COLD- NEUTRON SCAÏÏERING EXPERIMENTS
O N CYCLIC HYDROCARBONS
COLD. NEUTRON SCAÏÏERING EXPERIMENTS
O N CYCLIC HYDROCARBONS
WITH A ROTATING-CRYSTAL SPECTROMETER
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS DR. m . C. J.D.M. VERHAGEN, HOOGLERAAR IN DE AFDELING DER TECHNISCHE NATUUR-KUNDE, VOOR EEN COMMISSIE UIT DE SENAAT
TE VERDEDIGEN OP WOENSDAG 17 JANUARI 1968
TE 14 UUR
/^/y v/^
DOORLEENDERT ADAM DE GRAAF
NATUURKUNDIG INGENIEUR GEBOREN TE DORDRECHT
BIBLIOTHEEK
DER
TECHNISCHE HOGESCHOOL
DELFT
1967 "BKONDER-OFFSET" ROTTERDAMDIT PROEFSCHRIFT IS GOEDGEKEUBD DOOR DE PROMOTOR P R O F . DR. J . J . VAN LOEF
Aan mijn o u d e r s . Aan Ank, Gitta,
CONTENTS
page
1. INTRODUCTION 9
2. DESIGN AND CONSTRUCTION OF THE
ROTATING-CRYSTAL SPECTROMETER 13
2 . 1 . Introduction 13 2 . 2 . Resolution considerations 15
2 . 3 . Comparison with other s p e c t r o m e t e r s 20
2. 4. Monochromating p a r t 24 2 . 4 . 1 . Beam tube and collimators 24
2 . 4 . 2. Neutron filters 25 2 . 4 . 3 . Monochromator c r y s t a l s 27
2 . 4 . 4 . C r y s t a l rotor construction and driving 28
2 . 5 . Analyzing p a r t 30 2 . 5 . 1 . Neutron detectors 30 2 . 5 . 2 . Electronics 32 2 . 6 . P e r f o r m a n c e 33 2 . 6 . 1 . Resolution 33 2. 6. 2. Intensities 36 2 . 6 . 3 . Measurement of detector efficiencies 42
3. THEORETICAL BACKGROUND 45
3 . 1 . Introduction 45 3 . 2 . General theory of neutron scattering 46
3 . 3 . The L a r s s o n - B e r g s t e d t model 50 3 . 4 . Modifications of the Larsson-Bergstedt model 54
page
4. MEASUREMENTS ON CYCLIC HYDROCARBONS 60
4 . 1 . Introduction 60 4. 2. Experimental details 61
4. 3. Corrections applied to the measured s p e c t r a 63
4 . 4 . Quasi-elastic scattering 69
5. DISCUSSION 74 5 . 1 . Introduction 74 5 . 2 . Physical data of the investigated hydrocarbons 74
5 . 3 . Debye - Waller factors 79 5.4. Quasi-elastic line broadening 83
5 . 4 . 1 . Cyclohexane 86 5 . 4 . 2 . Cyclopentane 89 5 . 4 . 3 . Methylcyclohexane 91 5 . 5 . Inelastic scattering 93 5 . 5 . 1 . Cyclohexane 94 5 . 5 . 2 . Cyclopentane 96 5 . 5 . 3 . Methylcyclohexane 97 5 . 6 . Concluding r e m a r k s 98 APPENDIX 102 REFERENCES 103 SUMMARY 108 SAMENVATTING 110
1. I N T R O D U C T I O N
G a s e s , liquids and solids can be studied experimentally by means of various radiation scattering techniques. The available types of radiation include X - r a y s , infrared light, neutrons, electrons, micro waves and sound waves. The interaction of radiation with atomic or molecular s y s t e m s can be described as the scattering of a plane wave by a set of moving scattering c e n t r e s . As a result of the scattering p r o c e s s changes in direction and wavelength can be observed. F r o m ttiese changes a quantity S(K ,ou) is obtained, which is directly related to the differential scattering c r o s s section and reflects both the s t r u c t u r e and the thermal motion of the scattering c e n t r e s in the sample. The function S( K , u^ is called the " s c a t t e r i n g law" and it gives the probability that the energy of the scattered radiation changes by h m, while its momentum changes by h ic .
The function S(K^, (u) has two different forms depending on whether or not there is interference between the scattered waves. F o r electromagnetic radiat-ion interference i s important. This is not necessarily so for neutrons where the scattering amplitude depends upon the p a r t i c u l a r scattering nuclei and upon the orientation of the nuclear and neutron spins. In most c a s e s interference occurs only partly and then the intensity of the scattered neutrons is a mixture of two contributions, one for which the effective scattering amplitude and phase a r e the same for all nuclei and another for which the interference t e r m s a r e z e r o . The first will be r e f e r r e d to as the coherent p a r t , the second as the incoherent p a r t of the scattering.
Van Hove [ 1 ] has shown that the coherent p a r t of the scattering is p r o -portional to the four-dimensional F o u r i e r transform in space and time of the time-dependent correlation function G ( r , t ) . The incoherent p a r t is proportional to the F o u r i e r transform of the self-correlation function G ( r , t ) . The general p r o p e r t i e s and the behaviour of these functions have been described in some
detail by Van Hove. F o r a c l a s s i c a l system G ( r , t ) gives the probability of finding any p a r t i c l e at position r at time t when at time t=o a particle was found at the origin. In an analogous way G (r,t) defines the probability of finding at r and t the same p a r t i c l e that started at the origin at t=o.
The scattering, in general, can be elastic or inelastic. F r o m the uncertainty principle between time and energy it follows that the elastic scattering corresponds to the time-independent behaviour of the system, while the inelastic scattering is related to the time-dependent behaviour. The latter corresponds in molecular s y s t e m s to the rotational and vibrational motions of which the energy levels range from l e s s than 1 meV up to more than 100 meV.
Various types of radiation a r e not suitable to obtain information about time-o
dependent p r o c e s s e s . In the case of X - r a y s , with a wavelength of lA and an 4
energy of about 10 eV, it is impossible to detect energy changes on scattering of the order of some meV. On the other hand, in Mössbauer experiments using
4
gamma r a y s with an energy of 10 eV it is nearly impossible to detect energy
- 3 ° t r a n s f e r s l a r g e r than 10 meV. Neutrons with a wavelength of 1 A, obtainable
from nuclear r e a c t o r s , have an energy of 80 meV and energy changes of the order of 1 meV can be detected r a t h e r easily. Often It is desirable to use neutrons with a wavelength of about 4 A, corresponding to an energy of 5 meV and a " t e m p e r a t u r e " E/k_ of 60 K. Hence they a r e called "cold" neutrons.
Energy changes of 1 meV correspond to interaction times of about 10 s. This time scale can be reached both with neutrons and with infrared absorption and Raman scattering. As the wavelength of the electromagnetic radiation is some thousand A, the momentum transfer is always almost z e r o . Moreover, selection rules must be obeyed which is not the case for neutron scattering. Ultrasonic and dielectric-relaxation methods give information on a time scale
_7
of 10 s. In nuclear magnetic resonance (NMR) measurements it is possible to reach a time scale of 10 s.
Ever since intense neutron sources became available inelastic scattering experiments with neutrons have proven to be valuable for studying microscopic dynamics. Neutrons a r e sensitive both to high-frequency vibrational motions, and to low-frequency motions merging into diffusion. When diffusion is taking place the e l a s t i c - s c a t t e r i n g peaks will become broadened. In this case the scattering is called quasi-elastic.
As liquid and molecular dynamics have proven to be very complex, it is impossible to derive c r o s s section formulae from first principles. Therefore extensive use is made of simplifying models, some of which will be discussed 10
in chapter 3 of this t h e s i s . F r o m several investigations on molecular liquids it has become clear that with neutrons a mixture of the molecular c e n t r e - o f - m a s s translations and the atomic motions within the molecule is observed. In the plastic solids formed by globular organic compounds the translations will have vanished while rotations of the rigid molecules still can occur. An investigation of such compounds thus allows the study of the rotational motions in a solid separately.
Globular organic compounds have, since their discovery by T i m m e r m a n s about 30 y e a r s ago [2 ] , attracted considerable i n t e r e s t because of their peculair physical p r o p e r t i e s [ 3 , 4 ] . The molecules involved a r e almost spherical in shape either as a r e s u l t of molecular symmetry or of rotation of the molecule about its centre of gravity. This is the reason that they a r e called "globular". In the solid state phase transitions with l a r g e entropy and specific-volume changes occur from a low-temperature crystalline phase with low crystal s y m m e t r y , to a high-temperature phase with usually cubic structure and a certain degree of plasticity. The l a s t property has led to the name "plastic c r y s t a l s " for the solids above their phase transition point. The rather unusual behaviour of the plastic c r y s t a l s has been explained as resulting from practically unhindered rotation of the molecules about their lattice positions. Because of this rotation in the highly symmetrical molecular field the molecules get an opportunity to squeeze between other molecules so that molecular diffusion can occur at t e m p e r a t u r e s well below the melting point. As a consequence the plastic c r y s t a l s show a very small ent-ropy of melting.
Rotations and molecular diffusion in the solid can be detected by NMR since they will narrow the line width and reduce the second moment of the line. When reorientations occur with a frequency l a r g e r than the frequency of the NMR signal, however, it is impossible to derive the reorientation r a t e . Moreover it is difficult to distinguish between free rotations and r e s t r i c t e d rotations from one p r e f e r r e d orientation to another [ 5 ] . This distinction s e e m s to be important in understanding the phase behaviour of molecular c r y s t a l s [ 3 , 4 ] .
By using inelastic neutron scattering it is possible to distinguish between orientational and dynamical d i s o r d e r . The molecular rotations will show up as a line broadening of the elastic peak. Becka [6 ] showed the disorder to be dynamical in the globular compounds he investigated by means of neutron s c a t -tering. These r e s u l t s were of a qualitative nature only.
The aim of this thesis is twofold. In the first: place we want to show that low intensity inelastic neutron scattering experiments can be performed with a
well-designed s p e c t r o m e t e r even by using a low-flux r e a c t o r . Secondly, we want to investigate whether m o r e quantitative information about plastic c r y s t a l s can be obtained from neutron scattering experiments. In principle, it is possible to extract reorientation r a t e s and relaxation times of the molecular rotational motions from neutron scattering data. As an extension, also m e a s u r e m e n t s have been c a r r i e d out in the low-temperature crystalline phase and in the liquid phase.
The p a r t i c u l a r compounds selected for this study a r e cyclohexane and cyclopentane. In these compounds phase transitions in the solid state a r e known to exist as shown in NMR experiments [ 7 , 8 ] . F u r t h e r m o r e methylcyclohexane was studied in which the l a t e r a l methyl group stops the rotation, and no plastic phase is p r e s e n t . The methyl group, however, can perform torsional oscillations. The scattering of neutrons by these hydrocarbons is predominantly incoherent scattering by the protons.
A variety of neutron s p e c t r o m e t e r s has been constructed during the last fifteen y e a r s [see e . g . ref. 9 , 1 0 ] , In chapter 2 of this thesis the considerations leading to the construction of a rotating-crystal spectrometer (RKS) at the Delft swimming-pool r e a c t o r will be discussed. Also details of its construction a r e given and its performance is compared with r e s u l t s obtained with other neutron s p e c t r o m e t e r s . Scattering c r o s s - s e c t i o n formulae a r e derived in chapter 3 . F o r the quasi-elastic scattering a model recently proposed by L a r s s o n and Bergstedt [11] is used, in which some modifications will be introduced. After a description of the m e a s u r e m e n t s and a discussion of the corrections applied to the raw data in chapter 4, the obtained r e s u l t s a r e compared with the theory in chapter 5.
2. DESIGN AND CONSTRUCTION OF THE ROTATING-CRYSTAL SPECTROMETER
2 . 1 INTRODUCTION
The choice of a s p e c t r o m e t e r for inelastic scattering experiments with thermal neutrons depends on the resolution required for elastic and inelastic scattering, on the intensities, and on the complexity of the system also. In the scattering experiments on hydrocarbons we a r e mostly interested in the very small energy t r a n s f e r s broadening the elastic line, which gives information on diffusive and rotational motions. In o r d e r to resolve these small energy t r a n s f e r s a narrow spectrum of very slow neutrons incident on the sample must be used. F o r longer wavelengths, however, the intensities obtained from r e a c t o r s drop rapidly. A good compromise between intensity and wavelength can be made by the
o
selection of neutrons with a wavelength of approximately 4 A (5 meV energy). F o r the determination of the energy and the wavevector t r a n s f e r s the wavelengths of the neutrons incident on and scattered by the sample must be measured t o -gether with the scattering angle and the sample orientation.
A monochromatic neutron beam with small wavelength spread can be obtained by B r a g g - s c a t t e r i n g from a single-crystal monochromator or with a phased-chopper velocity selector. The wavelengths of the scattered neutrons can be m e a s u r e d by B r a g g - s c a t t e r i n g from a single c r y s t a l or, when pulsed monochromatic beams a r e used, by time-of-flight methods. In the latter the time a scattered neutron needs to travel a known distance, the flight path, is m e a s -ured.
Various types of neutron s p e c t r o m e t e r s can be constructed by a 13
combination of the different ways of incident wavelength selection and scattered wavelength determination [ 9 , 1 0 ] . In the triple-axis spectrometer e . g . both wavelength selection and analysis a r e performed with single c r y s t a l s [ 1 0 , 1 2 ] , The phased-chopper velocity s e l e c t o r s a r e composed of two or more choppers running in phase at very high rotational speed (up to 40000 rpm) [13-16]. In the crystal monochromator plus chopper system the wavelength selection is by B r a g g - s c a t t e r i n g from a single crystal while this monochromatic beam is pulsed by a chopper [17]. The incident wavelength selection in the rotating-crystal spectrometer is also performed with a single crystal [12,18 - 22] . By the rotation of this crystal with high angular speed the monochromatic beam is pulsed.
In the s p e c t r o m e t e r s with pulsed monochromatic beams the wavelength distributions of the scattered neutrons can be measured in several directions simultaneously. The counting rate in a triple-Eixis s p e c t r o m e t e r can be shown to be about a factor of ten lower than in time-of-flight s p e c t r o m e t e r s . Due to the low neutron flux at the Delft swimming-pool r e a c t o r (HOR) this type of apparatus could therefore not be considered.
The multi-chopper s p e c t r o m e t e r s a r e complex and expensive, and the beam c r o s s section is limited due to r o t o r - s t r e n g t h considerations. The two other types of time-of-flight s p e c t r o m e t e r a r e rather simple, thoi^h some neutron intensity is lost by the crystal reflection. On the other hand the background is lower as the monochromatic beam is reflected away from the reactorbeam. As the m a s s and the diameter of a rotating crystal a r e much s m a l l e r than those of a chopper, the driving and the mounting give l e s s difficulties. Moreover, large beam c r o s s - s e c t i o n s can be used without attenuation of the beam.
In the r o t a t i n g - c r y s t a l apparatus b u r s t focusing can be used, thus increasing the intensity without spoiling the resolution. In section 2. 3 we will show that for a given wavelength- and time resolution the rotating-crystal spectrometer yields a higher monochromatic neutron intensity than the other two time-of-flight spec-t r o m e spec-t e r s . Therefore ispec-t was decided spec-to build a rospec-taspec-ting-crysspec-tal specspec-tromespec-ter (designated RKS) at the HOR for the inelastic scattering experiments. This type of instrument was first suggested and built by Brockhouse, who also worked out the theory of the instrument [ 1 2 ] .
Section 2. 2 is devoted to a discussion of the elements contributing to the overall resolution. F r o m these considerations the different p a r a m e t e r s of the apparatus a r e fixed. The system uses a rotating single crystal to produce short b u r s t s of monoenergetic neutrons. A polycrystalline beryllium filter, located at the beam tube exit, eliminates unwanted crystal reflections and order
contamination, and s u p p r e s s e s background. A description of the monochromating p a r t is given in section 2 . 4 . The analysis of the neutron spectrum scattered from the sample i s done by the time-of-flight technique. The neutron detectors with their shielding and the electronics a r e discussed in section 2 . 5 . A discussion of the performance of the apparatus and a comparison with some other s p e c t r o m e -t e r s a r e given in sec-tion 2. 6. In -this sec-tion -the m e a s u r e m e n -t of -the de-tec-tor efficiencies is described also.
2 . 2 . RESOLUTION C ONSIDERATIONS
A schematic diagram of the rotating-crystal s p e c t r o m e t e r is shown in figure 2 . 1 . Neutrons p a s s through one of the beam tubes of the r e a c t o r and s t r i k e a single crystal rotating with high angular speed. Each time a set of c r y s t a l p l a n e s is satisfying the Bragg reflection condition
n \ .= 2 d sin 6 (2.1)
a b u r s t of monochromatic neutrons p a s s through the collimator C„ to the s a m p l e .
bervLLIum V filter , i::^ < collimator y C2 , 2 6 beam coUimotor tube I CI
€>-rotating crystal (moveable)
Fig. 2 . 1 , Schematic diagram of the rotating-crystal spectrometer.
In eq. (2.1) X is the wavelength of the reflected neutrons, n the order of reflection, d the interplanar spacing and 9 the Bragg angle. The diffraction angle, and thus the wavelength of neutrons incident on the sample, is variable by a movement of the monochromator along the r e a c t o r b e a m . Sample and detectors stay in fixed positions. The scattering plane is v e r t i c a l , due to space
limitations.
At the time the construction of the RKS was started the HOR was expected to be operated at a power level of 100 kW. This would give a neutron flux at the
11 2
beam tube entrance of about 4 x 10 n e u t r o n s / c m s. In order to get sufficient intensity at the detectors a moderate resolution must be accepted. As the power level would be increased in steps within some y e a r s , an improvement in r e -solution should be possible with only minor changes in the experimental s e t - u p .
In experiments with small energy t r a n s f e r s the wavelength resolution AVX and the time resolution At/t should be equal in order to obtain maximum r e -solution and intensity simultaneously. X is the most probable wavelength (4 A) in the incident neutron burst, and t is the time it takes these neutrons to travel
o
the distance between specimen and detector. Resolution and intensity depend on collimator divergences, the dimensions, mosaic spread and angular velocity of the c r y s t a l , the dimensions and orientation of the specimen and on the flight-path length.
The beam size is chosen to be 4 x 5 cm and the diameter of the cylindrical monochromator crystal is 4 cm. The (111) planes of lead have been used in most of the experiments so far, determining a scattering angle of 90 for neutrons
o
with 4 A wavelength. A calculation will be made of the resolution for elastically scattered neutrons, giving the p a r a m e t e r s for the apparatus. The influence of inelastic scattering is considered in section 2. 6 . 1 . A detailed discussion of the p a r a m e t e r s contributing to the resolution has been given by Brockhouse [ 1 2 ] .
The finite collimation a and the effective mosaic spread p of the rotating crystal contribute a time spread in the burst of
(2.2)
where uj is the rotational velocity of the crystal. The wavelength of the monochromatic neutrons follows from (2.1). The p r i m a r y wavelength resolution will then be given by
( ^ ^ ~ ^ j = c o t 9 d e = ^ cot 9 (2.3)
leading to a time spread
At^ = ^ [ ,^ jcot e (2.4)
for the elastically scattered neutrons with mean velocity v . L is the mono-chromator-specimen distance, Lg the specimen-detector distance. The detector thickness Ad contributes
At3 = ^ . (2.5) o
A neutron can be reflected anywhere within the rotating c r y s t a l , leading to an uncertainty in flight path. This t i m e - s p r e a d contribution can be separated into a p a r t due to the size of the crystal perpendicular to the reflecting planes
2 R sin e
At, = ^ . (2.6) ^ o
and into a p a r t due to the size p a r a l l e l to the reflecting p l a n e s . R is the effec-tive crystal r a d i u s . In favourable c a s e s the latter p a r t can be compensated partly by the Doppler effect due to the rotation of the crystal and it then contributes
2 R cos 9 m At, = 5 v (U m 2 - — (L^ + Lg) tg ( o (2.7)
When the c r y s t a l rotates in the opposite direction At_ changes into 2R cos e
Af^ - ~ ^ a > ^ ( L j + L 2 ) t g e . (2.8)
v o
The finite size of the specimen gives also r i s e to a time spread, which not only depends on thickness and geometry of the specimen but on the scattering angle also. These effects a r e partly correlated with At, and At^-. The overall spread is obtained approximately by adding the separate contributions r o o t m e a n -s q u a r e . The calculation i-s only approximate a-s not all the -s p r e a d -s have a Gau-s-sian shape.
If the second collimator C„ has a l a r g e r divergence than C neutrons reflec-ted early in the burst will have lower mean velocities than those reflecreflec-ted late in the burst. Thus focusing occurs and it is possible to increase the intensity
without a large i n c r e a s e in time spread. In the above formulas a will then be an average of the divergences of C^ and C„.
At 100 kW power level of the reactor we aimed at a time resolution of about 5-6 % for the elastically scattered neutrons. With equal contributions from wavelength spread and time spread (burst-time and flight-path uncertainties) this means
^1 "
S A X_
0.04. (2.9)The total wavelength spread is composed of the p r i m a r y effect (2.3), and the Doppler effect giving
VX„,^, V.
2 tu R sin 9
"^ "" (2.10)
• o ' ' 2 % \
This last contribution is practically compensated by the burst focusing, so that in condition (2. 9) only the p r i m a r y effect needs to be taken into account.
o °
The Bragg angle 9 =45 for 4 A neutrons reflected from the (111) lattice planes of a lead monochromator. Other p a r a m e t e r s chosen a r e p = 0 . 0 2 5 rad, L, = 60 cm, R = 1 . 4 cm, m = 1200 r a d / s , v = 1 0 c m / s . F r o m the condition
1 m m o
(2.9), together with (2. 3), we now find ff= 0. 040. When we choose ct, = 2a., we have a, = 0. 025 rad and a„ = 0. 05 rad. With this set of p a r a m e t e r s the contribut-ions to the time resolution of At. through At- a r e evaluated and shown in figure 2.2 as a function of u) for a detector at z e r o degree scattering angle. As the
W m i " TO rad/s
Fig. 2 , 2 , The different contributions to the time resolution for elastically scattered 4 A neutrons as calculated for a detector at zero degree scattering angle, versus the rotational speed of the lead m o n o -chromator,
detector is very thin At_ can be neglected. The total time spread At, obtained by taking the r . m . s. value of the various contributions is 65 ^i,s, so that the time resolution At/t = 0 . 0 5 3 .
o
When the influences of the finite size of the sample, of the scattering angle, and of the inelastic scattering respectively on the resolution a r e taken into account the formulae for the different contributions to the time spread a r e extended to those given in table II. 1.
TABLE II. 1. The different contributions to the line width in the RKS, 2 2 a + 2 P ^ 2tu At„ = At, = At, At, =
v^V2
Ad v' 2R sin m V 0 2R cos mr i
9 m ^m + L, At. sin t '2 V-)'}
cot e ,1-cot 9 I cot T f 1 ? cos 0 J + : ^ sin I
2.tan9^ [ -
, - ^ { L ,^ L , C ^ ) } .
o
f (^ 1 5 COS 0 J + - ? sin 0 I
0
At„ = channel length
At„ =
2tu_ - cot 9 ( L, + L„ m L 1 2
V . V ,
L I cot Y Tl - - 7 COS 0 ") + - r sin 0 \ 1)
1) At„ can be neglected, except in the case where shims a r e used in the
SoUer-o
slit collimators. These give r i s e to additional time s p r e a d s ,
Here 0 is the scattering angle of the neutrons, Y the angle between the mono-chromatic beam direction and the sample plane, V' the velocity of the scattered neutrons, and A the sample thickness. The use of shims in the collimators introduces an additional contribution (At„) to the line width, as has been shown
o
by Brockhouse [ 1 2 ] . The contribution of the sample thickness A is negligible for the thin hydrocarbon s a m p l e s . The resolution is further examined and compared with m e a s u r e m e n t s in section 2 . 6 . 1 .
2 . 3 COMPARISON WITH OTHER SPECTROMETERS
The resolution and intensity obtained with the three different time-of-flight s p e c t r o m e t e r s mentioned in section 2 . 1 will be compared in this section. The schematic d i a g r a m s of these s p e c t r o m e t e r s a r e given in the figures 2 . 1 and 2 . 3 .
collimotor
(b)
Fig. 2 . 3 . Schematic diagrams of (a) crystal plus chopper system, (b) two chopper system.
The intensity of monochromatic neutrons at the sample position is
I = ;^T„ 2 0 C-^J exp l - O ) 1 R,R T , (2.11)
s 4 n f o v \ y t ' L V x . ^ J X o c 'o o where
n =
F / L /
,
F = effective radiating surface of the neutron s o u r c e , L = source - monochromator distance,
o
T , = filter t r a n s m i s s i o n ,
0 = neutron flux at the s o u r c e , under the assumption of a Maxwellian velocity distribution,
X, = most probable wavelength,
X = mean wavelength of the monochromatic neutrons, R = AX/x = wavelength spread,
X o
R = crystal reflectivity,
T = duty cycle of choppers or rotating crystal.
We assume for all three systems equal values for 0 , L , T„, X. and X .
•^ ^ o o f t o
The burst frequencies a r e chosen equal also and such that no duty-cycle overlap will occur. The monochromatic beam intensity will be proportional to
P = n R, R T , (2.12) X o c
For approximately equal time resolutions we will evaluate the intensities for the different s p e c t r o m e t e r s .
a) Phased-chopper velocity selector.
The resolution and intensity for a system of two r o t o r s with curved slots have been derived by several authors [23,24]. The time spread at the detectors due to the wavelength spread is
'h ='-'^ <Ji) t ^ h ^ ^ ) ' <2.13)
where b is the slot width, lu the chopper angular speed, R the chopper radius, a the distance between the chopper c e n t r e s , L. the distance between the centre of the second chopper and the sample and L the flight path between sample and detector. The burst width is given by
AT = i - \ , (2, 14)
and the duty cycle by
T^ = 0.5 ^ . (2.15)
Here the factor 0 . 5 , proposed by Egelstaff [25] for a well-designed system, a r i s e s from the beam divergence.
b) Crystal monochromator in combination with a chopper.
The formulae for resolution and intensity a r e again taken from the l i t e r a t u r e [26]. The time spread due to the wavelength spread is
AX h ^ S ^1 ^ S
At, = T^ — = P cot 9 — . (2.16) 1 X V *^o V
0 0 o The burst width is given by
AT = " 5 ^ + - , (2.17) 2u)R u)
and the duty cycle is
T = 0.8 ^ A T . • (2.18) c 2Tr
H e r e the transmission of the chopper in "open" position is assumed to be 80%. c) Rotating-crystal s p e c t r o m e t e r .
The resolution has been calculated in section 2 . 2 . and can be found directly from figure 2 . 2 . The wavelength interval selected from the neutron beam is a r . m . s. addition of the p r i m a r y wavelength spread (2. 3) and the wavelength spread due to the Doppler effect (2.10).
Thus
2^ ^ \ 2 2(0 -"R ' /
- ^ » ffi— . (2.19) o /
The intensity in the burst is defined both by neutrons entering the crystal and by neutrons crossing the crystal while this turns into the Bragg reflection position, because the whole volume of the crystal is reflecting. With two reflections p e r revolution the duty cycle is
«) r 9 4 R ^ , ^
V
o
where At is given by eq. (2.2). Substituting this is (2.20) the duty cycle be-comes
r ^ po -^
, 4(1) R 2 m m2„ 2 1 I
(2.21)The burst width can be evaluated from a r . m. s. addition of the time s p r e a d s given in the equations (2. 2), (2. 6) and (2. 7) respectively, with L^ = o.
The numerical evaluation of intensities and resolutions is given in the Appendix. The pertinent data for the t h r e e s p e c t r o m e t e r s a r e summarized in table n . 2.
TABLE n . 2 Resolution and intensity in different types of time-of-flight s p e c t r o m e t e r . Spectrometer (a) P h a s e d -chopper system (b) Crystal + chopper (c) RKS At^ (M,S) 60 38 51 A T (us) 40 50 51 ^'tot^^^^^ Q (sterad) ^ o R, 78 63 64 - 4 3 . 3 X 10" 1 0 . 0 2 0 0 . 0 1 0 6. 6 X 10"^ 6 2 6 x 10 0 . 8 0 . 0 2 5 0 . 0 1 9 5 X 10" 7 6 4 . . 6 x 10 0 . 8 0 . 0 4 1 0 . 0 1 8 0 X 10" rx-4
Thus for equal time resolution the RKS gives a higher intensity at the sample than the other two s y s t e m s . When the intensity in the scattering s p e c t r a i s concentrated in a narrow interval around the wavelength of the incident
neutrons, a higher burst frequency can be tolerated without duty-cycle overlap. For system (b) a c r o s s - c h o p p e r , giving four b u r s t s per revolution, could be used, thus increasing the duty-cycle and the intensity in the monochromatic beam [ 2 7 ] . In the RKS other lattice planes can be used giving four or six reflections p e r revolution.
The intensity at the sample position, derived from eq. (2.11), for the RKS
2 1 1 2
i s I = 350 n / c m s, using 0 = 4 x 10 n / c m s at 100 kW r e a c t o r power, X. =
Sjj o t 1. 7 A and T , = 0. 5. With a sample thickness such that 20% of the incident
neutrons a r e scattered, a total count r a t e of about 15 counts/minute is to be 2
expected in a 120 cm detector assuming a counting efficiency of 30%. The back-ground will be lower than 5 counts/min. More details about the intensity calculations will be given in section 2. 6.2.
2 . 4 MONOCHROMATING PART
2 . 4 . 1 . B e a m t u b e a n d c o l l i m a t o r s
The RKS is installed at a 20 c m - d i a m e t e r radial beam tube of the HOR. A vertical section of the spectrometer is given in figure 2 . 4 . In order to suppress background radiation from fast neutrons and gamma quanta the radiating surface must be limited to the size being effective in illuminating the monochomator. With a beam c r o s s section of 5 x 4 cm and an entrance collimation of 1. 5 this
2
effective surface is about 60 cm . An inner beam tube with an inside diameter of 8. 8 cm has been constructed. The annular gap between the two tubes is filled with water. In the beam tube a 50 cm long stainless steel entrance cillimator is placed, limiting the beam to a 5. 2 x 4, 2 cm c r o s s section. The inner beam tube can be flooded with water to shut off the beam when changes in the set-up have to be
made. Normally the beam tube is evacuated, thus eliminating neutron scattering by air which otherwise would be about 15%.
In front of and behind the monochromator crystal two SoUer-slit collimators a r e placed, 40 cm in length with an inside c r o s s section of 5 x 4 cm. Effective collimator angles of 0.5 to 4 can be provided by the insertipn of 0. 015 cm thick steel s h i m s . In the r e a c t o r beam the maximum divergence is limited to 1.5 by the inner beam tube and the entrance collimator. The shielding around the neutron beam consists of boxes filled with berated paraffin and lead. The beam catcher consists of 15 cm lead and 30 cm paraffin.
2 0 b o r a t e d paraffin J 2 1 beam m o n i t o r 1 m 22 beam catcher
Fig. 2 , 4 , Vertical section of the Delft rotating-crystal spectrometer (RKS),
2 . 4 . 2 N e u t r o n f i l t e r s
Order contamination in the monochromatic beam is removed by using a 45 cm long polycrystalline beryllium filter. F o r neutrons with a wavelength g r e a t e r than twice the l a r g e s t interplanar spacing this m a t e r i a l i s nearly t r a n s p a r a n t due to the absence of coherent Bragg scattering. Beryllium h a s proved to be the most
o
convenient filter m a t e r i a l having a cut-off wavelength of 3.96 A, a small absorpt-ion c r o s s sectabsorpt-ion, a low incoherent scattering c r o s s sectabsorpt-ion and a high Debye t e m p e r a t u r e . By cooling the beryllium to liquid-nitrogen t e m p e r a t u r e a tenfold i n c r e a s e in intensity i s obtained due to the reduction of inelastic scattering. The
o
transmission of 4 A neutrons t h r o i ^ h the filter is 70%, while the attenuation of 5
fast neutrons and gamma radiation is a factor of 10 and 25 respectively [ 2 0 ] . A drawing of the beryllium filter is shown in figure 2 . 5 . The c r o s s section of the beryllium i s about 5. 5 x 4. 5 cm. The beryllium i s placed in a dewar
_3
evacuated to a p r e s s u r e of 10 t o r r through a 3 c m - d i a m e t e r vacuum line which
liquid N] in-outlet
'^ Be '-^/ss I /beryllium
boron carbide
dimensions in mm
Fig. 2 . 5 . The beryllium filter,
i s connected with a mechanical pump (C e n c o, Hyvac 28). The nitrogen is filled automatically from a dewar every hour. The liquid-nitrogen consumption i s 1.2 1/hour. The beryllium filter is located near the exit of the beam tube where it i s surrounded by a massive shielding of iron-shot borated paraffin and lead. In o r d e r to prevent slow neutrons, scattered from the filter, to produce capture gamma r a y s in the surrounding shielding, p a r t of the beryllium is enclosed in a boroncarbide shield.
For a further reduction of the fast neutron background a single-crystal filter consisting of several pieces of quartz has been placed in the entrance
col-o
limator. Quartz, 12 cm in length, gives 80% transmission of 4 A neutrons and reduces the fast neutron and the g a m m a - r a y intensities with a factor of 10 and 5 respectively ^ 0 ] . The neutron s p e c t r a behind the quartz and beryllium filters a r e shown in figure 2 . 6 .
2 3 ^ 5 6
neutron wavelength (A)
Fig, 2 . 6 , Neutron spectra at different positions in the RKS
(a) source spectrum,
(b) spectrum after quartz filter, (c) spectrum after quartz and beryllium
filters,
(d) monochromatic beam spectrum. 26
2 . 4 . 3 M o n o c h r o m a t o r c r y s t a l s
F r o m the crystal to be used on the RKS we r e q u i r e the following (a) l a r g e single crystal of cylindrical shape with 5 cm diameter and length, (b) sufficient strength for operation at high rotational speed,
o o
(c) interplanar spacing l a r g e r than 2.2 A so that for 4 A neutrons the B r ^ g angle is s m a l l e r than 65 ,
(d) high peak reflectivity,
(e) mosaic spread of 0. I*' to 0 . 5 ° ,
(f) low c r o s s sections for absorption, incoherent scattering and thermal diffuse scattering (high Debye temperature).
The scattering from extended single c r y s t a l s has been considered by Bacon and Lowde [28] and by Bacon [29]. Their reflectivity formulae can be used only approximately in this case because the effective crystal thickness depends on its rotational speed. Secondary extinction i s strong and the crystal-reflectivity func-tion loses its Gaussian shape. Most of the experiments have been performed using the (111) planes of a lead cylinder rotating about the [211] direction as a
*)
horizontal axis and giving two reflections per revolution . These planes have
Q O ^^
a spacing d = 2. 85 A which r e s u l t s in a Bragg angle of 45 for 4 A neutrons. Also the (200) planes of lead (d= 2.47 X, 9 = 55°) or the (111) planes of
alu-o alu-o
minium (d= 2.33 A, 9= 60 ) could be used. With a mosaic spread of 20 minutes of a r c , the reflecting power ("effective mosaic spread") of the lead crystal p = 0. 023 - 0. 025 rad, and the peak reflectivity R = 0. 92 - 0. 96, both depending on the rotational speed. The wavelength range within which the reflection may be considered to be complete is about 0.10 %, which matches the collimator diver-gences. These figures a r e valid for w values between 600 and 1500 r a d / s ,
The B r a g g - s c a t t e r e d neutrons a r e effectively used during a small fraction of the time only. The incoherently and inelastically scattered neutrons a r e p r o -duced all the time. The background caused by the latter is relatively high for a lead monochromator due to the low Debye temperature of lead ( 9 = 8 8 K). The inelastically scattered neutrons, mostly being scattered with neutron energy gain, can be removed from the monochromatic beam with a slab of poly-crystalline beryllium a few cm thick.
o o For aluminium the Bragg angle for 4 A neutrons is 60 , and therefore the
core-monochromator distance has to be increased. The integrated reflectivity
is a factor of three lower than in the case of lead. As a r e s u l t the overall inten-sity is reduced by a factor of four as compared with lead, while the resolution remains essentially the s a m e .
2 . 4 . 4 C r y s t a l r o t o r c o n s t r u c t i o n a n d d r i v i n g
The monochromator is mounted onto a support set parallel to the r e a c t o r beam. The crystal can be moved along the r e a c t o r beam, such that by a mechanical linkage between the centre of the crystal mount and the specimen position the collimator remains oriented. In this way the wavelength of the monochromatic neutrons can be varied continuously.
The lead c r y s t a l s have the shape of a knotted cone with 7 cm length and d i a m e t e r s of the end planes of 4, 0 and 3. 8 cm respectively. The crystal is tightly enclosed in an aluminium container as shown in figure 2. 7. a. The max-imum speed of the c r y s t a l , calculated from strength considerations is 18000 r p m . The c r y s t a l rotor is supported in air bearings and it is driven by a t h r e e -phase motor (AEG-AD 63k) with a simple belt drive. Rotational speed can be varied by a change of pulleys. The speed does not change by m o r e than 1% in a daily run, which is adequate for our purpose,
A m i r r o r with two faces is fixed to the rotor to sweep a light beam along a photocell (OAP 12) each time a crystal plane comes into reflecting position. The photocell pulses a r e amplified and a r e used to s t a r t the timing circuits of the electronics. The r i s e time of the light pulse is about 10 fis. Variations in rotational speed do not influence the measurements when light pulse and neutron burst a r e produced at the same moment, and when the resolution function does not change much with speed variations. These conditions a r e fulfilled.
Although aluminium monochromators have been run with speeds up to 24000 r p m , the heavy lead monochromator could be used with speeds up to 12000 rpm only due to the limited air p r e s s u r e available. After several failures in the c o m p r e s s e d - a i r system, which gave r i s e to damage of the r o t o r - s u r f a c e , a support using ball bearings has been constructed (figure 2. 7. b). These bearings do not need any servicing. The rotational speed has now been in-creased to 14500 rpm, while the m i r r o r has been replaced by a small cylinder with two diametrical s l i t s .
oLuminmm case lead singLe c r y s t o l 30 ^ m air gap m i r r o r for t r i g g e r systeem ( a ) 0 1 2 3 i 5 c m s c a l e
Fig. 2 . 7 . Drawing of crystal rotor with bearings, (a) air bearings,
(b) ball bearings. b o l l beoring Cylinder w i t h t w o d i a m e t r i c a l s l i t s p u l l e y aluminium cose ( b ) 0 1 2 3 i 5 c m scale 29
2.5 ANALYZING PART
2 . 5 , 1 N e u t r o n d e t e c t o r s
The r e q u i r e m e n t s for neutron detectors used in a time-of-flight spectro-m e t e r a r e
(a) high detection efficiency for thermal neutrons
(b) low detection efficiency for background radiation of fast neutrons and gamma r a y s ,
(c) thin detector to limit the flight-path uncertainty, (d) small time uncertainty.
Proportional counters filled with BF„ g a s , with boron enriched in B to 10 7
over 90%, and based on the B(n,(i) Li reaction, a r e often used. However, in o r d e r to obtain a high efficiency the detector thickness should be l a r g e which i s not suitable for our purpose. A BF„ counter with 2.5 cm diameter and a gas
^ o
p r e s s u r e of 70 cm Hg has a detection efficiency of about 17% for 1. 8 A neutrons, and of 34% for 4 A neutrons. The time-of-flight uncertainty is 25 ^j,s for the 4 A
3
neutrons. It is possible to use counters with higher gas p r e s s u r e , or He-filled proportional counters. At the moment of construction of the RKS these counters were not yet available. Lithium containing glass scintillators and lithium iodide c r y s t a l s , both based on the reaction Li(n, a) H, can not be used due to their high detection efficiency for gamma radiation.
We have used as detectors scintillator discs of 12. 5 cm diameter of the *)
same type as described by Stedman [30] . These scintillators a r e made by hotpressing finely divided lithium fluoride and silver activated zinc sulphide powders together with perspex as a transparant bonding medium. The lithium i s enriched in Li to 96%. The 0. 6 mm thick scintillator is mounted directly onto a 12.5 cm diameter photomultiplier (EMI 9583). Scintillator, multiplier, bleeder and preamplifier a r e assembled in a closed detector head.
The scintillator has a low transparancy for the light produced in the ZnS 3
(Ag) by the ionisation products of a- and H - p a r t i c l e s . This gives r i s e to a broad pulse-height distribution, while the detection efficiency is a complicated function of neutron wavelength. Light pulses produced by neutrons and gamma r a y s have decay times of about 300 ns and 20 ns respectively [ 3 1 ] . We use a preamplifier with rather large time constants such that the output pulses for
*)
neutrons and gammas have equal shape. Since the neutron-induced pulses a r e l a r g e r than the gamma-induced ones, pulse-height discrimination can be used to reduce the detection efficiency for gamma radiation. The intrinsic efficiency for neutron detection can often not be reached due to the broad pulse height d i s tribution and the r a t h e r high d i s c r i m i n a t o r levels n e c e s s a r y for gamma d i s -crimination.
Background, both due to gamma r a y s and to photomultiplier noise, can be reduced by the use of p u l s e s h ^ e discrimination (p. s. d . ) , as has been d e s -cribed by H a r r i s [ 3 1 ] and by Wraight [32]. At the same time an improvement in the neutron detection efficiency is obtained. Experiments on p . s. d. circuits a r e in p r o g r e s s [ 3 3 ] .
A maximum detection efficiency of nearly 60% has been reported by Sted-man [ 3 0 ] . This value s e e m s to be too high, however, a s suggested recently by Wraight et al [ 3 4 ] . T h e s e authors obtained efficiencies for 2. 8 A neutrons of 35% without, and of 45% with pulse-shape discrimination in the p r e s e n c e of a field of 5 mR/hour gamma radiation. We have found efficiencies of about 30% without pulse-shape discrimination for neutron wavelengths between 1.5 and 4
O
A. This will be described in section 2. 6. 3. Recently two BF„ counters, 2. 5 cm in diameter, 25 cm active length and 167 cm gas p r e s s u r e (Reuter Stokes RSN-90 A),have been put into operation, especially for the smallest scattering angles.
o o
Their efficiency for 1. 8 A and 4 A neutrons is 34% and 67% respectively. The shielding of the d e t e c t o r s against fast-neutron background has been optimized in o r d e r to limit the n e c e s s a r y amount of shielding m a t e r i a l [ 2 0 ] . Each detector is surrounded by a 40 cm long cylinder of a 80% B .C-20% paraffin mixture of 1.5 - 2 cm thickness. The inside wall of the cylinders is coated with 0.5 mm cadmium. The complete unit of detector plus shielding cylinder can be moved along a rail at a distance of 122 cm from the sample in o r d e r to cover scattering angles between 0 and 100 . Up to now seven detectors have been used simultaneously. Rail and detectors a r e positioned in a scattering chamber with walls built of boxes filled with a one-to-one mixture of borax and paraffin. The wall thickness v a r i e s from 20 - 40 cm (see figure 2 . 4 ) . All surfaces seen by the detectors a r e lined with 0.5 cm B.C-epikote (90%B.C) and 0.5 mm cadmium. The count r a t e due to neutron background has been reduced by this shielding with a factor af about 2000 to 3 counts/min (200 kW r e a c t o r power, monochromatic beam shut off with cadmium). This equals approximately the inherent background due to cosmic radiation and multiplier noise.
In the monochromatic beam two monitor counters a r e placed at distances of 31
20 cm and 140 cm from the monochromator. The first monitor m e a s u r e s the neutron flux incident on the sample, the second one can be used for the m e a s u r e -ment of the sample transmission. These monitors a r e 1 cm diameter BF„ counters with gas p r e s s u r e s of 70 cm Hg (LCT 0.2 NE 3/1) and 40 cm Hg (LCT 0 , 1 NE 3/1) respectively. Widths and relative positions of the time-of-flight peaks measured with the monitors can be used to determine the wavelength of the monochromatic neutrons, and to monitor the time resolution of the mono-chromatic beam.
2 . 5 . 2 E l e c t r o n i c s
All detectors and monitors a r e fed by the same high voltage supply (SELO SCM 406 R) via a balancing unit where the appropriate voltage for each detector can be selected. The electronic circuits associated with the counters a r e almost completely t r a n s i s t o r i z e d [ 3 5 ] . A 4096-channel time-of-flight analyzer with
5
channel capacity of 10 counts is used to s t o r e the data. The analyzer has been built by Technical Measurement Corporation according to our specifications. Afterwards the memory has been divided in two halves to facilitate the analyzer to accept pulses from two independent experiments. A block diagram of the electronics i s shown in figure 2. 8.
SELO HV supply
balancing unit
LiF-ZnS(Ag)
scintillator pre amplifier one of seven scattering detectors printer counter timer photo cell -12V supply three of sixteen inputs
L
pre amplifier pre amplifier 2 AO 81 units 16 linear amplifiers 8 Unes rotemeter TF441 time IE 161 master of flight unit input encoder- 1 5 V supply
T ©
|4.lin«s 2 monitors start pulse_r-"""
T F 4 4 2 time of flight unit I i lines eiines start pulse address format selector 12 lines, delay unit I — amplifier -15V supply1
address forrrtat selector -L-^-zL_SCEX. CN 6 6 0 - 2 TMC 6096-channel memory ( R K S 8x256 SCEX 6x512) oscilloscope display parallel p r i n t e r _ punched tape readout/readerUp to 16 signal inputs a r e available, whereas the number of time increments (time channels) can be varied from 16 to 4096 by factors of two. The channel length can be changed from 0.125 to 64 microseconds by factors of two. A variation of the channel length from 2.5 to 40, from 3 to 48, andfrom 3.5 to 56 microseconds, all by factors of two, has been made possible by a
*)
modification of the time-of-flight unit . Most of the time the RKS used 8 groups of 256 channels, while a r e a c t o r physics experiment utilizing a slow chopper (SCEX) used 4 groups of 512 channels. Channel lengths of 5 , 6 , 8 , 1 0 and 12 microseconds have been used in the experiments with the RKS. The clock of the time-of-flight unit is s t a r t e d twice p e r crystal revolution by pulses obtained from the optical trigger system. A digital delay unit has been built to p e r m i t delaying the s t a r t pulses without duty cycle overlap in order to optimize the performance.
An oscilloscope (Tektronix 503R) can provide an analog display of the stored information. Data readout is by p a r a l l e l p r i n t e r (Hewlett Packard) and by binary coded punched tape (Tally P a p e r Tape Punch 540R). Information can be read into the memory from punched tape (Tally P a p e r Tape Reader 550).
2.6 PERFORMANCE
2 . 6 . 1 R e s o l u t i o n
After preliminary t e s t s of p a r t s of the s p e c t r o m e t e r , e . g . beryllium filter, shielding, crystal driving, and electronics, the RKS has been put into operation during the first months of 1965. The first measurements have been performed with an aluminium monochromator which due to misorientation gives one reflection p e r revolution only. This crystal has been run at speeds up to 24000 r p m satisfactorily. With a r e a c t o r power of 100 kW and an angular speed of
2 18000 rpm the measured neutron flux incident on the sample was 25 n / c m s only. The full widths at half maximum of the monochromatic peak at the second monitor and of the elastically scattered neutron peaks at the detectors were 40 liS and 80 \i,s respectively.
Due to the limited a i r p r e s s u r e available for the air bearings the rotational speed of the lead monochromator has been limited to 12000 r p m . With the ball
The principle of this modification is due to R. Slegtenhorst, while C. A. v . d . Werve c a r r i e d out the necessary changes.
bearings the maximum speed i s 15000 rpm. The different contributions to the time resolution have been discussed in section 2. 2, and they a r e summarized in table H. 1.
Intensity and line width of the monochromatic beam have been measured with the second monitor, at various speeds of the monochromator. The monitor was placed at a distance of 140 cm from the lead c r y s t a l . The r e s u l t s a r e given in figure 2.9. The influence of the Doppler effect on the resolution can be seen
1500 r a d / s o focusing rotation • defocusing r o t a t i o n 660 rad/s 1B0 160 180 number of B ^ s channels
Fig. 2 . 9 . Measured line shapes at the monitor for various crysta^ speeds and for focusing and defocusing rotation. The measured points for defocusing rotation with 660 rad/s have been shifted by two time
clearly. T h e r e is no difference in integrated peak intensity between focusing and defocusing rotation. Using the formulae derived in section 2. 2, the different contributions to the line width can be computed. F r o m the divergence of the beam striking the monitor an cy-value of 0. 028 is determined. The calculated line widths at the monitor position and the measured values a r e compared in figure 2.10. F r o m the r a t h e r good agreement between the data we conclude that the r . m . s . addition is adequate. All line shapes in figure 2.9 a r e Gaussian shaped in the case of focusing rotation; for the defocusing rotation, however, the lines at higher speeds a r e definitely not Gaussian. In the latter case measured
Wm in 10 rad/5
Fig. 2 . 1 0 . The different contributions to the full width at half maximum of the burst of monochromatic neutrons as calculated for the monitor position, versus crystal rotor speed. Measured widths are indicated by the open circles.
and calculated widths of the elastic lines only a g r e e for the lowest speed. In figure 2. 2 the line width calculated for elastic scattering in a detector at zero degree scattering angle has been presented as a function of crystal speed. The resolution calculated for inelastic scattering at three different rotational speeds is given in figure 2 . 1 1 . The influence of the finite dimensions
100
50
600 rod/s
V i n l C m / s
Fig. 2 . 1 1 . Full width at half m a x i m u m of a burst of inelastically scattered neutrom versus their mean velocity as c a l c u
-o lated for a detectors at (? = 0 for different U) values. The velocity of the incident neutrons v = 1000 m / s .
o ^ 75 70 65 O 1 2 K in i,'^
Fig. 2 . 1 2 . The full width at half maxium of a burst of elastically scattered 4 A neutrons versus the wave vector transfer as calculated for the detectors, with different sample orientations. Measured widths for f = 50 , are indicated by open circles, U) = 1200 rad/s.
m
of the specimen i s demonstrated in figure 2.12 where the line width for elastic scattering at various scattering angles is shown. Some measured line widths a r e also included. It can be seen that there is a large influence from the sample orientation. Therefore the resolution function i s always r e m e a s u r e d after changes (or possible changes) in sample orientation have been made.
2. 6. 2 I n t e n s i t i e s
The monochromatic neutron intensity at the specimen position is given by eq. (2.11). The neutron flux 0 is r a t h e r difficult to e s t i m a t e . The water layer in front of the beam tube contributing effectively to the emerging beams has a thickness inversely proportional to the total neutron c r o s s - s e c t i o n . F o r water the total c r o s s section for slow neutrons i s proportional to their wavelength, and consequently a hardening of the emerging neutron spectrum will occur. C a s p e r s [36] has measured that the temperature of the neutron spectrum in-c r e a s e s from 300 K to 370 K at a water-vain-cuum boundary. This gives a d e in-c r e a s e
o o
in most probable wavelength X. from 1. 80 A to 1. 62 A, in good agreement with spectrum m e a s u r e m e n t s near a cadmium disc in water [37]. The beam tube has a finite c r o s s section, however, and neutrons scattering in the water around the tube will contribute to the flux at the beam tube entrance. We therefore estimate the effective X. to be about 1. 7 A.
36 k'
.
^^_
\
>rt
>
Sample JSo , ^os"/ / "°
^y
^ ^ ^ ^ 90°....-——"""'''^
The flux entering the beam tube is not isotropic. According to diffusion theory [38] the neutron c u r r e n t in the direction of the beam tube axis is 1. 25 times the neutron flux 0 , at the m o d e r a t o r - b e a m tube boundary. With a neutron
° 2
emitting surface of 60 cm , and a total transmission for cold neutrons in quartz, beryllium and construction m a t e r i a l s of T . = 0 . 5 , the neutron flux at the mono-chromator position will be
<^m = l ^ T j 2 . 5 0 ^ J ( ^ ) ' e x p [ - ( ^ ) ] f = 4 . 6 x 1 0 - ^ 0 ^ ^ . ( 2 . 2 2 )
With 200 kW r e a c t o r power we m e a s u r e d 0 « 3. 0 x 10 n / c m s; according 11 2"^
to (2. 22) we then find 0 = 6. 5 x 10 n / c m s. The maximum unperturbed
12 2 thermal flux at the core boundary has been measured to be 1. 7 x 10 n / c m s. This is in agreement with the value of 0 , estimated from the flux profile.
Combining (2.11) and (2. 22) we find for the intensity at the sample position
I = 1.2 X l O ^ R R , T , (2.23) s o X c ^
where R and T a r e defined by eq. (2.19) and (2. 21) respectively. The r e f l e c
-A C
tivity R is u) -dependent and v a r i e s from 0. 92 to 0. 96 (section 2 . 4 . 3 . ) . A Debye-Wallerfactor exp (-2W) = 0. 88 [39] reduces R somewhat. With the p a r a m e t e r s given in section 2. 2 (i. e. a = 0.040; m = 1200 r a d / s ) one finds
o "1 I = 600 n / c m s. With a sample transmission of 80% and a detector efficiency
of 0. 3 this intensity would give r i s e to a count r a t e in the detectors of about 25 counts/min. This count r a t e has been measured indeed.
The background in the detectors can be separated in a time-dependent and a time-independent p a r t . The first p a r t is caused by monochromatic neutrons scattered by the sample holders and by the surrounding a i r . The time-indepen-dent p a r t consists of .inherent background (2-4 c / m i n ) , fast neutron and gamma background (1. 5-4 c/min) and a background of slow neutrons scattered inelast-ically by the monochromator (2-3 c/min). Depending on the detector the total time-independent background amounts to 5-10 c/min.
With the formulae (2.19), (2. 21) and (2. 23) the intensity at the monitor position can be calculated as a function of crystal rotational speed. Due to the small c r o s s - s e c t i o n of the counter the divergence is reduced to a = 0. 028. R
K
h a s been corrected for the fact that p a r t of the wavelength interval around
o
X = 4. 05 A is beyond the Bragg cut-off wavelength for beryllium. The calculated intensity is compared with some m e a s u r e m e n t s in figure 2 . 1 3 . The theoretical 37
Fig. 2 . 1 3 . The intensity in the monochromatic beam as a function of crystal rotor speed as calculated at the monitor position, Measured points are indicated by open circles with error bars.
line has been normalized on the intensity measured at 975 r a d / s .
The resolutions and intensities of the RKS with a r e a c t o r power of 200 kW and monochromator speed of 1200 r a d / s a r e summarized in table H. 3. Even with this low power level we could obtain r a t h e r good statistics in the scattering spectra with running times of about 25 hours. In the meantime the r e a c t o r power has been increased to 500 kW and the monochromator speed to 1500 r a d / s , r a i s i n g the intensities at the sample and the detectors by a total factor of three. The shielding will be improved as the background nearly doubled with this power in-c r e a s e .
Brugger and B a r k e r [40] have recently made a compilation of intensities and resolutions of various s p e c t r o m e t e r s . The resolutions of the RKS a r e compared with some of their data in figure 2.14. In table II. 4 some pertinent p a r a m e t e r s of these s p e c t r o m e t e r s a r e presented. The RKS figures a r e based on a r e a c t o r power of 500 kW. The resolution of the RKS can be improved at the expense of intensity, and some figures with modified apparatus p a r a m e t e r s a r e included in fig. 2.14 and in table II. 4. These p a r a m e t e r s a r e given in table n . 5, where the count r a t e at the detectors has been calculated assuming 20% isotropic scattering and 30% detector efficiency»
38 r 2
TABLE n . 3. Intensities and resolutions of the r o t a t i n g - c r y s t a l s p e c t r o m e t e r (Delft).
Location: H O R Reactor Instituut, Delft Beam s i z e : 4 x 5 cm
Flight path: 1. 22 m
Source flux: 6.5 x 10 thermal n e u t r o n s / c m s Incident neutron energy: 5 meV
Intensity at sample position: 7 x 10 n/min 2)
Resolution (elastic): Energy distribution of b u r s t AE / E = 0. 08 Time distribution of b u r s t AT = 50 p,s B u r s t width at d e t e c t o r At = 65 LIS o '^ At / t = 0. 05 o o Resolution (inelastic):
Scattered neutron energy E , = 25 meV B u r s t width at detector At, = 45 |i,s
Atj/tj = 0. 08
Scattering angles:
Range 0 to 100 degrees Number of detectors 7
Solid angle p e r detector 0. 008 steradian Kind of detector ®Li F - ZnS (Ag)
Signal / background: Elastic peak 30
1) Reactor power 200 kW.
2) Lead monochromator with angular speed 1200 r a d / s .
TABLE n . 4 P a r a m e t e r s of the s p e c t r o m e t e r s from figure 2.14. Location Harwell Brookhaven Tt I s p r a Mol Chalk River Karlsruhe Delft ^) t l ?? Type^) I I I I II HI HI
m
i n i n Source flux 2 (n/cm s) 2 xl0^^6) 5 xlO^^ 5 x l 0 l 4 2 x i o " 9 xlO^^ 14 2 xlO 5.5x10^^ 1. 6x10^^ IT I t Flight path (m) 1.3 1.6 1.6 1.5 4 . 5 3 . 3 3 . 5 1.2 1.2 1.8 I 2) s (n/min) 4 xlO^ 2.4x10^7) 3 xio'^7) 3.6x10^ 1.3x10^ 2.5x10^ 3.5x10^ 2 xlO® 7 xlO^ 2.5x10^ ' o 0.074 0.04 0.025 0.078 0.03 0.017 0.015 0.055 0.035 0.02 'f 0.073 0.027 0.024 0.11 0.03 0.028 0.03 0.08 0.06 0.04 5)n
0. 008;0. 02 0.003;0. 06 0. 0033 0.006 0.001 0.008 0.008 0.0035 Mark in fig. 2.14 o x x + D * V A A A Ref. 31,41 16 16 40 40 40 40,421) I Cold-neutron choppers and phased-chopper velocity s e l e c t o r s II C r y s t a l monochromator plus chopper
i n Rotating-crystal s p e c t r o m e t e r s
2) Monochromatic neutron intensity at specimen position 3) Resolution for elastically scattered neutrons
4) Resolution for inelastically scattered neutrons with final energy E„ = 25 meV 5) Solid angle p e r detector (steradians)
6) 5 meV neutron intensity increased by a factor of 6 by liquid H„ source 7) Attenuation in beryllium filter and construction m a t e r i a l s not taken
into account
8) 8 meV incident energy instead of 5 meV like the others 9) 500 kW r e a c t o r power. 0.1 O01 + A A 0 A O K 11 • 10' 10° I5 (neutrons/minute)
Fig. 2 . 1 4 . The full width at half maximum of a burst of elastically scattered neutrons as measured at the detector, divided by their mean time-of-flight from the sample to the detector,At A ,
o o versus the intensity of the monochromatic beani, for the spectrometers given in table I I . 4 .
TABLE IL 5 Resolutions and intensities of the RKS with some different s e t s of apparatus p a r a m e t e r s . Lg (cm) a (rad) P^ (rad) u) (rad/s) F (cm^) T Q R. X I (n/min)
% \
Idet<*=/"'^''> I 120 0.040 0.025 1500 60 0.018 0.041 2x10^ 0.05-0.06 80 H 120 0.015 0.025 1500 30 0.016 0.032 7x10^ 0. 03-0. 045 25 III 180 0.010 0.020 1200 20 0.0125 0. 024 2. 5x10^ 0.02 4As has been discussed elsewhere [ 4 3 ] , the intensities at the specimen position and at the detectors of the RKS will be increased considerably in the course of 1968. At the s a m e time the background level will be reduced by i m -provements of the shielding, the use of pulse-shape discrimination circuits for the detectors, and the use of a 2 cm-thick beryllium filter in the monochromatic beam as suggested in section 2 . 4 . 3 . The increase in intensity will be obtained by a number of m e a s u r e s :
(a) i n c r e a s e of the r e a c t o r power from 500 kW to 2 MW;
(b) increase of the detector efficiencies by the p . s. d. circuits from 30% to about
o
40% (also BF„ counters with efficiencies of over 60% for 4 A neutrons will be used);
(c) the use of a slab of beryllium, 3 cm thick, in the core end of the beam tube
o
acting as a Maxwell demon, transmitting neutrons with X > 4 A and scattering the o t h e r s , which is expected to give a twofold gain in source flux intensity; (d) the installation of a cold neutron source using liquid methane at 100 K as a
moderator, which will increase the intensity of 4 A neutrons by a factor of four at least.
These improvements together will probably give a total i n c r e a s e in intensity at specimen and d e t e c t o r s of a factor of 30 to 40. Then our rotatingcrystal s p e c -t r o m e -t e r will compare very favourably wi-th o-ther s p e c -t r o m e -t e r s , as is c l e a r from figure 2 . 1 4 .
2 . 6 . 3 . M e a s u r e m e n t of d e t e c t o r e f f i c i e n c i e s
Because the scintillators a r e s e m i - t r a n s p a r a n t for the light pulses created by the charged p a r t i c l e s after neutron absorption, the neutron efficiency of the d e t e c t o r s cannot be calculated. This s e m i - t r a n s p a r a n c y also causes the broad height distribution for neutrons as shown in figure 2 . 1 5 . Both the pulse-height distributions due to the inherent background and to the gamma radiation from the r e a c t o r a r e displayed in this figure too. The arrow indicates the working bias of the detector.
S 50
Fig. 2 . 1 5 . Bias curves for one of the LiF-ZnS(Ag) detectors.
The efficiency can be described approximately by [ 3 1 , 3 4 ]
e(X) = AX „-aX -u e - e 1- aX (2. 24) 6 , . where a is the linear effective absorption coefficient for neutrons in Li, y, the absorption coefficient for the scintillator light, X the neutron wavelength, and A a proportionality factor. The absorption coefficient a can be calculated from the
c
thickness and the Li-amount of the scintillator, and from the c r o s s section for the Li(n, a) H reaction. A and p, must be determined experimentally.
The absolute detector efficiencies have been measured at some different neutron wavelengths by comparison with a lithium glass scintillator with a well-known efficiency [ 3 4 ] . A diagram of the experimental set-up is shown in figure 42
2.16 [33]. Monochromatic neutrons a r e selected by B r a g g scattering from a monochromator c r y s t a l . In this experiment the (111) planes of lead, aluminium
chopper with straight slits singLe crystal /. • second . , ' / collimator
^ ^ d£i«to?
Fig. 2 . 1 6 . Schematic diagram of the experimental set-up for the detector efficiency measurements.
and copper single c r y s t a l s were used to provide monochromatic neutrons with
o
wavelength of 4. 04, 3. 26, and 2.57 A respectively at the scattering angle of 90 . Higher o r d e r neutrons were p r e s e n t and they were used to m e a s u r e the efficiency at s h o r t e r wavelengths. In o r d e r to separate the different o r d e r s the incident beam is chopped by a simple Fermi-t5T>e chopper, and the time-of-flight spectrum of the scattered neutrons i s detected.
80 S? 60 >^ % 40 20
^^^
y ^
a "^ ^ j^6 1 2 3 4 5 neutron wavelength ( A )Fig. 2 . 1 7 . Efficiencies of different LiF-ZnS(Ag) detectors versus the neutron wavelength. (a) Stedman (with p . s . d . ) [ 3 0 ] , (b) (c) Glaser [42],
(d) Pietersz [ 3 3 ] .
F r o m a comparison of the s p e c t r a measured by the different LiF-ZnS(Ag) d e t e c t o r s with the spectrum m e a s u r e d by the g l a s s scintillator the efficiencies of the former could be computed. A typical r e s u l t for one of these detectors is shown in figure 2.17 together with r e s u l t s from Stedman [ 3 0 ] , and from Glaser [ 4 2 ] . As we have mentioned already in section 2 . 5 . 1 , the efficiencies stated by Stedman a r e too high. The A- and ^i,-values, given in table 11. 6, a r e determined by a l e a s t - s q u a r e fit of formula (2. 24) to the r e s u l t s . The efficiencies for the detection of 1. 8 A and 4 A neutrons of the different detectors a r e shown in table II. 6 also.
TABLE n . 6. Efficiency p a r a m e t e r s of the detectors (with a = 0.48 A" ^ D e t e c t o r no 1 2 3 5 6 7 8 A 0 . 3 7 0 . 3 1 0 . 3 1 0 . 3 5 0 . 3 5 0 . 2 9 5 0 . 3 3 M' 6 . 8 5 8 . 7 4 . 4 6 . 0 8 . 0 6 . 8 5 8 . 0 e ( 1 . 8 A ) 0 . 3 2 5 0 . 2 6 5 0 . 2 8 5 0 . 3 1 0 0 . 3 0 5 0 . 2 5 5 0 . 2 9 0 e (4 A) 0 . 3 0 5 0 . 2 3 0 0 . 3 0 0 0 . 3 0 0 0 . 2 7 5 0 . 2 4 0 0 . 2 6 0 44
