COLLISIONS OF A  MESONS WITH DEUTERONS
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Collisions of n  Mesons with Deuterons
V. B. BerestetskiY and I. ya. Pomeranchuk
Doklady Akademii Nauk SSSR, Volume LXXXI, No. b,
PP? 1019?1021. MoscOWLeningrad: 21 Dec 1951.
STAT
STAT
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coalsioNS Q !L MESONS WITH DEUTERONS
V. B. Berestetskiy
and I. Ya. Pomeranchuk.
he following is an article that appeared in the regular Physics section
of the thrice monthly Doklady Akad:emii Nauk SSSR, Volume 81, No, 6 (21 December
1951), pages 10191021. The article was submitted by academician L. D. Landau,
2~ October 1951,E
r
During collision of n mesons with deuterons, their scattering and also
their conversion into neutron mesons can occur, The crosssections of scattering
and conversions into a neutral meson were calculated (1) with the aid of the
theory of perturbations for various possible types of bond between nmesons and
nucleons. A series of data concerning the character of the interaction of
mesons with nucleons, in particular the dependence upon spin, can be analyzed
by comparison of the data on scattering (and on conversion into a neutral
meson) in hydrogen and deuterium, Theretidal considerations here do not re
quire any assumptions concerning the smallness of the interaction.
Let us apply the semiphenomenological method, employed earlier (2) to the
problem of the scattering of fast neutrons by deuterons and to the problem
of the capture of ,mesons by deuterons. Let a nmeson collide with a proton.
s
We shall calculate the familar amplitude of the scattered meson (charged or
neutral) for a given angle of scattering. If the meson possesses the spin 0,
then the amplitude of scattering must be scalar; the amplitude, however,
corresponding to the flight of the neutral meson must be scalar if the internal
"count" of the charged and neutral mesons are the same, and must be pseudo
scalar if they are opposite. .
1
STAT
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From this requirement we can determine the character of the possible
dependence of amplitude upon the nucleon's spin. Namely:
1) amplitude of scattering:
4J
? =a+b_O
It
where a is a scalar (which is generally a function of the angles), sigma Q
is the spin operator, and b is a pseudovector; here
(1)
~
?*
lock , (1a)
1
where b0 is a scalar function of the angles, and k and kare the initial and
final impulses (momenta) of the meson.
The same structure is possessed by the expression for the amplitude of
conversion of a n~meson into a neutral ?np for the same "count"
>_
h_
In the case, however, of different "count" we have;
m) A
IL ?R CO"
no
where t is a vector (also depending upon the angles).
Corresponding cross sections for collisions with hydrogen, averaged over
the orientations of the nucleon's spin, equal:
o _ /a/2 + /b/2 = c  + c'
TI a b
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/;o*?.. i' /','' ' 4?
.
(2)
(3)
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Let us now consider the collision of negative a meson with a deuterons
which collision leads to its conversion to a neutral meson. In so far as
the process proceeds only because of the interaction with the proton, the amp
litude of scattering will then be
?D = f*(p).e ~n D
toQ
())
M
where ?/2 is the radiusvector of the proton (in the system of the deuteron's
of inertia); '(p) is the wave function of the deuteron; ~.'() is the
center o ~ D F
wave function of the two neutrons forming as a result of the collision.
Let us derive the expression for the cross section averaged over the spin
states of the nucleons and summed over the states of the neutrons (the expres
sion is effected by the same method as in (1)):
0
Ito
1 (ak + a, cr)F +1(
~?  1B)F
2 A 3 g 2 A 3
where
(k , Y
/*( ik k ' ~ikk')p/2
)~/2 ( )
If we use for D the following expression
D
4)
where2 2/M is the deuteron's energy of bond, then we have
/ kk /
dD + ~. ? (+ i.) ..~ arc tan ?
f47. 2
.k ~
no A B A 3 ?B
~
i(kk')p/2
yApAp~Y{QI~
(5)
(6)
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Similarly in the case of different 'counts" of mesons, we have
no
or
no 3
/krc'/
arc tan  2 ..)
a~
(7)
(8)
We note the essential absence of cases of similar or different "counts' of it and n mesons. For k' close to k, then D (6) tends to 0 because (la)
B B kxk', but we have (arc tan x/x)
0
x=o
1, at the same time as r D 2rr
no
If there exists a connected state of the system of two neutrons (deuterons),
then its formation as a result of collision is possible. Here the system of
nitc3: onssmust pass from the triplet to the singlet state in order that, in
correspondence with Pauli's principle, the two neutrons might be found in the
Sstate.
The corresponding crosssections are:
/ e
~ D> ~s~ _ I. d' ..~ ~ arc tan 1kk' ..,~
no 3 n0 kelp 2 ( 2 (9 )
where O is or 6 , and 3~2 3 2/M is the dineutron's energy of bond, There
no B no
fore the amplitude of scattering is:

D m ei (k?k' )p/2 ' e?(1)/2) ' jad ..
It (p) Y
(10)
Here is the amplitude of scattering on neutron, and E' () i th ' final state
n P
of the neutron and proton. The scattering cross section summed over the neutron
and proton's states of motion possesses the following form:
~ D +0 + + cr + ? arc tan k  
tt a b a b /k"k' / 2o.
u
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1
( fT a(T a ? COS a
a, ? cos " ),
b b
where aat + c b are the crosssection of scattering on neutron, and Oa is
the phase difference of the quantities a and a1; similarly for
b
In the work (3) devoted to a similar review of the scattering of a nrrneson
on deutron, the presence of two types of scattering by nucleons (the types
corresponding to amplitudes a and b) was not taken into consideration.
For elastic scattering we have:
~ D ~ 16c~2 ~~ (arc tan ?i'/ a.) 2 La n _ ~/2 ~ a, a
(db +
3
+2 o~ o a ? cos +
a
+ 2 bb ? cos tb )1
Literature Cited
1. V. Berestetskiy and I. Pomeranchuk. Ibidem /Eok1ady7,9, ~olurne 77, page
803 (19f.1).
V. Berestetskiy and I. Shmushkevich. Zhurnal Eksper i teoret Fiziki,
Volume 21, No. 12 (1951).
2. I. Porrieranehuk. Ibidern ~okladil, Volume 78, page 249 (1951); Zhurnal
Eksper i teoret Fiziki, Volwne 21, page 1113 (1951); ibidera ok1ad7,
Volume 80, pages 47 (1951).
G. F. Chew. Physical Review, Volume 80, page 196 (1950).
3. S. Fernbach, T. A. Green, and K. M. Watson. Physical Review, Volume 82,
page 980 (1951).
END 
w 5 r
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