Neutron monochromators of BeO, MgO and ZnO single crystals

Nuclear Instruments and Methods in Physics Research A 747 (2014) 87–93

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Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima

Neutron monochromators of BeO, MgO and ZnO single crystals M. Adib a, N. Habib a, I.I. Bashter b, H.N. Morcos a, M.S. El-Mesiry a, M.S. Mansy b,n a b

Reactor Physics Department, NRC, AEAE, Cairo, Egypt Physics Department, Faculty of Science, Zagazig University, Egypt

art ic l e i nf o

a b s t r a c t

Article history: Received 23 June 2013 Received in revised form 8 February 2014 Accepted 9 February 2014 Available online 18 February 2014

The monochromatic features of BeO, MgO and ZnO single crystals are discussed in terms of orientation, mosaic spread, and thickness within the wavelength band from 0.05 up to 0.5 nm. A computer program MONO, written in “FORTRAN”, has been developed to carry out the required calculations. Calculation shows that a 5 mm thick MgO single crystal cut along its (2 0 0) plane having mosaic spread of 0.51 FWHM has the optimum parameters when it is used as a neutron monochromator. Moreover, at wavelengths shorter than 0.24 nm the reflected monochromatic neutrons are almost free from the higher order ones. The same features are seen with BeO (0 0 2) with less reflectivity than that of the former. Also, ZnO cut along its (0 0 2) plane is preferred over the others only at wavelengths longer than 0.20 nm. When the selected monochromatic wavelength is longer than 0.24 nm, the neutron intensities of higher orders from a thermal reactor flux are higher than those of the first-order one. For a cold reactor flux, the first order of BeO and MgO single crystals is free from the higher orders up to 0.4 nm, and ZnO at wavelengths up to 0.5 nm. & 2014 Elsevier B.V. All rights reserved.

Keywords: Neutron monochromators Single crystal BeO MgO ZnO

1. Introduction A neutron monochromator is a single crystal that selects neutrons with wavelengths according to Bragg's law. The range of wavelengths accepted by a monochromator depends on its crystal structure and mosaicity. Larger mosaicity increases the number of monochromatic neutrons that will make it to the sample, while reducing its wavelength resolution [1]. Common materials used as monochromator crystals are pyrolytic graphite (PG), silicon, copper, beryllium, iron, magnesium fluoride and heusler crystals. The choice of the monochromator depends on the range of incident energies required for the experiment and the desired energy resolution. However, a beam of monochromatic neutrons, selected from the spectrum of a nuclear reactor by means of diffraction by a monochromator crystal, will be in general contaminated with higher-order components. However, the silicon (1 1 1) reflection comes with the added advantage that second order reflection is forbidden [2]. In some cases, magnesium fluoride seems to be more preferable as a neutron monochromator, since its coherent scattering length is longer than that of silicon [3]. While all single-crystal monochromators used select wavelengths longer than 0.25 nm, a neutron filter is indispensable for the removal of unwanted contaminations from higher order reflections [4].

n

Corresponding author. Tel.: þ 20 1142173078; fax: þ 20 244620787. E-mail addresses: [email protected], [email protected] (M.S. Mansy). http://dx.doi.org/10.1016/j.nima.2014.02.022 0168-9002 & 2014 Elsevier B.V. All rights reserved.

Nowadays oxides single crystals of Be, Mg and Zn are commonly used as efficient neutron monochromators, since they have high scattering cross-sections [5] and are chemically more stable. Moreover, a bulk oxide single-crystal with particular orientation can be produced economically with low mosaic-spread and high purity. The second order reflections from BeO (0 0 2) and ZnO (0 0 2) are also forbidden. The use of large and perfect single-crystals of magnesium oxide (MgO) as a filter for thermal neutron beams has been studied by Thiyagarajan et al. [6] and Adib et al. [7]. However they did not study its use as a neutron monochromator. In the present work, a feasibility study of using BeO, MgO and ZnO single crystals cut along different planes as a neutron monochromator is conducted. Their characteristics are discussed in terms of crystal orientation, mosaic spread and thickness. A computer code MONO has been developed to calculate the neutron wavelength distribution of reflecting power P θhkl and reflected intensity I Ref from the single crystal at given orientation as a function of mosaic spread, thickness and reactor moderating temperature T. The MONO code is an adopted version of the computer code Mono-PG written by Adib et al. [8] and MgO-program by Adib et al. [7]. The adapted version can additionally provide the following calculations: 1. The reflecting power and intensity of the monochromatic neutrons from oxide single crystal set at glancing angle θ. 2. The energy and wavelength distribution of incident reactor neutron flux before and after reflection from the monochromator crystal was assumed to have 1/E for neutron energies E more than the epithermal. For lower energies the flux

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distribution is Maxwellian with neutron gas temperatures 293 K (R.T.) for thermal reactor while 20 K (L.H.) for cold one.

i. the wavelength band of the reflected neutrons, ii. the ratio of higher-order contaminations to the first one [9]. It is well known that the reflected neutrons from the ðh k lÞ planes satisfy Bragg's equation

2. Theoretical treatment

nλ ¼ 2dhkl sin θhkl

The main parameters determining the quality of a single crystal as a neutron monochromator with reasonable resolution are

Here, n is the order of reflection and θhkl is the glancing angle to the ðh k lÞ plane.

ð1Þ

Fig. 1. Flow chart of MONO code.

Table 1 Physical parameters of BeO, MgO, and ZnO. Physical property System and space group Molecular weight Lattice parameters (nm) No. of molecules/unit cell No. of unit cells/m3 Atomic positions

Coherent scattering length (m) Its structure factor F2hkl (m2) Total scattering cross-section (b) Absorption cross-section at 0.025 eV (b) a

Sears [3]. Adib et al. [7]. c Sabine and Hogg [14]. d Shein et al. [15]. e Kisi and Elcombe [16]. b

BeO

MgO a

(F.C.C.), Fm3m 40.3 ao ¼ 0.4213b

ZnO b

(H.C.P.), P63mc 25.011 ao ¼ 0.26984 co ¼ 0.42770a 2 2.22579E þ 28 Be: (1/3,2/3,0) (2/3,1/3,1/2)

4 1.3454E þ28 Mg: (0,0,0) (0,1/2,1/2)

O: (1/3,2/3,z) (2/3,1/3,1/2 þz) z ¼0.378d Be ¼0.7790E  14 O¼ 0.5805E  14e 0.3866E  27 11.863e 0.0076e

(1/2,0,1/2) (1/2,1/2,0) O: (1/2,1/2,1/2) (0,0,1/2) (0,1/2,0) (1/2,0,0)b Mg ¼0.5357E  14 O¼0.5805E  14e 1.993E  27 7.942e 0.063e

(H.C.P.), P63mca 81.389 ao ¼ 0.32501 co ¼ 0.52071c 2 2.07491E þ 28 Zn: (1/3,2/3,0) (2/3,1/3,1/2) O: (1/3,2/3,z) (2/3,1/3,1/2 þz) z ¼ 0.382d

Zn ¼ 0.5680E  14 O¼ 0.5805E  14e 0.2869E  27 8.363e 1.110e

M. Adib et al. / Nuclear Instruments and Methods in Physics Research A 747 (2014) 87–93

The reflected power of the ðh k lÞ plane at glancing angle θ within dθ is given by Bacon [9] as P θh k l dθ ¼

a dθ

ð2Þ

1 þ a þ ð1 þ 2aÞ1=2 coth ½Að1 þ 2aÞ1=2 

Here

89

As shown by Bacon [9], the integrated reflected power R þ1 Rθ ¼  1 P θhkl dθ from imperfect crystal of finite absorption reaches saturation for bulk crystal thickness of to. A stationary monochromator crystal in a white neutron beam is totally reflecting over a wavelength range given by Riste and Otnes [10]: Rλ ¼ Rθ λ cot θh k l

dθ ¼ dλ=2dhkl cos θhkl ; A ¼ μt o =γ o and a ¼ ðQ hkl =μÞWðθÞ

ð4Þ

Also the peak reflectivity is given by Shirane et al. [11] as

and μ is the linear absorption coefficient, t o is the crystal thickness and γ o is the direction cosine of the incident neutron beam relative to the inward normal crystal face. The Q hkl is crystallographic quantity given by Bacon [9] as Q hkl ¼ λ3 N 2c F 2 = sin 2θ

ð3Þ

WðθÞ has a Gaussian distribution with standard deviation η on mosaic blocks of single crystal. P θhkl is calculated for planes having Miller indices hkl the same as the cutting plane indicies hc kc lc i.e., αhkl ¼ 0, as well as for hc ;  kc ;  lc planes i.e., αhkl ¼ π.

Rp ¼

R0 Q t0 ; where R0 ¼ pffiffiffiffiffiffi c 1 þ R0 2π η sin θhkl

ð5Þ

The reflected intensity I Ref from crystals, when a neutron reactor beam having Maxwellian distribution ΦðλÞ is given by I Ref ¼ ΦðλÞP θhkl

ð6Þ

where ΦðλÞ for neutron gas temperature T is given by Gurevich and Tarasov [12] as ΦðλÞ ¼

constant 2 expð  h =2mkTλ2 Þ λ5

ð7Þ

Table 2 Input parameters for reflecting power P θhkl . Crystal

Cutting plane (h k l)

Glancing angle θ (deg)

Thickness to (mm)

λmin (nm)

λmax (nm)

Δλ (nm)

BeO MgO ZnO

002 200 002

43.20 45.00 35.01

4.84 5.00 4.05

0.1E  2 0.1E  2 0.1E  2

0.55 0.55 0.55

0.9E  3 0.9E  3 0.9E  3

0.0060

0.0060 BeO

0.0055

0.0040 0.0035 0.0030 0.0025 0.0020 0.0015

0.0045

0.0035 0.0030 0.0025 0.0020 0.0015 0.0010

0.0005

0.0005 0.26

0.27

0.28

0.29

0.30

0.31

0.32

0.33

0.34

0.0000 0.25

0.35

η=0.1° η=0.2° η=0.5° η=1.0° η=2.0° η=3.0° η=4.0° η=5.0°

0.0040

0.0010

0.0000 0.25

MgO

0.0050

η=0.1° η=0.2° η=0.5° η=1.0° η=2.0° η=3.0° η=4.0° η=5.0°

0.0045

0.26

0.27

Neutron Wavelength λ (nm)

0.28

0.29

ZnO

0.0055 0.0050 0.0045

η=0.1° η=0.2° η=0.5° η=1.0° η=2.0° η=3.0° η=4.0° η=5.0°

0.0040 0.0035 0.0030 0.0025 0.0020 0.0015 0.0010 0.0005 0.0000 0.25

0.30

0.31

0.32

Neutron Wavelength λ (nm)

0.0060

Peak Reflectivity Pθ002

Peak Reflectivity Pθ002

0.0050

Peak Reflectivity Pθ200

0.0055

0.26

0.27

0.28

0.29

0.30

0.31

0.32

Neutron Wavelength λ (nm) Fig. 2. Reflecting power P θhkl at various η.

0.33

0.34

0.35

0.33

0.34

0.35

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From Eq. (7), for constant pffiffiffiffiffiffiffiffiffiffiffiffiffi dλ, the peak of curve ΦðλÞ occurs at a wavelengthλ ¼ h= 5mkT . Thus for a thermal reactor with T ¼293 k, λ¼0.114 nm while, for a cold reactor λ ¼0.43 nm. A computer code Mono has been developed to calculate the distribution of reflecting power P θhkl as a function of neutron wavelength, and mosaic spread, setting glancing angle from crystal orientation and thickness t o . The code can also calculate the reflected intensity I Ref from both thermal and cold reactor fluxes. The flow chart of the computer code MONO is illustrated in Fig. 1. The flow chart contains two main subroutines: the first step in the code is to calculate the neutron reflectivity from the monochromator crystal while in the second step we calculate the reactor flux distribution within the selected wavelength from λmin to λmax within step Δλ. The applicability of the developed computer codes have been checked by Adib et al. [7,8,13]. The obtained agreement between the calculated transmission data for both PG and MgO single crystals with the experimental ones justifies the use of computer codes for calculations within accuracy sufficient for determining their neutron reflectivity characteristics. To get the code for scientific usage, contact the corresponding author via email address. The main physical parameters of BeO, MgO and ZnO used for the calculations are listed in Table 1.

3. Monochromatic features of single crystals From Eq. (2), the highest reflectivity is expected from the surface of a single crystal cut along ðhc kc lc Þ at glancing angle θ, with the

longest inter-planer distance dhkl and largest structure factor F 2hkl . From the crystal structure of used single crystals, (0 0 2) plane for BeO and ZnO and (2 0 0) plane for MgO satisfy these requirements. The wavelength distribution of the reflected neutrons P θhkl from BeO (0 0 2), ZnO (0 0 2) and MgO (2 0 0) single crystals, as a function of mosaic spread, as calculated assuming the input parameters given in Table 2 and physical parameters mentioned in Table 1. The result of calculation is displayed in Fig. 2, which, demonstrates that the highest reflectivity is from MgO (2 0 0), while the lowest one is from ZnO (0 0 2). This result is in agreement with their calculated structure factors given in Table 1. The result indicates that using of MgO as a neutron monochromator is preferred over the others. However, the distribution of its reflecting power is broader when compared with BeO and ZnO. According to Fig. 2 P θmax reaches a maximum value at mosaic spread η ¼0.51. This result is due to the assumed incident neutron beam having a divergence (0.41) which is comparable to the mosaic spread value. So, BeO (0 0 2) with 0.51 FWHM at mosaic spread η¼ 0.51 is preferred over the others when it is used as a neutron monochromator, due to smaller resolution value of the reflected beam ðΔλ=λ o 3%Þ. The integrated reflecting power Rθ ðnÞ of the monochromatic neutrons of first order reflection from single-crystal oxides (η ¼0.51) is calculated versus the crystal thickness with the same input parameters given in Table 2. The results of calculations are displayed in Fig. 3. As can be seen in Fig. 3, 5 mm is the optimum thickness for the beryllium and zinc oxides single crystals, as a neutron monochromator. Further, their second order reflection is forbidden and they are available at a

0.042

0.042 BeO

0.039 0.036

0.033

0.033

Integrated Refelctivity Rθ

0.036

0.030 0.027

θ st R 1 order θ nd R 2 order θ rd R 3 order θ th R 4 order

0.024 0.021 0.018 0.015 0.012

θ st R 1 order θ nd R 2 order θ rd R 3 order θ th R 4 order

0.027 0.024 0.021 0.018 0.015 0.012 0.009

0.006

0.006

0.003

0.003

0.000

MgO

0.030

0.009

0.000 0

2

4

6

8

10 12 14 16 18 20 22 24 26 28 30 32

0

2

4

6

8

10 12 14 16 18 20 22 24 26 28 30 32

Thickness t0 (mm)

Thickness t0 (mm) 0.042 0.039

ZnO

0.036 0.033

Integrated Refelctivity Rθ

Integrated Refelctivity Rθ

0.039

0.030 0.027 0.024 0.021

θ st R 1 order θ nd R 2 order θ rd R 3 order θ th R 4 order

0.018 0.015 0.012 0.009 0.006 0.003 0.000 0

2

4

6

8

10 12 14 16 18 20 22 24 26 28 30 32

Thickness t0 (mm) θ

Fig. 3. Integrated reflectivity R versus thickness t o : (a) BeO, (b) MgO and (c) ZnO.

M. Adib et al. / Nuclear Instruments and Methods in Physics Research A 747 (2014) 87–93

91

Fig. 4. Distribution of the reflected neutron intensities at various selected monochromatic wavelengths.

reasonable cost. However, the higher order contaminations may limit the use of MgO as a neutron monochromator. These contaminations are true when the incident neutron beam distribution is constant. The incident neutron beam distribution from steady state reactor obeys Maxwellian distribution with neutron gas temperature T. Therefore, the wavelength distributions of the reflected neutron intensities from BeO, MgO and ZnO were calculated at monochromatic neutrons with wavelengths of λ¼ 0.13 nm and 0.24 nm from thermal reactor flux assuming the same input parameters given in Table 2. The results of the calculations are displayed in Fig. 4a andb correspondingly. Fig. 4c displays the wavelength distribution of the reflected neutrons at λ¼ 0.36 nm from the cold one. The reflected neutron monochromatic intensities and higher orders contaminations from oxides single crystals were found to be dependent on both the reactor moderating temperature and the value of the selected monochromatic wavelength. For this reason, the integrated intensity of monochromatic neutrons from 5 mm thick crystals and the accompanying higher orders were calculated as a function of the neutron wavelength, λ, i.e. glancing angle θ. The results of the calculations are displayed in Fig. 5 for the incident thermal reactor neutron beam, and in Fig. 6 for the cold reactor one. Fig. 5 illustrates that MgO (2 0 0) is the best choice as a neutron monochromator without the need for a filter for monochromatic neutrons from a thermal reactor flux up to λ¼ 0.12 nm, while for BeO (0 0 2) and ZnO (0 0 2) λ is up to 0.24 nm. As shown in Fig. 6, if a cold reactor flux is available, the use of single-crystal oxides as neutron monochromators free from higher order contaminations at λZ0.24 nm is more appreciated. Additionally, ZnO (0 0 2) comes with the added advantage that one can select monochromatic neutrons with wavelengths longer than 0.42 nm, and up to λ¼0.5 nm.

To summarize the main results of BeO, MgO and ZnO oxide single crystals as neutron monochromators, the calculations that were carried out assume that the crystals have the same mosaic spared (η¼0.51), thickness to ¼5 mm and constant Δλ ¼0.05 nm. The results of the calculations of the integrated intensity of the reflected neutrons at λ ¼0.114 nm for the thermal reactor and at λ¼ 0.43 nm for the cold one along with the higher order contaminations, are listed in Tables 3 and 4 for the thermal and cold reactors, respectively. From Table 3, it is observed that the integrated intensity of the reflected monochromatic neutrons from MgO single crystal cut along (2 0 0) plane is much higher than the others at low contaminations (0.2%). Moreover, the glancing angle (15.71) is more suitable to carry out the diffraction experiments. In some cases BeO (0 0 2) is preferred as a neutron monochromator in a thermal reactor, since, the second order contamination is forbidden while it is observed in Table 4 that ZnO single crystal cut along its (0 0 2) plane is the most efficient neutron monochromator at long wavelength when installed at the exit of a channel of a cold reactor. Comparison of the performances of BeO, MgO and ZnO neutron monochromators are listed in Table 5 along with those reported by Riste and Otnes [10] for Be, Zn and PG. As can be seen in Table 5, it is obvious that MgO and BeO are preferred as neutron monochromators and as analyzers over beryllium and zinc, while PG is preferred as a selective neutron filter rather than a monochromator since the glancing angle (at λ¼ 0.127 nm) θhkl ¼111, which is not suitable to carry out the diffraction experiments. It is also observed that ZnO has low monochromatic performance (Table 5); however, if a cold reactor flux is available, ZnO (0 0 2) comes with the added advantage that one can select monochromatic neutron wavelengths up to λ ¼0.5 nm.

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Fig. 5. The integrated intensity of monochromatic neutrons and their higher orders from thermal reactor flux versus wavelength λ multiplied by the order of reflection.

6000

3000

Integratred reflected neutron intensity Σ ΙRef (n) count/sec.

Integratred reflected neutron intensity Σ ΙRef (n) count/sec.

BeO "Cold reactor"

5400 4800 4200 3600 3000 2400

λ ΣΙ

(1)

ΣΙ

(2)

ΣΙ

(3)

ΣΙ

(4)

1800 1200 600 3∗λ

0 0.12

0.16

0.20

0.24

0.28

4∗λ

2∗λ

0.32

0.36

MgO "Cold reactor"

2700 2400 2100 1800 1500

(1)

ΣΙ

(2)

ΣΙ

(3)

ΣΙ

(4)

λ

1200 900 600 300

2∗λ

0 0.15

0.40

ΣΙ

0.18

0.21

0.24

0.27

0.30

0.33

3∗λ

4∗λ

0.36

0.39

0.42

Neutron Wavlength λ (nm)

Neutron Wavlength λ (nm)

6000

Integratred reflected neutron intensity Σ ΙRef (n) count/sec.

ZnO "Cold reactor"

5400 4800 4200 3600 3000 2400

ΣΙ

(1)

ΣΙ

(2)

ΣΙ

(3)

ΣΙ

(4)

λ

1800 1200 600 4∗λ

3∗λ

2∗λ

0 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.40 0.44 0.48 0.52

Neutron Wavlength λ (nm) Fig. 6. The integrated intensity of monochromatic neutrons and their higher orders from cold reactor flux versus wavelength λ multiplied by the order of reflection.

M. Adib et al. / Nuclear Instruments and Methods in Physics Research A 747 (2014) 87–93

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Table 3 Integrated intensity of reflected monochromatic neutrons for thermal reactor. Oxides

Cutting plane (h k l)

θhkl at λ ¼0.114 nm deg

ΣI Ref ð1Þ (counts/s)

ΣIRef ð2Þ counts/s)

ΣI Ref ð3Þ (counts/s)

ΣIRef ð2Þ=ΣI Ref ð1Þ (%)

ΣI Ref ð3Þ=ΣI Ref ð1Þ (%)

BeO MgO ZnO

002 200 002

15.09 15.70 12.64

22,751 105,835 14,984

– 216 –

0.5 81.50 0.2

– 0.20 –

0.0020 0.0007 0.0013

Table 4 Integrated intensity of reflected monochromatic neutrons for cold reactor. Oxides

Cutting plane (h k l)

θhkl at λ ¼0.43 nm deg

ΣI Ref ð1Þ (counts/s)

ΣIRef ð2Þ (counts/s)

ΣI Ref ð3Þ (counts/s)

ΣI Ref ð2Þ=ΣI Ref ð1Þ (%)

ΣI Ref ð3Þ=ΣI Ref ð1Þ (%)

BeO MgO ZnO

002 200 002

79.19 – 55.67

4007 – 4243

– – –

9 – 1

– – –

0.22 – 0.02

Table 5 The monochromator performances at λ ¼0.127 nm. Riste and Otnes (1969)

Present work θ

θ

λ

Crystal

Reflection

η (min)

R (min)

R /η

R  10 (Å)

R

Crystal

Reflection

η (min)

Rθ (min)

Rθ/η

Rλ  102 (Å)

RP

Be Zn PG

002 002 002

22 34 68

11.0 13.6 58.0

0.50 0.39 0.86

1.1 1.9 8.7

0.42 0.31 0.74

BeO MgO ZnO

002 200 002

12.7 12.7 12.7

36.41 153.5 26.25

2.86 12.1 2.10

4.4 17.9 3.8

0.65 0.77 0.63

2

P

4. Conclusion The developed MONO-code, used to calculate the reflectivity from single crystals, is found to be sufficient for determining the neutron monochromatic characteristics. It is clear that BeO (0 0 2) with 0.51 FWHM on mosaic spread η ¼0.51 is more preferred than the others when used as a neutron monochromator, since the resolution of the reflected beam Δλ=λ is less than 3%. So, MgO (2 0 0) is the best choice as a neutron monochromator without the need for a filter for the neutrons from a thermal reactor flux up to λ ¼0.12 nm, while BeO (0 0 2) and ZnO (0 0 2) λ are up to 0.24 nm. If a cold reactor is available, then the use of the single-crystal oxides as neutron monochromators free from higher order contaminations at λ Z0.24 nm is appreciated. Moreover, ZnO (0 0 2) comes with the added advantage that one can select monochromatic neutrons with wavelengths up to 0.5 nm. References [1] S.W. Lovesey, Theory of Neutron Scattering from Condensed Matter, Oxford University Press Oxford, 1987.

[2] P. Mitchell, Solid State Physics: Magic Crystal Balls, 2006, P. 1 of 78 from lectures at Manchester University, Retrieved from 〈http:/porlhews.tripod.com/ sitebuildercontent/sitebuilderfiles/solidstatephysics.pdf〉. [3] V.F. Sears, Neutron News 3 (1992) 26. [4] E. Arzi, Introduction to Neutron Powder Diffractometry, University College, Cardiff Press, Wales, 2001. [5] J.G. Barker, D.F.R. Mildner, J.A. Rodriguez, P. Thiyagarajan, Journal of Applied Crystallography 41 (2008) 1003. [6] P. Thiyagarajan, R.K. Crawford, D.F.R. Mildner, Journal of Applied Crystallography 31 (1998) 841. [7] M. Adib, N. Habib, I. Bashter, M. Fathallah, A. Saleh, Annals of Nuclear Energy 38 (2011) 2673. [8] M. Adib, N. Habib, M.S. El-Mesiry, M. Fathallah, Energy and Environment Research 2 (2012) 35. [9] G.E. Bacon, Neutron Diffraction, third ed., Claredon, Oxford, 1975. [10] T. Riste, K. Otnes, Nuclear Instruments and Methods 75 (1969) 197. [11] G. Shirane, S.M. Shapiro, J.M. Tranquada, Neutron Scattering With a Triple-Axis Spectrometer, Cambridge University Press Cambridge, United Kingdom, 2002. [12] I.I. Gurevich, L.V. Tarasov, Low-Energy Neutron Physics, North-Holland Publishing Company, Amsterdam, 1968. [13] M. Adib, N. Habib, I. Bashter, H.N. Morcos, M. Fathallah, M.S. El-Mesiry, A. Saleh, Annals of Nuclear Energy 60 (2013) 163. [14] T.M. Sabine, S. Hogg, Acta Crystallography B25 (1969) 2254. [15] I.R. Shein, V.S. Kiĭko, Yu. N. Makurin, M.A. Gorbunova, A.L. Ivanovskiĭ, Physics of Solid State 49 (2007) 1067. [16] E.H. Kisi, M.M. Elcombe, Acta Crystallography C45 (1989) 1867.