Vladimir Bushuev Effect of the Thermal Heating of a Crystal on the Diffraction of Powerful Pulses of a Free-Electron X-Ray Laser Vladimir Bushuev M.V. Lomonosov Moscow State University Moscow, Russia e-mail: vabushuev@yandex.ru X International Symposium “Radiation from Relativistic Electrons in Periodic Structures” – RREPS-13 Armenia Yerevan – Sevan September 23-27, 2013

Short plan: 1. A few words about X-ray Free-Electron Laser (XFEL) and parameters of the pulses 2. Temperature dependences of the coefficients of specific heat capacity, the heat conductivity, and the linear thermal expansion coefficient 3. Thermal conductivity equation 4. Spatio-temporal distribution of the reflected and transmitted pulse intensities after diffraction of XFEL powerful pulses

Basic principles of X-ray free electron lasers 3 Basic principles of X-ray free electron lasers Long undulator (15-150 m) XFEL pulses Electron beam Eel  10-15 GeV Self-Amplified Spontaneous Emission (SASE) 3. With complete micro-bunching all electrons radiate in phase. This leads to a radiation power growth as N2. I  N2 2. The shot noise of the electron beam is amplified to complete micro-bunching. I  N 1. All N electrons can be treated as individually radiating charges, and the resulting spontaneous emission is proportional to N.

!! X-ray free electron laser starting from the shot noise in 4 X-ray free electron laser starting from the shot noise in the electron beam has been proposed by Derbenev, Kondratenko, and Saldin (1979, 1982); and also by Bonifacio, Pelegrini and Narducii (1984). Ratio of XFEL and SR brilliances: !!

XFEL Facilities 5 6.5 nm 0.15 nm SPring-8 L = 4 km 0.1-4 nm

Hard X-ray (SASE1 and SASE2) FEL radiation typical main parameters 6 Radiation wavelength 0.1 nm Bunch charge 0.02 nC 1 nC Pulse duration 10 fs 100 fs Source size (FWHM) 29 m 49 m S. divergence (FWHM) 1.9 rad 1.3 rad Spectral bandwidth 1.910-3 1.010-3 Coherence time 0.13 fs 0.23 fs Degree of the transverse coherence 0.95 0.71 Photons/pulse 0.31011 6.41011 Pulse energy 58 J 1260 J Low High !! Th. Tschentscher, XFEL.EU Technical Note TN-2011-001)

The radiation from a SASE XFEL is almost 7 The radiation from a SASE XFEL is almost transversely (that is spatially) coherent. Temporal (that is longitudinal) coherence is very low because of the start-up from the shot noise. XFEL pulses have a very irregular multispike temporal structure with a width of individual random subpulses about 0.1-0.2 fs and time intervals 0.3-0.4 fs between them.

General time structure of the EU XFEL Hard X-ray (SASE1 and SASE2) Wavelength: 0.05 – 0.16 nm Bunch charge: 0.02 – 1 nC !! Photons/pulse: (0.1 – 20)1011 Pulse Energy: 20 – 2500 J Power: 60 W/cm2 – 80 kW/cm2 10-5 Linac Coherent Light Source (LCLS) – одиночные импульсы, 120 Гц

Th.Tschentscher, XFEL.EU TN-2011-001

…For comparison: 1. Iron - 5 W/cm2 2. Thermal elements on atomic stations - 200 W/cm2 3. XFEL - 3-10 kW/cm2

There are three problems: 1. How powerful XFEL pulses heat up a crystal (in space and in time)? 2. How this heating influences on the diffraction of pulses? (monochromatisation, self-seeding) 3. On what distance we should place a crystal ? Main problem: I, T, d, B = f(r, t)

Tc  1-70 K T Simple estimations (for diamond): Tc = BctgB/2T, Admissible heterogeneity of distribution of temperature in different points of a crystal: Tc = BctgB/2T, Tc  1-70 K where T is linear thermal expansion coefficient. T Tc

2. Estimation of heat temperature by single pulse: T1 = Q0/(cr12). T1  0.05-20 K 0 = 0.1 nm, N = 2.81011, Q0 = 556 J, absorption part  = l/0<< 1,  = 3.15 cm-1, l = 50 m, Bragg geometry, C(400),  = 0.03, Q = Q0 = 15.57 J,  = 3.52 g/cm3 , z = 500 m, r1w = 597 m, c = (specific heat capacity) = 510 J/(kgК) at 300 К, T1  0.24 К, T1 = 0.24 K  2700 = 650 K !!??

3. Characteristic cooling time Typical values for diamond (heat exchange) T = r12c/4, where  is heat conductivity (diamond/Cu  14 – 4) !! Typical values for diamond T  1-700 s Distances between pulses in the packs t0 = 0.2  s However this time T is much less than interval of time between packs 0.1 с (z > 100 м).

Result such: Thermal parameters - linear expansion, a thermal capacity and heat conductivity behave in the "opposite" image (an example – a short blanket)........ Nevertheless, small times of heat exchange T and great values of critical temperature Tc at low temperatures testify to necessity of use of low initial temperature

Thermal conductivity equation cp(T/t) = div(gradT)  s(T  Ts) + F, where  - heat conductivity, s – coefficient of heat exchange, F(x,y,z,t) – the density of the heat sources.

Initial and boundary conditions: T(x, y, 0)=T0 , T(L/2, L/2, 0)=T0. Here T1 = Qp/(cpr12) – heat temperature at point x, y = 0 under the effect of one pulse, a2 = /cp is the thermal conductivity coefficient, qk = (2k  1)/L, k = m, n; i = x, y.

300 K 200 K 100 K 100 K Dependence of temperature of a crystal on time in a maximum of the pulse (x=y=0, solid curves) and on its edge (x=0, y=0.5 rp, dashed curves) at z1 =100 m (a) and z=500 m (b) and initial tempe- ratures T0 = 100 (1), 200 (2) и 300 K (3); Qp = 80   J.

L = 100 m Dependence of crystal temperature T(x,y=0) on time at different distances from XFEL (Qp=26 J, T0=200K) L = 100 m Interval between pulses is 0.2 s r1=300m, T=1.5s, T1=0.6K, Tmax=28 K r1=54m, T=0.05s, T1=18.9K, Tmax=49 K r1=33m, T=0.017s, T1=50K, Tmax=71 K

Effect of temperature T(x, y, t) on intensity of reflected and transmitted pulses The condition of “strong” diffraction:   2h Equation for the intensity of reflected (C = R) and transmitted (С = T) pulses:

Space distribution of the temperature T(x, 0, 600 s) 1. At T0 = 100 K Tmax = 38 K < Tc = = 75 К, 2. At T0 = 300 K It is inverse situation: Tmax = 71 K >> Tc = = 4 К. 300 K 200 K 100 K N =  0.31011 phot/pulse Space distribution of the temperature T(x, 0, 600  s) at initial temperature T0 = 100 K (1), 200 K (2) и 300 K (3); 4 – profile of the incident pulse intensity gx (right scale)

T0 = 200 K, t = 600 s Integral coefficients of reflection PR(x)  = /B: 1 - 2, 2 - 3, 3 - 3.5, 4 - 4, 5 - 4.5 6 -  5 7 - profile I0(x) T0 = 200 K, t = 600 s

Map of the intensity distribution of R-pulses IR(x, y, t,  = 0) during the various moments of time t N = 1011 photons/pulse, T = 200 K, diamond type IIа

Map of the intensity distribution of R-pulses IR(x, y, t,  = 0) during the various moments of time t N = 1011 photons/pulse, T = 200 K, diamond type IIа

Map of the intensity distribution of R-pulses IR(x, y, t,  = 0) during the various moments of time t N = 1011 photons/pulse, T = 200 K, diamond type IIа

Map of the intensity distribution of R-pulses IR(x, y, t,  = 0) during the various moments of time t N = 1011 photons/pulse, T = 200 K, diamond type IIа

Spectral reflection coefficient at different moments of time N =  2.81011, T0 = 200 К, Tc = 8.4 К.

Spectral reflection coefficient at different moments of time N =  2.81011, T0 = 200 К, Tc = 8.4 К.

Spectral reflection coefficient at different moments of time N =  2.81011, T0 = 200 К, Tc = 8.4 К.

Spectral reflection coefficient at different moments of time N =  2.81011, T0 = 200 К, Tc = 8.4 К.

Spectral reflection coefficient at different moments of time N =  2.81011, T0 = 200 К, Tc = 8.4 К.

Qp = 80 J Integrated coefficient of reflection Rint at initial temperatures T0 = 100 (1), 200 (2) and 300 K (3) on the distances z = 100 m (a) and z = 500 m (b) from XFEL , 4 – reflection coefficient an ideal crystal, 5 – a spectrum of the incident pulses with a width c = 2/c, where c  0.2 fs is the time of coherence.

line width of SASE X-ray FELs, DESY 10-053 (2010). G. Geloni, V. Kocharyan, E. Saldin, A simple method for controlling the line width of SASE X-ray FELs, DESY 10-053 (2010). R. R. Lindberg, and Yu. V. Shvyd'ko, Time dependence of Bragg forward scattering and self-seeding of hard x-ray free-electron lasers // ArXiv: 1202.1472v3 (9 Mar 2012) (Advanced Photon Source, Argonne National Laboratory, Argonne, IL 60439, USA). 1. SLAC National Accelerator Laboratory, Stanford, California 94309, USA, 2. Argonne National Laboratory, Argonne, Illinois 60439, USA, 3. Technical Institute for Superhard and Novel Carbon Materials, Troitsk, Russia 142190, 4. Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA.

1 2 Self-seeding scheme with wake monochromator for narrow-bandwidth X-ray FELs [1] G. Geloni, V. Kocharyan, E. Saldin, DESY 10-053 (2010) 1 2 900 m shot noise coher. X-ray signal Алмаз (400) 50-150 мкм Bandwidth down to 10-5 0.15 нм 10 фс 10 мкДж 120 Гц 20 0.4 eV LCLS

Short pulse transmission in the Bragg geometry Spectral transmission curve T()2 (1) and a spectrum of the incident pulse S() (2). Parameters: 0 = 0.15 nm, p = 0.15 fs; diamond, reflection (400), crystal thickness l = 100 m.

Integral spectral coefficient of transmission !?

Conclusions: 1. Within the limits of two-dimensional thermal conductivity equation with the distributed sources spatiotemporal distributions of the crystal temperature T (x, y, t) under action of XFEL pulses are calculated.   2. In the case of pulses with great and middle intensity the temperature of a crystal heating reach up to 100-300 K. However, the main problem is the big gradient of temperature exceeding critical values Tc  1-10 K. 3. For pulses of low power (N  1010 photons/pulse conditions of diffraction are carried out at the distances of order 800 meters. 4. It is desirable "to work" at initial temperature T0  100-200 K with optimal speed of heat removal. 5. It is necessary consideration of a problem taking into account temperature and time dependence of all thermal-physical parameters.

Acknowledgments 1. The Russian Foundation for Basic Research, Projects № 10-02-00768, 12-02-00924; 2. The German Federal Ministry of Education and Research (BMBF), project no. 05K10CHG.

Thanks for your attention

Проблема: T1(x, t) > T2  const Схематическое изображение кристаллов CM1 и CM2 в схеме двухкристального рентгеновского монохроматора. Очевидно, что температура T1(x, t) > T2  const.