A new approach to the three-body problem

Nuclear Physics AI41 (1970) 620--630; (~) North-Holland Publishin# Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

A NEW A P P R O A C H TO T H E T H R E E - B O D Y P R O B L E M W. GLOCKLE t

Service de Physique Thdorique, Centre d'l~tudes Nucldaires de Saclay, BP. no. 2-91-GIF-sur-YVETTE Received 10 July 1969

Abstract: A set of three integral equations for the wave functions of the three-body problem is proposed. One of these equations is the Lippmann-Schwinger equation, the other two are homogeneous equations and complete the definition of the boundary conditions. The two methods proposed to solve this system reduce to a Hilbert-Schmidt problem in a two-vector space.

1. Introduction The use of the Lippmann-Schwinger equation in the case where rearrangement channels exist is problematic, because the Lippmann-Schwinger equation does not define uniquely the solution for real energies. Therefore Faddeev's formulation 1) is now generally adopted. We try to formulate the supplementary conditions to the Lippmann-Schwinger equation to define the solution uniquely in the form of additional homogeneous integral equations. This is outlined in sect. 2 for the three-body problem. In sect. 3 we propose two algorithms for a solution and in sect. 4 we treat the case of identical particles. The whole numerical problem is presently under investigation, a small part of it, which might shed some light on the content of sect. 3. is presented in the appendix. We hope that in time a more rigorous mathematical justification of this new formulation can be given.

2. The Lippmann-Schwinger equations We consider three distinguishable spinless particles which interact by two-body forces. Their positions and masses are denoted by x 1, x2, x3 and ml, m2, ms, respectively. Corresponding to the three possibilities to have a pair of particles bound and the third particle moving freely we introduce the relative coordinates

r~ = x j - x k ,

R~ =

Xi--(~iXj'~-fltXk)

,

(1.1)

where (i,j, k) is a cyclic permutation of (1, 2, 3), 0q = mj/(m~+mk) and/~i = 1-~g. These coordinates are connected by r2 = - ~ l r x - R l ,

R2 = ( 1 - g ~ / / z ) r ~ - / / 2 R 1 ,

? On leave o f absence from the University Heidelberg, (West Germany). 62O

(1.1a)

THREE-BODY PROBLEM

621

and r 2 ~-- - - f l 3 / ' 3 , J t - R 3 ,

R 2 =

--(1--of2f13)r3--(z2R3,

(1.1b)

The other relations are given by cyclic permutations of the eqs. (1.1a) and (1.1b). The configurations of a bound pair and a free particle are described by the eigenfunctions

~i(ri, Ri ) = tp,(ri)eiQ,. R,,

(1.2)

of the Hamiltonians 1 H i ~

~

__

2#i

1

A--

Aa,+ V~,

i = 1, 2, 3,

(1.3)

with 1

gi

-

1

1

+--, mj mk

1

1

1

--=-+ - Mi mi mr + mk

and

Vi= V/k(lX~--Xk[).

The corresponding eigenvalues are given by Ei = (Q~2/2Mi)+ et, e~ being the binding energy of ~pi(rt). We fix now E~ to be the total energy Ein the c.m. system. Corresponding to these three configurations there exist three stationary solutions of the Schr~dinger equation

H~b} +) = (Hi+ Vi+ Vk)~l,}+) = EO~ +),

(1.4)

defined by 0}+) = lim i~ ~o E+ie-H

~i,

i = 1, 2, 3.

(1.5)

As is well known ~b}+) is a solution of the Lippma~n-Schwinger equation

~b~+) = ¢i + Gi( Vi + Vk)~b~+), with the resolvent G~ = l i m ( E + i e - H i )

(1.6)

-I. This equation (1.6) does not define

e"* 0

~b}+) completely. The functions ~b}+) and q/k+),j, k # i, are solutions of the corresponding homogeneous equation 1). Therefore one has to impose further boundary conditions which exclude in the solution of eq. (1.6) the admixture of~} +) and ~//k+). One possibility is to add the two following homogeneous equations, which ~ + ) satisfies as well as eq. (1.6)

~/}+) = Gj(Vk-k- V~)~t~+),

(1.7)

q,}+~ = G~(V,+ VjN +~.

(1.s)

The eq. (1.7) excludes in ~ + ) the admixture of q/~+), since every solution of eq. (1.7) has the following purely outgoing behaviour in the variables j, when the corresponding pair of particles is bound

¢}+>R~oo%(rj)~

<@jlVk+ V~l¢~+)>.

622

w . GL()CKLE

The analogue with respect to the asymptotic behaviour in the variables k can be inferred from eq. (1.8). There exists a further scattering solution of the Schrtidinger equation which is related to three incoming separated particles. It is given by ~b(o+) = lim is ~o, ~-.o E + i s - H

(1.9)

with ~o = e~k""eir"s' and E = k2/2#~+K2/2M~, i = I, 2, 3. The quantity ~o is a solution to the Hamiltonian H o = --A,J2prAR,/2Mi. Eq. (1.9) leads to the Lippmann-Schwinger equation g/o+ ) =

+ Go(V, +

+ vk) o+),

(1.10)

where Go = lira ( E + i e - H o ) -~. Introducing the scattering solutions of the Hamile~0

tonian Ht ~(+) O,l

----

el'" ~'~o~7)(ri),

(1.11)

the eq. (1.10) can be rewritten into the following Lippmann-Schwinger equations g/o+' =

,+ Gi(v + v )g/o+).

(1.12)

Thus ~bCo +) does not satisfy the system (1.6)-(1.8). The question arises now whether the system (1.6)-(1.8) defines the scattering solutions ~b~+)(i = 1, 2, 3) uniquely. Certainly a solution to the homogenous equations corresponding to the set (1.6)-(1.8) must be a solution to the SchrSdinger equation at the positive energy E (the zero of the energy will be taken at the lowest threshold) with the property to be purely outgoing in the variables Ri, when the solution is projected on the bound states g0i(ri) (i = 1, 2, 3) respectively. Further this solution is distinct from ~¢0+). As the scattering states {~+)(i = 1, 2, 3)~b~o+~} corresponding to all possible fragmentations in the initial channel together with the bound states form a complete set 2), one is lead to the conclusion, that in fact the homogeneous system corresponding to the equations (1.6)-(1.8) has no non-trivial solution for positive energies. Thus ¢~+)(i = 1, 2, 3) is uniquely defined by the system (1.6)-(1.8). The only solutions of the homogeneous system are the bound state solutions of the Schrfdinger equation at negative energies. These conclusions are outlined in the appendix A. One obtains the Faddeev equations when one applies the operator Go V1, Go V2, and Go V3 to the first, second and third equation, respectively, of the system (1.6)-(1.8) [ref. 6)]. By the same reasoning ~b~o +) is also uniquely defined by the three eqs. (1.12), with (6¢k) cyclic permutations of (1, 2, 3). However, the break-up amplitudes can be calculated independently of ff~o+).

623

THREE-BODY PROBLEM

3. Two proposals for solving the system (1.6)-(1.8) The first method uses different partial wave expansions for ~,~+). Corresponding to the three types of coordinates (1.1), we introduce three types of angular momentum operators in the relative motion 1

li

= 7r~xVr~,

'

Li

(2.1)

,i = .x_Rt × VR,. l

Then the operator L for the total angular momentum in the relative motion can be expressed as L = ll+L1 = / 2 + L 2 = Ia+La. (2.2) The eigenfunctions to the total angular momentum L can be introduced in three different ways

~I~(P,, R,) =

~

C(I,L,L, m,M,M)q/,,m,(P,)qlL,M,(I{,).

(2.3)

mi+Ml=M

We define three types of projection operators

Ps =

Ns

~

LM LM [q/zsLs)(q/~sLs[,

S = 1, 2, 3,

(2.4)

Is, L s = 0

where Ns are finite numbers which will be fixed later. As is proved in ref. z), the kernels Pi Gi(E+ ie)(Vj + Vk) are of the Hilbert-Schmidt type for e > 0. Thus multiplying the eqs. (1.6)-(1.8) by P~, Pj, Pk, respectively, we get the system

Pt~+) = Vt~iSu+PtG'Vk~[+)+P'OtVm~k}+)'

1 = 1,2,3 (1, k, m) cyclic"

One is now tempted to make the following approximation

P, Gt Vk¢ '~ P, G, VkPk ~O, PtGtVmO "~ PtGtVmPm¢.

(2.5)

In fact this can be justified by choosing suitable N values. Let us choose a number N~ and the corresponding operator P~ in such a way that the asymptotic parts of~J (+) which describe an ingoing and outgoing wave in the configuration k are containedin P~(+). The remaining part Vk(1-P~)¢/+) is a square-integrable function: Firstly, the outgoing waves o f ~ <+) in the other two configurations, different from k, are suppressed by the multiplicative factor Vk(rk) (see eqs. (1.1a) and (lAb) which express r k in terms o f r t and Rt, l # k), secondly, since the asymptotic behaviour o f t / + ) in the break-up channel is given by ei'/EJ2tLkrk2 + 2 M k R k 2

const

(#o[ Vt + 1/2+ V31$(+)),

(~/2#,r~ + 2Mk R~)¢t

624

w. GLOCKI.~

one is led to a product Vk(rk)(X/2ltkr2+2MkR~) -t, which is again square integrable. This asymptotic behaviour in the break-up channel can be directly derived from the homogeneous integral equation ~b}+) = Go ( Vz + V2 + Va)~b}+), which ~b~+) satisfies as well, and from the explicit expression for Go,

Go = i 4 ~ ' / Y f f ~ H~'(~/Ep), (4rr)2p 2

p2 = 2u~(r~_d)2+2Mk(kk_R~y.

Now for the square integrable function Z = Vk(1 _p~)~t+) a N~,' exists such that Z a~t,d.~(r,, ,M R k dt2,~,dt2.~,~V,~L,(rk,LM ' R,)Z(' r/,, R~) , N~,, )f

lk, Lk = 0

is arbitrarily small. Thus we can justify (2.5) by choosing Nk to be the larger of the numbers N~, N~'. It remains to know how large the finite numbers Ark have really to be chosen to justify the approximation (2.5). To get a first hint one can replace ~ by the inhomogeneous term ~. The quality of the approximation (2.5) in this case will be at least a necessary condition on the N k values. This is investigated in the appendix B. For the simple case considered in the appendix it turns out that N ~ 4 or 5 is sufficient. In addition we notice that in contrast to the relations (2.5) the replacement of Pt Gt Vk~kby Pl Gl VkPt~k would certainly not be allowed since the ingoing or outgoing wave of~k with a bound pair in the variables k cannot be represented by a finite number of partial waves with respect to the coordinates r~, Rz(l # k). Inserting the approximation (2.5) into the system (1.6)-(1.8) we get the following set of coupled equations for PI~kl, P2~ and P3@

V,~}+) = pt~,f,+p, GtVkVk~P}+)+p, GtVmP,n~(+),

l = 1, 2, 3

(lkm) cyclic"

(2.6)

As every kernel of this system is of the Hilbert-Schmidt type this is true for the total three-dimensional matrix kernel. Therefore with the exception of a discrete number of energies (in general complex) this system (2.6) has a unique solution. This refers to every choice of the numbers Nl (l = 1,2, 3). However a choice of too small Nl values will give solutions Pt~k, which are not the projections of a common wave function ~O. This will only be true if the Ns values are chosen large enough to justify the replacement (2.5). If one can solve the system (2.6) then it should also be possible to answer this question by enlarging the Nt values and looking for a stable behaviour of the T-matrix elements. The connection with the Faddeev equations is given by Vk~k = (E-Ho)~b (k), where ~ = ~kcl)+~b(2)+~ (a). This shows that the approximation Vk~b ~ VkPk~ corresponds to ~(k) ~ pk~k(k), which is usually assumed in practice when solving the Faddeev equations. In principle one can solve the system (2.6) by replacing each kernel by an approxi-

THREE-BODY PROBLEM

625

mation of finite rank. This can be done for instance by a method proposed by Weinberg 4) which makes use of the eigenfunctions of the kernel P t ' . - ' t ~,, k • r-Uk) v = ~0k)v(Zk) qv . t v = ~0k)o ,tv . ' t -v(tk) -v •

(2.7)

This integral equation is equivalent to a finite system of differential equations in two variables with the following boundary conditions:/-(zk) is bounded at the origin, the projection of F °k) on the bound state q~t is purely outgoing in the coordinate R t and F (~k) behaves like ei"/e/r t with r = ~/2/~trzz + 2M~R z if both r t and R t go to infinity. A numerical investigation of this problem is in progress. Now due to the Weinberg method one makes the following approximation M(lk)

t', as vk = Y~ ,iv-(lk)'"('~)',-, / v=l

1

/ u(lk)*l 1: I I'(kl)k

\iv

I VklXv


(2.8)

/

Thereby one neglects a residual kernel, whose eigenvalues are those ofPtGt Vk not included in the sum over v. These eigenvalues can therefore be made as small as is required to justify the approximation (2.8). Introducing the kernel of finite rank (2.8) into the system (1.6)-(1.8), the solution can be found by algebraic means. The T-matrix elements are of the type in the case of the break-up reaction or (OilVj+ Vk[IP~ + ) ) i n all other cases. Thus just the calculated portions VklP ~ VkPk~kenter into these matrix elements. Using the formal expressions found for the solution of the system (1.6)-(1.8) it is straightforward to write each T-matrix element as a ratio of two determinants. The determinant in the denominator is common to all T-matrix elements and the one in the nominator is dependent on the specific process. A second way of solving the system (1.6)-(1.8) can be applied if the operators V~ can be well defined. Multiplying t h e / t h equation by V~~ one gets a coupled system for the unknown functions Vl½~k, l = 1, 2, 3 Vt½~} +) = Vt~cI)ic~li+Vlk:GtV~V~I~+)+Vt~GtVSmV~m~b} +),

(lkm) cyclic.

(2.9)

The kernels are all of the type Vt~Gt V~(l # k) and are of the Hilbert-Schmidt type, too. Therefore one can again reduce the system to an algebraic one. The bound state wave function which is a solution of the homogeneous system corresponding to the set (1.6)-(1.8) can be found by the same two methods. The binding energies are given as the zeros of an energy dependent determinant, the determinant in the denominator, which was mentioned above in the context of the T-matrix elements.

4. Identical particles For spinless particles one can immediately infer from eq. (1.1) that an antisymmetric wave function q/a must have the following properties: ~ka(rl, R1) = ~ka(r2, R2) = ~k~(r3, R3), ~O,(r,, R,) = - O , ( - r i ,

R,).

(3.1) (3.1a)

626

w . OLOCKI.E

The homogeneous system for the bound state problem reduces to one equation, for instance ~b~u*d(rl, R1) = G1 r2""~,'bou*a'tr2, R2)+ G1 v,3 ~'aa'b°~*ar~r3 , R3).

(3.2)

The other two equations of the system are identical to eq. (3.2). This follows from the definition of the coordinates (1.1) and the fact that G~(rs, R~; r', R~) = G ~ ( - r , R~; -r~, R~). Now eq. (3.2) can be solved for the unknown functions P~ or V ~ by any of the two methods described in sect. 3. The condition (3.1a) can be taken into account by using only odd angular momentum states for the relative motion of a pair. The antisymmetric scattering wave function ~b.'° corresponding to a configuration with a bound pair in the incoming channel are defined by ~k~.° = lira ~¢~ a-.o E + i s - H = lim ie ,-.o E + i e - H

~ {¢p(rl)eiO •al+ q~(r2)e,Q. ,2+ q~(ra)e,~, a,}. ~3!

(3.3)

(The factor 2 in eq. (3.3) comes from the fact that the bound-state wave function of a pair must have negative parity because of the identity of the particles.) Introducing the solution ~b(+)(r,, R~) = lira i~ ~(r~, Rt), ,-,o E + i e - H eq. (3.3) yields ¢.° --

+

2 (G,(V2+ Va)~b(+)(rx, R,)+G2(Vx + Va)~k(+)(r2, R2) + G~(V1 + V2)~k(+)(r3, R3)). (3.4)

The antisymmetric solution ~b.~¢ can be expressed as

¢:

2

= ~.I (~(+)(rl'R1)"I-~(+)(r2'R2)+0(+)(r3'R3))'

(3.5)

where use is made of ~bf+)(-r~,R~) = -~b(r~,R~). This follows directly from the corresponding property of q~(ri, Ri). W e insert 0 l" into the right-hand side of eq. (3.4) and remember that ~,(+)(r~,R~) satisfiesthe system (1.6)-(1.8).One gets ~kl~ =

2 cI,(r,, R,)+ G,(Vj + Vk)~J~~, x/3!

i = 1, 2, 3.

(3.6)

Again the three equations are identical and we can use for instance =

2 ~(r~ R,)+G~V~C,7(r2,R2)+G,V~¢:(r3,R3), ,

which can be solved by the methods described in sect. 3.

(3.7)

THREE-BODYPROBLEM

627

Appendix A We investigate whether there exists a non-trivial solution Z in Hilbert space of the following homogeneous system corresponding to the eqs. (1.6)-(1.8) at energies E above the lowest threshold

GI(V2"FV3)x,

Z =

Z = G2(V3+

(A.I)

Vl)•,

z = c3(vi + v2)z, where X must certainlybe a solution of the Schr0dingcr equation ( H - E ) X = 0. Thcrcfore the general expansion of x into the complete set 2) ~-q~'*~l ~,~,t+)9 ,t,(+) °¢tQ2 , ,t,(+) IY~3 ~ J,(+) ~tlkjKj' .~, ~/SboundJ, where for simplicity of notation only the dependence of the basis functions on the wave numbers is indicated a n d j = 1, 2 or 3, is reduced to the following representation X

= ~, ; df2ft,A,(Q,)¢(+ + Jor z,,T~j~dK K JJ re df2.,df21~B/(k#, K.i)~,,~ .+. `.

(A.2)

Here every basis function is a solution of the SchrOdinger equation at the energy E. In the last integralwe have lkil = ~/2/~#(E-(K]/2M#)) and j = I, 2 or 3. Inserting this cxprcssion for Z into the firstequation of the systcm (A. l) and using the fact,that the scatteringsolutions obey the integraleqs. 0.6)-(1.8) and (I.12),one gets

f

d~o, AI(QX)~Q, +

f2

dK~K~

fy

df2k,dI2xtB1(kt, ~x,~'o,lr ~a~(+)k,K, = O.

(A.3)

The second and the third equations of the set (A. 1) lead to a rcsultanalogous to (A.3), where the index I is replaced by 2 and 3. The scalar product of cq. (A.3) with the bound state wave function ¢01(rl)yields

fdOe,

AI(Q,)e 'e'" R1 = O.

(A.4)

AI(Q1)and

Thus AI(Q1) = 0, as can be seen for instance by expanding e i0~'R' into spherical harmonics. The scalar product of eq. (A.3) with ¢p~,+)(rl) yields

f d~,

elKt " S ~ B a ( k l ,

K1) = O.

(A.5)

Again one concludes Bl(kl, RI) = 0 for all kl. Thus altogether Z = 0.

Appendix B We investigate the approximation (2.5) replacing ~k~+) by the inhomogeneous term • j. Thus we consider A -- IlV2*~l-

v2~e2~ d l / l l V ~ l l ,

(B.1)

628

w. GLOCKLE

as a function of the number N 2 of partial waves in P2 (see eq. (2.4)). In the numerical example given below V2 is chosen to be a Yukawa potential. Thus V2~ is well defined and it is introduced only to simplify the numerical calculation of the norm II v~,ll. In order to calculate the projection P2 we introduce states IriRfliL,(LM)> (i = 1, 2, 3) which have the coordinate representation ) - -1 5(r,-r,)q/,,L,ffir,), , L~t ,~, , = 1...~(ri_r~ ri ri R i R~ i

1

(B.2)

and the normalization

= r~r'~•

R,R,~I--~'f(R'--R~)St't"SL'L"SLL'fMM"(B.2)

The calculation of the matrix elements (riRiliLi (LM)IrjRjljL s (L' M')>, i # j, can be carried out in close analogy to 5). We find the result (rz R212L2(LM)Ir~ R~ l~ Lt(EM')> 2

rlRtfl2rzR2 x

-

1

x --~,

-

~

\

-

22

22

2flzr2R2

] ]

~(l-~,fld)

~

C(ItglL, m,MtM)C(12L2L, m2M2M)

2 L + l m m~+M,--U m,+M~=m

x Fl~m,FL,MjFlus, FL2M2exp -- im 2 arcos

2 R1

2 2

2

- - a t rl -- r e 20tt r 1r2 / , (S.4)

with

{V21+lx/(l-m)!(l+m)! (-)½"+") 2t

for

l + m even

for

l+modd.

The angles a r c o s ( . . . ) in (B.4) have to be chosen between 0 and re. To have a simple analytic expression for the bound state wave function qh in ~t for which the bound state energy e 1 and the potential parameters can easily be varied, we take in the numerical example below the lowest solution of the Hulth6n potential Vl(rt) = -- 2/2111a2[e"1/~- 11-1. Thus (without normalization) 1

= - - ( e rl

-K,I

-e

-~,,

• LM " )jL(QtR,)~/or(PlR1),

(B.6)

629

THREE-BODY PROBLEM

where K = x/-2/21el and ~ --- K+ 1/a. For simplicity we confine ourselves to L = 0 and equal masses. Then oo


N2

f f

12=0

J3

= E

t2

t

t2

r 2 dr 2 R 2

t

t

t

dR2(r2R21r 2 R 21212(00) ) ( r

t

2 R[ 12/2(00)1~1)

0 co

oo

-

r~dr~ R~drt(r2 Re 1212(00)1r~ Rt 00(00))

12=0 0

x Jo(Q~ R1) 1 (e_rr~_e_~,~). (a.7) rl

Inserting the corresponding expression (B.4) one finds after some calculation 1

Nz

(r2R21P2'~l) - - ~ ~ P,~(cos02)H,2(r2, g2),

(B.8)

[2=0

with

H12(r2, R2) = ½(212-1-1)___f~ldxP,~(X)jo(Q1 x/1-~r 2 +¼R 2 +¼r 2 R2x ) e-

K~/R22 + ~ r 2 2 - r 2 R 2 x _ e - 0~/R22 + ' ~ r 2 2 - r 2 R 2 x

×

,

4R +¼r -reR

(a.9)

and 02 = arc(R2, r2). Of course this result could have been written down directly without using the expression (B.4) due to the special case L = l~ --- L~ = 0. However in the general case for angular momenta different from zero the expression (B.4) will be very helpful. Now choosing V2 = e-U'2/r2 we have '

~,r(N2)

(B.10)

A(N2) : - VNexact--/¥approx , [

Nexac t

with Ne~t = - -2- ~ ~ d g ~ s i n E Q I R l e - " R ' f o RXdrl(e-2rrt-e-2~n)2 l s i n h / 2 r l /2Q1Jo R1 rl +

od R l ( e - e r R ~

-

e_2~R~)2

-1- e _ .R~

f? dr 1 sin2 Q l r l

R1

sinh/2rl,

(B.11)

rl

and oo

• ,tN2) = ~.2 lv.pp~x

l ff

12= 0 212 q- 1

r2dr2R2dR2 e -~'2 H22(r2, R2).

(B.12)

r2

0

In the following table the dependence of A is2) on N2 is shown for a few cases: we have varied the binding energies K 2 = -2/21el,/21 = ½m,e,t.... and the kinetic energies

w. GLOCKLE

630

el = Q 2 / 2 M t , M x = ~mneutron. T h e r e s u l t s i n t h e t a b l e 1 r e f e r t o p = 0.5 f m - 1 a n d a=

lfm. TABLE 1 The dependence o f A on N2

et = --8 MeV et = 1.5 MeV et = --8 MeV el = 37.5 MeV ei = --25 MeV et = 37.5 MeV K = 0 . 4 3 9 f m -1 Qi = 0.219 f m - t K = 0.439 fm -x Qx = 1-098 f m - I K = 1.098 fm - t Qx = 1.098 f m -x A t°) = 0.50 A o ) = 0.27 A t2) = 0.15 ACa) = 0.08 A t`*) = 0.02

A(O) = A tl) = L1t2) = ACa) = A t4) =

0.32 0.12 0.07 0.04 0.03

/Ito) _-- 0.39 At1) = 0.22 A t') = 0.14 A (a) ---- 0.09 ,d (4) = 0.05

References 1) 2) 3) 4) 5) 6)

L. D. Faddeev, J E T P (Soy. Phys.) 12 (1961) 1014 R. G. N e w t o n , Scattering theory o f waves and particles (McGraw-Hill Book C o m p a n y , 1966) W. Gl6ckle and D. Heiss, Nu¢l. Phys. A122 (1968) 343 S. Weinberg, Phys. Rev. 131 (1963) 440 K. Balian and E. Brezin, N u o v o Cim. submitted for publication J. Reval, private c o m m u n i c a t i o n