ISSN 10628738, Bulletin of the Russian Academy of Sciences. Physics, 2013, Vol. 77, No. 1, pp. 15–20. © Allerton Press, Inc., 2013. Original Russian Text © V.A. Bushuev, 2013, published in Izvestiya Rossiiskoi Akademii Nauk. Seriya Fizicheskaya, 2013, Vol. 77, No. 1, pp. 19–25.
Effect of the Thermal Heating of a Crystal on the Diffraction of Pulses of a FreeElectron XRay Laser V. A. Bushuev Faculty of Physics, Moscow State University, Moscow, 119991 Russia email:
[email protected] Abstract—Spatiotemporal dependences of the distribution of the crystal temperature under the effect of pulses of a freeelectron Xray laser are found using the solution of a thermal conductivity equation. The effect of temperature, its gradient, and the deformation of the crystal lattice on the diffraction reflection and the transmission of pulses in crystals of synthetic diamond is considered. DOI: 10.3103/S1062873813010061
INTRODUCTION Emissions of a freeelectron Xray laser (FEL) consist of pulses with wavelength λ ≈ 0.05–0.16 nm, duration τp ≈ 10–100 fs, and an angular divergence of ~1–3 μrad [1–3]. The pulses are characterized by an almost complete spatial coherence and a very moder ate temporal coherence, leading to a spectral width of pulses of ΔE/E ≈ 10–3. The authors of [4–6] suggested various fourchip and singlechip circuits to reduce the spectrum width to ΔE/E ≈ 10–5, allowing us to attain a selfseeding mode and better laser generation with the crystal placed between two undulators. The diffraction reflection of femtosecond pulses from crys tals and multilayered structures with the aim of their monochromatization and raising the degree of tempo ral coherence was considered in [7–11]. Pulse energies of the European FEL in channels SASE1 and SASE2, depending on the bunch charge, are 20–2500 μJ [3], leading to average energy flows of 60 W cm–2 to 80 kW cm–2 in the region of the first ele ments of Xray optics at distances of 500–800 m from the undulators. Allowance for and prevention of the strong thermal heating of the crystals and multilayered mirrors is one of our most serious problems. In this work, we consider the effect of such factors as the pulse energy; the temporal structure of the FEL radiation; the distance from the undulators; the initial and maximum temperatures of the crystal; the tem perature dependences of the coefficients of specific heat capacity; the heat conductivity; and the linear thermal expansion coefficient on the diffraction reflection and transmission.
outside the region of strong diffraction reflection with full width ΔθB, where Δθ = –(Δd/d)tanθB. As a result of crystal heating, lattice deformation Δd/d = αTΔT, where αT is the linear thermal expansion coefficient. We derive from condition |Δθ| < ΔθB/2 that admissi ble temperature deviations ΔT ≤ ΔTc, where ΔTc = ΔθBcotθB/2αT. The critical magnitude of ΔTc rises as the width of the Bragg reflection grows along with the Bragg angle, and expansion coefficient αT shrinks. For the sake of precision, we consider below a sym metric Bragg reflection (400) from a diamond crystal (λ = 0.1 nm, θB = 34.19°, ΔθB = 5.12 μrad, and μ = 3.15 cm–1 [12]). The steep temperature dependence of αT causes the critical values of ΔTc to fall from 75 K at T = 100 K to an extremely low value of 1.2 K at T = 600 K (see table). This imposes very rigorous condi tions on the temperature range within the limits of the crystal region into which the FEL pulses fall, espe cially at room and higher temperatures. FEL emission involves a series of pulses with dura tions τp ≈ 10–100 fs and energies Qp = ћ ωN, where N is the number of photons in a pulse (N ≈ (0.1–20) × 1011 [3]). The pulses are grouped into packets with dura tions of 0.6 ms and a repetition frequency of 10 Hz; the number of pulses in a packet is ≈2700, and the time interval between them Δt0 ≈ 0.22 μs. Let us now con sider crystal heating under the effect of these pulses. The spatiotemporal temperature distribution T(r , t) is determined from the parabolic thermal con ductivity equation (1) c pρ(∂T ∂t ) = div(κ grad T ) + F (r , t ), where cp is the specific heat capacity, ρ is density, κ is the heat conductivity, and F(r , t) is the density of the heat sources. Equation (1) must be supplemented by various initial and boundary conditions for temperature and its derivatives. The temperature in (1) and below is counted from the initial temperature T0(x, y, 0) = const. For a thin crystal, the fraction of absorbed
SPATIOTEMPORAL TEMPERATURE DISTRIBUTION Let us initially estimate the temperature difference ΔT at various points in the crystal, which does not affect the output of local Bragg angle θB(T) = θB + Δθ 15
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Linear thermal expansion coefficient αT × 106, specific heat capacity cp (J kg–1 K–1), heat conductivity of a type I diamond κ (W m–1 K–1) [13], and characteristic cooling time τT, depending on temperature T, K αT, K–1 c p, κ
100
200
300
400
600
0.05
0.45
1.0
1.80
3.09
29
214
514
854
1342
3050
1400
900
650
400
τT, μs
1.9
31.0
115.8
266.4
680.2
ΔTc, K
75.4
8.4
3.8
2.1
1.2
ΔT1, K
2.4
0.32
0.13
0.08
0.05
Note: ΔTc is the critical temperature, ΔT1 is the temperature of crystal heating under the effect of one pulse, and density ρ = 3.52 g cm–3. The wavelength is 0.1 nm, reflection (400); N = 2.8 × 1011, pulse energy W0 = 557 μJ, rs = 37 μm [3], αs = 0.76, z = 500 m, trans verse pulse size rp = 800 μm, and l = 50 μm.
energy η = μl/γ0 Ⰶ 1, where μ is absorption factor, l is crystal thickness, γ0 = cosθ, and θ is the incidence angle of the pulse; we therefore move from (1) to a twodimensional problem. Since the energy spectrum of FEL pulses is wider than the spectral width of the Bragg reflection by two orders of magnitude, we can ignore the effect of diffraction on quantity η. For a Gaussianshaped pulse, the density of sources F(x, y, t) = ηW(x, y)W(t), where W(x, y) = (γ0Qp/πlr12 )exp[–(x/rx)2 – (y/ry)2], (2) W(t) = (1/π1/2τ0)exp[–(t – ti)2/ τ 20 ]. (3) Here, rx = r1/γ0 and ry = r1. Transverse pulse radius r1 and its duration τ0 are associated with full size rp and duration τp at full width at half maximum (FWHM) by relations r1 ≈ 0.6rp and τ0 ≈ 0.6τp. The pulse radius at distance z from the FEL rp = rsM, where rs is the pulse size at the output of the FEL, M = [(1 + αsD)2 + D2]1/2, D = λz/(2.26rs2 ) and αs is the parameter that charac terizes the curvature of the wave front of a pulse in plane z = 0 [9–11]. It follows from the data in [3] that rp ≈ 500–1500 μm at distances from the undulators z ≈ 500–800 m. Let us now consider that pulse durations τp are much shorter than characteristic time τT of the temperature spreading due to heat exchange, τT = r12 /4a2 = r12 cpρ/4κ, (4) where a2 = κ/cpρ is the thermal conductivity coeffi cient. Solving (1) produces the following simple ana lytical expression for the temperature at point (x, y) and instant t: T ( x, y, t – t i ) = [ ΔT i / ( β x β y ) 2
2
2
2
1/2
× exp ( – x /r x β x – y /r y β y ) , ΔT1 = μQp/(πcpρ r12 ),
]
(5) (6)
γ 20 (t – ti)/τT, βy = 1 + (t – ti)/τT, –∞ < x,
where βx = 1 + y < ∞. Note that ΔT1 (6) is the heating temperature of the crystal at point x, y = 0 under the effect of one
pulse with no allowance for heat exchange. Tempera ture ΔT1 falls along with absorption factor μ, pulse energy Qp, and pulse radius r1; i.e., distance z increases. It is also independent of the crystal thick ness and angle of incidence. Temperature (5) dimin ishes over time as 1/(βxβy)1/2, while the size of the heat ing region on the crystal surface grows as rT(t) = rxβ1x 2 ; i.e., the rate of temperature drop is higher than the rate of heating region growth. Let t1, t2,… be the instants of pulse incidence on the crystal. Formula (5) is then transformed into n
T (x, y, t) =
∑T (x, y, t − t ),
(7)
i
i =1
where the number of pulses n is determined by condi tion t < tn. As the temperature rises from 100 to 600 K, the heat capacity of the diamond grows by a factor of almost 50, while heat conductivity falls by a factor of 7.6 (the table). Heating temperature ΔT1 (6) therefore also falls by a factor of almost 50, but the time of heat exchange (cooling) τT (4) also rises abruptly from 2 to 680 μs (the other parameters remain the same as in the table caption). It can be seen from the table that to shorten cooling time τT, it is desirable to operate at low temperatures, since the heat capacity is in this case low and the heat conductivity is high. Unfortunately, heat exchange times τT at T0 ≥ 100 K exceed time intervals Δt0 ≈ 0.2 μs between the pulses in a packet by 1– 3 orders of magnitude. The crystal temperature will therefore rise for the time of pulse packet incidence on the crystal, and the heat will have time to spread only in time intervals between the packets, since τT Ⰶ 0.1 s. Figure 1 shows the results from calculating the temperature of the crystal along axis x at y = 0 in the pulse peak (x = 0), at its edge (x = x1 = rp/2γ0), and the difference between these temperatures ΔT(t) = T(0, t) – T(x1, t) at initial temperature T0 = 300 K. It can be seen that at these parameters, the crystal is
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EFFECT OF THE THERMAL HEATING OF A CRYSTAL ON THE DIFFRACTION OF PULSES T, K 10
T, K 150
1
1
2
8 100
2
50
3
17
6 4 3 2
0
500
1000
1500
2000 t, μs
Fig. 1. Time dependence of the temperature of the dia mond crystal under the impact of a pulse packet at points (1) x = 0 and (2) x = x1; temperature difference ΔT(t) (3). Initial temperature T0 = 300 К, ΔTc = 3.8 K. All other parameters are the same as in the table caption.
quite strongly heated (164 and 104 K) until the packet is finished, and it then cools slowly. High temperature itself thus does not create problems, though it does lower the Debye–Waller thermal factor. It is important that the temperature difference in the region of a crys tal with size |x| ≤ x1 would not exceed critical value ΔTc at which the efficiency of diffraction reflection drops abruptly. In this case, however, the temperature differ ence (curve 3) greatly exceeds ΔTc. Prior to the arrival of the next packet, temperature T(0, t = 0.1 s) = 0.75 K; it returns to its initial value in anticipation of further heating of the crystal from the second packet, and so on. Temperature ΔT1 (6) can be reduced by selecting a high initial temperature and a greater distance of the crystal from the undulator. However, heatexchange time τT grows in this case; more important, critical dif ference ΔTc falls abruptly because of an increase in the linear thermal expansion coefficient αT (see table). Analysis of relation (7) allowing for the temperature dependence of all parameters shows that low initial temperatures are optimal. The high heat conductivity factor κ leads to rather short characteristic cooling times τT and, as a consequence, to a reduction in the maximum heating temperature Tmax and an increase in the duration of the plateau in the temporal depen dence of temperature difference ΔT(t), i.e., to a level ing of the temperature gradient (Fig. 2). Fulfilling condition ΔT(t) < ΔTc = 8.4 K allows us to hope for higher efficiency of diffraction reflection. It is also noteworthy that the abovementioned is possible only at rather low pulse energies Qp (i.e., at low bunch charges), and with an increase in the distance from the FEL to the crystal. Figure 3 shows the spatial distribution of the crystal temperature at various points in time. The tempera
0
500
1000
1500 t, μs
Fig. 2. Dependence of the crystal temperature at points (1) x = 0 and (2) x = x1; (3) temperature difference ΔT(t). Type IIa diamond crystal, N = 0.3 ⋅ 1011, Qp = 60 µJ [3], T0 = 200 К, сp = 214 J kg–1 K, κ = 4000 W m–1 K–1, rp = 800 µm, τT = 10.8 µs, ΔTc = 8.4 К, ΔT1 = 0.034 К, and Tmax = 10.1 K. Temperature difference ΔT(t) reaches a plateau beginning at ΔT ≈ 2 K.
ture rises in time intervals of 600 μs from the incidence of the pulse packet, and it simultaneously spreads along the crystal surface (curves 1–3), especially strongly in the intervals between the packets (curves 4, 5). At t = 1500 μs, the size of heated region rT(t) exceeds pulse size r1/γ0 on the crystal surface by a factor of almost 6. As the irradiation time grows, the size of the heated region rT(t) becomes comparable to the transverse sizes of the crystal. This means we must consider the edge conditions at the crystal–thermostat interface, which we will do in a separate work. It is evident that in order to prevent a large spatial temperature gradi ent, there are definite limitations on the rate of heat removal from the side faces of the crystal. T, K 3
10 2 8 6
1 4
4
5
2
0
–4000 –2000
0
2000
4000 x, μm
Fig. 3. Temperature distribution T(x, 0, t) along the crystal surface at various times t (µs): (1) 100, (2) 300, (3) 600, (4) 800, and (5) 1500. Parameters of the crystal and pulses are the same as in Fig. 2.
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DIFFRACTION OF RANDOM PULSES IN A CRYSTAL WITH A TEMPERATURE GRADIENT Let us consider the diffraction of FEL pulses in a crystal with spatially nonuniform timedependent temperature distribution T(r , t), where r = (x, y) is a coordinate on the crystal surface. The emission field at distance z from the FEL takes the form (8) E(ρ, z, t ) = A(t − z c)B(ρ, z)exp(ik0 z − iω0t ), where k0 = ω0/c, ω0 is the central emission frequency, and ρ = (ρx, ρy) is a coordinate in the transverse pulse section. The dependence of slowly varying pulse amplitudes (8) on the time and coordinate is deter mined by the following relations [9–11]:
∫
(9) A(t ) = As (t ) = As (Ω)exp(−iΩt )d Ω, 2 (10) B(ρ, z) = Bs (q )exp(iq ρ − iq z 2k0 )dq. where As(Ω) and Bs(q ) are the frequency and angular emission spectra at the FEL output in plane z = 0 (here and below, we omit the limits of integration performed from –∞ to +∞):
∫
∫ ∫
(11) As (Ω) = (1 2π) As (t )exp(iΩt )dt, (12) B s (q ) = (1 2π)2 B s (ρ)exp(−iq ρ)d ρ. It can be seen from (8) that the temporal structure of the pulses coincides with function As(t) (9) in the source plane and is maintained to the degree it propa gates in free space, while the transverse spatial distri bution of amplitude (10) depends on distance z, due to the diffraction spread of the pulse and curvature of its wave front. Field (8) is the totality of plane monochromatic waves k with amplitudes As(Ω)Bs(q ), wave vectors = (q, kz), and frequencies ω = ω0 + Ω, where kz = (k2 – q2)1/2 and k = ω/c. The slowly varying amplitudes of reflected (R) and transmitted (T) pulses on the crystal surface are determined by the relation AC (r , t ) = As (Ω)Bs (q )C(α)exp(iqr − iΩt )dqd Ω, (13)
∫∫
where C(α) = R, T represents the amplitude coeffi cients of diffraction reflection and transmission, respectively. The coordinate systems of ρ in the source plane and r on the crystal surface are associated by the simple relations ρx = xγ0 and ρy = y, z = z0 + xsinθ, where γ0 = cosθ. Quantity α = [k2 – (k + h )2]/k2 in (13) describes the local deviation from the exact Bragg condition α = 0, where h is the vector of the reciprocal lattice, h = 2π/d, and d represents the interplanar distances in the crystal. The region of strong diffraction reflection is determined by condition |α| ≤ 2|χh|, where χh is the Fourier component of the polarizability of the crystal. Since distances d depend on temperature T(r , t) because of thermal expansion, quantity α is generally
a function of coordinates x, y and time t. In addition, detuning parameter α depends on the angle Δθ of the crystal’s deviation from the Bragg angle θB, on fre quency Ω in energy spectrum S(Ω) = 〈|As(Ω)|2〉 of the incident pulse, and on its angular divergence q/k0. The Bragg angle is determined here from condition 2k0sinθB = h0, where h0 is the magnitude of the vector of the reciprocal lattice of an ideal crystal with uniform initial temperature T0. Allowing for expansion by small parameters Δθ/θB, Ω/ω0, αTT, and q/k0, we find the following explicit expression for function α: α(Δθ, Ω,T (x, y), q x ) (14) = 2 sin 2θ B[Δθ + (Ω ω0 + αTT ) tan θ B − q x k0]. Integration in (13) and the calculation procedure for R and T pulses IC = 〈|AC|2〉 depend substantially on the coherent properties of FEL pulses. In light of their high spatial coherence, we may consider functions B(ρ, z) (10) and Bs(q ) (12) to be practically deter mined. In addition, the slight angular divergence Δθs = Δqeff/k0 of FEL pulses compared to Bragg reflection width ΔθB = 2|χh|/sin2θB allows us to ignore the last summand in (14). As a result, integration over q in (13) leads to the spatial profiles of intensities of R and T pulses coinciding with the profile of incident pulse I0(r ) = |B(r , z)|2 multiplied by |C(α)|2 at qx = 0 in (14) (the approximation of local diffraction interaction). The condition for angular divergence Δθs Ⰶ ΔθB is equivalent to condition rs Ⰷ Λsin2θB/γ0 for source size rs, where Λ = λγ0/π|χh| is the depth of extinction. On the other hand, FEL pulses are characterized by very weak temporal coherence, since field ampli tude As(t) (9) is a random function with coherence time τc Ⰶ τp [2, 3]. Based on the data in [2, 3], the authors of [9–11] suggested a simple model for the time dependence of the amplitude of a random FEL pulse. The field amplitude is described in the form of a product of a smooth deterministic function of pulse envelope F(t) and random stationary signal a(t): As(t) = F(t)a(t). Function a(t) obeys the conditions 〈a(t)〉 = 0 and 〈|a(t)|2〉 = 1, and the function of tempo ral coherence g(τ) = 〈a(t)a*(t + τ)〉 is independent of time t; it is characterized by coherence time τc. Spec tral amplitudes a(Ω) are δ correlated; i.e., 〈a(Ω)a*(Ω')〉 = G(Ω)δ(Ω – Ω'), (15) where spectral density G(Ω), according to the Wiener–Khinchin theorem, is associated with coher ence function g(τ) by the Fourier transform:
∫
G(Ω) = (1 2π) g(τ)exp(−iΩτ)d τ.
(16)
It follows that the time dependence of statically averaged intensities of incident pulses is determined by the quadrate of the modulus of the envelope function: Is(t) = 〈|As(t)|2〉 = |F(t)|2. Since function F(t) varies slowly compared to a(t), the Fourier amplitude of the field in the narrow vicin
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ity Δt (τc Ⰶ Δt Ⰶ τp) near time t can be represented in the form of product As(Ω) ≈ F(t)a(Ω). Intensities of the reflected and transmitted random pulses are deter mined by double integral I C (r , t) = I 0(r ) C(α)C * (α') As (Ω)As*(Ω') (17) × exp[−i(Ω − Ω')t]d Ωd Ω'. Allowing for the δ correlation of random signal a(Ω) (15), the final expression for intensities R and T pulses depending on coordinate (x, y) and time t is (18) I C (r , t) = I 0(r )I s (t ) GC (α)d Ω,
∫∫
|C(α)|2G(Ω)
∫
where GC(α) = is the spectral density of reflected (C = R) and transmitted (C = T) pulses. In the context of this model, the temporal depen dence of the envelope of intensities of R and T pulses thus coincide in form with the incident pulse. Based on a more rigorous analysis of spectral correlator 〈As(Ω) As*(Ω')〉, it was shown in [9–11] that the shape of a reflected pulse I R(r , t) differs from Is(t) but only slightly, especially at τc Ⰶ τp. It follows from (18) that intensity IR Ⰶ I0, since spectral width ΔΩB of reflec tance |R(α)|2 is much less than width ΔΩc = 2/τc of the spectrum of incident pulse G(Ω). In contrast, transmit tance |T(α)|2 is almost constant over spectral region G(Ω), excluding the narrow deep valley in region |α| ≤ 2|χh|. This means that the transmitted pulse almost coincides with the incident pulse in intensity and shape, but an additional weak delayed and almost coherent pulse with intensity on the level of 10–2–10–3 of the incident pulse appears in temporal dependence IT(r , t) because of dispersive features of pulse propagation in the noted narrow region [10]. The authors of [5, 6] proposed using this to increase the temporal coher ence of FEL pulses as a result of selfseeding. According to (14) and (18), the intensities of R and T pulses in various regions of spectrum Ω are functions of several arguments: r , T(r , t), and Ω. In order to ana lyze the effect of the nonuniformity of thermal heating of a crystal on diffraction, we therefore consider the dependence of spectral intensities IC(r , Ω) = I0(r )GC(α) on the coordinate and frequency, with all other parameters being fixed. Figure 4 shows the integral (along axis y) depen dences of reflectance
∫
PR (x, Ω) = I R (x, y, Ω)dy
∫ I (x, y)dy 0
(19)
on coordinate x at various values of spectral detuning ε = Ω/ΔΩB, where ΔΩB = ω0|χh|/sin2θB is the spectral width of the Bragg reflection from a perfect crystal. There is almost no reflection in the central region of the spectrum (|ε| ≤ 1) because positive temperature contribution αTT(x, y) predominates, moving detun ing parameter α (14) beyond the limits of the Bragg condition. The broad symmetrically arranged maxima in curves 1 and 2 are explained by reflection from com
PR, rel. units 1.0
19
3
0.8 4 0.6
6
0.4 2 0.2 1
5
0
2000 x, µm
0 –2000
Fig. 4. Dependence of reflectance on coordinate x on the surface of a type IIa diamond crystal at various values of spectral detuning ε: (1) –2, (2) –3, (3) –4, and (4) –4.5. Curves 5 and 6 are profiles of I0(x) and T(x), respectively. Parameters: N = 2.8 × 1011, τp = 100 fs, τc = 0.17 fs [3], z = 800 m, rp = 1200 µm, T0 = 200 К, τT = 24.4 µs, βx = 2.6, Tmax = 70 К, l = 50 µm, and Δθ = 0.
PR, rel. units
4
1.0 0.8
3
1
2
0.6 0.4 0.2 0 –4
–2
0
2 Ω/ω0, 10–5
Fig. 5. Integral spectral reflectances PR(Ω) from (1) the ideal crystal and from type II diamond crystals with (2) Tmax = 70 K and (3) Tmax = 30 K; (4) is the spectrum of incident pulses G(Ω) normalized to the maximum; T0 = 200 K.
paratively cold regions of the crystal at edges x ≈ ±x1 of profile I0(x) of the incident pulse. Reflection intensity reaches its maximum (curve 3) in the ε ≈ –4 region of frequencies, since the effect of spectral deviation Ω/ω0 and temperature contribution to αTT in (14) are com pensated for, while the energy of the incident pulse is highest at point x = 0. It can be seen in Fig. 4 that the region of the crystal surface from which the pulses are effectively reflected, is smaller than region I0(x) (curve 5) onto which they fall. Figures 5 and 6 show the results from calculations of integrated (by x and y) spectral reflectances and transmittances with different maximum temperatures
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BUSHUEV PT, rel. units
4
1.0
parameters αT, cp, and κ not only on temperature but also on their temporal dynamics on the femtosecond scale.
0.8 1
2
0.6
3 0.4 0.2
ACKNOWLEDGMENTS The author thanks L. Samoylova and H. Sinn for their useful comments. This work was supported by the Russian Founda tion for Basic Research, project nos. 100200768 and no. 120200924; and by the FRG Ministry for Edu cation and Research, project no. 05K10CHG.
0 –4
–2
0
2 Ω/ω0, 10–5
Fig. 6. Integral transmittances PT(Ω) for (1) an ideal dia mond crystal and crystals with (2) T0 = 500 K and (3) T0 = 200 K; (4) is the spectrum of incident pulses, Tmax = 20 K.
of the crystal Tmax. The presence of a nonuniform tem perature field T(x, y) leads to diminished reflectance PR(Ω), to its broadening, and to the appearance of clearly pronounced asymmetry. It also shifts the reflection maximum to the negative spectral region (Fig. 5, curves 2, 3). As temperature Tmax declines (due, e.g., to a drop in pulse energy Qp), reflectance (curve 3) approaches that of a perfect crystal (curve 1). Figures 6 shows that spectral transmittance PT(Ω) is strongly distorted if we use a sufficiently high initial temperature (T0 = 500 K). The coefficient of linear thermal expansion is then greater than in Fig. 5 by a factor of 5.4, leading to more substantial broadening of the diffraction valley and reducing its depth (compare curves 2 and 3). Such behavior of function PT(Ω) can seriously complicate attaining the selfseeding mode. CONCLUSIONS In this work, we mainly analyzed the initial stage of crystal heating. It would be of interest to consider the quasisteady mode during prolonged irradiation, and to consider the dependences of thermodynamic
REFERENCES 1. Altarelli, M., et al., XFEL Technical Design Report no. DESY 2006097, Hamburg, 2006. http://xfel.desy.de/ tdr/index_eng.html 2. Geloni, G., Saldin, E., Samoylova, L., et al., New J. Phys., 2010, vol. 12, no. 3, p. 035021. 3. Tschentscher, Th., Proc. EUR. XFEL, EU TN2011 001, 2011, p. 1. 4. Saldin, E., Schneidmiller, E., Shvyd’ko, Yu., and Yurkov, M., Nucl. Instrum. Methods A, 2001, vol. 475, no. 2, p. 357. 5. Geloni, G., Kocharyan, V., and Saldin, E., Report DESY no. 10053, Hamburg, 2010, p. 1. 6. Geloni, G., Kocharyan, V., and Saldin, E., Report DESY no. 11224, Hamburg, 2011, p. 1. 7. Bushuev, V.A., Bull. Russ. Acad. Sci. Phys., 2005, vol. 69, no. 12, p. 1903. 8. Bushuev, V.A., J. Synchrotron Rad., 2008, vol. 15, no. 4, p. 495. 9. Bushuev, V.A. and Samoylova, L., Nucl. Instrum. Methods A, 2011, vol. 635, no. 4, p. 19. 10. Bushuev, V.A. and Samoylova, L., Cryst. Rep., 2011, vol. 56, no. 5, p. 819. 11. Bushuev, V.A. and Samoylova, L., Bull. Russ. Acad. Sci. Phys., 2012, vol. 76, no. 2, p. 153. 12. www.esrf.eu/UsersAndScience/Experiments/TBS/Sci Soft 13. Fizicheskie velichiny: Spravochnik (Physical Quantities. Handbook), Grigor’ev, I.S. and Meilikhov, E.Z., Eds., Moscow: Energoatomizdat, 1991.
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