Download Heat load estimates for XFEL beamline optics

Heat load estimates for XFEL beamline optics
Harald Sinn
Introduction
The European X-ray Free Electron Laser (XFEL) will provide hard X-ray radiation to user
experiments of unprecedented quality in terms of peak brightness and short pulse duration. The
electron bunch pattern will be – unlike at storage ring facilities – very inhomogeneous in time,
leading to a heat load of 10 kW and more during the 600 µs long X-ray pulse train. At the SASE 1
beamline this power will be concentrated on a spot with 0.5 mm diameter about 500 m away from
the undulator, where the first X-ray optical components like slits and monochromators will be
placed (Fig. 1). Even more dramatic is the situation during a single X-ray pulse, where the power
will be 20 GW for 100 fs. As previous estimates for LCLS [1], XFEL [2] and recent experiments at
FLASH [3] show, this power can be enough to cause under certain conditions evaporation or even
plasma formation on X-ray optical components. This report deals with the situation that is expected
for the SASE 1 beamline at the XFEL, producing X-rays of one Angstrom wavelength.
Figure 1: Schematic view of beamline components of the SASE 1 beamline at the XFEL. Numbers indicate
meters behind the exit of the undulator.
Instantaneous heat load
The X-ray pulse duration of 100 femtoseconds is short compared to phonon life times, so in first
approximation no thermal conduction will take place during the pulse. Photoelectrons generated by
the X-rays will carry their energy only over distances of less than a micrometer. This is short
compared to typical X-ray absorption lengths, unless one considers total reflection geometries for
X-rays, where this effect may become important [1]. For X-ray optical elements like shutters, slits,
and monochromators it is therefore a good approach to assume the instantaneous heat load of a
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single shot to be distributed according to the X-ray absorption depth profile. With a Gaussian
lateral beam profile, one gets a temperature distribution after the first pulse:
⎛
⎛ x2
⎛ −z ⎞
y2 ⎞⎞
⎟⎟ ,
T ( x, y, z ) = Tpulse exp⎜⎜ − 4 ln 2⎜⎜ 2 + 2 ⎟⎟ ⎟⎟ exp⎜⎜
⎝ x1/ 2 y1/ 2 ⎠ ⎠
⎝ z abs ⎠
⎝
(1)
where Tpulse is the maximum of the temperature distribution, x,y are the coordinates parallel and z
perpendicular to the surface, x1/2,y1/2 are the FWHM of the beam, and zabs is the absorption length
perpendicular to the surface. If the X-ray beam is circular with a FWHM of b and hits the surface
with an angle of incidence θ (rotation axis y), it follows:
x1 / 2 = b / sin θ , y1 / 2 = b, z abs = l abs sin θ .
(2)
The energy per X-ray pulse at 1 Angstrom laser wavelength at the XFEL is Q=2 mJ. The integral
Q=
∫c
p
ρ T ( x, y, z ) dV
(3)
Vol
yields
T pulse =
4 ln 2 Q
,
π c p ρ b 2 l abs
(4)
with the specific heat cp , the mass density ρ and the X-ray absorption length along the beam of labs.
A typical temperature rise per X-ray pulse is in the order of several Kelvin for the lighter elements
as shown in Table 1, well below their melting temperatures Tmelt. Note that Tpulse is independent of
the angle of incidence. Apart from melting, stress due to temperature gradients can also damage the
material. If these gradients occur on a small length scale compared to the beam dimensions, one can
approximate the condition for a minimum temperature gradient Tcrac where thermal cracks will
occur [4]:
Tcrac =
3(1 − ν )G
,
αE
(5)
with ν is the Poisson ratio, G the uniaxial yield strength, α the thermal expansion coefficient and E
the Young’s modulus. This condition is more restrictive for most materials than the melting
condition and it will further limit the number of feasible subsequent X-ray pulses on the optics.
Heat load during a bunch train
A pulse train consists of up to 3000 X-ray pulses with a spacing of 200 ns. If there was no removal
of heat during that time, the temperature increase per pulse would just add up and all materials in
Table 1, with the exception of beryllium, would be damaged during a pulse train. Unfortunately,
this is already a good approximation if the beam hits perpendicular onto the surface of the material.
However, by placing the optical elements in grazing incidence geometry, the X-ray penetration
perpendicular to the surface can be reduced, which leads to a steeper temperature gradient and thus
to a larger heat flow. To estimate a time evolution of the temperature distribution in equation (1),
one can simplify the problem to one dimension, since the beam footprint will be large compared to
zabs for small incidence angles and therefore the main component of the heat flow will be
perpendicular to the surface. One can further approximate the exponential part in (1) by a Gaussian
of the same width and calculate the heat diffusion perpendicular to the surface:
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T (t , z ) ∝
cpρ
⎛ − cpρ z2 ⎞
⎟,
exp⎜
⎜ 4λ (t + t 0 ) ⎟
4πλ (t + t 0 )
⎝
⎠
(6)
where
t0 =
2
c p ρ z abs
(7)
4λ
is a constant that describes the time it would take for a delta distribution to evolve into the Gaussian
that approximates the initial temperature distribution after the first X-ray pulse.
The idea is now to calculate for each material the minimum incidence angle, where 3000 pulses of
one pulse train will just lead to melting or cracking at the hottest part of the temperature profile.
This angle (last column in Table 1) gives an indication for the maximum angle under which a
particular material will survive a full pulse train.
With incidence angles of about 10 mrad it seems possible to use a selection of materials in Table 1
for shutters and slits, among them an aluminium-alloy and the copper based material GlidCop®,
which is currently a standard material for high heat load optics at storage rings. Another option
would be to coat these elements with e.g. glassy carbon, to increase the maximum tolerable angle.
Table 1: Material properties and behaviour under heat load for various materials for 2 mJ pulse energy, 12.3
keV and 3000 pulses with 200 ns spacing per pulse train at room temperature conditions. The last column
gives the maximum incidence angle under which the material would survive a full pulse train.
LiF
Be
B
BN
B4C
Cdiamond
Cglass
NaCl
MgO
Al
Al2014
Al2O3
AlN
Si
SiC
SiO2
Ti
GlidCop
g/cm3
cp
J/gK
labs,
(µm)
Tpulse
K
2.63
1.84
2.34
2.2
2.51
3.52
2.26
2.17
3.58
2.70
2.70
3.97
3.33
2.33
3.15
2.64
4.50
8.92
1.56
1.82
1.02
1.4
0.95
0.51
0.71
0.85
0.87
0.90
0.90
0.42
0.80
0.70
0.67
0.75
0.52
0.38
1172
13400
5958
3377
4730
2200
3420
208
350
265
265
283
303
237
242
371
35.5
8.88
1.46
0.16
0.50
1.0
0.62
1.78
1.28
18.31
6.42
10.96
10.96
15.0
8.74
18.18
13.82
9.57
85.0
231.5
0.326
0.20
0.2
0.12
0.207
0.07
0.15
0.252
0.18
0.37
0.37
0.30
0.22
0.266
0.183
0.16
0.361
0.343
G,
GPa
E,
GPa
,
10-6
Tcrac
K
Tmelt
°C
W/mK
mrad
0.011
1.5
1.8
0.1
0.155
1.2
0.2
0.002
0.138
0.1
0.415
0.28
0.29
0.124
0.138
0.048
0.250
0.224
64.97
300
441
25
440
1100
35
39.98
249
69
69
355
310
131
466
97.2
120.2
138.8
37
10.4
8.3
10
5.6
1.0
2.6
44
10.8
23
23
5.6
4.6
2.56
4.5
7.1
8.9
17
9.25
1154
1180
1056
149
3043
5600
3.06
126
119
494
296
476
814
161
175
448
476
870
1278
2079
3000
2450
3550
3550
801
2800
660
660
2050
2200
1410
2650
1476
1660
1083
4.01
201
27.4
30
60
1800
6.3
1.15
42
237
237
27.21
180
163.3
300
10.7
21.9
322
0.04
90°
24
12.0
2.15
17.7°
2.65°
-1.6
3.01
13.6
2.21
12.0
15.4
4.30
0.89
3.4
9.64
Heat load on the monochromator crystals
For the consideration of the monochromator crystals placed around 515 m behind the undulator
(Fig. 1), the above considerations are too rough, since no temperature dependence of the thermal
properties is considered. The thermal conductivity will increase dramatically at lower temperatures
for silicon and diamond. For a non-steady-state thermal situation, the thermal diffusivity is the
crucial quantity and it increases by several orders of magnitude when going to low temperatures
(see left part of Fig. 2). On the other hand, the heat capacity will also decrease and the temperature
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rise per X-ray pulse will be higher at low temperatures than at room temperature. For example, the
temperature increase per pulse for silicon at room temperature according to the Table 1 is 8 K,
while at 8 K base temperature the hot spot on the crystal will be at 126 K after the first pulse. It will
take longer to remove the heat from the centre of the beam footprint compared to areas on the edge
that still have very high thermal diffusivity. The temperature profile will be therefore different from
equation (1) and will qualitatively change its shape during the pulse train. To model this non-trivial
temperature behaviour, an IDL-program was developed that calculates the temperature distributions
in a monochromator crystal during a pulse train based on individual pulses. Two geometries were
considered: A transmission Laue-geometry for diamond monochromator crystals and a Bragggeometry, which would be more favourable for silicon monochromators, where large single crystals
are available. This is complementary to some previous work [5,6] with finite element calculations
on diamond and silicon monochromator crystals in these geometries. It turns out that a diamond
Laue-crystal cooled to a base temperature of 77 K would survive a pulse train, however with a peak
temperature of 1000 K at the end of a bunch train (upper solid curve in left Fig. 2). If the diamond
disc has a profile with 100 µm in the centre where the beam hits but much thicker (18 mm) at the
edges, the heat can be removed very efficiently which leads to a constant temperature around 200 K
for most of the pulse train (lower solid curve). This curve is very good agreement with reference
[5]. Natural silicon and 28Si were calculated in Bragg-geometry with a base temperature of 8 K,
similar to calculations in [6]. However, because in [6] a reduced electron bunch filling pattern and a
larger X-ray beam spot were assumed, the results deviate now significantly, showing that the
crystal would reach the melting condition after only 10% of the pulse train. In conclusion, as a first
monochromator, a diamond-based design with liquid nitrogen cooling seems to be the most
promising approach at this moment. A particular challenge will be a sufficiently good crystal
quality of the diamonds and the fabrication of profiled disc as suggested in [5]. Silicon
monochromators will be useful as secondary optical elements closer to the experimental stations.
Figure 2: Left: thermal diffusivity for silicon and diamond. The straight line is the high temperature limit
from phonon scattering in silicon. Right: temperature for different crystal monochromators during one pulse
train. The upper solid curve is a diamond disc, the lower solid curve is a diamond disc with a profile. The
two silicon curves are on top of each other. The dotted lines represent the base temperatures of 8 K (silicon)
and 77 K (diamond).
References
[1] J. Arthur et al., LCLS – Conceptual design report, SLAC-R-593, SLAC, Stanford (2002), and R. A.
London et al. SPIE 4500, 51-62 (2001)
[2] M. Altarelli et al., The European XFEL Technical Design Report, DESY, Hamburg (2006)
[3] S. P. Hau-Riege et al. Applied Phys. Lett. 90, 173128 (2007)
[4] D. D. Ryutov, Rev. Sci. Instrum. 74, 3722 (2003)
[5] J. Heuer, H. Schulte-Schrepping in TESLA, Technical Design Report, p.V248-V250 (2001)
[6] L. Zhang, A.K. Freund, T. Tschentscher, H. Schulte-Schrepping, SRI 2003 Conference, AIP Conf.
Proc. 705, 639 (2004)
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