The Two-Nucleon System in Three Dimensions

0.35 MB·English

by  J. Golak

The Two-Nucleon System in Three Dimensions
J. Golak1, W. Gl¨ockle2, R. Skibin´ski1, H. Wital a1, D. Rozpedzik1,
ι
K. Topolnicki1, I. Fachruddin3, Ch. Elster4, and A. Nogga5
1M. Smoluchowski Institute of Physics,
Jagiellonian University, PL-30059 Krak´ow, Poland
2Institut fu¨r Theoretische Physik II,
Ruhr-Universit¨at Bochum, D-44780 Bochum, Germany
0
1
3Departemen Fisika, Universitas Indonesia, Depok 16424, Indonesia
0
2
n 4Institute of Nuclear and Particle Physics,
a Department of Physics and Astronomy,
J
Ohio University, Athens, OH 45701, USA and
8

] 5Forschungszentrum Ju¨lich, Institut fu¨r Kernphysik (Theorie),
h
t Institute for Advanced Simulation and Ju¨lich Center
-
l
c for Hadron Physics, D-52425 Ju¨lich, Germany
u
(Dated: January 8, 2010)
n
[ Abstract

1 A recently developed formulation for treating two- and three-nucleon bound states in a three-
v
dimensional formulation based on spin-momentum operators is extended to nucleon-nucleon scat-
4
6 tering. Here the nucleon-nucleon t-matrix is represented by six spin-momentum operators accom-
2
panied by six scalar functions of momentum vectors. We present the formulation and provide
1
. numerical examples for the deuteron and nucleon-nucleon scattering observables. A comparison to
1
0 results from a standard partial wave decomposition establishes the reliability of this new formula-
0 tion.
1
:
v PACS numbers: 21.45.-v,21.30.-x,21.45.Bc
i
X
r
a
1
I. INTRODUCTION
A standard way to obtain scattering observables for nucleon-nucleon (NN) scattering is
to solve the Schro¨dinger equation either in momentum or coordinate space by taking advan-
tage of rotational invariance and introduce a partial wave basis. This is a well established
procedure and has at low energies (below the pion production threshold) a clear physical
meaning. At higher energies the number of partial waves needed to obtain converged results
increases, and approaches based on a direct evaluation of the scattering equation in terms
of vector variables become more appealing.
Especially the experience in three- and four-nucleon calculations [1, 2] shows that the
standard treatment based on a partial wave projected momentum space basis is quite suc-
cessful atlower energies, but becomes increasingly moretedious withincreasing energy, since
each building block requires extended algebra and intricate numerical realizations. On the
other hand for a system of three bosons interacting via scalar forces the relative ease with
which a three-body bound state [3] as well as three-body scattering [4] can be calculated
in the Faddeev scheme when avoiding an angular momentum decomposition altogether has
beensuccessfully demonstrated. Thus itisonlynaturaltostriveforsolvingthethreenucleon
(3N) Faddeev equations in a similar fashion.
Recently we proposed a three-dimensional (3D) formulation of the Faddeev equations
for 3N bound states [10] and 3N scattering [11] in which the spin-momentum operators are
evaluated analytically, leaving the Faddeev equations as a finite set of coupled equations
for scalar functions depending only on vector momenta. One of the basic foundations of
this formulation rests on the fact that the most general form of the NN interaction can
only depend on six linearly independent spin-momentum operators, which in turn dictate
the form of the NN bound and scattering state. Here we extend the formulation of the NN
bound state given in [10] to NN scattering and provide a numerical realization.
There have been several approaches of formulating NN scattering without employing a
partialwavedecomposition. AhelicityformulationrelatedtothetotalNNspinwasproposed
in [5], which was extended to 3N bound state calculations in [6]. The spectator equation
for relativistic NN scattering has been successfully solved in [7] using a helicity formulation.
Aside from NN scattering, 3D formulations for the scattering of pions off nucleons [8] and
protons off light nuclei [9] have recently been successfully carried out.
In Section II we introduce the formal structure of our approach starting from the most
general form of the NN potential. We derive the resulting Lippmann-Schwinger equation
and show how to extract Wolfenstein parameters and NN scattering observables. Numerical
realizations of our approach that employ a recent chiral next-to-next-leading order (NNLO)
NN force [12–14] as well as the standard one-boson-exchange potential Bonn B [15] are
presented in Section III. The scalar functions, which result from the evaluation of the spin-
momentum operators and have to be calculated only once are given in Appendices A and
B. Finally we conclude in Section IV. The more technical information necessary to perform
calculations with the chiral potential is given in Appendix C. In Appendix D the Bonn B
potential is presented in the form required by our formulation.
II. THE FORMAL STRUCTURE
We start by projecting the NN potential on the NN isospin states tm , with t = 0,m =
t t
| i
0 being the singlet and t = 1,m = 1,0,1 the triplet. We assume that isospin is conserved,
t
−
2
butallowforchargeindependence andchargesymmetry breaking, andthusforadependence
on m ,
t
ht′m′t | V | tmti = δtt′δmtm′tVtmt (2.1)
Furthermore, the most general rotational, parity and time reversal invariant form of the
off-shell NN force can be expanded into six scalar spin-momentum operators [17], which we
choose as
w (σ ,σ ,p′,p) = 1
1 1 2
w (σ ,σ ,p′,p) = σ σ
2 1 2 1 2
w (σ ,σ ,p′,p) = i (σ· +σ ) (p p′)
3 1 2 1 2
w (σ ,σ ,p′,p) = σ (p p′)· σ ×(p p′)
4 1 2 1 2
w (σ ,σ ,p′,p) = σ ·(p′×+p) σ ·(p′×+p)
5 1 2 1 2
w (σ ,σ ,p′,p) = σ ·(p′ p) σ ·(p′ p) (2.2)
6 1 2 1 2
· − · −
Each of these operators is multiplied with scalar functions which depend only on the mo-
menta p and p′, leading to the most general expansion for any NN potential
6
Vtmt vtmt(p′,p) w (σ ,σ ,p′,p) (2.3)
≡ j j 1 2
j=1
X
The property of Eq. (2.1) carries over to the NN t-operator, which fulfills the Lippmann-
Schwinger (LS) equation
ttmt = Vtmt +VtmtG ttmt, (2.4)
0
with G (z) = (z H )−1 being the free resolvent. The t-matrix element has an expansion
0 0
−
analogous to the potential,
6
ttmt ttmt(p′,p) w (σ ,σ ,p′,p) (2.5)
≡ j j 1 2
j=1
X
Inserting Eqs. (2.3) and (2.5) into the LS equation (2.4), operating with w (σ ,σ ,p′,p)
k 1 2
from the left and performing the trace in the NN spin space leads to
A (p′,p)ttmt(p′,p) = A (p′,p)vtmt(p′,p)
kj j kj j
j j
X X
+ d3p′′ vjtmt(p′,p′′)G0(p′′) ttjm′ t(p′′,p) Bkjj′(p′,p′′,p). (2.6)
Z jj′
X
The scalar coefficients Akj and Bkjj′ are defined as
A (p′,p) Tr w (σ ,σ ,p′,p) w (σ ,σ ,p′,p) (2.7)
kj k 1 2 j 1 2
≡
(cid:16) (cid:17)
Bkjj′(p′,p′′,p) Tr wk(σ1,σ2,p′,p) wj(σ1,σ2,p′,p′′) wj′(σ1,σ2,p′′,p) (2.8)
≡
(cid:16) (cid:17)
Hereall spindependencies areanalytically evaluated, andthecoefficients onlydependonthe
vectors p, p′, and p′′. The explicit expressions for the coefficients are given in Appendix A.
3
Thus we end up with a set of six coupled equations for the scalar functions ttmt(p′,p),
j
which depend for fixed p on two other variables, p′ and the cosine of the relative angle
| | | |
between the vectors p′ and p, given by pˆ′ pˆ.
·
Since Eqs. (2.3) and (2.5) are completely general, any arbitrary NN force can be cast into
this form and serve as input. Finalizing the formulation, we only need to antisymmetrized
in the initial state by applying (1 P ) p m m tm , and consider the on-shell t-matrix
12 1 2 t
− | i| i| i
element for given tm :
t
m
Mtmt (2π)2 ttmt (p′,p)
m′1m′2,m1m2 ≡ − 2 m′1m′2,m1m2 on−shell
m
= − 2 (2π)2 hm′1m′2| ttmt(p′,(cid:12)(cid:12)(cid:12)p)+(−)t ttmt(p′,−p)P1s2 |m1m2i . (2.9)
(cid:16) (cid:17)
(cid:2) (cid:3)
Here Ps interchanges the spin magnetic quantum numbers for the initial particles, m rep-
12
resents the nucleon mass.
For the on-shell condition, characterized by p′ = p the vectors p p′ and p + p′
are orthogonal. Under this condition, the opera|tor|σ |σ| can be represe−nted as a linear
1 2
·
combination of the operators w , j = 4 6 [18], i.e.
j
−
1
σ σ = σ (p p′) σ (p p′)
1 · 2 (p p′)2 1 · × 2 · ×
×
1
+ σ (p+p′) σ (p+p′)
(p+p′)2 1 · 2 ·
1
+ σ (p p′) σ (p p′) (2.10)
(p p′)2 1 · − 2 · −
−
We can use the relation of Eq. (2.10) for internal consistency checks of the calculations.
However, in order to keep the most general off-shell structure of Eq. (2.5), we need to keep
all six terms. We will come back to the numerical implications of this fact below.
From Eq. (2.9) we read off that the scattering matrix is given by
6
m
Mtmt = (2π)2 ttmt(p′,p) m′m′ w (σ ,σ ,p′,p) m m
m′1m′2,m1m2 − 2 j h 1 2| j 1 2 | 1 2i
j=1 (cid:20)
X
+ ( )tttmt(p′, p) m′m′ w (σ ,σ ,p′, p) m m (2.11)
− j − h 1 2| j 1 2 − | 2 1i
(cid:21)
On the other hand the standard form of the on-shell t-matrix for given quantum numbers
tm [18] reads in the Wolfenstein representation
t
Mtmt = atmt m′m′ m m
m′1m′2,m1m2 h 1 2| 1 2i
ctmt
i m′m′ w (σ ,σ ,p′,p) m m
− p p′ h 1 2| 3 1 2 | 1 2i
| × |
mtmt
+ m′m′ w (σ ,σ ,p′,p) m m
p p′ 2h 1 2| 4 1 2 | 1 2i
| × |
(g +h)tmt
+ m′m′ w (σ ,σ ,p′,p) m m
(p+p′)2 h 1 2| 5 1 2 | 1 2i
(g h)tmt
+ − m′m′ w (σ ,σ ,p′,p) m m (2.12)
(p p′)2 h 1 2| 6 1 2 | 1 2i
−
4
Due to the action of Ps in Eq. (2.9), which interchanges m with m , the two parts
12 1 2
of Eq. (2.11) yield different results. Again, standard relations [18, 19] must be applied to
extract the Wolfenstein parameters:
1
atmt = Tr(M)
4
1 w (σ ,σ ,p′,p)
ctmt = i Tr M 3 1 2
− 8 p p′
1 (cid:18) w (σ ,|σ ×,p′,p| ) (cid:19)
mtmt = Tr M 4 1 2
4 p p′ 2
1 (cid:18) w (σ| ,×σ ,p| ′,p)(cid:19)
(g +h)tmt = Tr M 5 1 2
4 (p+p′)2
(cid:18) (cid:19)
1 w (σ ,σ ,p′,p)
(g h)tmt = Tr M 6 1 2 (2.13)
− 4 (p p′)2
(cid:18) − (cid:19)
It isstraightforward towork out thoserelations starting fromEq. (2.11). Inorder to simplify
the notation we write t ttmt(p′,p), t˜ ttmt(p′, p), and x pˆ′ pˆ and obtain
j ≡ j j ≡ j − ≡ ·
1 3 1
atmt = t +( )t t˜ + t˜ + p4(1 x2)t˜ +p2(1 x)t˜ +p2(1+x)t˜
1 1 2 4 5 6
− 2 2 2 − −
ctmt = ip2√1 xh2 t ( )tt˜ i
3 3
− − −
1 1 1
mtmt = t +p4(1 x(cid:0)2)t +( )t(cid:1) t˜ t˜ + p4(1 x2)t˜
2 4 1 2 4
− − 2 − 2 2 −
p2(1 x)t˜ p2(1+x)ht˜
5 6
− − −
i 1 1 1
gtmt = t +p2(1+x)t +p2(1 x)t +( )t t˜ t˜ p4(1 x2)t˜
2 5 6 1 2 4
− − 2 − 2 − 2 −
htmt = p2(1+x)t p2(1 x)t +( )t ph2(1 x)t˜ +p2(1+x)t˜ i (2.14)
5 6 5 6
− − − − −
h i
It remains to consider the particle representation. For the proton-proton or neutron-neutron
system the isospin is t = 1. Thus the above given Wolfenstein parameters are already the
physical ones and enter the calculation of observables. In the case of the neutron-proton
system both isospins contribute and the physical amplitudes are given by 1(a00 + a10),
2
1(c00 +c10), etc.
2
Once the Wolfenstein parameters are known, all NN observables can readily be calculated
taking well defined bilinear products thereof [18]. For example, the spin averaged differential
cross section I is given as 1Tr MM†.
0 4
For completeness, we also give the derivation of the deuteron which carries isospin t = 0
and total spin s = 1. We employ the operator form from Ref. [5],
1
p Ψ = φ (p)+ σ p σ p p2 φ (p) 1m
h | mdi 1 1 · 2 · − 3 2 | di
(cid:20) (cid:18) (cid:19) (cid:21)
2
φ (p) b (σ ,σ ,p) 1m , (2.15)
k k 1 2 d
≡ | i
k=1
X
where 1m describes the state in which the two spin-1 states are coupled to the total spin-1
| di 2
and the magnetic quantum number m . The definition of the operators b can be easily read
d k
off the first line of Eq. (2.15). The two scalar functions φ (p) and φ (p) are related in a
1 2
5
simple way to the standard s- and d-wave components of the deuteron wave function, ψ (p)
0
and ψ (p) by [5]
2
ψ (p) = φ (p),
0 1
4p2
ψ (p) = φ (p). (2.16)
2 2
3√2
Next we use the Schro¨dinger equation in integral form projected on isospin states,
Ψ = G V00Ψ . (2.17)
md 0 md
Inserting the explicit expression of Eq. (2.15) we obtain
1
φ (p) + σ p σ p p2 φ (p) 1m =
1 1 2 2 d
· · − 3 | i
(cid:18) (cid:19)
h 6 i
1
d3p′ v00(p,p′) w (σ ,σ ,p,p′)
E p2 j j 1 2
d − m Z Xj=1
1
φ (p′)+ σ p′ σ p′ p′2 φ (p′) 1m , (2.18)
1 1 2 2 d
× · · − 3 | i
(cid:18) (cid:19)
h i
where E is the deuteron binding energy. We remove the spin dependence by projecting
d
from the left with 1m b (σ ,σ ,p) and summing over m . This leads to
d k 1 2 d
h |
1 2
1md bk(σ1,σ2,p) φk′(p)bk′(σ1,σ2,p) 1md =
h | | i
mXd=−1 kX′=1
1 6
1
d3p′ v00(p,p′)w (σ ,σ ,p,p′)
E p2 j j 1 2
d − m mXd=−1Z Xj=1
2
φk′′(p′) bk′′(σ1,σ2,p′) 1md . (2.19)
× | i
k′′=1
X
Defining the scalar functions
1
Adkk′(p) ≡ h1md|bk(σ1,σ2,p)bk′(σ1,σ2,p)|1mdi (2.20)
mXd=−1
and
1
Bkdjk′′(p,p′) ≡ h1md|bk(σ1,σ2,p)wj(σ1,σ2,p,p′)bk′′(σ1,σ2,p′)|1mdi, (2.21)
mXd=−1
we obtain for Eq. (2.19)
2 6 2
1
Adkk′(p)φk′(p) = E p2 d3p′ vj00(p,p′) Bkdjk′′(p,p′)φk′′(p′) . (2.22)
kX′=1 d − m Z Xj=1 kX′′=1
6
Note that Ad and Bd are both independent of the interaction. Therefore, these coeffi-
kk′ kjk′′
cients can be prepared beforehand for all calculations of the deuteron bound state, which
consists of two coupled equations for the functions φ (p) and φ (p). The summation over
1 2
m guarantees the scalar nature of the functions Ad (p) and Bd (p,p′), which are given in
d kk′ kjk′′
Appendix B. The azimuthal angle can be trivially integrated out, leading to the final form
of the deuteron equation
2
Adkk′(p)φk′(p) =
k′=1
X
2π 2 ∞ 1 6
E p2 dp′p′2φk′′(p′) dx vj00(p,p′,x)Bkdjk′′(p,p′,x), (2.23)
d − m kX′′=1Z0 Z−1 Xj=1
where x pˆ′ pˆ.
≡ ·
III. NUMERICAL REALIZATION
A. The deuteron
For a numerical treatment of Eq. (2.23), it is convenient to first define
1 6
Zk,k′(p,p′) ≡ dx vj00(p,p′,x)Bkdjk′(p,p′,x) (3.1)
Z−1 j=1
X
and then assume that the integral over p′ will be carried out with some choice of Gaussian
points and weights (p ,g ) with j = 1,2,...,N. This leads to
j j
2 N p2 2 1
gjp2jZkk′(pi,pj) + δij2mjπAdkk′(pi) φk′(pj) = Ed 2πAdkk′(pi)φk′(pi). (3.2)
k′=1 j=1 (cid:18) (cid:19) k′=1
X X X
Eq. (3.2) can be written as a so-called generalized eigenvalue problem
Rξ = E Yξ, (3.3)
d
or
2N 2N
Rll′ξl′ = Ed Yll′ξl′, (3.4)
l′=1 l′=1
X X
where
l = i+(k 1)N,
−
ξl′ = φk′(pj), l′ = j +(k′ 1)N,
−
p2
Rll′ = gjp2jZkk′(pi,pj) + δij2mjπAdkk′(pi)
1
Yll′ = δij2πAdkk′(pi). (3.5)
7
TABLE I: The parameters of the chiral potential of Ref. [13] in order NNLO. The LEC’s are given
for the cutoff combination Λ= 600 MeV and Λ˜= 700 MeV. Thepion decay constant F and masses
π
are given in MeV. The constants c are given in GeV−1, C and C in GeV−2 and the other C in
i S T i
GeV−4
gA Fπ mπ0 mπ± m c1 c3 c4
1.29 92.4 134.977 139.570 938.919 -0.81 -3.40 3.40
C C C C C C C C C
S T 1 2 3 4 5 6 7
-112.932 2.60161 385.633 1343.49 -121.543 -614.322 1269.04 -26.4880 -1385.12
TABLE II: Meson parameters for the Bonn B potential [15]. The σ parameters shown in the table
are for NN total isospin 0. For NN total isospin 1 they should be replaced by m = 550 MeV,
σ
gα2 = 8.9437, Λ = 1.9 GeV and n =1.
4π α
meson m [MeV] gα2 fα Λ [GeV] n
α 4π gα α
π 138.03 14.4 1.7 1
η 548.8 3 1.5 1
δ 983 2.488 2 1
σ 720 18.3773 2 1
ρ 769 0.9 6.1 1.85 2
ω 782.6 24.5 0 1.85 2
Since Ad = 0, Ad = Ad = 0 and Ad = 0, the matrix Y is diagonal and can be easily
11 6 12 21 22 6
inverted, we encounter an eigenvalue problem
Y−1R ξ = E ξ, (3.6)
d
which is of the same type and dimen(cid:0)sion as(cid:1)is being solved for the deuteron wave function in
a standard partial wave representation, where one calculates the s- and d-wave components,
ψ (p)andψ (p). Theconnectionbetweenthetwosolutions, (φ (p),φ (p))and(ψ (p),ψ (p)),
0 2 1 2 0 2
given by Eqs. (2.16) provides a direct check of the numerical accuracy.
As a first example we use a chiral NNLO potential [13], which for the convenience of
the reader is briefly described in Appendix C. For the specific calculation performed here
we take the neutron-proton version of this potential and employ the parameters listed in
Table I.
We consistently use these potential parameters in the 3D and the PW calculations. In
the first case we solve Eq. (3.6) for φ (p) and φ (p) and then use Eqs. (2.16) to obtain ψ (p)
1 2 0
and ψ (p). In the second case we employ the standard partial wave representation of the
2
potential and solve the Schro¨dinger equation directly for ψ (p) and ψ (p). Both methods
0 2
give the same value for the deuteron binding energy, namely E =-2.19993 MeV and s-state
d
probability P =95.291 %. The wave functions are identical as can be seen in Fig. 1.
s
As second NN force we choose the Bonn B potential [15], which has a more intricate
structure due to the different meson-exchanges and the Dirac spinors. The operator form
of this potential, corresponding to the basis of Eq. (2.2) is derived in Appendix D and the
parameters are given in Table II. In this case the nucleon mass is set to m= 939.039 MeV.
8
Againwe haveanexcellent agreement between the3Dandthepartialwave basedcalculation
for the deuteron binding energy, E =-2.2242 MeV, the s-state probability (P = 95.014 %)
d s
and the wave functions, which are displayed in Fig. 2.
In summary, we confirm that the 3D approach gives numerically stable results, which are
in perfect agreement with the calculations based on standard partial wave methods.
14
0
12
-0.05
2] 10 2]
3/ 3/
m 8 m -0.1
p) [f 6 p) [f
(0 (2 -0.15
ψ 4 ψ

-0.2
2
0 -0.25
0 0.5 1 1.5 2 0 1 2 3 4
-1 -1
p [fm ] p [fm ]
FIG.1: Thes-wave(left)andd-wave(right)componentofthedeuteronwavefunctionasafunction
of the relative momentum p for the chiral NNLO potential specified in the text. Crosses represent
results obtained with the operator approach and solid lines are from the standard partial wave
decomposition.
B. NN scattering observables
The inhomogeneous LS equation (2.6) for the six components ttmt can be solved for a
j
fixed value of p. For the vectors pˆ and pˆ′ we choose the explicit representation
pˆ = (0,0,1)
pˆ′ = ( 1 x′2,0,x′)
−
pˆ′′ = (p1 x′′2cosϕ′′, 1 x′′2sinϕ′′,x′′) (3.7)
− −
p p
so that the scalar products become
pˆ′ pˆ = x′
·
pˆ′′ pˆ = x′′
·
pˆ′ pˆ′′ = x′x′′ + 1 x′2 1 x′′2cosϕ′′ y. (3.8)
· − − ≡
p p
Let us now calculate the integral term on the right-hand-side of Eq. (2.6) for a positive
energy of the NN system, E p20:
c.m. ≡ m
p¯
1
S (p′,p,x′) dp′′p′′2 f (p′′;p′,p,x′), (3.9)
k ≡ p2 p′′2 +iǫ k
Z 0 −
0
9
14
0
12
-0.05
2] 10 2]
3/ 3/
m 8 m -0.1
p) [f 6 p) [f
(0 (2 -0.15
ψ 4 ψ

-0.2
2
0 -0.25
0 0.5 1 1.5 2 0 1 2 3 4
-1 -1
p [fm ] p [fm ]
FIG. 2: The same as in Fig. 1 but for the Bonn B potential [15].
where
f (p′′;p′,p,x′) f (p′′)
k k
≡ ≡
1 2π
6
m dx′′ dϕ′′ Bkjj′(p′,p′′,p,x′,x′′,ϕ′′) vj(p′,p′′,y) tj′(p′′,p,x′′) . (3.10)
jX,j′=1−Z1 Z0
Here the index tm for the t-matrix element is omitted for simplicity. For the momentum
t
integration in Eq. (3.9) an upper bound p¯ is introduced, since the contributions to the
integral for larger momenta are insignificant, the potential and the t-matrix are essentially
zero. Then the integral of Eq. (3.9) can be treated in a standard fashion and one obtains
p¯
p′′2f (p′′) p2f (p ) 1 p¯+p
S (p′,p,x′) = dp′′ k − 0 k 0 + p f (p ) ln 0 iπ . (3.11)
k p2 p′′2 2 0 k 0 p¯ p −
Z 0 − (cid:18) − 0 (cid:19)
0
It is tempting to solve Eq. (2.6) by iteration and then sum the resulting Neumann series
with a Pad´e scheme. The determinant of the 6 6 matrix A(p′,p,x′), which appears on
×
both sides of (2.6), can be easily calculated with the result
2 4
det(A) = 65536p8p′8 p2 p′2 1 x′2 . (3.12)
− − −
(cid:16) (cid:17) (cid:16) (cid:17)
In particular, this determinant is zero for p′ = p and x′ = 1. However, by a careful choice
±
of the p, p′ and x′ points, it is possible to work with non-zero values of det(A), so that the
matrix A can be inverted. In this case Eq. (2.6) can be written as
t(p′,p,x′) = v(p′,p,x′)+A−1(p′,p,x′) S(p′,p,x′), (3.13)
where t(p′,p,x′), v(p′,p,x′) and S(p′,p,x′) denote now six-dimensional vectors with compo-
nents t , v and S . Note that S(p′,p,x′) contains the unknown vector t(p′,p,x′). We arrive
j j j
at the following iteration scheme:
t(1)(p′,p,x′) = v(p′,p,x′)
t(n)(p′,p,x′) = v(p′,p,x′)+A−1(p′,p,x′)S(n−1)(p′,p,x′), for n > 1, (3.14)
10