 0
and
(E,,,--H*)~o~,*=
0
(15)
are multiplied b y ~p~* and ~oa, subtracted and integrated over a certain volume which contains the compound nucleus. Owing to the smallness of the parameter (3) the wave functions ~Pa are relatively small outside the nucleus and the corresponding integrals depend weakly on the choice of 3. Due to Ea being complex, the functions ~Px exponentially increase at infinity, ~a ~ e x p ( - - I m ka r) at r --~ oo in channel ~, and therefore the volume v must be finite. Let us select the radii R H of the hypersurface limiting • so that the ratio R/R H (R is the radius of the nucleus) is of the order of the parameter (3). Then R << R H, and at a distance R H from the nucleus, the tails of wave functions corresponding to the virtual many-particle processes and channels closed at the given energy will make an insignificant contribution. At the same time the modulus of Im kx RH will be smaller than unity and in the integrals over dr the main contribution will come from the integration over the internal part of the compound system, r < R. The application of the Green theorem yields
(E~,--Ex)f
open
,p~*~0xdv = ~ 7~cTae[(Le)Ex-- (Lc)Ej,].
C
(16)
The quantities y depend on the radius of the hypersurface and are determined according to (13). The real part of the logarithmic derivatives changes slowly with energy, and the derivative 8 Re LdSE equals, in order of magnitude, ER-X (see ref. ~), p. 352). This means that in the approximation of a small parameter (3), in eq. (16) only the imaginary parts Pc of the logarithmic derivatives are essential. It is clear that for the levels not overlapping the thresholds, the functions ,p~ and W~ are mutually orthogonal in the approximation I']lE--Et~e~aJ << 1. At # = ~ we obtain the following relation containing the norm of functions ~a: Na =
f
open
[Wa[~dz = -- (Im Ex) -1 ~ [uae[2.
e
(17)
On the other hand, with the aid of the conventional procedure and applying Green's theorem to the analytic continuation of the functions V(-)" and OW(+~/OEnear the pole Ea, the following equality m a y be established: f~o~*~pxdv :
1 °~ o ~ (OLc~ ~/Ex
rL.
(18)
In the same approximation Le = const, and the integral (18) is equal to unity. Now it can easily be found that ½f
I~oa--~p212dv: N a - 1.
(19)
252
V. I. SERDOBOLSKY
It is clear that Na > 1. The equations for ~oa and yr~, accurate to the ratio
Fa/[ E--Emresh[, coincide, and the dependence of solutions describing compound states on the choice of limiting conditions is weak; therefore, for isolated levels, ~va approximately equals ~r~, and Na ---- 1. If the levels overlap, even small changes in the initial equation and limiting conditions lead to an essential change in the shape of eigenfunctions. Let us assume than n levels of the compound nucleus lie in a small region of energies of the order o f / ' . The respective wave functions ~oa are linearly independent and in the approximations made satisfy the biorthogonality condition. Variation of limiting conditions will lead to a shift of eigenvalues by a quantity of the order of/1 and to the formation of new eigenfunctions which will be linear combinations of the former. Applying this argument to the functions ~px we come to the equality =
(20)
/t
The coefficients a~a are determined from the biorthogonality condition =
ax; -- fw;wad
,
= axx = N x.
(21)
Hence open
U.ue UA e
at,;~ = 2i c E,,* -- E x
(22)
We multiply (20) by ~;* and integrate the result; the equation thus obtained imposes a relation between the coefficients a ~ :
a~,xam, = da~.
(23)
/t
The relation (20) between the functions ~va is approximately satisfied in the internal region; on the surface of the nucleus, eq. (20) holds good only when the effects connected with the permeability of the potential well are inessential. It is noteworthy that eq. (23) with coefficients (22) can be deduced from the principal equation (11) if it is demanded that the approximate S-matrix with a certain number of selected levels satisfies the unitarity condition. Specific applications and derivation of dispersion formulas for certain particular cases of practical interest is given in a separate paper 12).
6. Averaged C r o s s - S e c t i o n s and Correlations in the Signs of 7ac If the signs of Yac do not correlate, it is clear from (23) that the cross terms over levels with/z ~ 2 vanish in averaging with respect to levels and to channels alike. The quantities Tac are real and the Fa are equal to --2 Im Ea. Recently mention has often been made of the need for taking into account the
D I S P E R S I O N FORMULAS
~5~
correlation in the distribution of the signs of the amplitudes of the reduced halfwidths of the compound-nucleus levels 4, 13) .Brown and De Dominicis admitted that it is possible to obtain the matrix elements of direct processes by taking account of correlations over sufficiently large energy intervals, Let us assume that the average )'~e'~'h~ is not equal to zero. Just as in ref. e) we write down the cross-section averaged with respect to energy: ,,c,c ~
Isc,cl 2
= Isc,~12+ Is,%mPI2-- S~ ~
2.
(2,4)
The diagonal terms with ff = ~. in the right-hand side of (24) yield the wellknown Weisskopf-Ewing equation for processes passing through the formation of the compound nucleus. The contribution of the remaining terms depends on the correlation of the quantities )'he- Analysis of eq. (11) shows that for P >> D a certain positive additional term is obtained. This term m a y be interpreted as a cross-section of a certain rapid process evidently involving a small number of particles. Yet for _r' << D the correlation addition to the cross-section proves to be negative. This result contradicts the assumption made in ref. 4) concerning the nature of the terms describing direct interactions. The procedure of summing up the influence of remote levels was used by Bloch 3), Brown and De Dominicis 4) as well as Lane 13). It is characteristic that in all these papers use was made of the closure relation for the wave functions kwh in the Kapur-Peierls theory. After summation terms of the type (1) were obtained. It can easily be seen, however, that all values connected with the resonances escape from the theory at the moment one uses the closure relation. The fact that, for example, a considerable part of the nuclear interaction is described by the optical potential shows that the system of physical wave functions of the compound nucleus can by no means be regarded as a closed one. Thus, the correlations evolved in refs. 3,4,1a) are merely correlations between the amplitudes of the formal mathematical levels of the Kapur-Peierls theory; as for the )'he related to the real compound states of the nucleus, it is natural to retain for them the Bethe assumption.
7. A p p e n d i x : Wave F u n c t i o n s 4 , ~P(i), ~Po(+) in the R e p r e s e n t a t i o n with Quantum Numbers asLdM To simplify the formulas we use the Dirac notations. For the spherical function depending on the radiusvector of the coordinate we shall use the notation [lm). As before 2)
(25)
Similar relations hold for the spinor ]sv) and the spin-angular function of the total momentum. The bracket ]~) will designate the wave function of the inter-
254
V.I.
SI~RDOBOLSKY
nal motion in the channel ~ normalized so that the integral <~]~> with respect to internal variables is unity. The aggregate of the quantum numbers ~slJM will be designated b y the index c as before. The wave function $ describes the motion of non-interacting particles with the quantum numbers 0~v and the relative momentum at infinity ?ik. In normalizing
<4"1~> = %~%,..,,~(k'--k).
(26)
is equal to
I~> = i ~
~ Y*,,
(s~lmlJM)Je-Oc W Ic>.
(27)
JM
Here the ket Ic> designates the ket I~lJM> and the remaining notations are conventional 2). The interaction leads to the appearance of divergent waves in other channels. Outside the range of forces
[~0(+)> = i
o,
v'I r'
Ic'>.
(28)
vqr'
(c'l.
(29)
JM
Acoording _to (4)
,,,2-~k.I ~Y,m
(svlm[JM) ~,
Ztr~
,
Cp
]M
The transition amplitude from the state ~b to state 4' is expressed b y the matrix element <~'lV']~o(+)>,Replacing V' b y t h e difference (E--H'o)- (E--H), the integration reduces to a surface part, as a result of which <¢']V']~(+)> _-- i~.(2zt)-l(m'mk'k)-½
× ~ Yvm"
Y*,~
(s'vTm'JJM)(s~,lmlJM)(S~c--6c,c). (30)
JM
In the representation ~slJM the wave functions of scattering on the optical potential (central without spin-orbit coupling) are equal to [~o(+)> =
~Y*
(svlmlJM)ut(kr)l~slJM>,
(31)
JM
<~Vo(-)1 = i ~
Y,,, -~ (svlmlJM)ut(kr)
(32)
JM
uz being equal to v-½r-l(I,--S~°O~) outside the region of interaction. Sometimes it is convenient to express the optical scattering S-matrix through the logarith-
DISPERSION FORMULAS
255
mic derivative L°:
F ~2 2Le°--Lc* ~2e2 exp(2iJe°), Sc° = c Le0__Le-
te 0 = L.
yU c ..I R"
(33)
Here ~2e and L e are the same as in ref. 2). In conclusion I avail myself of the opportunity to thank Professor A. S. D a v y d o v for his attention and stimulating advice. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)
E. P. Wigner a n d L. Eisenbud, Phys. Rev. 72 (1947) 29 A. M. Lane and R. G. Thomas, Revs. Mod. Phys. 30 (1958) 257 C. Bloch, Nuclear Physics 4 (1957) 503 G. E. Brown and De Dominicis, Annals of Physics b (1959) 209; and others H. Feshbach, Annals of Physics 5 (1958) 357 M. Sano, S. Yoshida and T. Terasawa, Nuclear Physics b (1958) 20 P. L. K a p u t and R. E. Peierls, Proc. Roy. Soc. A l b b (1938) 277 F. L. F r i e d m a n and V. F. Weisskopf, Niels Bohr and the Development of Physics (Pergamon Press, London, 1955) V. I. Smirnov, A Course of Higher Mathematics (Moscow, 1959) vol. V, p. 576 T. Teichman and E. P. Wigner, Phys. Rev. 87 (1952) 123 H. Feshbach, C. F. Porter and V. F. Weisskopf, Phys. Rev. 9b (1954) 448 V. I. Serdobolsky, J E T P 38 (1960) 1903 A. M. Lane, Nuclear Physics 1 1 (1959) 625
Editor's Note: A point of view akin to t h a t o1 the present paper is outlined, on a somewhat more rigorous basis, in m y lectures a t the Summer Meeting of Physicists a t Hercegnovi (Yugoslavia) (September 1959). This t r e a t m e n t is a n extension of a n investigation b y J. H u m b l e t published in 1952, and will shortly been presented in more detail b y J. H u m b l e t and myself in this journal.
D I S P E R S I O N F O R M U L A S IN T H E T H E O R Y OF N U C L E A R REACTIONS V. I. S E R D O B O L S K Y
P. N. Lebedev Physical Institute, Moscow, U S S R Received 4 March 1960 A b s t r a c t : The p a p e r offers a t h e o r y of resonances in nuclear reactions, in which the concept of t h e R - m a t r i x is n o t used. U n d e r consideration is a certain p a r a m e t e r whose smallness m a k e s it possible to select, o u t of t h e total n u m b e r of formal levels obtained in the theory, the resonance levels a t t r i b u t a b l e to the f o r m a t i o n of the long-lived c o m p o u n d nucleus. F o r the levels which do n o t overlap the nuclear-reaction thresholds the c o m p o u n d - n u c l e u s resonances are singled o u t b y tile poles of a Green's function at c o m p l e x energies. The dispersion f o r m u l a s t h u s o b t a i n e d are linear w i t h respect to resonances and contain the complex reduced half-widths. These f o r m u l a s do n o t depend on auxiliary " h y p e r s u r f a c e radii" a n d take into a c c o u n t the tails of w a v e functions responsible for multi-particle processes occurring close to the surface of the nucleus.
1. I n t r o d u c t i o n
The invariably successful application of the Breit-Wigner resonance formulas for the analysis of a great variety of nuclear reactions has contributed to the evolution of a number of general mathematical theories justifying and specifying the simplest resonance formulas. In 1947 Wigner suggested the well-known theory 1) based on the application of the R-matrix. In a number of subsequent papers this theory was perfected and extended to embrace various particular cases. The success of the theory for the past ten years has been summarized in an extensive paper 2). Yet at present it has been clear that the Wigner theory is unsatisfactory in many respects. A large number of auxiliary parameters without physical meaning figure in the R-matrix formalism. These are the energy levels and halfwidths of formal "remote levels" and arbitrary "radii of nucleus hypersurface". It is not clear how the processes involving the emission of two or more real or virtual particles should be described. On the other hand, practical computation has shown that in many cases the apparatus of the Wigner theory proves cumbersome and unjustifiably complex. Accumulation of experimental data has shown that processes occurring without the formation of a compound nucleus bulk large in nuclear reactions. Papers concerned with the "unified theory of nuclear reactions" 3-6) have appeared. It was found that scattering on an optical potential as well as direct i:~teractions and processes involving the formation of a compound nucleus may conveniently be described b y a single mathematical apparatus. Dispersion 245
246
v.I.
SERDOBOLSKY
formulas in the framework of the unified theory of nuclear reactions have been studied b y Feshbach 5) as well as Sano, Yoshida and Terasawa 6). The formulas obtained in ref. 5) have a limited range of application. In ref. 6) the permeability coefficients in dispersion formulas are represented as infinite series which makes their practical application rather difficult. The present paper has two basic aims: to overcome the most essential theoretical drawbacks inherent to the current formal theory and to develop a procedure for evolving dispersion formulas with allowances for optical interaction and other processes weakly depending on energy. Sec. 2 introduces a parameter, defined b y eq. (3), whose smallness makes it possible to regard the formal quantities Ez and Yxe as compound-state resonance parameters. Sec. 3 investigates the analytic properties of the wave function and the Green's function ( E - - H ) -1 of the general problem of collision with rearrangement of particles. The expansion (8) for the Green's function near the poles is established. Sec. 4 discusses a dispersion expansion (11) independent of arbitrary radii and taking account of the possibility of multiparticle decay. Under consideration is the problem of the reality of the quantities ~'ac included in practical formulas. These quantities prove to be complex as a rule. Sec. 5 demonstrates that the compoundstate wave functions in the region of energies far from the threshold form a biorthogonal system. Hence follows eq.. (23) for parameters entering into the dispersion formulas. Sec. 6 discusses the problem of correlations in the signs of Yac and allowances for the contribution of the "remote levels".
2. Selection of Compound States As in the previous papers concerned with the unified theory of nuclear reactions, we shall proceed from the integral equations of the formal theory of rearrangement collisions. Let ¢ be the wave function of free motion of particles of the ~ type with the spin quantum numbers sv and the momentum hk in the centre-of-mass system. Let H be the total Hamiltonian of the problem, and H o the Hamiltorfian of free motion in the channel ~, H being equal to Ho+V. Two solutions ~vc+) and ~v~-) are obtained from eq. (4) for Im E --~ i 0 . (see below). Let us introduce an optical potential V ° and the wave functions ~0~+J and ~o ~-~ describing scattering on an optical potential in each of the channels. The indices and operators relating to some other channel with other quantmn numbers will be designated b y dashes. Let ~ be equal to V--V°; then the following identity (see refs. ~,e)) can be written:
+(~o~-~']¢"(E--H+ie)-~Yrl~o(+~).
(1)
The first term in (1) describes an optical interaction and vanishes for ~' ¢ ~. The second term represents the amplitude of direct processes in the distorted
DISPERSION
FORMULAS
247
wave approximation method. In ref. 6) the authors regarded the third term of the right-hand side of eq. (1) as a resonance one and applied the Kapur-Peiefls theory 7). As is well known, only some of an infinite number of formal levels correspond to those really observed. Let us determine a parameter whose smallness allows of regarding a given level as conditioned b y the formation and disintegration of a compound nucleus. We shall proceed from the time description of a nuclear reaction suggested in ref. 8) and isolate compound states b y their large lifetimes. Average reaction times are closely connected with the nature of the energy dependence of wave functions and amplitudes: according to the uncertainty principle A E ~ ?i/t, where A E is the interval over which the wave functions change essentially. The characteristic periods of potential scattering and direct processes are, in general, determined b y the time needed b y the particles for their flight across the nucleus R/v. For particles with the highest orbital momenta this time increases, due to slowing down near the centre, b y a factor (kR)/Po, where Pc is the permeability coefficient determined in ref. 2). The decay time of the compound state ~ with respect to the channel ~ s l J M (index c) is equal to ~/2Pe[y~e[ 2. The ratio tnight :
tdecay
[7~c12 ER '
~2 where E R - - - - - - - , 2mR 2
(2)
is at the same time the ratio of the width of the compound-nucleus level to that of the optical resonance width -Pc ER- The parameter describing a compound state m a y be determined more accurately with the help of wave functions. The ratio of the square of the wave function modulus of a decaying state averaged over the hypersurface of the nucleus to the internal volume average is open
(4~) - z E I(CJkr-/~.)R[2 c
open ~, 2 I N" J¢~cl
The sum over c includes only open channels. The quantity (3) describes a relative time which the system spends in one-particle states possible at the given energy. This time should be small for the compound nucleus. Treatment of experimental data performed in ref. 10) shows that even for light nuclei the parameter (3) rarely exceeds 0.1. 3. A n a l y t i c
Properties
Analytic dependence of the wave function of the stationary problem on complex energy calls in principle for special proof. With real E this proof can probably be performed with the aid of general theorems on the regular dependence of the solutions on the form of the equation and the boundary conditions.
248
V. I. S E R D O B O L S K Y
With Im E ~ 0 the proof of analyticity offers no difficulties. The spectral analysis 1) shows that the Green's function G -= ( E - - H ) -1 is analytical with respect to E at all points which do not coincide with the points of the H operator spectrum. The wave function ~v is connected with ( E - - H ) -1 through the Gell-Man-Goldberger equation ~v = ¢ + ( E - - H ) - I V ~ .
(4)
The continuous spectrum of the operator lies on the half axis of real energies E > El. For Im E -+ +0, E > E~, G passes into the Green's funtions G~*), and ~vinto the ~vc±) of the physical problem. The asymptotic form of the wave functions (by the radiation condition) contains wave numbers proportional to ~/~,, where E , is the energy of the nth threshold of tile reaction. Tile analytic continuations of ~oc*) ,therefore, are unique analytic functions only of the arguments ~ / ~ 1 . . . . ~ / E - - E . . . . . . The region of determinations of these functions constitutes a system of Riemann sheets with brarmh points at E~ . . . . E . . . . . . From the continuity equation it follows that in the principal Riemann sheet in which all Im v / ~ n > 0, the Green's function G may have poles only at real E. These poles describe the stationary states of the system. Apart from them, complex poles lying in other Riemann sheets are possible. If the pole approaches the half-axis E > E~ of the principal sheet on which a continuous spectrum lies, resonances can be observed ill the cross sections. Tile time reversibility of quantum mechanics leads to a certain symmetry in the analytic continuations G and ~. Let 0 be a time reversibility operator, such that H* = OHO +. (5) It is convenient to introduce the operation B which, in acting on the point E in the system of Riemann sheets, leads to reflection with respect to the real halfaxis oi the principal sheet E < E 1 . For example, tile result of the action of B on E is E*, B ~ V / E - - E , is equal to -- ( ~ / E - - E , ) * , etc. For real energies it is obvious that OG~±~O+ ---- BG(±~'. (6) The operation B* changes the signs of all roots in the arguments of analytic functions and consequently does not violate analyticity. The relation (6) therefore holds good for the complex E as well. It is clear that the poles of G are paired, a pole ~ at the energy B E a corresponding to each pole 2 at the energy E a. Physical meaning can evidently be attributed only to the first order poles. The function ERes VJEa satisfies the Schroedinger equation with complex energy E a and contains only divergent waves in all channels; therefore tile residue [Res(JM]v)]E ~ can be regarded, except for a proportionality factor, as the wave function ,q,~of the quasi-stationary state with given integrals of motion
DISPERSION
249
FORMULAS
J and M. The constant phase of ~ should naturally be chosen so that 1PX*= (-- )J-M O~a (--M) •
(7)
With the help of the relations (4), (6) and (7) the resonance part of the Green's function can be expanded near the poles over the quasi-stationary state wave functions. In Dirac's notations ~ is a ket I~) and ~pX*the bra <~tI. The expansion has the form
IX>O.I _]_Gsmootla.
(8)
It will be noted that if the compound-nucleus level overlaps the nuclearreaction threshold, the optical wave function changes essentially over energy intervals of the order o f / ' owing to the permeability coefficient. The isolation of compound states according to energy dpendence therefore becomes impossible. In refs. 11) and 4) the approximation / ' << ]E--EtkreBh I is actually used, where Emre8h is the energy of the threshold closest to E. In the same approximation, the second term ill (8) which remains after the subtraction of the polar part of G can be regarded as the Green's function of direct and optical processes.
4. Dispersion Expansion As has been established, the compound-nucleus resonances are contained in the last term of eq. (1). To obtain a dispersion formula it is sufficient to substitute (8) and pass to the representation with quantum numbers otslJM. Employing the formulas given in the appendix, we obtain
sL =
i
E--E~
J + (so,o),mooth.
(9)
With the help of relations like (5), (7) and (25) we can easily see that the resonance part of (9) is symmetrical with respect to the indices c' and c. For the sake of convenience we introduce the notation
The expansion (9) assumes the form
s L = (sL),moot,,+2i 2: .,o,u.o E~--E"
(")
An essential theoretical advantage of eq. (11) is that it was derived without the assumption of the orthogonality of the wave functions of the channels on the nucleus hypersurface. Thus, there arises no "problem of cutting off the tails of wave functions" characteristic of the current theories. It is clear
250
V. I. SERDOBOLSKY
that multi-particle processes occurring on the nucleus surface do not distort the shape of the dispersion formula and the influence of closed channels (in the energy region far from threshold) is inessential. The decay of the compound nucleus into three or more particles can be described with eqs. (9) and (11) through a proper choice of the function u c with allowance for a continuous spectrum of outgoing particles. Eq. (11) does not contain the radii of the channels, the choice of which is largely arbitrary. If, on the other hand, it is assumed that there exists a hypersurface on which the wave functions [c> are orthogonal, the usual results will be obtained. In the energy region far from threshold the values of Uc depend weakly on energy and it m a y be assumed that u c = ue(k~r). Let us replace ~e" in the brackets b y the difference H - - H o - - V ° and make use of Green's theorem. We find that uae = Oe Pe½7xe •
(12)
Here £2e and Pe are the same as in ref. 2). The amplitudes of the reduced halfwidths are as usual equal to
[~2 R \ ½
(li' R ~½
7ae = t-2-m--m)
(13)
The quantities 7ac are complex, in general. It wall be noted that the 7ac in the Kapur-Peierls theory ~) are also complex. In the R-matrix theory, with its procedure of eliminating the infinite background of levels, the dispersion formula contains, not the Yac, but certain modified quantities Otaesmoothly depending on energy. In the simplest case of the equation with one level (see ref. 2)) •A =
(1--R°(L--B))-I~tA
•
(14)
If the remaining part of the S-matrix corresponds to scattering on an optical potential, R ° has only diagonal elements equal to (Le°--Be) -1. In the energy region far removed from optical resonances, [Le°l >> 1 (an infinitely high potential wall) and ~ae reduce to the real ?xe of Wigner's theory. Thus, the quantities ?ae in (12) are real only when there is one isolated level of the compound nucleus, while other processes are well described b y scattering on an infinitely high potential wall. 5. P h a s e s of the R e d u c e d H a l f w i d t h s
The complex character of Yae leads to doubling the indefinite parameters and at a first glance considerably depreciates the dispersion equation (11). It turns out, however, that the phases of these values can be determined from certain general considerations. To begin with, it will be noted that if the halfwidths of levels are considerably less than the distance to the nearest threshold of the nuclear reaction, the wave
DISPERSIONFORMULAS
251
functions of the compound states ~0a and VrXform a mutually orthogonal system. To show this, the equations (Ea--H)~px = 0
and
(E,,,--H*)~o~,*=
0
(15)
are multiplied b y ~p~* and ~oa, subtracted and integrated over a certain volume which contains the compound nucleus. Owing to the smallness of the parameter (3) the wave functions ~Pa are relatively small outside the nucleus and the corresponding integrals depend weakly on the choice of 3. Due to Ea being complex, the functions ~Px exponentially increase at infinity, ~a ~ e x p ( - - I m ka r) at r --~ oo in channel ~, and therefore the volume v must be finite. Let us select the radii R H of the hypersurface limiting • so that the ratio R/R H (R is the radius of the nucleus) is of the order of the parameter (3). Then R << R H, and at a distance R H from the nucleus, the tails of wave functions corresponding to the virtual many-particle processes and channels closed at the given energy will make an insignificant contribution. At the same time the modulus of Im kx RH will be smaller than unity and in the integrals over dr the main contribution will come from the integration over the internal part of the compound system, r < R. The application of the Green theorem yields
(E~,--Ex)f
open
,p~*~0xdv = ~ 7~cTae[(Le)Ex-- (Lc)Ej,].
C
(16)
The quantities y depend on the radius of the hypersurface and are determined according to (13). The real part of the logarithmic derivatives changes slowly with energy, and the derivative 8 Re LdSE equals, in order of magnitude, ER-X (see ref. ~), p. 352). This means that in the approximation of a small parameter (3), in eq. (16) only the imaginary parts Pc of the logarithmic derivatives are essential. It is clear that for the levels not overlapping the thresholds, the functions ,p~ and W~ are mutually orthogonal in the approximation I']lE--Et~e~aJ << 1. At # = ~ we obtain the following relation containing the norm of functions ~a: Na =
f
open
[Wa[~dz = -- (Im Ex) -1 ~ [uae[2.
e
(17)
On the other hand, with the aid of the conventional procedure and applying Green's theorem to the analytic continuation of the functions V(-)" and OW(+~/OEnear the pole Ea, the following equality m a y be established: f~o~*~pxdv :
1 °~ o ~ (OLc~ ~/Ex
rL.
(18)
In the same approximation Le = const, and the integral (18) is equal to unity. Now it can easily be found that ½f
I~oa--~p212dv: N a - 1.
(19)
252
V. I. SERDOBOLSKY
It is clear that Na > 1. The equations for ~oa and yr~, accurate to the ratio
Fa/[ E--Emresh[, coincide, and the dependence of solutions describing compound states on the choice of limiting conditions is weak; therefore, for isolated levels, ~va approximately equals ~r~, and Na ---- 1. If the levels overlap, even small changes in the initial equation and limiting conditions lead to an essential change in the shape of eigenfunctions. Let us assume than n levels of the compound nucleus lie in a small region of energies of the order o f / ' . The respective wave functions ~oa are linearly independent and in the approximations made satisfy the biorthogonality condition. Variation of limiting conditions will lead to a shift of eigenvalues by a quantity of the order of/1 and to the formation of new eigenfunctions which will be linear combinations of the former. Applying this argument to the functions ~px we come to the equality =
(20)
/t
The coefficients a~a are determined from the biorthogonality condition =
ax; -- fw;wad
,
= axx = N x.
(21)
Hence open
U.ue UA e
at,;~ = 2i c E,,* -- E x
(22)
We multiply (20) by ~;* and integrate the result; the equation thus obtained imposes a relation between the coefficients a ~ :
a~,xam, = da~.
(23)
/t
The relation (20) between the functions ~va is approximately satisfied in the internal region; on the surface of the nucleus, eq. (20) holds good only when the effects connected with the permeability of the potential well are inessential. It is noteworthy that eq. (23) with coefficients (22) can be deduced from the principal equation (11) if it is demanded that the approximate S-matrix with a certain number of selected levels satisfies the unitarity condition. Specific applications and derivation of dispersion formulas for certain particular cases of practical interest is given in a separate paper 12).
6. Averaged C r o s s - S e c t i o n s and Correlations in the Signs of 7ac If the signs of Yac do not correlate, it is clear from (23) that the cross terms over levels with/z ~ 2 vanish in averaging with respect to levels and to channels alike. The quantities Tac are real and the Fa are equal to --2 Im Ea. Recently mention has often been made of the need for taking into account the
D I S P E R S I O N FORMULAS
~5~
correlation in the distribution of the signs of the amplitudes of the reduced halfwidths of the compound-nucleus levels 4, 13) .Brown and De Dominicis admitted that it is possible to obtain the matrix elements of direct processes by taking account of correlations over sufficiently large energy intervals, Let us assume that the average )'~e'~'h~ is not equal to zero. Just as in ref. e) we write down the cross-section averaged with respect to energy: ,,c,c ~
Isc,cl 2
= Isc,~12+ Is,%mPI2-- S~ ~
2.
(2,4)
The diagonal terms with ff = ~. in the right-hand side of (24) yield the wellknown Weisskopf-Ewing equation for processes passing through the formation of the compound nucleus. The contribution of the remaining terms depends on the correlation of the quantities )'he- Analysis of eq. (11) shows that for P >> D a certain positive additional term is obtained. This term m a y be interpreted as a cross-section of a certain rapid process evidently involving a small number of particles. Yet for _r' << D the correlation addition to the cross-section proves to be negative. This result contradicts the assumption made in ref. 4) concerning the nature of the terms describing direct interactions. The procedure of summing up the influence of remote levels was used by Bloch 3), Brown and De Dominicis 4) as well as Lane 13). It is characteristic that in all these papers use was made of the closure relation for the wave functions kwh in the Kapur-Peierls theory. After summation terms of the type (1) were obtained. It can easily be seen, however, that all values connected with the resonances escape from the theory at the moment one uses the closure relation. The fact that, for example, a considerable part of the nuclear interaction is described by the optical potential shows that the system of physical wave functions of the compound nucleus can by no means be regarded as a closed one. Thus, the correlations evolved in refs. 3,4,1a) are merely correlations between the amplitudes of the formal mathematical levels of the Kapur-Peierls theory; as for the )'he related to the real compound states of the nucleus, it is natural to retain for them the Bethe assumption.
7. A p p e n d i x : Wave F u n c t i o n s 4 , ~P(i), ~Po(+) in the R e p r e s e n t a t i o n with Quantum Numbers asLdM To simplify the formulas we use the Dirac notations. For the spherical function depending on the radiusvector of the coordinate we shall use the notation [lm). As before 2)
(25)
Similar relations hold for the spinor ]sv) and the spin-angular function of the total momentum. The bracket ]~) will designate the wave function of the inter-
254
V.I.
SI~RDOBOLSKY
nal motion in the channel ~ normalized so that the integral <~]~> with respect to internal variables is unity. The aggregate of the quantum numbers ~slJM will be designated b y the index c as before. The wave function $ describes the motion of non-interacting particles with the quantum numbers 0~v and the relative momentum at infinity ?ik. In normalizing
<4"1~> = %~%,..,,~(k'--k).
(26)
is equal to
I~> = i ~
~ Y*,,
(s~lmlJM)Je-Oc W Ic>.
(27)
JM
Here the ket Ic> designates the ket I~lJM> and the remaining notations are conventional 2). The interaction leads to the appearance of divergent waves in other channels. Outside the range of forces
[~0(+)> = i
o,
v'I r'
Ic'>.
(28)
vqr'
(c'l.
(29)
JM
Acoording _to (4)
,,,2-~k.I ~Y,m
(svlm[JM) ~,
Ztr~
,
Cp
]M
The transition amplitude from the state ~b to state 4' is expressed b y the matrix element <~'lV']~o(+)>,Replacing V' b y t h e difference (E--H'o)- (E--H), the integration reduces to a surface part, as a result of which <¢']V']~(+)> _-- i~.(2zt)-l(m'mk'k)-½
× ~ Yvm"
Y*,~
(s'vTm'JJM)(s~,lmlJM)(S~c--6c,c). (30)
JM
In the representation ~slJM the wave functions of scattering on the optical potential (central without spin-orbit coupling) are equal to [~o(+)> =
~Y*
(svlmlJM)ut(kr)l~slJM>,
(31)
JM
<~Vo(-)1 = i ~
Y,,, -~ (svlmlJM)ut(kr)
(32)
JM
uz being equal to v-½r-l(I,--S~°O~) outside the region of interaction. Sometimes it is convenient to express the optical scattering S-matrix through the logarith-
DISPERSION FORMULAS
255
mic derivative L°:
F ~2 2Le°--Lc* ~2e2 exp(2iJe°), Sc° = c Le0__Le-
te 0 = L.
yU c ..I R"
(33)
Here ~2e and L e are the same as in ref. 2). In conclusion I avail myself of the opportunity to thank Professor A. S. D a v y d o v for his attention and stimulating advice. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)
E. P. Wigner a n d L. Eisenbud, Phys. Rev. 72 (1947) 29 A. M. Lane and R. G. Thomas, Revs. Mod. Phys. 30 (1958) 257 C. Bloch, Nuclear Physics 4 (1957) 503 G. E. Brown and De Dominicis, Annals of Physics b (1959) 209; and others H. Feshbach, Annals of Physics 5 (1958) 357 M. Sano, S. Yoshida and T. Terasawa, Nuclear Physics b (1958) 20 P. L. K a p u t and R. E. Peierls, Proc. Roy. Soc. A l b b (1938) 277 F. L. F r i e d m a n and V. F. Weisskopf, Niels Bohr and the Development of Physics (Pergamon Press, London, 1955) V. I. Smirnov, A Course of Higher Mathematics (Moscow, 1959) vol. V, p. 576 T. Teichman and E. P. Wigner, Phys. Rev. 87 (1952) 123 H. Feshbach, C. F. Porter and V. F. Weisskopf, Phys. Rev. 9b (1954) 448 V. I. Serdobolsky, J E T P 38 (1960) 1903 A. M. Lane, Nuclear Physics 1 1 (1959) 625
Editor's Note: A point of view akin to t h a t o1 the present paper is outlined, on a somewhat more rigorous basis, in m y lectures a t the Summer Meeting of Physicists a t Hercegnovi (Yugoslavia) (September 1959). This t r e a t m e n t is a n extension of a n investigation b y J. H u m b l e t published in 1952, and will shortly been presented in more detail b y J. H u m b l e t and myself in this journal.