On the neutron diffraction in a crystal in the field of a standing laser wave

Nuclear Instruments and Methods in Physics Research B 268 (2010) 2539–2543

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On the neutron diffraction in a crystal in the field of a standing laser wave K.K. Grigoryan a, A.G. Hayrapetyan b,*, R.G. Petrosyan a a b

Yerevan State University, 1 Alex Manoogian Str., 0025 Yerevan, Armenia Institute of Applied Problems of Physics, National Academy of Sciences of Republic of Armenia, 25 Hr. Nersisyan Str., 0014 Yerevan, Armenia

a r t i c l e

i n f o

Article history: Received 18 April 2010 Received in revised form 7 May 2010 Available online 15 May 2010 Keywords: Neutron scattering Neutron diffraction Schrödinger equation for neutron Standing laser wave Crystal

a b s t r a c t The possibility of high-energy neutron diffraction in a crystal is shown by applying the solution of timedependent Schrödinger equation for a neutron in the field of a standing laser wave. The scattering picture is examined within the framework of non-stationary S-matrix theory, where the neutron–laser field interaction is considered exactly and the neutron–crystal interaction is considered as a perturbation described by Fermi pseudopotential (Farri representation). The neutron–crystal interaction is elastic, and the neutron–laser field interaction has both inelastic and elastic behaviors which results in the observation of an analogous to the Kapitza–Dirac effect for neutrons. The neutron scattering probability is calculated and the analysis of the results are adduced. Both inelastic and elastic diffraction conditions are obtained and the formation of a ‘‘sublattice” is illustrated in the process of neutron–photon–phonon elastic interaction. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction Neutron diffraction with X-ray, electron and positron diffraction methods is productively used in condensed matter research due to its specific interaction with matter. In contrary to X-rays and charged particles neutrons barely interact with atoms forming a crystal, they interact via nuclear and magnetic forces which endows neutrons with some essential advantages to the neutrons diffraction in comparison to the other diffraction methods. Neutrons can identify properties and structure of crystalline matter in a deeper and completely different level. For instance, an establishment of magnetic properties of a magnetic medium or examination of a crystalline matter consisting of light atoms. Great number of scientific literature is devoted to neutrons and their application in condensed media [1–5]. Especially note book [5], where the main theoretical apparatus of quantum mechanics of neutrons interacting with a matter is introduced in a very refined and simple manner. In recent years researches in this very intensively developing area continue and both theoretical and experimental new results appear [6–19]. Recently for the first time a problem of short-wave (high-energy) neutron diffraction is discussed in theory [11–13], where the probability of short-wave neutron diffraction in a crystal is shown within the framework of S-matrix theory when an external laser or hypersonic field exists. In the conventional case, when any of external fields are absent, only the thermal neutrons give a dif* Corresponding author. Tel.: +374 93342623. E-mail addresses: [email protected], [email protected] (A.G. Hayrapetyan). 0168-583X/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2010.05.052

fraction pattern. The external fields’ application essentially and qualitatively affects diffraction intensity and enables to observe theoretically high-energy neutrons diffraction, whose wavelengths are less than a lattice period even for two–three orders. External fields give opportunity to consider the neutron–photon–photon resonant interaction due to the interference laws and derive the condition under which the diffraction occurs under the large angles (numerical estimations are adduced below). Present paper is a continuation of discussed problem in [11–13], here both thermal and high-energy neutrons diffraction is considered in a crystal under the influence of a standing laser wave, application of which allows to examine neutron both elastic and inelastic diffraction with respect to a field; and in the case of elastic diffraction one can illustratively talk about the neutron diffraction on a ‘‘sublattice” which is formed in the process of multiphoton neutron–photon–phonon elastic interaction. In fact the situation is very similar to Kapitza–Dirac effect applied for neutral particles (see also [13]). Here we consider non-magnetic materials and restrict our examination with a crystal consisting of an ordinary cubic lattice with single isotope elements on vertices, and the interaction of the neutron with a crystal is described by means of Fermi pseudopotential.

2. Neutron–laser and neutron–crystal interactions Neutrons diffraction in a crystal under the influence of a standing laser wave is considered in the basis of non-stationary S-matrix theory, which in a refined manner illustrates the dynamics of quantum mechanical three-particle system. In our case the

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neutron–photon–phonon system’s behavior is discussed, and the interaction is realized via multiphoton channels. Generally the neutron–photon–phonon system’s time-dependent Hamiltonian is given in the form

Hðt Þ ¼ H0 ðt Þ þ V ðt Þ;

ð1Þ

which includes two terms. The first one is the basic (non-perturbed) neutron–field interaction. And the second one being a perturbed part describes the neutron–crystal interaction. In order to construct the jwð~ r; tÞi wave function of non-perturbed system Schrödinger time-dependent equation must be solved for a neutron moving in the space which is filled by the laser radiation (Farri representation). For the neutron which interacts with the laser field by means of anomalous magnetic moment l  jlj ¼ 1; 91lB (lB is the Bohr magneton) Schrödinger non-stationary equation can be written in the form

! 2  h @jwð~ r; t Þi ~ ~  D þ lrH jwð~ ; r; t Þi ¼ i h @t 2m

ð2Þ

where  h is the Planck’s constant, m is the neutron mass, D ¼ @ 2 =@x2 þ @ 2 =@y2 þ @ 2 =@z2 is the Laplace operator, ~ r¼ ~ ¼ ð0; 0; H0 cosð~ ðrx ; ry ; rz Þ are Pauli matrices, H k~ r  xtÞ þ H0 cosð~ k~ r þ xtÞÞ is the magnetic strength vector of standing laser ~ 0, ~ wave, H k and x are the amplitude, wave vector and frequency of a standing laser wave respectively. The solution of Eq. (2) we look in the form ~ jwð~ r; tÞi ¼ ehðP~rEtÞ F 1 ðu1 ÞF 2 ðu2 Þjw0 i; i

ð3Þ

where u1  ~ k~ r  xt, u2  ~ k~ r þ xt are the phases of a standing wave, F 1 and F 2 are slowly varying functions, and jw0 i is a constant spinor. One can see that Eq. (2) can be reduced to the following equation:

" 2 ! ! ~ ~  k2  00 P~ k P~ k h ~  x F 01 F 2 þ ih þ x F 1 F 02 F 1 F 2 þ 2F 01 F 02 þ F 1 F 002 þ ih m m 2m ! # ~ P2 F 1 F 2  lH0 r2 ðcos u1 þ cos u2 ÞF 1 F 2 jw0 i ¼ 0; ð4Þ þ E 2m where prime means a derivative with respect to each argument of F 1 and F 2 . Now note that ~ P and E, which are arbitrary constants, are neutron momentum and energy parameters. In the exponent in (3) if we add a vector in the form const~ k and a scalar constx to the ~ P and E respectively one can see that the form of jwi function is not changed, and a notation of F 1 or F 2 functions is only needed. Thus, a condition should be imposed on these parameters. Let us take ~ P 2 =ð2mÞ ¼ E. When switching off the laser field ~ P and E become a free neutron momentum and energy respectively. Eq. (4) cannot be solved exactly. Generally exact solutions of Schrödinger equation are obtained only for some cases. This concerns neutrons as well. Nevertheless an exact solution for a neutron in the external field can be established in case of neutron’s motion in the field of circularly polarized electromagnetic field [20]. The solutions of Schrödinger equation for neutrons in external fields can be found in [21–26,5,12]. In fact we work in eikonal approximation. Taking into account that the coefficient before the second derivatives of F 1 and F 2 is small quantity, and F 1 and F 2 are slowly varying functions the first term in Eq. (4) can be dropped. Hence the following system of linear differential equations is obtained:

(

F 01 ¼ ib1 rz cos u1 F 1 ; F 02 ¼ ib2 rz cos u2 F 2 ;

ð5Þ

where the notation b1;2  lH0 =ð h~ P~ k=m  h  xÞ is introduced. Substituting the solutions of Eqs. (5) in (3) one can find the solution of Schrödinger equation (2) i

~

jwð~ r; t Þi ¼ ehðP~rEtÞ eib1 rz sin u1 eib2 rz sin u2 jw0 i:

ð6Þ

Constants of integrals (5) are included in jw0 i. Wave function (6) describes the neutron–standing laser wave system. Contribution of neutron–photon system to the non-stationary S-matrix theory is considered exactly. In Farri representation we consider a space filled by electromagnetic radiation in which neutrons scattering occurs. Then a crystal is located; its interaction with neutrons is considered as a perturbation to the neutron–laser interaction. In other words we examine the scattering of neutron–photon system on a crystal, meanwhile neglecting the crystal–laser interaction. The perturbation Hamiltonian generally depends on time. Nevertheless we assume that the state of a crystal is not changed during the interaction process and consider elastic diffraction of neutrons with respect to a crystal; excited states of a crystal do not appear. This can be obviously illustrated later when we choose the wave functions of a crystal before and after interaction in such a way that the resonant condition (energy conservation law) does not include crystal excitation energy quanta. For a simplicity of physical picture of scattering process it is convenient to examine neutron coherent scattering (neutron diffraction). For this purpose we consider a crystal with primary cubic lattice consisting of single isotope element atoms with zero spin and located in the vertices of a cube. If the crystal consists of nuclei with different isotope content or with non-zero spin, then a part of a scattering will be incoherent, so called isotopic and spin incoherence. In the absence of external fields more complete theory of thermal neutron scattering is given in [27–29]. Neutron’s time of passing through a characteristic distance is comparable with the period of propagation of excitation which arises in the crystal. For this reason it can be accepted that neutron wave interacts not with an individual atom, but with aggregation of atoms. For the neutron’s interaction with the crystal which consists of single isotope element atoms with zero spin and has single nucleus in elementary cell can be written in the form

V ðt Þ ¼ 

2  2ph A X  d ~ r ~ Rn ; m n

ð7Þ

which is known as Fermi pseudopotential. Here A is the scattering amplitude of a neutron from the nucleus, ~ Rn ¼ ~ n þ~ nn is the nucleus ~ ~ ~ ~ position near the cell n ¼ n1 a1 þ n2 a2 þ n3 a3 , ni are integers (jni j ¼ 1; 2; . . . ; N) and ~ ai are the vectors of basic translations (i ¼ 1; 2; 3), N is the number of elementary cells in crystal. The shift of atoms from their equilibrium position is considered within the frame of second quantization, and the atom shift operator from the ~ n cell in harmonic approximation is given in the form

~ nn ¼

X S;~ q

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i h y ~ eS ð~ qÞ bqS ei~q~n þ bqS ei~q~n ; ~ 2MNXS ðqÞ

ð8Þ

where M is the nucleus mass, ~ q, XS ð~ qÞ and ~ eS ð~ qÞ are phonon wave vector, frequency and polarization vector from the S branch respecy tively; bqS and bqS are the phonon annihilation and creation operators respectively, satisfying Bose commutation relations y ½bk ; bk0  ¼ dkk0 , ½bk ; bk0  ¼ 0. Such an approach is used for the investigation of thermal neutron diffraction in a crystal in the absence of any external field in [30]. And for the first time it is productively applied both for high-energy and thermal neutrons diffraction in a

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crystal in the presence of external field (laser or hypersonic) in [12,11,13]. Note that the second quantization method allows to consider neutron diffraction in a crystal under the processes of multiphoton (multiphonon) emission or absorption from laser (sound) wave. This fact is very important since only multiboson channels allow to reveal high-energy neutron diffraction. 3. Neutron–crystal–laser interaction We consider the neutron–photon–phonon interaction within the framework of non-stationary S-matrix theory. For this purpose we must use the given dynamical state (6), take its known value at the moment t 0 and generate from this the dynamical state at the moment t. Consequently the problem is to construct, preferably in a most exact way, a unitary operator U 0 ðt; t0 Þ, which describes the time evolution of a dynamical state in Schrödinger representation. Then we pass to the usual procedure [31] and construct so called intermediate (interaction) representation in order to obtain the whole neutron–photon–phonon system’s time evolution operator, which finally yields the neutron diffraction probability. The unitary operator U 0 ðt; t0 Þ satisfies the Schrödinger equation

ih

@U 0 ðt; t0 Þ ¼ H0 U 0 ðt; t 0 Þ @t

Z

U 0 ðt; t

t0

H0 ðsÞU 0 ðs; t0 Þds:

ÞU y0 ðt; t 0 Þ

U y0 ðt; t 0 Þ

¼

¼ U y0 ðt; t 0 ÞU 0 ðt; t0 Þ 0 U 0 ðt 0 ; tÞ ¼ U 1 0 ðt; t Þ;

U 0 ðt; t 0 Þ ¼ U 0 ðt; t 00 ÞU 0 ðt 00 ; t 0 Þ:

¼ 1;

ð11Þ ð12Þ

ð13Þ

It is very convenient to pass to the interaction representation since the exact dynamical state of the whole system cannot be established from Schrödinger equation. S-matrix theory simply solves the problem of a scattering of quantum mechanical full system. Nevertheless the dynamical state of the full system is chosen in the form:

Y

jmSq i;

where S;q jmSq i is the wave function of the crystal oscillation state with mSq phonons from the S branch and ~ q wave vector. Here we have taken into account that the distribution of oscillating states of a crystal is independent from the distribution of a dynamical state of the basic neutron–laser field system. The time evolution operator U I ðt; t 0 Þ in the interaction representation obtains the same properties (11)–(13) and satisfies the same Eqs. (9) and (10) written for the perturbation Hamiltonian in the interaction representation

V I ðtÞ  U y0 ðt; t 0 ÞVðtÞU 0 ðt; t0 Þ:

ð15Þ

Using the iteration approach the whole dynamical system’s time evolution operator Uðt; t0 Þ, which also satisfies the conditions (11)–(13) and the equations (9) and (10) written for the Hamiltonian H, can be represented in the form

n¼0

where

@R ; @ðDtÞ

W ~Pf ;~Pi ¼ lim

Dt!1

ð18Þ

2

where R ¼ jMfi j is the square of modulus of probability amplitude. The bar on R means statistical averaging and the matrix element Mfi has the form

D E Mfi ¼ Wf jU ð1Þ ðt; t 0 ÞjWi   Z t dsU 0 ðt; sÞU 0 ðs; t 0 ÞjWi : ¼ ðihÞ1 Wf jV

ð19Þ

t0

Based on the evolution law jwðtÞi ¼ U 0 ðt; t 0 Þjwðt0 Þi one has i

~

0

~

~

0

~

U ðnÞ ðt; t 0 Þ;

0

U 0 ðt; t 0 Þ ¼ ehEðtt Þ eib1 rz ½sinðk~rxtÞsinðk~rxt Þ eib2 rz ½sinðk~rþxtÞsinðk~rþxt Þ : Now, we introduce the following notations:

Rs1 s2  hw0 jJs1 ððb1f  b1i Þrz ÞJs2 ððb2f  b2i Þrz Þjw0 i;

ð16Þ

ð21Þ

k=m  hxÞb1fif g;2fif g ¼ lH0 ; ðh~ Pfif g~ ~ ~ P ~ P þ ðs þ s Þh~ k; hQ

ð22Þ

hXs1 s2  Ei  Ef þ ðs1  s2 Þhx:

ð23Þ

s1 s2

i

1

f

2

Here the numbers s1 and s2 obtain both positive and negative values. They show a number of photons in the standing laser wave. Illustratively one can talk about s1 photons which propagate in the opposite direction of s2 photons. Then substituting (14) and (20) in (19) we find the following form of matrix element:

Mfi ¼  

Q

1 X

ð17Þ While obtaining the expansion (16) the relation U I ðt; t0 Þ  U y0 ðt; t 0 ÞUðt; t0 Þ between the free unitary operators is taken into account. We examine the first approximation of the expansion (16). This illustrates the refined picture of quantum dynamical system’s behavior during the scattering period Dt  t  t0 . The neutron scattering probability is given via the formula

ð14Þ

S;q

Uðt; t0 Þ ¼

 Vðsn1 Þ . . . U 0 ðs2 ; s1 ÞVðs1 ÞU 0 ðs1 ; t 0 Þ:

ð20Þ ð10Þ

and the composition law

jWð~ r; tÞi ¼ jwð~ r; tÞi

dsn . . . ds1 U 0 ðt; sn ÞVðsn ÞU 0 ðsn ; sn1 Þ

t

The time evolution operator takes the following properties: 0

t>sn >...s1 >t0

ð9Þ

with the initial condition U 0 ðt0 ; t0 Þ ¼ 1, which is equivalent to the following integral equation:

U 0 ðt; t 0 Þ ¼ 1  ði hÞ1

U ð0Þ ðt; t 0 Þ ¼ U 0 ðt; t 0 Þ; Z U ðnÞ ðt; t 0 Þ ¼ ðihÞn

2phA im * Y

þ1 X

mSq j

S;q

Rs1 s2

s1 ;s2 ¼1

XZ

~ d~ reiQ s1 s2~r dð~ r ~ Rn ÞjmSq

n

+Z

t

dseiXs1 s2 s ;

t0

where we have taken into account the expansion of the exponent with respect to the Bessel function. ~ Pi , ~ P f and Ei , Ef are the neutron momentum and energy in the initial and final states respectively. After integrating one has þ1 X ~ ~ 2phA X 0 Rs s X1 ðeiXs1 s2 t  eiXs1 s2 t Þ eiQ s1 s2 n m s ;s ¼1 1 2 s1 s2 n 1 2 Y ~~  hmSq jeiQ nn jmSq i:

Mfi ¼ 

ð24Þ

S;q

The statistical average of last term in (24) can be written in the form

8 9 < X = Y h  1 2 ~ ~ ~s s ~ Sq Þ ðQ hmSq jeiQ nn jmSq i ¼ exp  qÞÞ ð þ m e ð~ 1 2 S : ; 2 qÞ 2MN XS ð~ S;q S;~ q  ews1 s2 ; ð25Þ

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which is obtained by making use of the Weil identity [30] and the Sq ¼ ½expð Wick’s theorem [32]. Here m hXS ð~ qÞ=ðkTÞÞ  11 is the average number of phonons. Further for the square of modulus of the matrix element (24) one can find 2

ð2pÞ2 h A2 R¼ m2

X s0 Þt 1 2

e

j Rs1 s2 j e

e

s1 ;s2 ;s01 ;s02

iðXs1 s2 Xs0



2 ws s 1 2

Xs1 s2 Xs01 s02

ws0

s0 1 2

2 X ~s s ~ iQ n e 12 n

  iX 0 0 Dt iðXs1 s2 Xs0 s0 ÞDt 1 2 1  eiXs1 s2 Dt  e s1 s2 þ e :

ð26Þ

Due to the large number of N the square of the modulus of the exponent in (26) can be replaced by the d-function in the following way:

2 X  ð2pÞ3 N  ~ ~s s ~ n iQ 1 2 d Q s1 s2 þ ~ e g ; ¼ n V

ð27Þ

b1 þ g 2~ b2 þ g 3~ b3 is where V is the volume located by a crystal, ~ g ¼ g 1~ the reciprocal lattice vector, g i are integers, ~ bi are reciprocal lattice elementary vectors. The case ~ g ¼ 0 is excluded since we consider neutrons scattering, i.e. the deviation of an initial neutron from its initial direction is obligatory. The exponent exp½iðXs1 s2  Xs01 s02 ÞDt in (26) is strongly oscillating function and because of finite duration of real processes its maximal value is obtained under the natural condition s1 ¼ s01 , s2 ¼ s02 . Thus the derivative over Dt in (18) gives

2 lim

  sin Xs1 s2 Dt

Dt!1

Xs1 s2

¼ 2pdðXs1 s2 Þ:

ð28Þ

Substituting (26)–(28) in (18) finally we obtain the following form for the neutron scattering probability:

W ~Pf ;~Pi ¼

2 þ1 ð2pÞ6 h A2 N 2w X ~s s þ ~ Rys1 s2 Rs1 s2 dðQ g ÞdðXs1 s2 Þ: e 1 2 2 m V s ;s ¼1 1 2

ð29Þ Generally S-matrix theory allows to consider three-particle interactions in condensed media which are of a great interest. Besides neutron–photon–phonon [12] and neutron–phonon–phonon [11,13] interactions electron–photon–phonon interaction in polar semiconductors is examined as well [33,34].

bye–Waller factor obtains its maximal value. In contrary to this it can lead to the damping of elastic coherent scattering for all scattering angles h–0 depending on the temperature and crystal properties. The Debye–Waller factor decreases with a decrease of nucleus mass and increase of scattering angle, neutron energy and crystal temperature. Such kind of negative and undesirable consequences are inevitable. The main difference between the sound and electromagnetic waves’ applications is that in case of sound waves the Debye–Waller factor can be changed and tuned and for some appropriate choice of sound parameters it obtains its maximal value even in case of light nuclei matter and even in case when T–0 [11,13]. Note, the negative influence of nuclei thermal motion in the process of neutron–photon–phonon interaction can vanish as well when the photon–phonon interaction is taken into account. Now we turn to the examination of energy and momentum conservation laws. d-functions in (29) give

Ei  Ef þ ðs1  s2 Þhx ¼ 0; ~ Pf þ ðs1 þ s2 Þh~ k þ h~ g ¼ 0: Pi  ~

~ Pf þ 2sh~ k þ h~ g ¼ 0; Pi  ~

2

Summarizing, the obtained expression (29) for neutron scattering probability demonstrates the possibility of both thermal and high-energy neutron diffraction in a crystal under the influence of a standing laser wave. The d-functions allow to obtain energy– momentum conservation laws which in their turn establish diffraction conditions both for elastic and inelastic scattering with respect to a field. The term Rys1 s2 Rs1 s2 is the amplitude of the scattering probability including information about neutron polarization. A discussion of polarization effects is the aim of our forthcoming papers. It is well known that the thermal motion of nuclei forming a crystal negatively impacts diffraction intensity, and as a result it decreases [30,35]. The term

e

8 9

= < Xh  ð~ g~ eS ð~ qÞÞ2 1 Sq þm ¼ exp  : ; MNXS ð~ qÞ 2 S;~ q

ð32Þ

Due to the choice of (14) the phonon part is absent in the expression (31). Positive values of s1 and s2 correspond to the process of photons absorption from the laser wave and the negative values correspond to photons emission. Being resonant conditions the conservation laws allow to examine neutrons both elastic and inelastic diffraction with respect to the field of standing laser wave. It become obvious from energy conservation law that in case when s1 ¼ s2  s the photons energy is not changed, Eiðs1 ¼s2 Þ ¼ Ef ðs1 ¼s2 Þ . This is very similar to Kapitza– Dirac effect [36–38], exactly the same number of photons participate in the absorption and emission processes. The similar situation arises in [13], where in case of neutrons diffraction in a crystal under the influence of a standing sound wave the Kapitza–Dirac effect is considered. A standing laser wave can be considered as a set of photons with the momenta h ~ k and  h~ k, and the neutron scattering occurs under the process of an absorption of the photon with the momentum h ~ k and the stimulated emission of the photon with the momentum  h~ k. As a result, the neutron momentum increases by 2s h~ k in addition to the part  h~ g , and is directed along the h angle without changing its value (h is the angle between the initial and final directions of a neutron). In case of elastic diffraction one has

which can be written in the form

4. Diffraction condition

2w

ð31Þ

ð30Þ

is the well-known Debye–Waller factor, a measure of the influence of thermal motion on observing violation of lattice periodicity. In case of motionless lattice (when the temperature T ! 0) the De-

2

4d sin

h 1 2 2 2 2 1 2 ¼ s k kn d þ k2n þ 2 s~ k~ g k2n d ; 2 p2 p

ð33Þ

where kn is the neutron wavelength, d ¼ 2p=j~ g j is the interatomic plane distance. On the other hand the relation (33) can be written in the form

2deff sinðh=2Þ ¼ kn ;

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 where deff  d= 1 þ s2 k d =p2 þ s~ k~ g d =p2 is an effective period of a certain ‘‘sublattice” which is formed in the process of neutrons’ interaction with the crystal and the standing laser wave. Finally let us consider the neutron inelastic diffraction. For this case from the conservation laws we obtain

" 1 mðs1  s2 Þxk2i cos h ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 1 þ 4p2 h mðs s Þxk 1 þ 12p22h i #  k2  2  i 2 ðs1 þ s2 Þ2 k þ g 2 þ 2ðs1 þ s2 Þ~ k~ g ; 8p

ð34Þ

pffiffiffiffiffiffiffiffiffiffiffi where ki ¼ 2p h=Pi ¼ 2ph  = 2mEi is the de Broglie wavelength of the initial neutron. Note, the traditional Bragg diffraction condition is

K.K. Grigoryan et al. / Nuclear Instruments and Methods in Physics Research B 268 (2010) 2539–2543

obtained from (33) and (34) when the laser field is turned off, 2d sinðh=2Þ ¼ kn . Relations (33) and (34) allow us to theoretically predict values of certain physical quantities measured in the process of neutrons diffraction. They can be used both for heavy and light nuclei matter for all scattering angles h–0. Particularly, for the laser frequency 1015 Hz, laser wavelength 105–104 cm and lattice period 108 cm we have the following numerical estimations for photon numbers and scattering angle. For thermal neutrons we have s1 þ s2 6 103 , s1  s2  101 , and for high-energy neutrons we find s1 þ s2 6 104 , s1  s2  10 and s1 þ s2 6 105 , s1  s2  103 for 109 cm and 1010 cm wavelengths respectively. The optimal interval of h is 20–60° depending on the wavelength of neutrons. These estimations show that the diffraction conditions (33) and (34) can be proved in experiments. To our knowledge in the scientific literature there are no papers devoted to the experimental investigation of high-energy neutrons diffraction in crystals. We hope that such works will appear in near future. Note, that only a few papers are known devoted to high-energy particle diffraction in solids, particularly in [39–41] reflection high-energy positron diffraction from Si(1 1 1) surface is considered with 20 keV positron beam. References [1] W. Marshall, S.W. Lovesey, Theory of Thermal Neutron Scattering, University Press, Oxford, 1971. [2] G.E. Bacon, Neutron Diffraction, Clarendon Press, Oxford, 1975. [3] I.I. Gurevich, L.V. Tarasov, Fizika Neytronov Nizkikh Energij (Low Energy Neutron Physics), Nauka, Moscow, 1965. [4] Yu.Z. Nozik, R.P. Ozerov, K. Hennig, Neytroni i Tverdoe Telo: Strukturnaya Neytronografiya, tom 1 (Neutrons and Solids: Structural Neutron Diffractometry, vol. 1); Neytroni i Tverdoe Telo: Neytronografiya Magnetikov, tom 1 (Neutrons and Solids: Neutron Diffractometry of Magnetics, vol. 2); Neytroni i Tverdoe Telo: Neytronnaya Spektroskopiya, tom 3 (Neutrons and Solids: Neutron Spectroscopy, vol. 3), Atomizdat, Moscow, 1979. [5] V.K. Ignatovich, Neytronnaya Optika (Neutron Optics), Fizmatlit, Moscow, 2006. [6] R. Michalec, P. Mikula, M. Vrana, J. Kulda, B. Chalupa, L. Sedlakova, Physica B + C 151 (1988) 113–121. [7] J. Kulda, M. Vrana, P. Mikula, Physica B + C 151 (1988) 122–129.

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