- Email: [email protected]
Multiple scattering in an inelastic neutron scattering experiment with liquid methane
Physica 54 (1971) 393-401 o North-Holland Publishilzg Co.
MULTIPLE SCATTERING
SCATTERING
IN AN INELASTIC
EXPERIMENT A. TIITTA
WITH
LIQUID
NEUTRON METHANE
and E. TUNKELO
Reactor Laboratory, Defiartment of Technical Physics, Helsinki University of Technology, Otaniemi, Finland Received 26 January 1971
Synopsis The influence of multiple scattering on an inelastic-scattering experiment with liquid methane has been studied. The amount of multiple scattering has been calculated theoretically, and theoretical and measured scattering spectra compared. Correction for multiple scattering is found to improve the agreement between the theoretical scattering model and experimental results.
1. Introduction. The dynamics of liquid methane has been the subject of several investigations, both theoretical and experimental, since it is one of the simplest molecules having rotational degrees of freedom. However, agreement between theory and experiment has not yet been reached11 2). Experimentally a much larger inelastic intensity is obtained than the best and most reasonable models could predict; also the rotational peaks, which should be observed assuming that the rotations are free, have not yet been found. The same kind of difficulties has been encountered in trying to cxplain theoretically the measured scattering spectra in the case of solid and gaseous methane a). Very little attention has been paid to the influence of multiple scattering on the inelastic scattering experiments with thermal and cold neutrons. A very general presentation of this subject was given by Slaggiea), where multiple scattering was studied by solving the integral equation for the vector flux in the sample. Detailed knowledge of the scattering cross section of the substance being studied was, however, needed. Another method, where prior knowledge of the scattering cross section is not needed, has been given by Pfeiffer and Shapiros). In this paper use is made of the invariant imbedding principle to derive halving relations to neutron transmission and reflecting functions by use of which transmission and reflection functions for infinitesimal sample thickness can be calculated starting from values of these functions which have been measured using a sample of 393
A. TIITTA
394
AND E. TUNKELO
finite thickness. With an infinitesimal
sample thickness multiple scattering
is assumed to be absent and the transmission and reflection functions can be converted into the cross section. A measurement of the transmission and reflection
function
and scattered
over the entire range of incident
angles needed is, however,
energies and incident
very time consuming
and may
not be possible with all neutron spectrometers. Since the theoretical scattering model for liquid methane seemed reasonable it was considered feasible that the discrepancy between the experimental and theoretical results was mainly due to the fact that previous investigations had employed thick samples, up to 1 mm, without correcting for multiple scattering. A new scattering experiment was therefore performed using a thinner sample, the influence of multiple scattering being analyzed on the basis of the theoretical model of Dasannacharya and Venkataramans). 2. Theoretical. The theoretical model used for the sacttering cross section of liquid methane assumes the rotations of the molecule to be free and the molecules to be in their vibrational ground state. The translational motion of the molecules is explained by the diffusion model. Only the incoherent scattering by the protons was considered. The double-differential scattering cross section per molecule was calculated from the expressions) :
4a; mv
d2a
------_=_----_
d.f-2d7
2xfi
_x8yv
74 2DK2
e(fim+B)12kT 2J
(DG)2 + (CO+ b/h)2
I,J=O
+
2j +
*
BT(j)
‘ST
1
m=If-J1
j;(Kb),
(1)
where ap is the scattering length of a bound proton, m is the neutron mass. TO is the time-of-flight per unit length of the incoming neutron, T is the time-of-flight per unit length of the scattered neutron, yv is the DebyeWaller factor of the vibrational motion of methane, D is the self-diffusion constant of liquid methane, T is the absolute temperature of the sample, j is the initial rotational quantum number of the methane molecule, J is the rotational quantum number after scattering, b is the carbon-proton bond length in the methane molecule and in(x) is the spherical Bessel function of nth order. K = k - ko, where ko is the wave vector of the incoming neutron and k is the wave vector of the scattered neutron. The energy loss of the neutron in the scattering event is given by: fro = Eo - E = $m(l/Ti -
l/G).
(2j + 1)s exp[-@i(i BT(1’) =
2 (292 + VI
1)2 exp[--fi%(n
+
1)/21kT] + 1)/21kT]
’
MULTIPLE NEUTRON SCATTERING IN LIQUID METHANE
is the Boltzmann statistical quantum number is j, and
B = (fi2/2W(i + 1) -JU
factor
for an initial
rotational
state
395
whose
+ I)19
is the rotational energy loss of the methane molecule in the scattering event. I denotes the moment of inertia of the molecule. The analysis of the multiple scattering was based on that of Slaggied). The main problem was to solve the integral equation for the vector flux in an infinite slab sample of thickness a:
z denotes the coordinate perpendicular to the plane of the sample, 52 is the unit vector to the direction of propagation of neutrons, no is the molecular density in the sample, a(~‘, T, W.32) is the double-differential cross section per molecule, @J!is the total cross section of the methane molecule and ~1= cos 6, where 6 is the angle between the direction of propagation of neutrons and the positive z axis. The primed coordinates correspond to neutrons before scattering and the unprimed after scattering. The sample was placed such that it lay between z = 0 and z = a. The limits of the integration over .z’ were taken from 0 to z or from z to a depending on whether y was positive or negative. The A?’ integration was taken over all space and the 7’ integration from zero to infinity. The unscattered flux in the sample could be represented by the equation: +O(&
7, a) =
F(T) exp[-nOaT(T)z/pO]
d(9
.- fiO),
(3)
where F(T) is the time-of-flight spectrum of the incoming neutrons, ~0 is the cosine of the angle between the direction of the incoming neutrons, 520, and the z axis, and 6(51 - L&J)is the Dirac delta function. The incoming spectrum was normalized so that f&T)
dT =
1.
0
In eq. (2) the absorption in the sample has been neglected. Eq. (2) was solved by an iterative method in terms of the number of times the neutron scatters in the sample. The procedure of solution was to insert the m times scattered flux in the sample into the integral part of the right-hand side of eq. (2) and to obtain the m + 1 times scattered flux by numerical integration. The scattering model was tested by comparing the measured flux &(T, S)
396
A. TIITTA
AND E. TUNKELO
to the sum of the fluxes, $112(~,In), where
dm(7* sl) =
+n(& 7, Q),
when
C#lVL(O, 7, fi),
when
P > 0, #U
(5)
)....
3. Calculation of the multiple scattering. An Algol programme was written for an Elliott 503 computer in order to calculate the fluxes, (bm, to the measured angles covering the measured energy range. The parameters used for methane were those of Griffing6). The macroscopic value of 2.7 x 10-5 cmsjs was used for the self-diffusion constant7). The fluxes were calculated up to four collisions (m = 4). From the calculated spectra the relative share of the multiply scattered flux was determined (fig. 1). The convergence of the iterative solution could be approximated on the basis of table I, where the ratio &+i/& is expressed for two values of T. In the inelastic part of the scattered spectra the twice-scattered intensity seems very large but as m increases the ratio c#B~+&,, appears to reach a constant value which depends on the energy of the scattered neutrons. The ratio #+,J+i is expressed in fig. 2 as a function of m for a scattering angle of 14.5”. The convergence
1.0
- 29O
0.5
11O 1-60~
I
500
1000
Fig. 1
1500
(clw
1
2
3
1
n-l
Fig. 2
Fig. 1. The ratio of the multiply scattered neutron flux to the singly scattered flux The transmission of the sample is 86.7%. The scattering angle increases in the direction of the arrows. The quasi-elastic scattering due to diffusion has not been included in this stage of the calculations. Fig. 2. The ratio $,,J+r versus 112.The scattering angle is 14.5”. The points belonging to the same energy of scattered neutrons are connected with a line.
MULTIPLE
NEUTRON
SCATTERING
IN LIQUID
METHANE
397
TABLE I The ratio &+I/+~
for two values of time-of-flight
from the inelastic part of the
calculated spectrum 740 ps/m
340 ps/m m
14.5”
29”
44”
60”
14.5”
29”
44”
60”
1
1.15
0.99
0.83
0.66
0.62
0.48
0.37
0.32
2
0.36
0.37
0.37
0.37
0.28
0.28
0.28
0.28
3
0.31
0.30
0.31
0.30
0.26
0.26
0.27
0.26
of the iteration seems to be very rapid in the elastic part of the spectrum, whilst in the inelastic region the four times scattered intensity seems to be approximately 7% of that scattered only once, independent of the scattered neutron energy. The maximum error that arises from limiting the iteration to four collisions can be estimated to be less than 3%. The numerical integrations were calculated using 18 energy groups and 8 scattering angles. In the calculations the quasi-elastic peak of the cross section was handled as a delta function in order to be able co perform the numerical integration over the time-of-flight range. With the programme used the calculation of the theoretical spectra took 8 hours of computer time. The diffusion broadening was taken into account by fitting an effective diffusion constant to the spectrum of the largest scattering angle. The resolution broadening was also calculated using the same programme. The best fit was obtained with an effective diffusion constant of 3.7 x 10-s ems/s, which shows that the multiple scattering also affected the diffusion broadening of the quasi-elastic spectrum despite the relatively low multiple scattering
in the elastic region of the spectra.
4. Experimental.
The
scattering
experiment
was performed
using
the
time-of-flight spectrometer of the cold-neutron facility of the Triga Mk II reactor in the Reactor Laboratory of the Helsinki University of Technology. The incoming spectrum was the beryllium-filtered cold-neutron spectrum. The spectrometer had four detector banks at four different scattering angles, 14.5”, 29”, 44” and 60”, the time-of-flight spectra from each being simultaneously registered. The sample used was 99.95% methane, provided by Messer-Griesheim G.m.b.H, at a temperature of 91.7 f 0.4 K. As a result of the low intensity of cold neutrons it was not possible to use a sample of thickness smaller than 0.35 mm, this corresponding to 86.7% transmission for cold neutrons. Since a sample of area 4 x 6 cm2 was placed in a beam having an area of 3 x 4 cm2, the sample could be regarded as an infinite plate. The sample was placed at right angles to the beam so that the incident direction angle,
A. TIITTA
398
AND E. TUNKELO
@(L) sr.ps/m
500
1000
Fig. 3. The experimental and theoretical neutron spectrum scattered from a liquidmethane sample as a function of the time-of-flight. The scattering angle is 14.5”.
-5 ,(‘o)
29O
sr.ps/m
,_
l-
500
1000
c (Wml
Fig. 4. The experimental and theoretical neutron spectrum scattered from a liquidmethane sample as a function of the time-of-flight. The scattering angle is 29”.
MULTIPLE
NEUTRON
I
SCATTERING
IN LIQUID
1o-5
2( sr+s/m 1
5
METHANE
399
LLO
500
1000
e
b/m)
Fig. 5. The experimental and theoretical neutron spectrum scattered from a liquidmethane sample as a function of the time-of-flight. The scattering angle is 44”.
60' S-
._
500
1000
'c Wm-Jl
Fig. 6. The experimental and theoretical neutron spectrum scattered from a liquidmethane sample as a function of the time-of-flight. The scattering angle is 60”.
400
A. TIITTA
AND
60, was equal to zero. The experiment
E. TUNKELO
ran for 50 hours with the sample in
the beam and 25 hours with an empty sample holder, in order to measure the background
intensity.
5. Results.
In figs. 3-6
the measured
and theoretical
spectra
including
multiple scattering are shown. The usual corrections for background, detector efficiency and air scattering have been applied. The measured spectra were normalized by use of a monitor counter and the measured beam intensity such that it corresponded to eq. (4). The independent normalization of the measured and theoretical spectra succeeded very well with the smallest three scattering angles; with the angle of 60” a correction factor of 0.73 (probably due to a different type of neutron counters used in this angle) was needed to match the experimental spectrum. The fitting was applied in the interval of 1400-1500 ,us/m. In figs. 3-6 the scale of the spectra corresponds to that of the theoretical ones. Since only 18 energy groups were used, no conclusion can be drawn concerning the details of the spectra. Consequently the separate rotation peaks may have been smeared out by multiple scattering as well as by hindrance to the rotations. The agreement in the inelastic region of the spectra is much better than the earlier results where correction for multiple scattering was ignoreds). More scattering occurs, however, with a very small energy transfer than the theoretical model predicts; also the very high inelastic peak due to rotations, which should be seen according to the theory, is not observed in the measured spectra. A more rigorous analysis of the diffusion broadening could, possibly, give better agreement without making any changes in the scattering model. This study shows rather clearly the importance of taking into account the effect of multiple scattering in the inelastic scattering experiments, since by smearing the details of the scattered spectra, multiple scattering diminishes the obtainable information about the scattering cross section of the substance being studied. No general and practical solution to the problem of multiple scattering has, however, been found. The multiple scattering and possibilities to eliminate its troublesome influence on inelastic scattering measurements are the subject of a further study, now in progress. Acknowledgements. The authors wish to express their thanks to Dr. A. Palmgren and Mr. H. Pijyry for their help in carrying out the experimental work.
MULTIPLE
NEUTRON
SCATTERING
IN LIQUID
METHANE
401
REFERENCES I) Harker, Y. D. and Brugger, R. M., J. them. Phys. 42 ( 1965) 275. 2) Dasannacharya, B. A. and Venkataraman, G., Phys. Rev. 156 (1967) 196. 3) Bajorek, A., Natkaniec, I., Parlinski, K., Sudnik-Hrynkiewicz, M., Janik, Janik, J. M., Otnes, K. and Tunkelo, E., Physica 41 (1969) 397. 4) Slaggie, E. L., Nuclear Sci. Engng. 30 (1967) 199. 5) Pfeiffer, W. and Shapiro, J. L., Nuclear Sci. Engng. 38 (1969) 253. 6) Griffing, G. W., Phys. Rev. 124 (1961) 1489. 7) Nagaziadeh, J. and Rice, S. A., J. them. Phys. 36 (1962) 2710.
J. A.,
MULTIPLE SCATTERING
SCATTERING
IN AN INELASTIC
EXPERIMENT A. TIITTA
WITH
LIQUID
NEUTRON METHANE
and E. TUNKELO
Reactor Laboratory, Defiartment of Technical Physics, Helsinki University of Technology, Otaniemi, Finland Received 26 January 1971
Synopsis The influence of multiple scattering on an inelastic-scattering experiment with liquid methane has been studied. The amount of multiple scattering has been calculated theoretically, and theoretical and measured scattering spectra compared. Correction for multiple scattering is found to improve the agreement between the theoretical scattering model and experimental results.
1. Introduction. The dynamics of liquid methane has been the subject of several investigations, both theoretical and experimental, since it is one of the simplest molecules having rotational degrees of freedom. However, agreement between theory and experiment has not yet been reached11 2). Experimentally a much larger inelastic intensity is obtained than the best and most reasonable models could predict; also the rotational peaks, which should be observed assuming that the rotations are free, have not yet been found. The same kind of difficulties has been encountered in trying to cxplain theoretically the measured scattering spectra in the case of solid and gaseous methane a). Very little attention has been paid to the influence of multiple scattering on the inelastic scattering experiments with thermal and cold neutrons. A very general presentation of this subject was given by Slaggiea), where multiple scattering was studied by solving the integral equation for the vector flux in the sample. Detailed knowledge of the scattering cross section of the substance being studied was, however, needed. Another method, where prior knowledge of the scattering cross section is not needed, has been given by Pfeiffer and Shapiros). In this paper use is made of the invariant imbedding principle to derive halving relations to neutron transmission and reflecting functions by use of which transmission and reflection functions for infinitesimal sample thickness can be calculated starting from values of these functions which have been measured using a sample of 393
A. TIITTA
394
AND E. TUNKELO
finite thickness. With an infinitesimal
sample thickness multiple scattering
is assumed to be absent and the transmission and reflection functions can be converted into the cross section. A measurement of the transmission and reflection
function
and scattered
over the entire range of incident
angles needed is, however,
energies and incident
very time consuming
and may
not be possible with all neutron spectrometers. Since the theoretical scattering model for liquid methane seemed reasonable it was considered feasible that the discrepancy between the experimental and theoretical results was mainly due to the fact that previous investigations had employed thick samples, up to 1 mm, without correcting for multiple scattering. A new scattering experiment was therefore performed using a thinner sample, the influence of multiple scattering being analyzed on the basis of the theoretical model of Dasannacharya and Venkataramans). 2. Theoretical. The theoretical model used for the sacttering cross section of liquid methane assumes the rotations of the molecule to be free and the molecules to be in their vibrational ground state. The translational motion of the molecules is explained by the diffusion model. Only the incoherent scattering by the protons was considered. The double-differential scattering cross section per molecule was calculated from the expressions) :
4a; mv
d2a
------_=_----_
d.f-2d7
2xfi
_x8yv
74 2DK2
e(fim+B)12kT 2J
(DG)2 + (CO+ b/h)2
I,J=O
+
2j +
*
BT(j)
‘ST
1
m=If-J1
j;(Kb),
(1)
where ap is the scattering length of a bound proton, m is the neutron mass. TO is the time-of-flight per unit length of the incoming neutron, T is the time-of-flight per unit length of the scattered neutron, yv is the DebyeWaller factor of the vibrational motion of methane, D is the self-diffusion constant of liquid methane, T is the absolute temperature of the sample, j is the initial rotational quantum number of the methane molecule, J is the rotational quantum number after scattering, b is the carbon-proton bond length in the methane molecule and in(x) is the spherical Bessel function of nth order. K = k - ko, where ko is the wave vector of the incoming neutron and k is the wave vector of the scattered neutron. The energy loss of the neutron in the scattering event is given by: fro = Eo - E = $m(l/Ti -
l/G).
(2j + 1)s exp[-@i(i BT(1’) =
2 (292 + VI
1)2 exp[--fi%(n
+
1)/21kT] + 1)/21kT]
’
MULTIPLE NEUTRON SCATTERING IN LIQUID METHANE
is the Boltzmann statistical quantum number is j, and
B = (fi2/2W(i + 1) -JU
factor
for an initial
rotational
state
395
whose
+ I)19
is the rotational energy loss of the methane molecule in the scattering event. I denotes the moment of inertia of the molecule. The analysis of the multiple scattering was based on that of Slaggied). The main problem was to solve the integral equation for the vector flux in an infinite slab sample of thickness a:
z denotes the coordinate perpendicular to the plane of the sample, 52 is the unit vector to the direction of propagation of neutrons, no is the molecular density in the sample, a(~‘, T, W.32) is the double-differential cross section per molecule, @J!is the total cross section of the methane molecule and ~1= cos 6, where 6 is the angle between the direction of propagation of neutrons and the positive z axis. The primed coordinates correspond to neutrons before scattering and the unprimed after scattering. The sample was placed such that it lay between z = 0 and z = a. The limits of the integration over .z’ were taken from 0 to z or from z to a depending on whether y was positive or negative. The A?’ integration was taken over all space and the 7’ integration from zero to infinity. The unscattered flux in the sample could be represented by the equation: +O(&
7, a) =
F(T) exp[-nOaT(T)z/pO]
d(9
.- fiO),
(3)
where F(T) is the time-of-flight spectrum of the incoming neutrons, ~0 is the cosine of the angle between the direction of the incoming neutrons, 520, and the z axis, and 6(51 - L&J)is the Dirac delta function. The incoming spectrum was normalized so that f&T)
dT =
1.
0
In eq. (2) the absorption in the sample has been neglected. Eq. (2) was solved by an iterative method in terms of the number of times the neutron scatters in the sample. The procedure of solution was to insert the m times scattered flux in the sample into the integral part of the right-hand side of eq. (2) and to obtain the m + 1 times scattered flux by numerical integration. The scattering model was tested by comparing the measured flux &(T, S)
396
A. TIITTA
AND E. TUNKELO
to the sum of the fluxes, $112(~,In), where
dm(7* sl) =
+n(& 7, Q),
when
C#lVL(O, 7, fi),
when
P > 0, #U
(5)
)....
3. Calculation of the multiple scattering. An Algol programme was written for an Elliott 503 computer in order to calculate the fluxes, (bm, to the measured angles covering the measured energy range. The parameters used for methane were those of Griffing6). The macroscopic value of 2.7 x 10-5 cmsjs was used for the self-diffusion constant7). The fluxes were calculated up to four collisions (m = 4). From the calculated spectra the relative share of the multiply scattered flux was determined (fig. 1). The convergence of the iterative solution could be approximated on the basis of table I, where the ratio &+i/& is expressed for two values of T. In the inelastic part of the scattered spectra the twice-scattered intensity seems very large but as m increases the ratio c#B~+&,, appears to reach a constant value which depends on the energy of the scattered neutrons. The ratio #+,J+i is expressed in fig. 2 as a function of m for a scattering angle of 14.5”. The convergence
1.0
- 29O
0.5
11O 1-60~
I
500
1000
Fig. 1
1500
(clw
1
2
3
1
n-l
Fig. 2
Fig. 1. The ratio of the multiply scattered neutron flux to the singly scattered flux The transmission of the sample is 86.7%. The scattering angle increases in the direction of the arrows. The quasi-elastic scattering due to diffusion has not been included in this stage of the calculations. Fig. 2. The ratio $,,J+r versus 112.The scattering angle is 14.5”. The points belonging to the same energy of scattered neutrons are connected with a line.
MULTIPLE
NEUTRON
SCATTERING
IN LIQUID
METHANE
397
TABLE I The ratio &+I/+~
for two values of time-of-flight
from the inelastic part of the
calculated spectrum 740 ps/m
340 ps/m m
14.5”
29”
44”
60”
14.5”
29”
44”
60”
1
1.15
0.99
0.83
0.66
0.62
0.48
0.37
0.32
2
0.36
0.37
0.37
0.37
0.28
0.28
0.28
0.28
3
0.31
0.30
0.31
0.30
0.26
0.26
0.27
0.26
of the iteration seems to be very rapid in the elastic part of the spectrum, whilst in the inelastic region the four times scattered intensity seems to be approximately 7% of that scattered only once, independent of the scattered neutron energy. The maximum error that arises from limiting the iteration to four collisions can be estimated to be less than 3%. The numerical integrations were calculated using 18 energy groups and 8 scattering angles. In the calculations the quasi-elastic peak of the cross section was handled as a delta function in order to be able co perform the numerical integration over the time-of-flight range. With the programme used the calculation of the theoretical spectra took 8 hours of computer time. The diffusion broadening was taken into account by fitting an effective diffusion constant to the spectrum of the largest scattering angle. The resolution broadening was also calculated using the same programme. The best fit was obtained with an effective diffusion constant of 3.7 x 10-s ems/s, which shows that the multiple scattering also affected the diffusion broadening of the quasi-elastic spectrum despite the relatively low multiple scattering
in the elastic region of the spectra.
4. Experimental.
The
scattering
experiment
was performed
using
the
time-of-flight spectrometer of the cold-neutron facility of the Triga Mk II reactor in the Reactor Laboratory of the Helsinki University of Technology. The incoming spectrum was the beryllium-filtered cold-neutron spectrum. The spectrometer had four detector banks at four different scattering angles, 14.5”, 29”, 44” and 60”, the time-of-flight spectra from each being simultaneously registered. The sample used was 99.95% methane, provided by Messer-Griesheim G.m.b.H, at a temperature of 91.7 f 0.4 K. As a result of the low intensity of cold neutrons it was not possible to use a sample of thickness smaller than 0.35 mm, this corresponding to 86.7% transmission for cold neutrons. Since a sample of area 4 x 6 cm2 was placed in a beam having an area of 3 x 4 cm2, the sample could be regarded as an infinite plate. The sample was placed at right angles to the beam so that the incident direction angle,
A. TIITTA
398
AND E. TUNKELO
@(L) sr.ps/m
500
1000
Fig. 3. The experimental and theoretical neutron spectrum scattered from a liquidmethane sample as a function of the time-of-flight. The scattering angle is 14.5”.
-5 ,(‘o)
29O
sr.ps/m
,_
l-
500
1000
c (Wml
Fig. 4. The experimental and theoretical neutron spectrum scattered from a liquidmethane sample as a function of the time-of-flight. The scattering angle is 29”.
MULTIPLE
NEUTRON
I
SCATTERING
IN LIQUID
1o-5
2( sr+s/m 1
5
METHANE
399
LLO
500
1000
e
b/m)
Fig. 5. The experimental and theoretical neutron spectrum scattered from a liquidmethane sample as a function of the time-of-flight. The scattering angle is 44”.
60' S-
._
500
1000
'c Wm-Jl
Fig. 6. The experimental and theoretical neutron spectrum scattered from a liquidmethane sample as a function of the time-of-flight. The scattering angle is 60”.
400
A. TIITTA
AND
60, was equal to zero. The experiment
E. TUNKELO
ran for 50 hours with the sample in
the beam and 25 hours with an empty sample holder, in order to measure the background
intensity.
5. Results.
In figs. 3-6
the measured
and theoretical
spectra
including
multiple scattering are shown. The usual corrections for background, detector efficiency and air scattering have been applied. The measured spectra were normalized by use of a monitor counter and the measured beam intensity such that it corresponded to eq. (4). The independent normalization of the measured and theoretical spectra succeeded very well with the smallest three scattering angles; with the angle of 60” a correction factor of 0.73 (probably due to a different type of neutron counters used in this angle) was needed to match the experimental spectrum. The fitting was applied in the interval of 1400-1500 ,us/m. In figs. 3-6 the scale of the spectra corresponds to that of the theoretical ones. Since only 18 energy groups were used, no conclusion can be drawn concerning the details of the spectra. Consequently the separate rotation peaks may have been smeared out by multiple scattering as well as by hindrance to the rotations. The agreement in the inelastic region of the spectra is much better than the earlier results where correction for multiple scattering was ignoreds). More scattering occurs, however, with a very small energy transfer than the theoretical model predicts; also the very high inelastic peak due to rotations, which should be seen according to the theory, is not observed in the measured spectra. A more rigorous analysis of the diffusion broadening could, possibly, give better agreement without making any changes in the scattering model. This study shows rather clearly the importance of taking into account the effect of multiple scattering in the inelastic scattering experiments, since by smearing the details of the scattered spectra, multiple scattering diminishes the obtainable information about the scattering cross section of the substance being studied. No general and practical solution to the problem of multiple scattering has, however, been found. The multiple scattering and possibilities to eliminate its troublesome influence on inelastic scattering measurements are the subject of a further study, now in progress. Acknowledgements. The authors wish to express their thanks to Dr. A. Palmgren and Mr. H. Pijyry for their help in carrying out the experimental work.
MULTIPLE
NEUTRON
SCATTERING
IN LIQUID
METHANE
401
REFERENCES I) Harker, Y. D. and Brugger, R. M., J. them. Phys. 42 ( 1965) 275. 2) Dasannacharya, B. A. and Venkataraman, G., Phys. Rev. 156 (1967) 196. 3) Bajorek, A., Natkaniec, I., Parlinski, K., Sudnik-Hrynkiewicz, M., Janik, Janik, J. M., Otnes, K. and Tunkelo, E., Physica 41 (1969) 397. 4) Slaggie, E. L., Nuclear Sci. Engng. 30 (1967) 199. 5) Pfeiffer, W. and Shapiro, J. L., Nuclear Sci. Engng. 38 (1969) 253. 6) Griffing, G. W., Phys. Rev. 124 (1961) 1489. 7) Nagaziadeh, J. and Rice, S. A., J. them. Phys. 36 (1962) 2710.
J. A.,