9.2 Generalization to Many-Nucleon Projectiles
433
Therefore, one has a | ˆ
U 0 |k c 0 and b | ˆ
U 0 |k d 0 in Eq. (9.99). As one has
k a , k b k max , one can have k c k max or k c k max (same for k d ) in Eq. (9.96).
Let us then consider the different values that k c and k d can bear. On the one hand,
one obtains from Eq. (9.101)
k c , k d k max ⇒ K
CM 2 k max .
(9.103)
Consequently, δ(K CM − K
CM ) = 0 in Eq. (9.100) in this situation. On the other
hand, having k c k max , k d k max , or k c k max , k d k max leads to vanishing matrix elements of ˆ
V res due to cluster approximation. Therefore, the matrix
elements a k b | ˆ
V res − ˆ
U 0 |k c k d are negligible if ∃(i, j) ∈ {a, b, c, d} | k i
k max and k j k max . Eq. (9.96) is then fulfilled in a basis of plane waves.
9.2.6.2 Cancellation of High Energy Matrix Elements with Real-Energy
Neutron States
Let us consider a one-body neutron state of real energy, of orbital angular momentum , and linear momentum k, denoted by |i. The asymptote of |i is that of
Eq. (2.7), so that |i can be written as
|i = C|kk B + |i
+
,
(9.104)
where |kk B is a state of Bessel function, C is a constant and |i
+
is a state satisfying
an outgoing wave condition. As |i is generated by a smooth finite-range potential,
one can assume that the basis-generating potential vanishes identically if k ≥ k
(H)
max
(see Sect. 9.2.6), as results will be the same up to an arbitrarily small precision.
Equation (2.2) expressed in momentum space [56] then implies that
+
|k B = 0
for k ≥ k
(H)
max .
Let us write the matrix elements of ˆ
V res − ˆ
U 0 using Eq. (9.104):
ab| ˆ
V res − ˆ
U 0 |cd =
f a ,f b ,f c ,f d
a f b | ˆ
V res − ˆ
U 0 |f c f d ,
(9.105)
where |f i = {|k i i m i B , |i + } with i = {a, b, c, d}. The matrix elements
a f b | ˆ
V res − ˆ
U 0 |f c f d are expressed using the Bessel function expansion of |f i
states:
a f b | ˆ
V res − ˆ
U 0 |f c f d =
a a m a k
b b m b | ˆ
V res − ˆ
U 0 |k
c c m c k
d d m d B
× ×f a |k
a a m a B b |k
b b m b B
× ×f c |k
c c m c B d |k
d d m d B dk
a dk
b dk
c dk
d ,
(9.106)
where (9.96) is supposed to be verified.
433
Therefore, one has a | ˆ
U 0 |k c 0 and b | ˆ
U 0 |k d 0 in Eq. (9.99). As one has
k a , k b k max , one can have k c k max or k c k max (same for k d ) in Eq. (9.96).
Let us then consider the different values that k c and k d can bear. On the one hand,
one obtains from Eq. (9.101)
k c , k d k max ⇒ K
CM 2 k max .
(9.103)
Consequently, δ(K CM − K
CM ) = 0 in Eq. (9.100) in this situation. On the other
hand, having k c k max , k d k max , or k c k max , k d k max leads to vanishing matrix elements of ˆ
V res due to cluster approximation. Therefore, the matrix
elements a k b | ˆ
V res − ˆ
U 0 |k c k d are negligible if ∃(i, j) ∈ {a, b, c, d} | k i
k max and k j k max . Eq. (9.96) is then fulfilled in a basis of plane waves.
9.2.6.2 Cancellation of High Energy Matrix Elements with Real-Energy
Neutron States
Let us consider a one-body neutron state of real energy, of orbital angular momentum , and linear momentum k, denoted by |i. The asymptote of |i is that of
Eq. (2.7), so that |i can be written as
|i = C|kk B + |i
+
,
(9.104)
where |kk B is a state of Bessel function, C is a constant and |i
+
is a state satisfying
an outgoing wave condition. As |i is generated by a smooth finite-range potential,
one can assume that the basis-generating potential vanishes identically if k ≥ k
(H)
max
(see Sect. 9.2.6), as results will be the same up to an arbitrarily small precision.
Equation (2.2) expressed in momentum space [56] then implies that
+
|k B = 0
for k ≥ k
(H)
max .
Let us write the matrix elements of ˆ
V res − ˆ
U 0 using Eq. (9.104):
ab| ˆ
V res − ˆ
U 0 |cd =
f a ,f b ,f c ,f d
a f b | ˆ
V res − ˆ
U 0 |f c f d ,
(9.105)
where |f i = {|k i i m i B , |i + } with i = {a, b, c, d}. The matrix elements
a f b | ˆ
V res − ˆ
U 0 |f c f d are expressed using the Bessel function expansion of |f i
states:
a f b | ˆ
V res − ˆ
U 0 |f c f d =
a a m a k
b b m b | ˆ
V res − ˆ
U 0 |k
c c m c k
d d m d B
× ×f a |k
a a m a B b |k
b b m b B
× ×f c |k
c c m c B d |k
d d m d B dk
a dk
b dk
c dk
d ,
(9.106)
where (9.96) is supposed to be verified.