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LM*

¡Mi

Bi

'#}.

THE SCATTERING OF SLOW NEUTRONS BY ORTHOHYDROGEN

L. OLIVI

w

l i l i »

wÊ=

ir

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Joint Nuclear Research Center Ispra Establishment - Ita.y Reactor Physics Department Experimental Neutron Physics

(2)

uiwSufcit

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(3)

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.

(4)
(5)

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

(6)

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

(7)

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.

(8)

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

(9)

Ί'ν.T.M.ι», Ι*»")

d)

M

' y M

d n w

u ^ V O - . - C . - K V ^ L

9 li-

H W ^ . j t ë V

I , d r

for 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

(**)

(10)

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^-ιη

where 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

3fmj

ce^)d«;

The expression (5) is developed, by using the Clebsch-Gordan expansion of * y,.)ro (BX) ym{QX) and the property of orthogonality

(11)

(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)

(12)

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 {™:) )

ULÌ

rXi^YAj (f)y[9.,y.)-r^-

m

TY

z

μί)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

(13)

/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

a

C , ^ . 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)

(14)

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 , ^ . ^ , ^ ^ ^

β

t

m

T^«n*t

T

^^y«i

ff

*fii=a^^^^:{ f Wô oorver* w«^, ert

yfao-α) 7il«-«0l

- ^ σ ^ , , ο ^ ] ^ K-^

2

fa-

2

-^V-

i2

^j

d

d

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

r

W-

tfn

l

2x

5+H

ot

-

or

«(

3ll

r^-

zu

'T^

2

(15)

II

Where :

2X

Cin(2x)= (!i££Ë dt;

(16)

V ' U.

u

M V

J

(MXU2) < V ¿. ik.iXk^2) V \

J

Η-o l

^W=2xjWjw­(2M)j*(x)

^(xi=X

2

[l­(l^)^X)

+

2X¿W^­£(2k

1 +k+1)|

1)^(x)]

.2 k=o

In 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 (. ]

(16)

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

(17)

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,

(18)
(19)

barn sterad

d f f

Ï K , (*»*.«>

T I T

barn 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

(20)

16

barn sterad

d°S?,,,(*î·')

d n

bara

sterad

3

9 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.

(21)

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

(22)

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

(23)

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

(24)

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

(25)

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líe !*■ i!' «Uil

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