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THE SCATTERING OF SLOW NEUTRONS BY ORTHOHYDROGEN
L. OLIVI
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Joint Nuclear Research Center Ispra Establishment - Ita.y Reactor Physics Department Experimental Neutron Physics
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EUR 4496 e
T H E S C A T T E R I N G O F S L O W N E U T R O N S BY O R T H O H Y -D R O G E N . by L. OLIVI
Commission of The European Communities
Joint Nuclear Research Center - Ispra Establishment (Italy) Reactor Physics Department - Experimental Neutron Physics Luxembourg, June 1970 - 20 Pages - 7 Figures - FB 40.—
The differential and total slow neutron cross sections by ortho-hydrogen have been derived by taking into account the removal of the degeneracy of the rotational states of the molecule.
The result shows that the cross sections depend on the direction of quantization as it is expected.
A numerical calculation for application has been performed in some typical cases and the result is shown in the graphs.
EUR 4496 e
T H E S C A T T E R I N G O F S L O W N E U T R O N S BY O R T H O H Y -D R O G E N , by L. OLIVI
Commission of The European Communities
Joint Nuclear Research Center - Ispra Establishment (Italy) Reactor Physics Department - Experimental Neutron Physics Luxembourg, June 1970 - 20 Pages - 7 Figures - FB 40,—
The differential and total slow neutron cross sections by ortho-hydrogen have been derived by taking into account the removal of the degeneracy of the rotational states of the molecule.
The result shows that the cross sections depend on the direction of quantization as it is expected.
EUR 4496 e
COMMISSION OF THE EUROPEAN COMMUNITIES
THE SCATTERING OF SLOW NEUTRONS BY ORTHOHYDROGEN
by
L. OLIVI
1 9 7 0
Joint Nuclear Research Center Ispra Establishment - Italy
ABSTRACT
The differential and total slow neutron cross sections by ortho-hydrogen have been derived by taking into account the removal of the degeneracy of the rotational states of the molecule.
The result shows that the cross sections depend on the direction of quantization as it is expected.
A numerical calculation for application has been performed in some typical cases and the result is shown in the graphs.
KEYWORDS
EPITHERMAL NEUTRONS THERMAL NEUTRONS CROSS SECTIONS HYDROGEN
QUANTUM MECHANICS ROTATIONAL STATE MOLECULES
1. Introduction *)
Solid hydrogen shows a λ anomaly in the specific heat at
temperatures below 1.6 °iCf which is caused by the orthoH2
molecules /~1 7· The attempts to explain this anomaly assume
that the degeneracy of the rotational ground state of the
orthoH molecule is removed by a crystalline field. Consequently,
the orthoHp molecules are split up in two groups vhich have
a different ζ component of the rotational momentum, namely
m = 0 and m = + 1. Furthermore, a cooperative ordering pro
cess leads to the creation of an ordered state between these two groups ¿~2, 3, 4_7·
The differential scattering and the total cross section of an
orthoH„ molecule have been derived for these two groups of
molecules in order to evaluante the possibility of obtaining
information about the structure of solid orthohydrogen below
the X temperature by neutron scattering experiments. The result
of this evaluation and the proposal for such an experiment have
been published elsewhere /~5_7·
The theory given by Schwinger and Teller for the scattering
of slow neutrons by para and orthoH molecules in their
ground state /~6_7 has been modified in order to obtain for the
two groups of ortho molecules separate expressions of the total
cross section, of the elastic differential cross section and of
the differential cross section with deexcitation of the first
rotational state. For the derivation of these expressions it is
necessary to introduce as additional variable the angle between
quantization axis and momentum transfer vector and to evaluate
separately the transition probabilities from the initial states,
m = 0 and m. = + 1 for the elastic and inelastic scattering
of a neutron by an ortho molecule. The obtained expressions have
been evaluated numerically and some typical results are given in
the graphs of this report.
2. Derivation of the Cross Sections
A. Basic relationships
A scattering process, in which a slow neutron with initial
momentum p_ is scattered through an angle -v1, into the solid
angle A by a H molecule in the center of mass system is consi
dered. The H molecule is initially in a state with the vibrational
quantum number ν , the resultant spins s of the two protons, the
rotational momentum J and the magnetic quantum number m .
This state is described by the wave function φ , (r), where
ν, it m j —
r is the distance between the two protons.
The neutron will change the momentum from p_ to £ by exciting the molecule into the state defined by the quantum numbers ν',
Τ m ,, and described by the wave function cb T, (r).
I ~v'J,rriji '
In the chosen system of coordinates the z-axis coincides with the direction of ρ (fig. 1). The direction of quantization £ of the molecule is defined by the angular coordinates and
the direction of the momentum ρ by the
scattering angle & and <p . (fig. 1).
The scattering cross sec tions will be derived for all possible transitions for the initial states with the quantum numbers mT = 0, and mT = + 1 se
parately.
The neutron differential cross section is therefore obtained by summing the final states over all possible m ,
values. According to previous derivation cross sections are:
fig. 1
Ί'ν.T.M.ι», Ι*»")
d)
M' y M
d n wu ^ V O - . - C . - K V ^ L
9 li-H W ^ . j t ë V
I , d rfor the transition without spin change and:
for the transitions with spin change; a and a., are the singlet
(*)
and triplet state scattering lengths for the free proton
Both (ï) and (2) obey the condition (1) = (1) imposed by the
Pauli exclusion principle; as a consequence the transition
selection rule (1) = (1) + S has to be observed.
In the temperature range of interest here the vibrational quantum
number ν is always zero. In addition the considered neutron ener
gies are too low to excite vibrations. For these reasons the
eigenfunction of the molecule is given by the expression:
(3)\
m^
%HS
6(r-
re]ì
ìy^
W)where r,θ , % are the molecule coordinates with respect to a system, where zaxis coincides with the direction of quanti
zation q; r is the equilibrium nuclear separation; Υ, (θ , % )
e α^τ- .-i^ χ , m
is the spherical harmonic of 1 degree and m order, normalized
to 1 (**).
(*) a = 2,373 10 1 2 cm
O _<] 2 af = 0,537 10 cm
(**)
If we write E and E, , i w u « xnicnial energy of the molecule
in the state belonging to the quantum numbers J and J' with
ν = v' = 0 , the conservation of energy provides:
with
·. 1
η η
M is the neutron mass.
After expanding the functions cos [¿V (ϋ·"-!1)»!] and
sin ["n^-"^*^ o f t h e expressions (ï) and (2) in spherical
harmonics and integrating over r, we obtain :
( 4 ) d o" r , s - ; j , s , m3^i« - )
d η. rnfz-γ 1--0 mr-1
Ί.Τ
w i t h
CSiS. = |-kl-lA5l)C3ara0)2 + (arQo)2[s(s+1Xi-|As|) + (3-2s)|A5|]
νΨ
k = -Μ^-ιηwhere i, ( T ) is the spherical Bessel function of fractional ord< of first kind of the argument χ ; 90 and yo are the angular
coordinates of the _k vector with respect to a system, where the z-axis coincides with the direction of quantization and:
(5)
ΠΤ
(i,
m;
mi,
mj.)= y
lilB(fl.5o
,y
s^(e.x)y
3fmjce^)d«;
The expression (5) is developed, by using the Clebsch-Gordan expansion of * y,.)ro (BX) ym{QX) and the property of orthogonality
(6)
Hl
(i,mirn.mr)=(H)mïp3îIKiMlCTr(lA0,0)CTT(l,m^
L 471(21+1)J JJ J'J J J
with the conditions:
m = m — mT , r - j 1 J +j'
By remembering that C , (1,0,0,0) = 0 unless (1)
J » J
the relation (4) is rewritten as:
1 + T + T »
1.
( 7 ) dgflto,^(·»,«.)_ ATO ρ (2Tti)(2ni)Τ LH'CitfWkT(fl%»ttfri
d n " cc. ' fø [(2itl)(2l'tn]* *i fli'
The following coefficients and functions are introduced:
,(*i
L 2 λ + 1 J
S ^ i ( L , U ) = CLiL<i>%m";rn;m'')CLiL,(j>vm'';m',m")
tóíncfl.'fti=yi;mce..?.iyi:mC9.,ç.)
w h i c h h a v e t h e p r o p e r t i e s :
(λ) λ-L+L'
eL L, =0 , unless (-1) =1
f
,,n
(W)=/'*W)
The expression of the differential cross section for all the
permitted transitions from the initial state belonging to J,
s, M into the final state belonging to J', s' is now given
by:
< 8 > * ^ £ H « > f f ^
The cross section (8) depends on the direction _q_, through the
angles θ„ and <p° and has the same form for the initial states
mT and m .
Β. The formula of the differential cross section for the
orthoH molecule
For the reasons explained in the introduction, the further deri
vation will be limited to the cross section of the orthoH,.
molecule in the ground state, i.e. J = 1.
The condition J = 1 enables us to transform the expression (8)
by writing the Clebsch Gordan coefficients explicitly:
(9)
d {™:) tø )
ULÌ
rXi^YAj (f)y[9.,y.)-r^-
mTY
zμί)ν(9.,ΐ.η
JV1 ^=_T L 2JVJ V-1 y-^V 2 1 V 3 T*1 TAmT -I
J_2_u
21
We r e c a l l that / 7 7
8it¿ m2 y (θ,?)
m = - l 't,m
/ΙΓΛ Υ rAi^Ü^Xl^lliki^ji^*y ω «ny (o?iJJy(Bi; V:
m =-L · ι. L,y
il;
By developing (9); taking into account (10),· one gets:
("Ά Q.
The expression (11) is the differential slow neutron cross-sec tion in the center of mass system for an ortho-hydrogen molecule
in the ground state with the magnetic quantum number m .
The most interesting transitions from the ground state J = 1 are into the states J' = 1, and τ« = o.
In the case of elastic scattering, i.e. J = J' = 1 the cross section is
(rtij! , t\ ■.
and in the case of i n e l a s t i c scattering, i . e . J = 1 and J ' = 0
.(■Ml
(.,3)
d^ > U
aC , ^ . i-o
K|-
2X3co5*a..i)]¿>);
in this process, the neutron gains an energy of 0.023 eV.
C. The formula of the total cross eection for the orthoH
molecule
From the definition of y , 0C„ , and 0Ç it follows (fig. 1)
I O
cos$=JL_ (OÍ2 αΐ_4^2); s J d ^ - i i d if
20(0,, V °T U ' ' orn, 0 aw.
Since ( f i g . 1 ) sinfín■-Ss. sin^
i t i s easy to find:
a n d cos(k>)=^^_ÇosjT
n
cos90--L (of sin | sin^cos(y-ri) + (a cos^-cO cos&
2t (
■2u
( 3 c o s2M dfr J — ( a c o s ^ - i f e K ^ V ^ K2^ «2) ^ ^ ^ ]
4 if«,
Therefore the total cross section is obtained by the integration of (11) over the whole solid angle described by 3" and f ;
cr
( 1 4 )
- f a · * * · ) -(«ien
^ - í , ^ . ^ , ^ ^ ^
βt
mT^«n*t
T^^y«i
ff*fii=a^^^^:{ f Wô oorver* w«^, ert
yfao-α) 7il«-«0l
- ^ σ ^ , , ο ^ ] ^ K-^
2fa-
2-^V-
i2^j
dd
The previous integral can be solved with the help of the
recurrence formulae /~8_7 for the Bessel functions. The final result is:
^ ( « , ^ Σ ^ Β
σι'
^M-^^^ia^^l^^^.^í^*
(ot-ΌΓο
+
6CT*f-
1|i
rW-
tfnl
2x5+H
ot-
or«(
3llr^-
zu'T^
2
II
Where :
2X
Cin(2x)= (!i££Ë dt;
(16)
V ' U.
uM V
J(MXU2) < V ¿. ik.iXk^2) V \
JΗ-o l
^W=2xjWjw(2M)j*(x)
^(xi=X
2[l(l^)^X)
+2X¿W^£(2k
1 +k+1)|1)^(x)]
.2 k=oIn the most interesting cases J' = 1 and j« = O the total cross
sections are:
1,1¡U K ' CT*
2Γί2,-Ν I2
4 Cin(2ot.)a.[Ìe(a.VÌ ( ο φ ζ σ ί . ^ ο Ο ί , ^ . Η j (<0
¿CO (. ]
12
and
W / χ "rt Ci,o 0.0;1,1 \ ' Ό; ØJ 2
2 . .2
4 ein(2x)-X2[¿e(xW,ÍX)]l·
-2- 3 X2)
-( 1 8 )
4<S0 2 (L
i r(«o-«T][i(x)>Í J J>2Ti!L> i,2
1(xi]^2x2[l-2^(x)]-2(3«1-cx:-6)e¡n(2x
J| ^ - « o )
D. Conclusions
The differential and total cross section for the two types of
molecule m = 0 and m = 1 have been numerically calculated and
the most significant results are shown in the figures.
According to these results one may conclude that the most
promising experiment is the measurement of the differential
cross section for a single crystal, because one can analyse
one of the two processes: Τ = 1, τ * =1 ; J = 1 , J ' = 0 ; for the
total cross section the two processes overlap and no real
difference is detectable.
According to these results it is possible to obtain information
about the solid orthohydrogen below the λ point by measuring
the differential cross section of a single crystal of oH^ for
temperatures above and below the λ temperature. The implications
and the possible results have been discussed elsewhere. /~5 7
Acknowledgements
The author wishes to thank Dr. Peretti and Dr. Blässer for the
13
REFERENCED
1. R. Hill and Β. Ricketson,
Phil. Mang. 45, 277-282 (1954)
2. G.M. Bell and W.M. Fairbarn,
Mol. Phys. 4, 481 (i 961 ), 5, 605 (1962), 7, 497 (1964) 3. Α. Danielian,
Phys. Rev. 138 A, 282 (1965) 4. J.C. Raich and H.M. James,
Phys. Rev. Letters 1_6, 173 (1966) 5. R. Misenta and L. Olivi,
paper submitted for publication to "Solid itate communications" ( n o t a v a iia b i e)
6. J. Schwinger and E. Teller, Phys. Rev. 52, 286 (1937) 7. L. Olivi,
Atti Sem. Fis. Univ. Modena XIX (1970) 8. M. Hamermesh and J. Schwinger,
barn sterad
d f f
Ï K , (*»*.«>
T I Tbarn sterad
θο = 0
FÍO. 2 - Ortho-H2 differential elastic cross-section
3=1,3=1 as function of incoming neutron
energy, for a scattering angle of JL,
and for the angle θο = 0 and QQ-1L
2 o rrij=0
16
barn sterad
d°S?,,,(*î·')
d n
bara
sterad
39 o z X
FlCj.3" Ortho-H2 differential inelastic cross - section
j*=1,j"=0 as function of incoming neutron
energy, For a scattering angle of JL and for
the angle θο = 0 and Θ„=Χ.
barn
sterad θο=0
.(mj)
Ί ' ι '> ; ι
d a
8-barn sterad
6-
A-
2-η
UH
C ) Ä. ι
mj = o
\ v TTljzil
Λ *
θ θ = ί
2
Fig.4 " 0rtho-H2 differential elastic cross-section
3 =
ì, 3 = 1 as function of the scattering angle
$
for a neutron of incoming energy E = 20 meV,
for the anale θο =0 and θο = *
J 2
O ïor m j ; 0
barn sterad
dgffi\,(»,E:20n»V)
""dn
bara
sterad
θο =0
Fia.5-Ortho - Η2 differential inelastic cross-section
Jr0,^'=0 as function of the scattering angle
£, for a neutron of incoming energy E=20meV
end for the anale θο = 0 end Q0=^·
J 2
barn
meV
FÍO.6~ Ortho -H2 elastic and inelastic total cross-section as function of ¡nco =
ming neutron energy, for an angle of the direction of guantzation,
with the incoming direction, ξ = 0
100
barn
ao
Ό>
60
40
20
mev FiCf.7~ Ortho - H2 elastic and inelastic total cross - section as function of
incoming neutron energy, for an angle of the direction of quan:
tization, with the incoming direction, 1 =
·^-o 3 = i , 3'=1 , mj=·^-o · 3 = 1, y = o , m3=o
PBÉ
líe !*■ i!' «Uil
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