ЯДЕРНАЯ ФИЗИКА, 2011, том 74, № 2, с. 324-329
ЯДРА
SPIN STRUCTURE OF THE "FORWARD" NUCLEON CHARGE-EXCHANGE REACTION n + p ^ p + n AND THE DEUTERON
CHARGE-EXCHANGE BREAKUP
©2011 V. L. Lyuboshitz, V. V. Lyuboshitz*
Joint Institute for Nuclear Research, Dubna, Russia Received May 28, 2010
The structure of the nucleon charge-exchange process n + p ^ p + n is investigated basing on the isotopic invariance of the nucleon—nucleon scattering. Using the operator of permutation of the spin projections of the neutron and proton, the connection between the spin matrices, describing the amplitude of the nucleon charge-exchange process at zero angle and the amplitude of the elastic scattering of the neutron on the proton in the "backward" direction, has been considered. Due to the optical theorem, the spin-independent part of the differential cross section of the process n + p ^ p + n at zero angle for unpolarized particles is expressed through the difference of total cross sections of unpolarized proton—proton and neutron—proton scattering. Meantime, the spin-dependent part of this cross section is proportional to the differential cross section of the deuteron charge-exchange breakup d + p ^ (pp) +n at zero angle at the deuteron momentum k^ = 2kn (kn is the initial neutron momentum). Analysis shows that, assuming the real part of the spin-independent term of the "forward" amplitude of the process n + p ^ p + n to be smaller or of the same order as compared with the imaginary part, in the wide range of neutron laboratory momenta kn > 700 MeV/c the main contribution into the differential cross section of the process n + p ^ p + n at zero angle is provided namely by the spin-dependent term.
1. ISOTOPIC STRUCTURE OF NUCLEON-NUCLEON SCATTERING
Taking into account the isotopic invariance, the nucleon-nucleon scattering is described by the following operator:
â(p, p') + b(p, p' )f (1) f (2).
f (p, p' )
(1)
fpp^pp(p, P ) - /nn^nn(p, p ) -
- à(p, p')+b(p, p'); /np^np(P, p') - à(P, p') — b^ p');
(2)
meantime, the matrix of amplitudes of the charge transfer process is as follows:
fnp^pn(P, p') = 2b(p, p') = (3)
= f pp^pp(p, p ) _ fnp^np(p,
Here, f(1) and f(2) are vector Pauli operators in the isotopic space; a(p, p') and b(p, p') are 4-row matrices in the spin space of two nucleons; p and p' are the initial and final momenta in the c.m. frame, the directions of p' are defined within the solid angle in the c.m. frame, corresponding to the front hemisphere.
One should note that the process of elastic neutron-proton scattering into the back hemisphere is interpreted as the charge-exchange process n + + p — p + n.
According to (1), the matrices of amplitudes of proton-proton, neutron-neutron and neutronproton scattering take the form:
E-mail: Valery.Lyuboshitz@jinr.ru
Formulas (2), (3) for the matrices of scattering amplitudes may be rewritten also in the representation of states with total isotopic spins T = 1 (t(1)f(2) = 1) and T = 0 (f (1)f(2) = -3). In accordance with ( 1 ), the matrices a(p, p'), b(p, p') and the matrices f (T=1)(p, p'), f (T=0)(p, p'), describing the nucleon—nucleon scattering in the states with T =1 and T = 0, are connected by the relations:
â(P, po = ^(/(t=0)(p, po + 3/(t=1)(p, p')); (4) &(p,p') = ^(/(t=1)(p,po - /(t=0)(p,p'));
so, we obtain:
fpp^pp(p, p ) = f nn^nn (p, p' )= (5)
= f (T=1)(p, p' );
/„( p,po = ¿(/^(p.p') + /(t=0)(p,p')); f„{p,p') = I(/(^=i)(p,pO - /(T=0)(p,p'))-
It should be stressed that the differential cross section of the charge-exchange reaction defined in the front hemisphere 0 < 9 < n/2, 0 < 0 < 2n (here, 9 is the angle between the momenta of initial neutron and final proton, 0 is the azimuthal angle), should coincide with the differential cross section of the elastic neutron—proton scattering into the back hemisphere
by the angle 9 = n — 9 at the azimuthal angle 0 = = n + 0 in the c.m. frame. Due to the antisymmetry of the state of two fermions with respect to the total permutation, including the permutation of momenta (p' ^ —p'), permutation of spin projections, and permutation of isotopic projections (p ^ n), the following relations between the amplitudes fnp^pn(p, p') and f nP^nP(p, — p') hold [1-4]:
/(r=1)(p, p') = — P(1'2)/(T=1) (p, —p'), (6) f(T=0)(p, p')= P (1'2)I(T=0)(p, —p'),
f np^pn(p, p ) - P( ' )fnp^np(p, p ), (7)
where P(1'2) is the operator of permutation of spin projections of two particles with equal spins; the matrix elements of this operator are [5]: (m1 m2|P(1'2) |m1m2) = Sm[m26m'2mi. For particles with spin 1/2 [1-5]
p( 1.2) = ±(/(1,2) +
r(1) A(2)
(8)
Taking into account relations (7), (8), and (9), the following matrix equality holds:
/rap^pra(P> P ) fnp^pn{p, P ) — fnP^nP(P, p )fnp^np(p, -p0-
(10)
— f+
As a result, at any polarizations of initial nucleons the differential cross sections of the charge-exchange process n + p — p + n and the elastic neutronproton scattering in the corresponding back hemisphere coincide:
da<
np^pn
dQ
(P, P')
da,
np^np
dQ
(P, -P')- (11)
However, the separation into the spin-dependent and spin-independent parts is different for the amplitudes
fnp^pn(p, p') and fnp^np(P, — p')!
2. NUCLEON CHARGE-EXCHANGE PROCESS AT ZERO ANGLE
Now let us investigate in detail the nucleon charge transfer reaction n + p — p + n at zero angle. In the c.m. frame of the neutron—proton system, the amplitude of the nucleon charge transfer in the "forward" direction fnp^pn(0) has the following spin structure:
where I(1'2) is the four-row unit matrix; ^(1), ^(2) — vector Pauli operators. The eigenstates of operator (8) are three symmetric triplet states (the total spin S = 1) with the eigenvalues equaling to +1, and one antisymmetric singlet state (the total spin S = 0) with the eigenvalue equaling to —1. It is evident that P (1'2)
is the unitary and Hermitian operator: P(1,2) = P(1,2) +, P(1,2) P(1,2)+ = I(1,2). (g)
Let us emphasize the following circumstance. In the case of the amplitude of the charge-exchange reaction fnp^pn(p, p'), the two-row matrices with the index 1 act between the spin states of the initial neutron with the momentum p and the final proton with the momentum p', and the two-row matrices with the index 2 act between the spin states of the initial proton with the momentum —p and the final neutron with the momentum —p'.
In the case of the amplitude of elastic scattering into the back hemisphere f np^np(p, — p'), the two-row matrices with the index 1 act between the spin states of the neutrons with the momenta p and —p', and the two-row matrices with the index 2 act between the spin states of the protons with the momenta —p and p'.
/np^pn(0) — Cif( ' ) +
+ C2^(1)&(2) - (a(1) 1)(^(2) 1)] + + cs(^(1) 1)(^(2) 1),
(12)
where l is the unit vector directed along the incident neutron momentum. In so doing, the second term in Eq. (12) describes the spin-flip effect, and the third term characterizes the difference between the amplitudes with the parallel and antiparallel orientations of the neutron and proton spins.
The spin structure of the amplitude of the elastic neutron—proton scattering in the "backward" direction fnp^np(n) is analogous:
fnp^np(n) - c1f ( ' ) +
+ C2^(1)¿(2) - (¿(1) 1)(^(2) 1)] +
+ c3(&(1) 1)(^(2) 1).
(13)
However, the coefficients c in Eq. (13) do not coincide with the coefficients c in Eq. (12). According to Eq. (7), the connection between the amplitudes
fnp^pn(Q) and fnp^np(n) is the following:
/np^pn(0) - P( ' ) fnp^np(n),
(14)
where the unitary operator P(1,2) is determined by
Eq. (8).
326
V.L. LYUBOSHITZ, V.V. LYUBOSHITZ
As a result of calculations with Pauli matrices, we obtain:
1 „ ci = —-(ci +2 c2 + c3);
(15)
+ \\ci -2C2 + C3\2 = |?i|2 + 2|ïï2|2 + |ïï3|2.
Thus,
donp^pn ^ donp^np / \
■(°) = —^—w.
dQ dQ
just as it should hold in accordance with relation (11).
3. SPIN-INDEPENDENT AND SPIN-DEPENDENT PARTS
OF THE CROSS SECTION OF THE REACTION n + p — p + n AT ZERO ANGLE
It is clear that the amplitudes of the proton—proton and neutron—proton elastic scattering at zero angle
have structure ( 12) with the replacements c1,c2,c3 — c(pp) c(pp) c(pp) c c c(np) c(np) c(np)
spectively. It follows from the isotopic invariance (see Eq. (3)) that
_ „(pp) c(np) ci — „1 „1 )
(17)
_ c(pp) c(np) c2 _ c2 — „2 I
_ c(pp) c(np) c3 _ c3 - c3 •
In accordance with the optical theorem, the following relation holds, taking into account Eq. (17):
4пт П (pp) T (np)\
— Imci = —(Im с) J - Im с I J) =
k k 1 1
(18)
!)We use the unit system with h = c =1.
in the "forward" direction for unpolarized nucleons can be presented in the following form, distinguishing the spin-independent and spin-dependent parts:
~
c2 =--(ci - c3); c3 = —-(ci - 2c2 + c3).
Hence, the "forward" differential cross section of the nucleon charge-exchange reaction n + p — p + n for unpolarized initial nucleons is described by the following expression:
^»(0) = |Cl|2 + 2|C2|2 + |C3|2= (16) = i|ci + 2c2 + c3|2 + i|ci - c3|2 +
do.
np^pn
dQ
(0)_ |d|2 + 2|c212 + |сз|2 _
(19)
do(si) do(sd) U>Unp—ypn ^-Q-J ^ UUfip^pn
dQ
dQ
(0)-
In doing so, the spin-independent part
da'.
(si)
dQ
-(0)
in Eq. (19) is determined by the difference of total cross sections of the unpolarized proton—proton and neutron—proton interaction:
do
(si)
np^pn
dQ
(0)_ |C112 _
(20)
16n2
(opp - Onp)2(1 + a2),
where a = Re ci/Im c1. The spin-dependent part of the cross section of the "forward" charge-exchange process is
do
(sd)
np^pn
dQ
(0) _ 21„212 + |сз|2•
(21)
Meantime, according to Eqs. (13), (15), and (16), the spin-dependent part of the cross section of the "backward" elastic neutron—proton scattering is
do
(sd)
np^np
dQ
(n) _ 2|C2|2 + „312
(22)
We see that
dO(sd) dO{SA) d°np^pn/„n / d°np^np/ \
"(°) +-^—W-
dQ
dQ
Further it is advisable to deal with the differential cross section ^\t=0, being a relativistic invariant (t = -(pi - p2)2 = (p - p')2 - (E - E')2 is the square of the 4-dimens
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