CC BY
17Planar X-ray Waveguide-Resonator and Some Aspects of its Practical Application
Egorov V.K. ( [email protected] )(1), Egorov E.V. (1), T.V. Bil'chik (2)
(1) IPMT RAS, Chernogolovka, Moscow District, 142432 Russia (2) Moscow Engineering an Physical Institute, Moscow, 115409 Russia
ABSTRACT
Features of planar X-ray waveguide-resonator (PXWR) functioned on base of an X-ray total reflection phenomenon and distinguished by an air slit presence as the transportation spaces are discussed. It is shown that the slit size interval characterized by a resonance manner of an X-ray beam guiding across the waveguide exists. The high efficiency of X-ray beam transportation systems based on planar X-ray waveguide-resonators allows these devices to complete in guiding abilities with polycapillary optics. Characteristics of an X-ray diffractometer and TXRF-spectrometer equipped with PXWR are analyzed. The experimental data been carried out at a thin film testing by applicating the diffractometer and the TXRF spectrometer built with using PXWR are presented. The idea of creation the planar X-ray resonance waveguide-monochromator is considered, in passing.
Key words: Total X-ray reflection; planar X-ray waveguide-resonator (PXWR); X-ray standing wave (XSW); diffractometry; TXRF spectrometry; resonance waveguide-monochromator (RWM).
INTRODUCTION
Problems of an X-ray radiation efficient focusing and a transportation of X-ray beams on long distances without attenuation are valid for the optics of an X-ray radiation range, as before. It is generally agreed that a polycapillary optics based on a phenomenon of an X-ray beam multiple total reflection on an inner surface of the hollow glass capillary is one of the most attractive line been promoted to attain these ends [1]. X-ray capillary fibers contain, usually, hundreds of individual channels with inner diameter between 5 and 50 ^m. Its real efficiency for X-ray CuKa is near 4^15% [2].
At the same time, alternative avenues of an X-ray beam transportation and an enhancement of a beam radiation density are existed. One such line is a building of a planar X-ray waveguide (PXW). Planar X-ray waveguide been created firstly [3] was a combination of a light material film been transparent core between two reflecting layers. Waveguides with a material core was studied by many authors [4-10]. They show that the devices are distinguished by interesting properties [5] but are typical of significant absorption losses. Therefore, PXW with a material transporting core can not complete with polycapillary optics. However, if the transportation layer
* E-mail: [email protected]
will be carried out as an air slit, the attenuation of an X-ray beam carried forward by the slit can be brought to minimum.
The first planar X-ray waveguide with an air space as a guiding medium, in form of a slitless X-ray collimator, was elaborated by Mingazin group [11,12]. The slitless X-ray collimator consists from two quartzes polished plate's intimate mated together. It was shown that such collimators are beautiful X-ray radiation conductors, and can be used for an X-ray beams creation in form of thin strokes.
Further development of the branch was defined by scientific activities two science groups from IPMT RAS [13-20] and Van der Waals-Zeeman Institute [21-24]. In practical relation, this branch is more productive. They developed the mechanism of an X-ray radiation compression and suggested practical application fields of this advanced X-ray device. The present work is devoted to a physics discussion of the planar X-ray waveguide-resonator (PXWR) and its practical use in diffraction and TXRF investigations. The way of a compression and transportation of X-ray beams by means of PXWR can be consided as a real alternative to a polycapillary optics in an X-ray practice.
X-RAY STANDING WAVE FORMATION UPON TOTAL REFLECTION OF A PLANE WAVE ON DIELECTRIC PLANE
Let be an electromagnetic monochromatic plane traveling wave with the a-polarization (i.e. vector of electric field E0 is perpendicular to the x-z plane in Fig. 1a), wavelength k0 and wave vector k0=1/k0 impinge on the boundary separating two materials. If the materials have different refraction indices, part of the wave energy is reflected and the remainder passes to the second material or is refracted. An interference field appears in the first material irrespective of the remainder value. The interference field area value depends on the width of an incident plane wave and the incidence angle 9. The interference intensity in the area is directly determinated by the reflection factor on a boundary between materials and peaks out at the total reflection of an incident radiation beam. Referring to Fig. 1a, the incident and reflected traveling E-field plane waves can be described as [25]:
A 0 (r;t )= E 0ei [rat - 2n( k*x - (1)
and
A R (r;t ) = E Rei [rat - 2n( k*x+^ (2)
where ra is the angular velocity of the wave, ER is the electric field in a reflected wave. For the sake of convenience, let z=0 corresponds to the reflector surface.
By locating the intersections of the crests and troughs for the two traveling plane waves in Fig. 1a, one can easily show that the interference between the two coherent waves generates an X-ray standing wave (XSW) with planes of maximum and minimum intensity parallel to the boundary surface. The period of the standing wave D is defined by expression [26]:
k
D (3)
2sin 9 v 7
In a simplest case amplitude relation between electric fields of the incident, reflected and refracted (Ej) waves are described by the Fresnel equations [25]:
sin 9 - n • sin 9 - ±
E ^ =
R
sin 9 + n • sin 9 2 sin 9
(4)
sin 9 + n • sin 9
where 9 is the refraction angle and n is the relative refraction index. Equations (3) and (4) describe the reflection phenomenon for any electromagnetic plane wave on a plane interface
between two materials. They can also be used to describe in a first approximation a total external reflection of an X-ray beam limited in the transverse direction on a vacuum - material interface [27,28]. For this conditions the phase change for the electric vector E^ in reflected
wave relative to incident one y is defined by the expression:
Vcos2 9-n2
(5)
♦ y
tan — = ■
2
sin 9
Figure 1. Classical scheme of the standing wave field formation at a total reflection of a plane monochromatic wave on the dielectric surface (a). Angle of a phase shift at a total reflection T as a function of the incident angle for CuKa beam on SiO2 without (1) and with (2) taking into account on absorption factor (b). Electromagnetic field intensity in reflectors surface layers with (2) and without (1) taking into account the modulation by a standing wave interference field (c).
It is necessary to notice that the total reflection of an X-ray beam on a dielectric plane is no equivalent to a specular reflection of a plane wave. The real total reflection phenomenon is characterized by a wave penetration into a reflecting plane and an appreciable attenuation. The parameter of the X-ray penetration depth is equal to distance reducing the X-ray wave intensity in "e" times and is defined by the expression:
(Az)2 =
A2
1
8n
-92)2 + 4p2 +(92 -92)
(6)
where 9c is the critical angle of a total reflection, P is the imaginary part of the refractive index n equal to unity for vacuum and less than unity for most materials. It can be written as [29]:
n = 1 -5-ip (7)
where 5 is the real part of the refraction index deviation from unity and reflects the material polarization degree upon X-ray excitation. The imaginary part P characteristics the degree of radiation attenuation in the material. Values of 5 and P for various media are listed in [30] and do not exceed 1 • 10-5. The polarization factor magnitude is directly connected with the critical angle of total X-ray beam reflection 9c and can be expressed by the wavelength of incident radiation k0 and common material parameters [29]:
92
S = — =
2
e2NZ'pA2
2 (8) 2nmc2 A
where e and m are the charge and the mass of an electron, c is the light velocity, N is the Avogadro number, Z' and A are the effective charge and atomic mass for the reflector material,
and p is its density. The attenuation factor can be expressed across the linear coefficient of material absorption ^ [29]:
k
p=^ (9)
The phenomenon of an X-ray wave total reflection is accompanied by the Goos-Hanchen wave front displacement of the beam reflection position about the point of the beam incoming place [31,32]:
k 1
Ax =--,. , - (10)
n
7(2 -e2)+2ip
Minimum and maximum displacements are arrived at e=0 and e=ec, correspondingly:
k
Ax„
A
nec' k i
(ii)
xmax —
For the nonabsorption medium, the displacement is tended to infinity, and the lateral wave appears [33]. When P^0, the displacement transforms into a nonstationary parameter.
In the conjunction of the secular reflection for an X-ray radiation amplitude relation between electric vectors in the incident, reflected and refracted waves can be written in the forms [26,29]1:
E1 =
R
e-Ve2 - 25-2ip
e+Ve
E1 (z = o) =
2 -25-2ip
2e
(12)
e+Ve1
- 25- 2ip
In this case, reflectivity and formal transmissivity factors can be represented by [29,34]:
R1 =
Ti =
yT(z)
(e-a)2 + b2 (e+a)2 + b2 ;
dz
(13)
and phase shift for the electric vector in the reflected wave relative to the incident one y will be represented by following equations:
2eb „ _
tgy=^—2—rr for e<er
e2 - a2 -tgy= o for e>e(
where a and b are determined by formulae [29]:
(14)
2
0
1
2
2
0
0
1
description of a total X-ray wave reflection demands to define more precisely those expressions.
a2 =
b2 =-
e2 - 25) + 4ß2 +(e2 - 25) ^(e2 - 25)2+4ß2 -(e2 - 25)
The function y=f(e) in shown on Fig. 1b for cases ß=0 and ß^ü.
It is important to notice that the transmissivity factor reflected, in reality, reservation and absorption energy effects is determinated by an integral expression. Such form of the expression is connected with the infinite propagation of an electrical field in the reflector volume. The common expression for the field in a reflector material has the form [35]:
2e ^
-e X
Et (z,x,t) =
Ha+ib)
e + a + ib
1i(cot - 2nkxx)El
(16)
The term beyond brackets describes the wave motion along the x axis direction. The term in brackets defines the field expansion along z-axis. The expression (16) has a complex form. At the absence of the absorption (P=0 and b=0, too) the energy supplied by a reflector volume at the initial moment of an incident X-ray beam interaction with the reflector will leave the volume fully after the beam shutting-off. The absorption presence leads to an irrevocable loss's appearance at the beam-reflector interaction. The value of irrevocable losses is defined by the square of the module of the term in brackets in the expression (16):
ET
4e2
4nzb
1
(e + a )2 + b2
(17)
The diagram of this function is shown in Fig. 1c. The total value of losses T is described by the expression:
4q2 <x> _ 4nzb
T = (Q \2 u2 J e" * dz (18)
(0+ a)2 + b2 0
As mentioned above, the reflected wave appearance excites the interference field in vacuum. Nevertheless, the interference field will appear in the reflector volume near the plane interface in form of a standing wave, too. The standing wave period induced in a reflector volume D will be differed from the value defined by expression (3) owing to a refractive index difference. A standing wave field appearance in a near interface reflector area leads to an irrevocable loss decreasing in owing to the periodical modulation appearance in an electric field distribution. The modified expression for the total value of losses T can be represented as2:
T* =
4e2
(e+a)2 + b2
cos I y
2nz
4e2
(e+a )2 + b2
cos I y
D
4nez X
4nzb
X dz:
4nzb
X dz
(19)
X
*
2Expression (19) is valid for the infinite beam size. In this case, the interference field area in a reflector volume expands longwide of z-axis boundlessly. Narrow beams will be characterized by the finite interference field area in the reflector volume and will demand the fixed value for a top integration limit.
V
■ : \ I 3
where y is the phase shift angle been calculated by expression (14). The function of the integration element in (19) is presented as a diagram in Fig. 1c. The magnitude of T is smaller as the value T. By this means we can state with accuracy that including into account an interference field in a reflector volume leads to an increase of the reflection factor in case of an absorbing reflector. The expected magnitude for the reflection factor will be described by expression:
R * = R + (T - T*) (20)
Pointed correction is not great for a single reflection but it shows a noticeable win in the case of a consecutive multiple X-ray beam - reflector interaction. Fig. 2 shows curves of multiple reflections, as an illustration when the standing wave inside reflector materials is taken or not into consideration.
Because of the interference field is presence in the vacuum over the interface, the intensity of a reflected beam in the standing wave area usually
defined as |Ao + Ar can be represented by next formula [26]:
R*i
___i____
_ ±___L__
J-
Angle reflection 9/e0
Figure 2. Curves of X-ray beam reflection factors for multiple reflections with (R*±) and without (Ri) taking into account the interference field in the reflector volume. CuKa radiation reflection on SiO2 (P*0). Indexes 1, 2, 3, 4 correspond to 1, 10, 100 and 1000 consecutive reflections.
i(e,z)=|e
= |2
1 + R* +
2Vr
cos y +
2nz
D
(21)
where D is the standing wave period expressed by (3).
The interference field can be observed both at 0<0c and 0>0c, but in the latter angle range its intensity decreases abruptly [30]. The standing wave period D achieves its minimum at 0=0c in the O<0<0c range. As the incident angle decreases, the period D increases to become infinitely large at the grazing incident angle (0=0). However, this is not the case in practice, because the coherence of incidence and reflected beams is broken. The most obvious factor causing the interference field erosion is the width finiteness of incident radiation lines AX [25]. It is generally accepted, that the interference field does not become smeared, if the condition [29]:
X
(22)
AX < — 4
holds.
STANDING WAVE FORMATION IN AN AIR GAP OF AN X-RAY PLANAR
WAVEGUIDE
Consecutive reflections of a plane electromagnetic wave in the slit formed by two parallel plate's results in the formation of several interference field areas in it (Fig. 3a). Varying the slit size can lead to overlapping the areas to create the uniform interference field zone within the total clearance of the slit (Fig. 3b)3. Therefore, it can be expected that an XSW excitation can be
3It is very important to notice that the uniform interference field zone will be appear, in the context of the specular reflection model, for some specific reflection angles, only. The uniform interference field zone appearance for any
formed in the plane slit, when its width falls within a certain range. The minimum slit size for this can be evaluated from expression (3) for the critical angle of total reflection in condition of the one standard wave period allocation in a clearance size:
X X nmc2A smin = Dmin = 2 sin0c * 207 V 2e2NZ'p (23)
The minimum slit size promoting the XSW formation is independent of an X-ray incident radiation wavelength. The material structure density p of reflectors is a real factor influencing the minimum size smin. Its magnitude for quartz reflectors (Z=10; A=20) is 21 nm. In practice the s min value is smaller because an XSW is characterized by a visible intensity up to the X-ray penetration depth.
The average value of the penetration depth parameter for an X-ray slitless collimator
introduced in the work [14] as Az = Azjj-^j is 3.6 nm for MoKa radiation impinging on a quartz
plate. So, the real size of the minimum clearance between the quartz plates' s*min is found to be approximately 14 nm. This value is comparable with size of the double roughness of optical polished reflectors 10 nm.
a The upper restriction for the slit in an X-ray
waveguide can be evaluated using ratio given by the Eqn.(22) and proportional relation:
X X2
= D„„x =
fAX^ 4AX
ÍAX
(24)
For MoKa radiation, the wavelength is X=0.707-10-1 nm and AX=0.29-10-4 nm [36]. Substitution these value into Eqn.(24) gives smax=434 nm. In practice, its magnitude is s max=36 nm. Similar value for CuKa radiation is equal to s max=91 nm. It should be stressed that the X-ray spectra of the incident and emergent beams can not tally between them. For example, if we shall try to form the beam excited Figure 3. Scheme of a standing wave field by X-ray tube with Mo anode using an X-ray waveguide
formation at multiple consecutive reflections of a plane monochromatic wave (a), the principle scheme explaining the formation of the standing wave uniform zone upon trapping monochromatic plane wave radiation by a planar extensive waveguide-resonator (b). Az - penetration depth of a radiation, Ax -Goos-Hanchen front wave displacement.
with the slit size s>smax, we find a small characteristic deposit in the spectrum of an emergent beam. But if the size does not excel the value smax (smin<s<smax), the characteristic radiation MoKa will be present in the emergent beam X-ray spectrum with a great intensity. It can be expected that the slit width varying in a waveguide would cause the X-ray spectrum modification in the emergent beam at transportation of the white X-ray radiation.
It is vital to notice that the expression (24) defines in addition to the upper value of a slit width admitted the waveguide-resonance penetration of an X-ray beams the specific angle restriction. It introduces the minimum incident angle 9min in the next form:
reflection angle can be obtained by taking into consideration the reflector interface erosion and the Goos-Hanchen
displacement.
4The important practical parameter influencing on the upper slit size is a degree of the plate parallelity in an X-ray waveguide-resonator.
2AA
9min =— (25)
These relations denote that the resonance transportation mechanism can be realized for the narrow extent slit in the slit width interval smin<s<smax and for the incident angles area 9min<9<9c
for the every reflector of the waveguide. At the using of an X-ray tube with a spot width F (near 0.2 mm for our tube) we get that the radiation penetrates by the resonance manner in the waveguide from edges of the spot, only. The radiation emitted by the central spot area of the tube can pass across the slit in form of the direct beam, only. 9min for MoKa radiation is equal 0.046° and for CuKa one - 0.043°. This parameter does not depend on the material density of reflectors. By this mean, we see that the part of the radiation emitted by tube can be transported by the resonance manner in PXWR, only.
The phenomenon of the interference field area expansion on the total slit space is characterized by an additional effect. Interference field areas excited in reflector volumes at multiple successive total reflection (Fig. 3a) are distinguished by a small size, because the width of an X-ray beam captured by the slit is restricted. In consequence, the magnitudes R and R defined by equations (13) and (20) correspondingly will be slightly differed. By contrast, the standing wave expansion on the total slit space leads to an enhancement of the interference field areas up to infinity, and the difference between R and Rf becomes to be evident. This fact is one of reason, why PXWR is characterized by an advance efficiency in comparison with a polycapillary optics been peculiar the standing wave conception. But it is necessary to point that the X-ray emergent beam for a circular capillar must be calculated on base of expression (20) because small interference field areas are always formed in the capillar walls and the difference between R and Rf can lead to a significant discrepancy for experimental and calculated intensity magnitudes.
ATTENUATION OF AN X-RAY BEAM RADIATION IN AIR GAP OF A PLANAR WAVEGUIDE-RESONATOR
The excitation of a standing wave in a waveguide-resonator slit gives rise to a stationary distribution of the interference field intensity both along the slit channel (along axis x) and crosswise (along axis z). This distribution is pictured in Fig. 4 to fit the input of a PXWR (x=0). The distribution is plotted for a plane X-ray beam (CuKa) impinging into a slit of the quartz waveguide-resonator under the angle 9=0.92-9c on the reflector surface. The reflection conditions correspond to the value of the phase variation y=45° and the standing wavelength D«1.1-Dmin. The X-ray incident wave is characterized by the penetration depth Az=8.6 nm. Hence, the distribution shown in Fig. 4 is consistent with the expression Az^0.4-Dmin.
The standing wave within the slit is characterized by the intensity distribution described by expression (21). Outside the slit, the distribution of an X-ray beam intensity will be defined by the product:
J (9,z) = I(9,z)e-' ^ (26)
where I(9,z) corresponds to the expression (21) and the penetration depth Az submits to the expression (6). The total integral intensity of an X-ray beam in the inlet slit area (x=0) can be represented in form of two terms:
W(9,x = 0) = L(9,x = 0)+ M(9,x = 0) (27)
where L(0,x=0) will characterize the total integral intensity portion to concentrate in the slit space, and M(0,x=O) is connected with the portion percolating into waveguide reflector's volumes. The functions L(0,x=O) and M(0,x=O) can be represented by the expressions:
+X
L(0,x = 0)= J I(0,z)dz;
PXWR gap
(28)
'Az
dz
M(0,x = 0) = 2 J I(9,z)<
X
where s is the slit width. At the X-ray moving in the slit along x-axis its total integral intensity W(0,x) together with its components L(0,x) and M(0,x) will demagnify. But the ratios between the components and between every component and the total integral intensity W(0,x) must be saved at any x-magnitude because the standing wave interference field produces the equality of boundary conditions between the slit space and the reflector volumes along the whole slit. Consequently, we can write: L(0,x = 0) L(0,x)
a =
W(0 ,x = 0) W(0 ,x
= const
-s/2 0 s/2
Crosswise ordinate z (tim)
Figure 4. Intensity distribution function for an XSW in a waveguide-resonator slit and top layers of quartz reflectors for an X-ray beam impinging on the slit under a certain angle of a total reflection 0 for quartz reflectors. The function without absorption upon total reflection is shown by a dashed line. The function reflects the picture for ^=0.1541 nm (CuKa); 0=0.92-0c; s=97 nm.
(29)
If the total integral intensity of an X-ray beam for the slit inlet position (x=0) is represented by expression (27), the intensity for the distance x=Sx from the inlet will be described as:
W(0, x = Sx) = L(0, x = Sx) + M(0, x = Sx) X-ray beam does not suffer an attenuation in the slit space. The X-ray radiation undergoes absorption and scattering inside the reflector's volumes, only. Hence, the common beam attenuation can be represented in the form:
AW(0, x = Sx) = W(0,x = 0) - W(0, x = Sx) =
= M(0,x = 0) - M(0,x = Sx) = M(0,x = 0)(1 - e-^Sx) (30)
Dividing the left and right terms in expression (30) on W(0,x=0) and directing Sx^0, we can get a simple differential equation:
dW M(0,x = 0)
"W=-W(^rdx=-0^dx (31)
The integration of the equation with the initial condition Wtotal=W(0,x=0) leads to the inference of the main waveguide-resonator equation:
W(x) = W0e "a^x (32)
The magnitude of a depends on the wavelength of incident radiation, the reflector material properties, the angle of radiation impinging and the width of the waveguide slit. The dependence of a on the incident angle can be evaluated by calculating its values for some incident angles of CuKa radiation reflected on one-quartz waveguide-reflectors. This gives the values a(0c)=0.8 and
a(9c/2)=0.05. Using formula (32) with the values of a for the incident angle 9=0.929c and the condition sslit~4D, we can calculate the total intensity attenuation for an X-ray beam of CuKa radiation in the planar quartz extensible waveguide-resonator. The total intensities corresponding to an X-ray beam after passing the way ¿=10 mm in the waveguide are W(9c)=0.3W0 and
W(9c/2)=0.6W0. It is interesting to compare these values with the magnitudes characterized for the model of a multiple total X-ray reflection. Such calculations require the knowledge of the total X-ray beam reflection quantity N in the long slit [1] that can be evaluated on base of expression:
£
N = ----(33)
Ax + s • ctan 9
where I is the waveguide length, Ax is Goos-Hanchen shift (10) and s is the size of the clearance. For the quartz reflectors length ¿=10 mm, the clearance size s=97 nm and CuKa radiation we can
get N(9c)«200, N(9c/2)=100. Using of this magnitudes we can calculate the real intensity for the model of the multiple X-ray total reflection: W(9c)=0.008W0; W(9c/2)^0.1W0. At the sizes characterized for a slitless collimator, the difference between the values calculated for those models is higher.
PLANAR X-RAY RESONANCE WAVEGUIDE-MONOCHROMATOR
Preceding sections are devoted to discussion of the planar X-ray waveguide-resonator using the total X-ray reflection phenomenon for the standing wave generation in the slit space of the
device. However, it is well known that the excitation of an X-ray standing wave is possible for the Bragg geometry, too [37]. The X-ray standing wave can be excited in the narrow slit between two parallel polished reflectors, if the reflectors are perfect monocrystals been orientated mutually. In this case, we shall get a new device: resonance waveguide-monochromator (RWM). The mechanism of the standing wave formation in Bragg reflection conditions is in somewhat differed from one in the case of a total X-ray reflection (Fig. 5), and the standing wave period will be equal to the lattice plane distance dcryst satisfied to Bragg conditions. A formal consideration of a standing wave formation in Bragg planar X-ray waveguide (resonance waveguide-monochromator) requests the replacement of the X-ray depth penetration parameter Az by the parameter of the primary extinction length zext [38]:
1
Figure 5. Principal scheme of a standing wave picture formation in the planar slit of a resonance waveguide-monochromator. DSW (standing wave length) is equal dcryst. Intensity of an emergent beam is equal one for an incident beam (Ii=AiAXi=Ie=Ae8Xe).
sin 9b
mc v
zext _ _ _ '
2X| c
e2 Fh
(34)
where a is the extinction factor, c is the polarization factor been equal to unity for a a-polarization, c is the light velocity, m and e are the electron characteristics, v is the unit cell volume for the reflector material, Fh is the relative structure factor of the chosen reflection. The zext magnitude defines the crystal thickness attenuated the X-ray beam intensity falling on the crystal under Bragg angle with "e" factor. It is significantly that the primary extinction length
does not depend on the wavelength of the X-ray radiation. The magnitudes of the primary extinction length are usually two orders higher as values of the depth penetration at an X-ray total reflection.
For example, the magnitude of zext for (200) NaCl reflection is equal to 660 nm [38]. Because of this, the practical size of a resonance waveguide-monochromator slit may be visibly differed from one calculated by the procedure adopted for a planar X-ray waveguide-resonator, and its efficiency must be lower as compared to a PXWR built on the total reflection phenomenon. Nevertheless, the X-ray radiation density in its emergent beam will be significantly higher as one for the conventional Bonse-Hart monochromator [39,40]. The planar X-ray resonance waveguide-monochromator function is conceptually identical with the Borrmann effect manifested to perfect crystals [41,42]. The minimum clearance size for the Bragg reflection case can be evaluated on base of the expression:
Dmm = dcryst (35)
2 sin 0b
b
where 0b is the Bragg reflection angle. Intensities of incident and emergent X-ray beams will be approximately equal, but the energy width of the line will be different. The line narrowing will be accompanied with the corresponding enhancement of lines amplitude (Fig. 5):
Aj AA j = Ae SA e (36)
where Ai and Ae are the amplitudes of the incident and emergent X-ray beams, AAI and SAe are FWHM of radiation on the inlet and outlet for the resonance waveguide-monochromator.
EXPERIMENTAL STUDY OF PXWR FEATURES
The energy spectrum of the beam formed by PXWR is of prime interest for both practical purposes and an understanding of an X-ray waveguide-resonator physics. The spectrum of an energy distribution was analyzed by using the modified diffractometer HZG-4 and a perfect monocrystal SiO2 (101) as a disperger in the angle interval 2°<0<20°. Figure 6 shows diffractometer pattern of an X-ray energy dispersion for the beam formed by a planar X-ray waveguide-resonator. The geometry of measurements is sketched on an insertion. Since the angle interval chosen contains a single reflection set of the crystal disperger (101), this area is suitable for dispersion analysis of the energy spectrum for the emergent waveguide-resonator beam. These measurements were carried out at the open discriminator window. The vertical axis represents reflection intensity. It is particularly remarkable that the real intensity for the peak (101) CuKa in the chosen geometry was over 100 000 counts/s. The horizontal axis represents the dispersion angle scale together with scales for the wavelength and the energy for the radiation analyzed. Diffractometer pattern (energy diagram) shows that the intensity of "white" X-ray radiation in the spectrum of an emergent waveguide-resonator beam is not high, although one can notice some intensity increase in the high edge of the energy area.
Data offered in Fig. 6 provided indirect proof for the standing wave concept showing the ability of the waveguide-resonator to transport X-ray radiation without losses. Authors of work [18] choose the PXWR slit width s=88 nm for calculations about transport of the characteristic radiation CuKa,p. The concept of standing wave formation in a PXWR clearance asserts that the slit size s=88 nm assists the standing wave excitation in the X-ray energy region E<9 keV. This limit is due to radiation length coherence. Characteristic and white radiations are characterized by similar values of a length coherence [30]. At the same time there is no step on the intensity, overline in the region of E=9 keV. This states that the standing wave interference field disappears
7011 - ---,I"m"..'
F ^-^
Wav
g 600 - resoi
SiO, crystal riOll
3guide- \ \ \
iator 1 \ CuKp
f 500 ~ jk I 400 - I \
o 301» ~ \
~ 2(10 -J \
0 T^^—I-1-1-1-1-1-1-1-1-1-1-
2 4 6 8 10 12
0.350 0.583 0.815 1.047 1.277 1.505
53.3 26.6 17.75 13.33 I0.6S 8.92
Figure 6. Diffractometer pattern for SiO2 crystal obtained by HZG-4 precise X-ray instrument equ by a planar X-ray waveguide-resonator ^PXWR=10C s=88 nm. Copper anode tube regime U=40 keV, mA. 4=75 mm, 4=60 mm, 4=235 mm, Si=0.1 mm
radiation penetration into top layers of reflector flats explain, in principle, the form modification of the overline in the case of the PXWR.
The density evaluation for an X-ray beam inside a waveguide slit may be carried out by comparing the total intensity of an X-ray beam formed by waveguide-resonator and slit-cut collimation systems. Calculation shows that the total intensity of the beam formed by PXWR is equivalent to one of the beam formed by a conventional system equipped by a single cut-slit with size near Si=0.15 mm. Line width of this beam in the space position been corresponded to the waveguide-resonator inlet place is equal to 0.6 mm. In this manner, one can calculated that the X-ray radiation density in the beam formed by PXWR exceeds one leaked across the cut-slit system by the factor 6800. The compression factor of the beam on the waveguide-resonator inlet is near 1200.
The geometry of our experiment's execution with the PXWR application was not optimal, and the total intensity obtained had not the highest possible value for the chosen X-ray tube regime. Maximum intensity is possible at the special waveguide-resonator positioning about the focal spot of an X-ray tube. Such position will correspond to the waveguide X-ray capture angle A^=29c (A9=0.42° for CuKa and SiO2 reflectors) and the distance between the focal spot and the waveguide inlet ^1=25 mm. The position optimization will lead to the intensity increasing by 20
times in comparison with the geometry chosen. In this case, the beam divergence will increase as well to the value of A^=29c.
There is the special interest in the experimental researches of waveguides with different sizes of the air clearance. These measurements were carried out in geometry been distinguished by an enhancing angle resolution as comparative ones shown on Fig. 7. The distance between the detector slit and outlet of a waveguide was equal 460 mm at the same value of a detector slit. In that case, the angle slit size was smaller as a using goniometer step and the distance between waveguide-resonator outlet and the detector slit was larger. Therefore, the total peak intensity reduced by order of magnitude, approximately, in a >;s;,
comparison with the intensity distribution for the beam formed by PXWR in geometry of Fig. 7 conditions. Figure 8b shows six X-ray intensity distributions belonging to waveguides with next sizes of the clearance: 30, 88, 154, 210, 2000 and 10000 nm. First size value corresponds to a case of the slitless X-ray collimator. The value is the middle distance size between inner surfaces of reflectors and varies on the collimator length from zero up to 60 nm. In common, one may assert that the clearance size in the X-ray waveguide-slitless collimator is nonstationary parameter. However, an experimental intensity of the slitless collimator emergent beam is characterized by a significant magnitude. In according to model of X-ray standing waves, the case of an X-ray slitless collimator presents an intermediate state between an X-ray beam transportation absence and its transportation on the resonant guiding mechanism. This
□
PX WR-in
s—3(>±.10 ii
PXWR-in
■Vi V
A
Figure 8. (a) Experimental geometry for study of the intensity distribution in PXWR emergent beam at various PXWR air gap (clearance) sizes. ^=75 mm, ^PXWR=100
mm, ^2=60 mm, 4=400 mm, S2=0.1 mm. (b) Overlines obtained by experimentally for waveguides with various clearance size. The waveguide-slitless collimator is
characterized by a non-stationary magnitude of a clearance size. function for the waveguide characterized by a clearance size s=88 nm corresponds to the pure resonant transportation of an X-ray radiation. Just the same, the comparison of its intensity
distribution with the slitless case one shows that the total intensities for both cases are approximately equal.
It is interesting to compare a peak FWHM for beam intensity distributions in geometries of Fig. 7 and Fig. 8 after PXWR with slit width 88 nm. The angle FWHM in Fig. 8 is smaller but free propagation of the beam must be characterized by an invariable FWHM. The showing effect is connected with the diffraction phenomenon and will be discussed in other place.
The air clearance size s=154 nm characterized for third testing waveguide is higher as the maximum clearance value promoted for a standing wave arising in a guiding space. In that event, the standing wave does not exist in all clearance space, and the resonance transportation can be not come true by the uniform manner. Waveguides with clearance size s=210 and 2000 nm regard to the intermediate case, too. The double intensity increasing at the clearance size widening in 10 times (from 210 up to 2000 nm) is explained by an essential intensity enhancing of the X-ray direct beam penetrated across clearance. Last overline of an X-ray beam intensity distribution (s=10^m) is corresponding to the pure mechanism of a multiple total reflection. Double central peak is the result of the direct X-ray beam propagation across the clearance. Complex form of the peak is corresponded to the distorted form of an X-ray tube focal spot. Satellites present the peaks connected with the total X-ray reflection on an every reflector. The elementary consideration shows that the total reflection in the clearance leds to an appearing the satellite's quantity up to four.
Experimental data presented is the base for the statement that the resonant manner of an X-ray beam transportation accomplishes by the regime of the nondiscrete mode excitation. According to the theory of the discrete mode excitation [4,21,23] the waveguide-slitless collimator must characterize by the sole mode, and the waveguide with a size clearance s=88 nm must distinguish by existence of three allowing modes. Incident beam presents a total set of incident angles 9<9c but the integral intensities for the waveguide-slitless collimator and the waveguide with s=88 nm are approximately equal. We plane to discuss this question in other place with a detail criticism of an experimental result's interpretation produced in [21,23].
APPLICATION OF PXW FOR DIFFRACTION RESEARCHES
A high total intensity and acceptable divergence of the X-ray beam formed by a planar X-ray waveguide-resonator enables use of this new X-ray optical elements for practical purposes. Phase and structural analysis of thin film coating provide the excellent example to show the practical importance of planar X-ray waveguide-resonators. Figure 9 displays fragment of the diffraction pattern obtained at the study of the epitaxial thin film target Ge0.23Si0.77/Si [400]. The film had the thickness 100 nm. Sample was in a partial strain state after thermoimplantation treatment by Si+ ions with energy E=200 keV [43]. The pattern is presented Kp and Ka1,2 X-ray reflexes [400] for the film and substrate. In spite of a trifle of the waveguide-resonator slit size (s=88 nm), the reflection's intensity is considerable. Measurement radius is not great, but the pattern displays a beautiful angle resolution for the substrate reflex. Poor resolution for the film reflex is explained by a high value of the residual structure strain. The pattern demonstrates wide possibilities of the planar X-ray waveguide-resonator application for the diffraction investigations of monocrystals and thin film epitaxial structures.
Figure 10 shows the diffractometer pattern of the Au(115)/SiO2 (100)/Si target obtained by diffractometer with PXWR, using the average X-ray tube regime (U=30 keV; I=35 mA). The geometry of a measurement is shown on insertion. Peaks presented on the diffractometer pattern correspond to the FCC structure of Au. The relative intensity of the peak (111) is larger than the
one on the standard random diffractometer pattern of Au. Therefore, one can propose the existence of a small texture. A sublayer SiO2 does not show on the pattern; there appears to be an amorphous structure in it. The wide halo presents in an angle area 6-33^35°. Its area belongs to the silicon (400) reflection. The substrate reflection has a minimum deposit because of a special choice for the target position. (If it arranges to the reflection position, the peak intensity will exceed 100 000 count/s.) The diffractometer pattern has of high quality owing to high radiation density in the beam formed by PXWR, although the size of the target diffraction zone did not exceed, on average, 3 mm2. Maximum diffraction intensity and optimal diffractometer pattern quality were achieved at a distance between the target and detector collimator slit 4=160 mm. This value does not correlate with a goniometer radius (R=235 mm) and a distance between the waveguide outlet and target position (4=60 mm). It shows that the problem of a measurement geometry optimization for PXWR application remains unsolved, but it is very interesting that the half from R and 4 sum is approximately equal to 4.
В 5000
Scattering geometry 1, . . 1..........1
1
0 L
60
64
Dctector angle 29°
Figure 9. Diffractometer pattern of the epitaxial thin film target Ge0.23Si0.77 (100 nm)/Si [100] after treatment by Si+ ions. Geometry of measurements (^PXWR=100 mm,
s=88 nm) are shown on insertion. Peaks KB and Ka12 [400]. Copper anode tube regime U=20 keV, I=10 mA. ¿i=75 mm, ¿2=60 mm, ¿3=160 mm, S2=0.1 mm.
Scattering geometry 1. 1„™ 1,
Target
-Л D
Figure 10. Diffractometer patterns of the multilayer structure Au (115 nm)/SiÜ2 (100 nm)/Si [100], obtained with using the HZG-4 diffractometer equipped by PXWR (4xwr=100 mm, s=88 nm). Copper anode
regime U=30 keV, I=35 mA. ¿1=75 mm, ¿2=60 mm,
¿,=160 mm, S2=0.5 mm.
TXRF SPECTROMETRY WITH PXWR
Development of an X-ray slitless collimation technology for TXRF spectrometry [14-17] shows wide perspectives for the PXWR application in TXRF analysis. Incident beam in TXRF spectrometer must have a high X-ray radiation density, low divergence and small thickness of the beam. The planar X-ray waveguide produces just the same beams. Because of the X-ray radiation density on the outlet of PXWR in the diffractometer geometries described above is higher as a conventional one about 3 000 times, it is possible to expect that the detection limit for TXRF spectrometer equipped by PXWR will be lower up to two order in comparison with a standard TXRF instrument at the same intensity of an X-ray source. Figure 11 shows the TXRF spectrum of the thin film target Si077Ge023/Si were the same to study in diffraction experiments (Fig. 9). The geometry of measurements is presented on insertion.
at/cm2; N(Ca)=2.4101 at/cm2; N(Ni)=1.3 1011 at/cm2; N(Zn)=0.61011 at/cm2
Planar
waveguide-resonator Si077Ge023 (100 nm) \Si
1 "„",,"„"
<xxx.
,, " II " ,, " II
Planar X-ray waveguide (¿PXW=70 mm, s=35
nm) transports the white and Mo characteristic radiation from tube anode up to the target studied. The characteristic and soft white radiation propagate by the resonant manner, where as the white radiation with X<XMo is guided by the manner of a multiple total reflection. The distance between the anode spot and the inlet of the waveguide is near the optimal value (¿=40 mm, Aein=2ec=0.21°). The treatment of
this spectrum shown that the contents of chief admixture elements are the next: N(K)=2.71011
11 at/cm2; N(Fe)=3.5-1011 at/cm2; N(Cu)=1.41011 The detection limit for any elements is calculated on base of a partial background value in the energy interval been
uniquely those elements [16]. The magnitudes of
10 2
those parameters are near (2^5)-10 at/cm , and it is good enough for successful analytical using of TXRF with PXWR. This parameter is better as it for this equipped by slitless collimator [15]. It is need to notice optimized.
CONCLUSION
The comparison between consequences of the resonant X-ray radiation transport model and experimental testing data obtained of real quartz planar waveguides shows that the resonance model is true, in common. Moreover, there are the experimental evidences today [44] to believe that the conception of an X-ray standing wave will be beautiful base on the way to create waveguide-resonators with managing properties by means, for example, of the X-ray - acoustic modulation [45].
The model of the standing wave forming discussed in the work is simplest. It ignores longitudinal and transverse shifts of an X-ray beam at its reflection on a plane interface. The model is not taking into account the beam size finitude. Fresnel reflection equations are used in the simplest form but they must be applied in form counted the monotonous variation of a refractive index in the interface area [46] in line with, for example, Sivukhin's presentation [47]:
Secondary photon energy hv (keV)
Figure 11. TXRF spectrum of target Sio.77Geo.23 (100 nm)/Si [100] collected in geometry shown on insertion. The target excites by an X-ray beam irradiated by molybdenum anode tube and transported by a planar X-ray waveguide-resonator (^pxwr=70 mm, s=35 nm). Regime of registration
U=25 keV, I=10 mA, t=300 sec, l=40 mm. Channel step 16 eV/channel.
target obtained by TXRF spectrometer
that the measurement conditions were not
R ,=
sin" (e-w) i4nd
V 7 1 +-Yy
sin
R
:(e + w)
= tan2 (e-w) tan2 (e + w)
cos e
1+
i4nd yx cos2 w - yz cos2 e
2
(37)
cos2 w - cos2 e
cos e
were 0 and 9 are incidence and refraction angles, d is the lattice parameter, yx, yy and yz are the specific lattice constants depended from the refractive index form representation [48].
The problem of an X-ray beam polarization after waveguide-resonator remains to be opened today.
2
The improvement of the model is planned in next publications.
PXWR can have a greatest practical significance as a transporter and concentrator of X-ray radiation beams. It will be used with success in arrangement of conventional X-ray instruments: X-ray diffractometers, TXRF and X-ray fluorescence spectrometers, X-ray microscopes and microfluorescent tomography. The application of a resonance waveguide-monochromator, using an ideas hybridization of the Bonse-Hart monochromator and the X-ray standing wave, attracts peculiar attention. One of possibilities to use this device, for example, is the creation of an X-ray resonant accumulator for a hard electromagnetic radiation. Devices can be used for a neutron beam formation, both.
ACKNOWLEDGEMENTS
The authors thank Dr. A.V. Okhulkov and Dr. E.V. Shulakov for useful discussion of the problems, assistant O.S. Kondratiev for the help in calculations, Dr. A.T. Volkov for the waveguide-resonators preparation.
The work was prepared at a partial financial support of RFFI (project #01-03-33167), of Ital Structure Company (president Mr. F. Pasty) and of plastic deformation research group MISiS (manager Zvonkov A.O.).
REFERENCES
[1] Kumakhov M., Komarov F., Physics Report, Review section of Physics Letters, 1990, 191, #5, 289-352.
[2] Bly P., Gibson D., Laser Focus World, 1996, 3, 26-28.
[3] Spiller E., Segmuller A., Appl. Phys. Letts., 1974, 24, #2, 60-61.
[4] Marcuse D., Theory of Dielectric Waveguides, Academic, San Diego, 1991, 540.
[5] Feng Y. and et. al, Phys. Rev. Let., 1993, 71, #4, 537-540.
[6] Zheludeva S.I., Kovalchuk M.V., Novikova N.N. and et. al, Kristallografiya, 1995, 40, #1, 145-158 (in Russian).
[7] Lagomarsino S. and et. al, Appl. Phys., 1996, 79, #8, 4471-4473.
[8] Di Fonzo S., Jark W., Lagomarsino S., Clanninl C., De Caro L., Cedola A., Muller M., Nature, 2000, 403, 638-640.
[9] Jark W., Cedola A., Di Fonzo S., Fiordelisi M., Lagomarsino S., Kovalenko N.V., Chernov V.A., Appl. Phys. Let., 2001, 78, #9, 1192-1195.
[10] Kovalenko N.V., Mytnichenko S.V., Chernov V.A., Proceeding of Russian National Conference "X-ray Optics-2001", IFM RAS Nizhniy Novgorod, 2001, 291-303 (in Russian).
[11] Mingazin T.A., Zelenov V.I., Lejkin V.I., Instruments and Experimental Techniques, 1981, 24, #1, part 2, 244-247.
[12] Lejkin V., Mingazin T., Zelenov V., Instruments and Experimental Technology, 1984, 27, #6, part 1, 1333-1336.
[13] Zuev A., Shabelnikov L., ElectonnajaPromishlennost, 1990, 5, 69-71 (in Russian).
[14] Egorov V.K., Zuev A.P., Maljukov B.A., Isvestiya VUZov, TsvetnayaMetallurgiya, 1997, 5, 54-69 (in Russian).
[15] Egorov V., Kondratiev O., Zuev A., Egorov E., Advances in X-ray Analysis, 2000, 43, 406417.
[16] Egorov V.K., Zuev A.P., Egorov E.V., Zavodskaja Laboratorija, 2001, 67, #3, 3-11 (in Russian).
[17] Egorov V.K., Zuev A.P., Kondratiev O.S., Egorov E.V., Surface (X-ray, Synch., Neutron Invest.), 2001, 6, 27-36 (in Russian).
[18] Egorov V.K., Egorov E.V., Discovery of a Planar Waveguide for an X-ray Radiation, Los Alamos e-print archive: http://arxiv.org/list/physics.ins-det/recent/#0101059, 2001.
[19] Egorov V., Bil'chik T., Egorov E., Proceeding of 7th Annual International Conference on High Technology in Russian Industry, MGU: Moscow, 2001, 230-235 (in Russian).
[20] Egorov V.K., Egorov E.V., Thin Solid Films, 2001, 398-399, 405-412.
[21] Zwanenburg M.J., Peters J.F., Bongaerts J.H.H., De Vries S.A., Abernathy D.L., van der Veen J.F., Phys.Rev.Letts., 1999, 82, 1696-1699.
[22] van der Veen J.F., Zwanenburg M.J., Huisman W.J., Synchrotron Radiation News, 1999, 12, 49-54.
[23] Zwanenburg M.J., van der Veen J.F., Ficke H.G., Neerings H., Rev.of Sci.Instr., 2000, 71, 1723-1732.
[24] Bongaerts J.H.H., Zwanenburg M.J., Zontone F., van der Veen J.F., J.Appl.Phys, 2001, 90, 94-100.
[25] Born M., Wolf E., Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, Pergamon Press: New York, 6 ed, 1993, 808.
[26] Bedzyk M., Bommarito G., Schildkraut J., Phys. Rev. Let., 1989, 69, #12, 1376-1379.
[27] Brekhovskikh L.M., Uspekhi Fiz. Nauk, 1953, 50, #4, 539-576. (in Russian).
[28] Lotsch H.K.V., JOSA, 1968, 58, #4, 551-561.
[29] Blochin M., Physic der Runtgenstrahlen, Verlag der Technic: Berlin, 1957, 535.
[30] Klockenkamper R., Total Reflection X-ray Fluorescence Analysis, Wiley: New York, 1997, 245.
[31] Goos F., Hanchen H., Ann. der Phys., 1947, 6, #7-8, part 1, 334-345 (in German).
[32] Snyder A., Love J., Applied Optics, 1976, 15, #1, 236-238.
[33] Tamir T., Oliver A.A., J. Opt. Sos. Ameri., 1969, 59, #8, 942-949.
[34] Brekhovskikh L.M, Waves in layered media, Academic press: New York, 1980, 345.
[35] Schaefer Cl, Einfuehrung in der Theoretische Physik, 1949, 3, part 2, Optik, 540. (in German).
[36] Mirkin L., Handbook of X-ray analysis of Policrystal materials, Consultans Burean: New York, 1964, 862.
[37] Batterman B., Phys. Rev., 1964, 133, #3a, 759-764.
[38] James R., Solid state Phys., 1963, 15, 53-147.
[39] Bonse U., Hart M., Apply Phys. Let., 1965, 7, #9, 238-240.
[40] Sanchez del Rio M., Ferrero C., Freund A.K., Rev. Sci. Inst., 1995, 66, #2, 2206-2209.
[41] Borrman J., Phys. Z., 1950, 127, 297-316.
[42] Azarov L. and et. al, X-ray Diffraction, McGraw-Hill: New York, 1974, 664.
[43] Vyatkin A.F., Egorov V.K., Egorov E.V., Mat. Res. Soc. Synp. Proc., 2000, 585, 183-189.
[44] Egorov V.K., Egorov E.V., Report on MRS Spring Meeting, April 1-5, 2002, San Francisco, Report #B4-15, p. 3 of Program Symposium B.
[45] Roshcupkin D.V., Tucoulou R., Brunel M., Appl. Phys. Let., 1999, 75, #5, 639-640.
[46] Gasgnier M., Nevot L., Phys. Status Solidi (A), 1981, 66, 525-540.
[47] Sivukhin D.V., Zh. Exp. Teor. Fiz, 1948, 18, #11, 976-994. (in Russian).
[48] Kizel V.A., Otrazhenie sveta, Nauka: Moscow, 1973, 351. (in Russian).
