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Synchrotron monochromator heating problem, cryogenic cooling solution
Nuclear Instruments and Methods in Physics Research A 456 (2001) 163}176
Synchrotron monochromator heating problem, cryogenic cooling solution P. Carpentier *, M. Rossat, P. Charrault , J. Joly , M. Pirocchi , J.-L. Ferrer , O. KamK kati , M. Roth Laboratoire de Cristallographie et Cristollogene& se des Prote& ines (LCCP), Institut de Biolologie Structurale Jean-Pierre Ebel, 41 Rue Jules Horowitz, 38027 Grenoble Cedex 1, France European Synchrotron Radiation Facility (ESRF), 6 Rue Jules Horowitz, BP 220, 38043 Grenoble-cedex, France Received 21 March 2000; accepted 15 May 2000
Abstract We report on a new prototype of a cryogenic cooling system which is running on the beamline BM30-A/FIP at the ESRF Grenoble. This system has been specially designed to cool the "rst crystal of a double crystal monochromator subject to the white beam delivered by a bending magnet X-ray source on a third generation synchrotron. The proposed cryogenic cooling loop represents an alternative solution between the usual water cooling method and the very e$cient but expensive cryogenic system already existing for insertion devices. The new concept of this cooling loop, an open thermodynamic system, is `optimala from the point of view of the economical versus performance ratio to dissipate a heat load up to 1 kW. For such a heat load range the e$ciency of classical water cooling system begins to fail, and the thermal dilatation of silicon a!ects the monochromating process. Some details of the loop and the running principle are exposed in this paper. A particular attention has been paid to the conception of the crystal and the heat exchangers. The distortion of the crystal has been veri"ed in practice from the measurements of the double crystal rocking curves under di!erent experimental conditions and compared with the expected theory. 2001 Elsevier Science B.V. All rights reserved. PACS: 07.85.Qe; 07.20.Mc; 41.50#h Keywords: Synchrotron radiation; X-ray; Monochromator; Heat load; Thermal distortion; Cryogenic cooling
1. Introduction The performance of a cylindrically bent double crystal X-ray monochromator, based on the Bragg law, is related to the geometrical quality of the crystallographic di!racting lattices of the two crystals. For symmetric re#ection monochromators, * Corresponding author. Tel.: #33-47-688-2789; fax: #3347-688-5122. E-mail address: [email protected] (P. Carpentier).
the crystallographic di!racting planes of the "rst crystal must be as #at as possible to obtain an optimal monochromating process. The second crystal is curved with the best cylinder shape possible for the sagittal focusing process to obtain a beam spot of minimum size at the sample. However, the monochromator performance is always degraded by unwanted distortions of the crystallographic lattices. Those deformations are caused on the one hand by the heat load on the "rst crystal exposed to the `white beama, and on the other
0168-9002/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 0 ) 0 0 5 7 8 - 7
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hand by defects of mechanical bending of the second crystal. As there are no known faultless solutions to those two problems, development of new cooling systems and new benders is still a major preoccupation in the "eld of research of optical elements to optimize the monochromators performance on synchrotron X-ray radiation sources. This work is dedicated to the improvement of the re#ection quality of the "rst crystal of a double crystal X-ray monochromator. The aim of this paper is to present the principles and the "rst tests of a new cryogenic cooling system. This system has been adopted as a `solutiona concerning the thermal distortion problem of the monochromator "rst crystal for `FIPa, the French Collaborating Research Group beamline for Investigation of Protein structures.
2. Monochromator heat load 2.1. Source and beamline description The station FIP is located at the front-end BM30-A of the European Synchrotron Radiation Facility (ESRF), a 6 GeV energy electron storage ring. The BM30-A X-ray source consists of a bending magnet with a "eld of 0.4 T (E "9.576 keV). A Multiwavelength Anomalous Di!raction (MAD)
X-ray experiment in biocrystallography requires a monochromatic beam with a variable wavelength chosen between 5 and 25 keV and a narrow relative energy resolution *E/E+10\. The remaining energy emitted by the source is useless and leads to heat loads on the optical devices of the beamline. The optics presented schematically in Fig. 1 consists of two platinum-coated mirrors and a double silicon crystal monochromator. The monochromator has been designed to work either with a pair of Si(1 1 1) or Si(3 1 1) crystals, with the di!racting crystallographic planes parallel to the re#ecting face. The energy of the quasi-monochromatic beam is determined by the modi"ed Bragg law
d 2d sin(H )"mj 1# , FIJ sin(H )
(1)
where d is the spacing of crystallographic planes FIJ of Miller indices hkl, and the refractive index of silicon is considered to be n"1!d!ib. The re#ected angle H is shifted from the Bragg angle H such as H "H #2d/sin(2H ) (H "H when n"1). The absorption of the full power emitted by the source is distributed between the "rst mirror, the "rst crystal of the monochromator, and the di!erent beryllium windows. The "rst mirror is
Fig. 1. Schematic view of the optic of the beamline FIP.
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a cut-o! "lter of high energy and then is used as a harmonics rejector (relieving the beam from its harmonic components, m'1 in Eq. (1)), its re#ectivity can be simpli"ed to a drastic step falling down from 1 to 0 above the cutting energy E . The cut-o! angle a is related to ! ! E through the platinum-coated mirror constant ! a E "83(mrad keV). The level E above which ! ! ! the energy is absorbed by the mirror is in practice chosen at of the value of the desired monochro matic beam energy E [1]. The remaining lowerH energy dissipated by the "rst crystal is then expressed as
P(E )"10 H
#H
e\I#R (E) dX dE.
(2)
Fig. 2. Power and #ux versus wavelength on the "rst crystal of the monochromator of FIP.
X
The brightness (E) is the number of photons emitted by the source per solid angle and time unit in 0.1% of spectral band width *E/E, k(E) is the X-ray absorption through a thickness `ta of beryllium. The solid angle is determined from the horizontal divergence of the source W "2 mrad, & and from the vertical divergence which can be approximated as W +mE\ . The total power 4 emitted by the source, 140 W for a current of 200 mA, is calculated using expression (2) over the whole energy spectrum (E "R) and with a zero H thickness of beryllium window. The power load on the "rst crystal P(E ) (with 1.5 mm of beryllium) is H presented in Fig. 2. The #ux U received by the "rst crystal is de"ned by the ratio of the power P(E ) H over the footprint area, which corresponds to the beam divergence with a photon energy running from 0 to 4E /3 after the "rst mirror. Fig. 2 shows H those #uxes versus wavelength for the two possible crystallographic orientations Si(1 1 1) and Si(3 1 1). The power of the white beam and the beam spot area on the "rst crystal are both increasing with the desired energy, and those antagonist e!ects lead to a maximum of #ux at about 0.7 As . 2.2. Thermal distortions, cryogenics versus water cooling In the case of an ideal crystal, the quasi-monochromatic beam extracted from the incident paral-
lel white beam is re#ected with a unique angle H and a spectral spreading corresponding to the Darwin width D. However, the high #uxes (above 10 W/m in Fig. 2) produce a thermal distortion which spreads the beam over an angular area *H detailed as *H"*H #*H #*H . U
(3)
The "rst and second terms on the right-hand side correspond to the spreading produced by the slope error in the crystallographic planes, whereas the last one is related to the thermal variation of re#ecting planes spacing. Those e!ects can be quickly evaluated for a non-thick crystal with respect to the spot size (c+10\ m) by a simple one-dimensional model, the heat is #owing from the re#ected face to the cooled back side with no volumic spread. A temperature gradient takes place between those two faces according to Fourier law: *¹"Uc/k. It produces strains in respect to the thermal expansion coe$cient a. The di!erential dilatation between the re#ecting and the back faces is responsible of the general bowing of the crystal. If ¸ is the beam foot-print size, the total spreading angle due to that thermal curvature is
Ua *H "¸ . k
(4)
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Table 1 Linear expansion and thermal conductivity of silicon extract from Ref. [2] Temperature (K)
a(k\)
k (W/mK)
"a"/k (W/mK)
80 125 300
!5;10\ 0 2.6;10\
1340 600 148
3.75;10\ 0 1.76;10\
The variation of #ux from 0 to U in the front face produces a thermal bump at the center of the spot. The height of this bump is approximated by summation of the dilatation lengths along the crystal thickness f"acU/2k. It is fair to consider a Gaussian shape to this bump with ¸ as full-width at half-maximum, the spreading corresponding the maximum slope is then
c *H "1.43 2¸
Ua . k
(5)
The spreading due to the thermal variation of dspacing, is given by the di!erential form of the Bragg law (Eq. (1)): *H "tan H(*d/d). The U relative lattice variation *d/d under the thermal gradient up to the X-rays attenuation length in silicon is weak, thus *H is negligible with BU respect to the other terms (4 and 5). The pertinent parameter describing the thermal distortion of the crystal is the ratio a/k (expressions (4) and (5)). Thus, the only way to minimize the thermal distortion is to increase the value of k and reduce the value of a. Those physical characteristics of the materials vary with temperature. The values presented in Table 1 [2] show that the thermal distortion of silicon is strongly reduced at low temperature. The idea of the advantage of using monochromators at cryogenic temperature was "rstly proposed by Rhen [3] and Bilderback [4] and the "rst test on synchrotron source has been done by Joksch [5]. The optimal temperature 125 K for which the ratio "a"/k falls down to zero is di$cult to reach in practice. However, a refrigeration below 125 K brings a very signi"cant gain of distortion of at least a factor of 40 compared to the room temperature (Table 1). Liquid nitrogen (LN2) is the most
Fig. 3. Comparison beween thermal angle spreading and Darwin widths (silicon 3 1 1 and 1 1 1), in the water and cryogenic cooling cases.
appropriate cryogenic #uid which is able to ful"ll this condition [6]. Some numerical estimations of the above model in di!erent cases demonstrate the usefulness of a cryogenic cooling. The values of the #uxes are extracted from Fig. 2, the expansion coe$cient and thermal conductivity from Table 1, the beam size ¸ is estimated from vertical divergence. The representation of the re#ecting spreading width curves versus wavelength (from 0 to 3 As ) are drawn in Fig. 3 and compared to the Darwin widths D for both Si(1 1 1) and Si(3 1 1) crystals. It clearly reveals that a cryogenic cooling system restricts the distortion to a negligible level compared to D. A water cooling system produces a spreading angle at least comparable to D around 1 As in the case of the Si(1 1 1), and the distortion of the Si(3 1 1) is unfavorable up to 1.5 As . The e$ciency of usual monochromator cooling systems reach its limit with the power supply by a bending magnet source at the ESRF. The thermal bump proportional to c (expression (5)) can be reduced using a very thin crystal, but it becomes too #imsy and subject to mechanical deformations. There is anyway nothing to do against the bowing e!ect by the way of usual cooling (Eq. (4)). The solution of the thermal problem requires necessarily a cryogenic coolant, and the considerable gain brought justi"es clearly the use of such a method.
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3. Cooling machine, principle and description 3.1. Thermodynamic considerations The running principle of the proposed cooling system is based on the force convection heat exchanges. The heat supplied to the crystal (hot point) is dissipated by boiling in a LN2 pool (cold point). The heat is carried between those two points by LN2 moved by a pump in a refrigeration loop (see the photograph in Fig. 4b). The main trouble is the mechanical vibrations of the "rst crystal that can be induced by LN2 #owing or boiling during transport, with dramatical consequences on the quality of the beam at the focus point. Thus, to avoid any boiling the heat must be stored in the LN2 as sensible heat (Table 2). The thermodynamic cycle of this system is schematically described in Fig. 4a, and involves four distinct phases. (1) Adiabatic LN2 compression +P , ¹ ,P +P #*P, ¹ ,. The LN2 is pumped from the cryogenic tank and pressurized inside the cooling loop. This adiabatic e!ect increments the #uid enthalpy in respect to the work supplied *h"=3/od (Table 2), but without signi"cant change of volume and temperature. (2) Isobaric heat absorption +P #*P, ¹ ,P +P #*P, ¹ #*¹,. The thermal load Q of the monochromator is transfered to LN2 through the exchanger as sensible heat P"odC *¹ (Eq. (2), . Table 2). (3) Isobaric heat rejection +P #*P, ¹ # *¹,P+P #*P, ¹ ,. The heat Q stored in the LN2 is evacuated through the sub-cooling coil by LN2 boiling in the bath with an evaporation rate P/¸o (Eq. (2), Table 2). (4) Isenthalpic LN2 expansion +P #*P, ¹ ,P +P , ¹ ,. The LN2 is expanded back in the cryogenic tank through a laminar expansion valve without exchanging energy. It produces an excess of enthalpy *h dissipated through boiling, but the evaporated fraction of expanded LN2 *h/¸+10\ is negligible. The LN2 is stored in the tank at the liquid}gas melting line (P "1.013 bar, ¹ "77.4 K). This open loop needs nitrogen mass transfer across the boundaries of the system: The evaporated nitrogen
Fig. 4. (a) Schematic representation of the thermodynamic open system; (b) Photograph of the cryogenic loop, with details: (A) pump; (B) LN2 transfer lines; (C) heat exchanger; (D) sub-cooler; (E) expansion valve; (F) LN2 bath; (H) monochromator; (RX) beam; (Q) heat load and (W) work supply.
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Table 2 Thermophysical properties of LN2 at 77.4 K and 1.013 bar extracted from Ref. [8] Density o (kg/m) Speci"c heat C (J/kg K) . Vaporization latent heat ¸ (J/kg) Viscosity k (N s/m) Thermal conductivity k (W/m K)
809 2064 199;10 1.52;10\ 0.135
in the tank is automatically replaced. The theoretical LN2 consumption only due to the heat dissipation in phase 4 is calculated between 0.5 to 2 l/h. In practice, there is in addition about 100 W of thermal losses along the loop, thus the LN2 real consumption of the machine is situated between 3 and 4 l/h. The e$ciency of the system is de"ned by the cryogenic circulator, a centrifugal pump BNCP43 for LN2 #uid manufactured by Barber}Nichols (Denver, CO, USA) [7]. The LN2 pressure increment in the loop *P in phase 1 is controlled by the head #ow curves of the pump (rotation frequency and #ow rate). The phase diagram of nitrogen [8] allows to know *¹ in phase 2, the
maximum increase of temperature without boiling for the given *P. The power dissipation e$ciency of the loop in phase 2 for di!erent running modes is presented in Fig. 5.
Fig. 5. Performance of the cooling loop at di!erent pump rotation frequencies and as a function of the #ow rate.
3.2. Heat absorber [9,10] The heat absorbers consist of two copper cooling jackets clamped to the crystal side faces via an indium foil in order to ensure a good thermal contact and to relax the thermal di!erential dilatation strains with silicon. Those absorbers are each composed of 28 cooling "ns regularly spaced and carved from the cooper block (see Fig. 6). The #ow is then divided in the ducts between the "ns to reduce the #uid velocity and then ensure a laminar #owing mode preventing the crystal from the turbulent #ow vibrations. The low e$ciency of heat transfer in laminar #owing mode is compensated by the extended "n surface, the e!ective heat exchange coe$cient of the crystal}exchanger interface is multiplied by 8. The heat transfer features
Fig. 6. Schematic representation of the crystal geometry and crystal-absorbers assembly.
obtained by the usual methods are summarized in Table 3. The indium foil and of the copper absorber plate produces a negligible gradient of temperature. The gradient induced by the heat exchange process at the LN2/absorber interface *¹" P/(2)S)h ) is estimated at about 5 K for an average heat load of 100 W. The limit of the e$ciency of this system is situated around 1 kW (*¹(50 K and Re(2100).
P. Carpentier et al. / Nuclear Instruments and Methods in Physics Research A 456 (2001) 163}176 Table 3 Heat exchanger characteristics Cooling "ns with size (section * length mm) 1*7*50 Rectangular duct size between "ns (mm) 1*7 Re: Reynold number (at 100 l/h or 0.136 m/s) 1280 Nu: (Nusselt number for rectangular duct 1;7 mm) 6 Prandtl number Pr"kC /K 2.32 H: Heat exchange coe$cient (W/m K) 460 h : E!ective heat exchange coe$cient (W/m K) 3600
4. Crystal design 4.1. Geometrical description The crystal is bulky but it is proposed to evacuate the heat in the volume and to decrease the
169
where q(r) is the heat generated at the point r in the crystal per unit of volume and time, which is nearly a surface e!ect owing to the X-ray absorption of silicon, q(r)+UH(a !"x")H(b !"y")d(z!c) (d and H are the Dirac and Heaviside functions). The boundary conditions are k ) urad(T) ) n" 0 in the free faces and symmetry planes, k ) urad(T) ) n"h(¹!¹ ) in the cooling faces, and k ) urad(T) ) n"U in the beam spot area. The vector n is the outward normal to the considered face. The method of separation of variables leads to the following solution to Eq. (6): ¹"X(x)>(y)Z(z). The homogeneous boundary conditions provide a solution expended in terms of Fourier cosine series for X and of Sturm}Liouville cosine series for >, whereas the non-homogeneity along z leads to an hyperbolic cosine series for Z,
U A cos(mpx/a)cos(k py/b)cos h(((m/a)#(k /b)pz) K L L L ¹"¹ # . k p((m/a)#(k /b)sin h(((m/a)#(k /b)pc) K L L L
gradient of temperature in the re#ecting face. Following the system of axes proposed in Fig. 6 with the origin at the center of the bottom face, the crystal is a parallelepiped delimited by !a(x(a, !b(y(b, and 0(z(c, with two planes of symmetry; x"0 and y"0. The beam spot is rectangular de"ned by !a (x(a ,!b (y(b and z"c. b is given by the vertical divergence of the beam, and a by the horizontal divergence and the Bragg angle. The heat is evacuated through the lateral faces y"$b, all the other surfaces are free.
The temperature gradient through the crystal in cryogenic conditions with the concerned #ux is small enough (*¹+Uc/k) to accurately describe the thermal steady state by the linear Poisson equation (k constant ) [13]. *¹"!q(r)/k,
In expression (7) m, n are positive integers, and the value k is the nth positive solution of the equation L resulting from the Robin boundary conditions (for large values k +n) L k (k p)"cotan(k p). L hb L The Fourier/Sturm}Liouville coe$cients are given by 2a sin(k pb /b) L A " , L ak p(sin(2k p)/2k p#1) L L L 4 sin(mpa /a)sin(k pb /b) L A " K L mk p(sin(2k p)/2k p#1) L L L
4.2. Thermal problem, analytic solution [11,12]
(6)
(7)
if m'0.
The distribution of cryogenic temperature in a quarter of the re#ecting face is depicted in Fig. 7 with a contour level plot drawn with steps of 0.1 K. The heat #ows (rather along y) from the center of the crystal to the exchangers, with "rst a spherical shape which becomes progressively planar. The
170
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Fig. 7. Surface temperature pro"le in the re#ecting face for the cryo-cooled Si(1 1 1) at j"1 As .
Table 4 Gradient of temperature along x, y and z, with di!erent cooling processes and crystallographic orientations
LN2 LN2 Water Water
Si (hkl)
¹ (K)
¹ (K)
¹ (K)
*¹ (K) V
*¹ (K) W
*¹ (K) X
(1 1 1) (3 1 1) (1 1 1) (3 1 1)
77.4 77.4 293 293
81.3 81.3 293.4 293.4
82.46 82.7 302.0 304.2
0.46 0.74 4.09 6.60
0.54 0.73 6.47 8.35
1.02 1.26 7.78 10.0
rectangle in the base represents the limits of a quarter of the beam spot. Expression (7) is applied to compare di!erent cooling modes and crystallographic orientations at 1 As . The results summarized in Table 4 demonstrate the high reduction of thermal gradient through the crystal brought by cryo-cooled silicon. The thermal gradient distribution inside the crystal leads to a bulk distortion. The slope error in the re#ecting face results in a combination of the bowing and bump e!ects (Eq. (3)), treated independently. The slope along the `ya-axis plays a secondorder role, and will be neglected in the calculation. The bump pro"le f (x, y) may be approximated XA as the simple summation along the `za-axis of all the length elements thermally expanded a(¹(x, y, z)!¹ ) dz from the node (x, y, z) to the node (x, y, c). The estimated slope relative to the
thermal bump in the re#ecting face is given by *f (x, y) *H +! XA *x Ua " k K L A mp sin(mpx/a)cos(k py/b) L ; K L . a((m/a)#(k /b))p L
(8)
The bowing e!ect results from the thermal di!erential expansion of the di!erent `xya sheets under the thermal gradient. The expansion f (y, z) along V to the `xa-axis, of a segment of initial length x is a approximated as the summation of the thermally expanded length elements a(¹(x, y,z)!¹ ) dx from the "xed node (0, y, z) to the node (x, y, z). The
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Fig. 8. Re#ecting face slope error along the `xa axis for the Si(1 1 1) case at j"1 As .
bowing slope error of the crystal along `xa may be evaluated as the "rst derivative of this expansion versus z, *f (y, c) *H + V *z Ua " k
A a mpx k py K L sin cos L . mp a b K L (9)
The distribution of the absolute value of the slope error *H""*H #*H " for a Si(1 1 1) crystal along the `xa-axis is drawn in Fig. 8 as a threedimensional map with a logarithmic scale, for both cryogenic and ambient cooling process, in a quarter of the re#ecting face. The calculation has been carried out on the basis of a beam spot corresponding to a wavelength of 1 As . The cryogenic cooling process provides a concave slope error nearly two order of magnitude below the convex one produced by the ambient cooling mode. This latter spreading angle is comparable to the Darwin width.
From 293 to 77.4 K the thermal linear expansion *¸/¸ of the copper exchangers is one order of magnitude above the silicon crystal. It produces, however, a negligible additional mechanical deformation. Firstly, the exchangers are clamped to the crystal via the faces `ya, whose dilatation has only a secondary e!ect on the slope error along the direction of beam propagation `xa (Fig. 6). Moreover, the indium foil between copper and silicon #ows and relaxes the di!erential dilatation between those two materials. Some simulations by Finite Element Analysis (FEA) have con"rmed those assumptions.
5. Monochromator performance 5.1. Theoretical consideration We can evaluate the consequence of the "rst crystal slope error *H(x, y) on the performance of the monochromator by ray tracing calculation, with a second crystal assumed without defects. The central ray (x"0, y"0) is de#ected by 2H and
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!2H inside the monochromator, to select the wavelength j . Any other ray falling at the point (x, y) on the "rst crystal area dx dy gives a slightly di!erent wavelength j with the glancing angle H "H #*H(x, y). The "rst crystal re#ected power inside a spectral band dj centered on j is dP(j)"I R(j!j ) dj. It is proportional to the re#ectivity R(j!j ) equated in the zero absorp tion approximation as [14]
4 D , R(H!H )" H!H $ (H!H )! D 4 (10) where D is the Darwin width de"ned as 2r jC"F(hkl)"e\+ D(hkl)" . (11) R (x, y,q)" R(g)R(g#*H(x, y)#q) dg. 4D \ (12)
If each ray (x, y) has an equal contribution to the intensity of the whole beam, the rocking curve R(q) may be expressed as the integration of the double power ratio carried out over the whole region of the beam spot.
? @ 1 R(q)" R (x, y,q) dx dy. (13) 4a b \? \@ Fig. 9a and b present the numerical results of the rocking curves estimated from Eq. (13) for both the Si(1 1 1) and Si(3 1 1) at a wavelength of 1 As . The three-dimensional graphical representations depict the evolution of the rocking curves for an absorbed typical power running from 0 to 200 W in the ambient temperature cooling case. Under the cryogenic cooling conditions, owing to the crystal
low distortion, the heat load has no visible e!ects on the rocking curves. The latter curves correspond strictly to those obtained under the ambient cooling conditions at the zero heat load limit. The single plot located in the back ground of each graph is the re#ecting power ratio of a perfect unique crystal with respect to the Darwin solution (Eqs. (10) and (11)), drawn for comparison. A contour level plot by step 0.1 relative to the re#ecting power ratio map is situated in the base of each graphs. Some typical numerical results of the rocking curves deterioration with the increasing power are summarized in Table 5; decrement of the maximum of re#ectivity and spreading of Full-Width at HalfMaximum (FWHM). The ideal rocking curves have been calculated using the Xop program [15] and a simulation based on Eq. (13) at 0 W, both results agree within 4%. 5.2. Experimental performance The tests of the present cryogenic cooling system and the performance of the monochromator have been conducted directly on the beam line FIP in its "nal working order, with the pair of Si(1 1 1) crystals and with 200 mA of current in the synchrotron ring. The veri"cation has "rstly concerned the beam position stability at focus point under di!erent working states of the cooling machine. The beam has been monitored through a direct X-ray camera and displayed on a video screen. We have noticed that there is no beam vibration up to a pump rotation speed of 4000 rpm. This observation has been con"rmed by rocking curves measurement. The curves of double re#ectivity remain unchanged up to 4000 rpm, and begin to broaden out above this limit. The weak vibrations appearing are due to the cryogenic pump which begins to work hard and to shudder. A perfect operating cycle, working smoothly, requires a pressure stability inside the cooling circuit kept better than 0.005 bar around the desired value, as it is clearly the case in a wide running range of the cryogenic circulator [7]. The practical running modes of the cooling machine which have been adopted on the beam line FIP are as soft as the thermal load requires, 2400 rpm near the low power density limit (at 1.75 As ) and 2700
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Fig. 9. Simulated rocking curves as a function of the power load at j"1 As : (a) Si(1 1 1); (b) Si(3 1 1).
rpm near the high #uxes limit (at 0.7 As ). The LN2 #ow rate is estimated from head #ow curves around 150 l/h. The stability of the beam position over a long period is also excellent owing to the constant
temperature of the cooling #uid (LN2) stored exactly at 77.4 K. The temperature of the crystal is continuously controlled by a thermocouple placed in the body of the crystal, the observed value
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Table 5 Comparison of rocking curves of Si(111) and Si(311) at di!erent heat loads (Values in parentheses: calculation from Xop [15])
Si(1 1 1) Si(3 1 1)
Unique crystal
Double crystal
0W
0W
100 W
200 W
D (lrad)
FWHM (lrad)
R
*
FWHM (lrad)
R
*
FWHM (lrad)
R
*
FWHM (lrad)
R
*
20.9 8.9
22.2 (22.9) 9.5 (9.6)
1 (0.97) 1 (0.97)
29.2 (30.4) 12.6 (12.8)
0.82 (0.79) 0.82 (0.76)
34 28
0.68 0.39
48 34
0.51 0.26
Fig. 10. Rocking curves of Si(1 1 1) as a function of wavelength: (䉫) experimental points; (- - -) computation for non-distorted crystal.
displays a perfect stability ((1 K) around 80 K. The pressure of LN2 inside the optic loop is also recorded by a captor located at the monochromator outgoing pipe. The absence of LN2 boiling is therefore carefully checked, a bubble formation is signed by two sudden consecutive jumps of pressure of opposite sign. Fig. 10 presents a comparison between the theoretical and the experimental Si(1 1 1) rocking curves. The double re#ectivity data has been collected at the focus point (see Fig. 1) with an ion chamber or/and with a photo-diode placed just behind. The measurements have been recorded at eight energies
sampled in the working range of the monochromator (see column 1 of Table 6). After the o!set subtraction from the whole experimental curves, the maxima positions have been scaled on the theory (see columns 2 and 3 of Table 6). One can unambiguously a$rm that the experimental double re#ectivity of the monochromator FIP takes the exact shape of the self-convoluted curve of a simple perfect silicon crystal re#ectivity at all energies. A slight discrepancy is however visible in the wings of the peaks, the observed values are lower than the theoretical ones. For the low values of intensity the used X-ray photons counters are probably less
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Table 6 Comparison between characteristics of theoretical and calculated Si(1 1 1) rocking curves at di!erent wavelengths FWHM(1): Darwin solution; FWHM(2): calculation from Xop [15]; FWHM(exp): measurement j (As )
R
*
H !H (lrad)
FWHM (1) (lrad)
FWHM (2) (lrad)
FWHM (exp) (lrad)
0.72 0.98 1.10 1.20 1.28 1.40 1.50 1.75
0.786 0.767 0.756 0.751 0.745 0.733 0.726 0.705
28.8 39.2 43.9 48.5 52.0 56.6 61.2 71.6
21.0 28.6 32.4 35.3 37.8 41.4 44.7 52.5
21.6 29.7 33.6 36.9 39.6 43.7 47.2 56.1
23.6 29.1 34.8 36.9 39.0 43.9 47.6 53.2
e$cient and unable to discriminate "ne variations, this e!ect is particularly pronounced at 1.75 As where the beam is less brilliant. Fig. 11 and Table 6 allow to compare the observed FWHM of the rocking curves to the expected ones. The observations agree with theory within 10% of deviation. The FWHM of double re#ectivity curves is (2 times the width of the simple re#ectivity curves in the Gaussian approximation. This latter width is itself 1.06 times the width of total re#ection D (Eq. (11)). Finally, one "nds that the double rocking curves width is given by, FWHM+1.5D. Taking Eq. (11) in the small Bragg angle approximation (sin(2H )+j/d ), D and then the FWHM of FIJ double re#ectivity are assumed to vary linearly with the wavelength. As depicted in Fig. 11, the experimental data (points and dashed line) behave as predicted (full line) with a "tted slope of 30 lrad/As close to the theoretical coe$cient. Moreover, the extrapolated residual value of the experimental FWHM at the zero wavelength is nearly null. The experiment has been repeated with a variable slits aperture (crystal heat load) and does not give a result signi"cantly di!erent. We can conclude that thermal and/or mechanical mounting distortion of the "rst crystal of the monochromator FIP is lower than its Darwin width. The re#ection Si(1 1 1) is not a!ected by the weak residual slope error (if it exists), and the double re#ectivity is set to the maximum. The geometry of the beamline FIP does not allow to pass the beam
Fig. 11. Rocking curves FWHM of Si(1 1 1) as a function of wavelength: (- -䢇- -) experimental points; (*-) computation for non-distorted crystal.
directly through the monochromator without a re#ection on the "rst mirror. Thus, it has not been possible to go further and to measure the "rst allowed harmonic Si(3 3 3), a sharper re#ection which is sensitive to "ne crystal deformations.
6. Discussion The analytical point of view shows that the slope error of the monochromator "rst crystal is driven by the expression U;+geometrical factor,;+a/k,. The #ux U emitted by a bending magnet source on a third generation synchrotron (up to 10 W/m) produces the heat load limit on optical elements for the e$ciency of usual cooling systems. For the Si(1 1 1) crystal, the deterioration of the monochromating process appears at short wavelengths expressed as a loss of intensity and energy resolution. As a typical example of protein crystallography X-ray experiment at the maximum #ux (around 0.7 As ), we quoted the use of the anomalous signal near the LIII edge of scatters belonging to the actinide series of atoms (Th, Pa and U). In an experiment requiring a monochromatic beam with a high-energy resolution (as X-ray spectroscopy
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studies) the monochromator can be equipped with crystals of higher Miller indices as the Si(3 1 1). The thermal distortion then becomes very problematic in the whole energy range, and "ne experimental details are lost due to the poor resolution of the apparatus. Up to now, one plays with the geometrical factor by devising more e$cient crystal shapes to maintain a reasonable thermal distortion, so as to go on working with an ambient thermalisation system. However, this restricted strategy comes up quickly to unrealistic complex or fragile silicon crystals leading to worst mechanical distortions. For the "rst time on a bending magnet source, we have chosen to act on the factor +a/k, using liquid nitrogen as coolant, to solve de"nitively the heat load problem. Thus, our monochromating crystal has a simple bulky parallelepipedic shape (nearly cubic) and then is free of mechanical distortion. For economical reasons, we have devised a new cryogenic cooling loop with a novel optimal design, made up of a reduced number of elements compared to the existing systems. The running principle is radically di!erent, based on an open thermodynamic system: the liquid nitrogen closed cooling circuit passes via the cryogenic tank, which is open to the external media and allows to evaporate nitrogen during the energy dissipation process and to feed the system with fresh #uid. The heat dissipation e$ciency is limited on one hand by the maximum #ow rate in laminar mode, and on the other by the maximum pressurisation o!ered by the pump which prevents no monochromator vibrations. This e$ciency limit is situated at about 1 kW, one order of magnitude above our needs. The experimental rocking curves recorded on the Si(1 1 1) re#ection con"rm that the "rst crystal slope error is less than the Darwin width. As expected, the performance of the cryocooled monochromator FIP is then optimum from the thermal load point of view in its full working range. The two crystals of the monochromator maintained at 80 and 294 K, respectively exhibit a di!erence in crystallographic plane spacing *d such as *d/d"!2.3;10\. It leads to the only nuisance that the two crystals are then not parallel according to the Bragg law. Thus, the direction of the monochromator exit beam walks with the Bragg angle by twice *H"!tan(H )*d/d. It corresponds to a shift of
the beam from 1.5 to 5 mm at the focus point which is, however, easily corrected by the second mirror to keep a "xed output. A prototype of this cryogenic cooling system is running on the beamline FIP (BM30-A) at the ESRF, Grenoble, since January 1999, and gives full satisfaction. Acknowledgements The authors would like "rstly to acknowledge G. Ardin from J.L.S. Inc. for the cryogenic realization of excellent quality. This work was performed with the technical support of the SERAS CNRS, we are grateful to J.P. Roux for his assistance, D. Grand for the technical drawings of the machine and P. Jeantey for the FEA simulation. From the ESRF optics group, we would like also to thank Dr. J. Hoszowska for fruitful discussions and J. P. Vassali for the perfect silicon crystals preparation. References [1] J.-L. Ferrer, J.-P. Simon, J.-F. Berar, B. Caillot, E. Fanchon, O. KamK kati, S. Arnaud, M. Guidotti, M. Pirocchi, M. Roth, J. Synchrotron Rad. 5 (1998) 1346. [2] Y.S. Touloukian, R.W. Powell, C.Y. Ho, P.G. Klemens (Vol. 1); Y.S. Touloukian, R.K. Kirby, R.E. Taylor, T.Y.R. Lee (Vol. 13), Thermophysical Properties of Matter, Vol. 1 and 13, IFI/Plenum Press, New York (recommended values), 1975. [3] V. Rhen, Proc. SPIE 582 (1985) 238. [4] D.H. Bilderback, Nucl. Instr. and Meth. A 246 (1986) 434. [5] S. Joksch, G. Marot, A. Freund, M. Krisch, Nucl. Instr. and Meth. A 306 (1991) 386. [6] G. Marot, Opt. Eng. 34 (2) (1995) 426. [7] Barber-Nichols (Denver, CO, USA), Cryogenic Circulator BNCP43 for LN2 #uid, 1998. [8] B.A. Younglove, J. Phys. Chem. Ref. Data 11 (Suppl 1) (1982), 62. [9] F.D. Michaud, Nucl. Instr. and Meth. A 246 (1986) 444. [10] G. Marot, M. Rossat, A. Freud, S. Joksch, H. Kawata, L. Zhang, E. Ziegler, L. Berman, D. Chapman, J.B. Hastings, M. Larocci, Rev. Sci. Instr. 63 (1) (1992) 477. [11] B.X. Yang, M. Meron, Y. Ruan, W. Schildkamp, Proc. SPIE 1997 (1993) 302. [12] G.S. Knapp, M.A. Beno, C.S. Rogers, C.L. Wiley, P.L. Cowan, Rev. Sci. Instr. 65 (9) (1994) 2792. [13] H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, Oxford University Press, Oxford, 1959. [14] R.W. James, The Optical Pricinples of the Di!raction of X-rays, Ox Bow Press, Woodbridge, CT, 1982. [15] M. Sanchez del Rio, R.J. Dejus, Proc. SPIE 3152 (1997) 14.
Synchrotron monochromator heating problem, cryogenic cooling solution P. Carpentier *, M. Rossat, P. Charrault , J. Joly , M. Pirocchi , J.-L. Ferrer , O. KamK kati , M. Roth Laboratoire de Cristallographie et Cristollogene& se des Prote& ines (LCCP), Institut de Biolologie Structurale Jean-Pierre Ebel, 41 Rue Jules Horowitz, 38027 Grenoble Cedex 1, France European Synchrotron Radiation Facility (ESRF), 6 Rue Jules Horowitz, BP 220, 38043 Grenoble-cedex, France Received 21 March 2000; accepted 15 May 2000
Abstract We report on a new prototype of a cryogenic cooling system which is running on the beamline BM30-A/FIP at the ESRF Grenoble. This system has been specially designed to cool the "rst crystal of a double crystal monochromator subject to the white beam delivered by a bending magnet X-ray source on a third generation synchrotron. The proposed cryogenic cooling loop represents an alternative solution between the usual water cooling method and the very e$cient but expensive cryogenic system already existing for insertion devices. The new concept of this cooling loop, an open thermodynamic system, is `optimala from the point of view of the economical versus performance ratio to dissipate a heat load up to 1 kW. For such a heat load range the e$ciency of classical water cooling system begins to fail, and the thermal dilatation of silicon a!ects the monochromating process. Some details of the loop and the running principle are exposed in this paper. A particular attention has been paid to the conception of the crystal and the heat exchangers. The distortion of the crystal has been veri"ed in practice from the measurements of the double crystal rocking curves under di!erent experimental conditions and compared with the expected theory. 2001 Elsevier Science B.V. All rights reserved. PACS: 07.85.Qe; 07.20.Mc; 41.50#h Keywords: Synchrotron radiation; X-ray; Monochromator; Heat load; Thermal distortion; Cryogenic cooling
1. Introduction The performance of a cylindrically bent double crystal X-ray monochromator, based on the Bragg law, is related to the geometrical quality of the crystallographic di!racting lattices of the two crystals. For symmetric re#ection monochromators, * Corresponding author. Tel.: #33-47-688-2789; fax: #3347-688-5122. E-mail address: [email protected] (P. Carpentier).
the crystallographic di!racting planes of the "rst crystal must be as #at as possible to obtain an optimal monochromating process. The second crystal is curved with the best cylinder shape possible for the sagittal focusing process to obtain a beam spot of minimum size at the sample. However, the monochromator performance is always degraded by unwanted distortions of the crystallographic lattices. Those deformations are caused on the one hand by the heat load on the "rst crystal exposed to the `white beama, and on the other
0168-9002/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 0 ) 0 0 5 7 8 - 7
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hand by defects of mechanical bending of the second crystal. As there are no known faultless solutions to those two problems, development of new cooling systems and new benders is still a major preoccupation in the "eld of research of optical elements to optimize the monochromators performance on synchrotron X-ray radiation sources. This work is dedicated to the improvement of the re#ection quality of the "rst crystal of a double crystal X-ray monochromator. The aim of this paper is to present the principles and the "rst tests of a new cryogenic cooling system. This system has been adopted as a `solutiona concerning the thermal distortion problem of the monochromator "rst crystal for `FIPa, the French Collaborating Research Group beamline for Investigation of Protein structures.
2. Monochromator heat load 2.1. Source and beamline description The station FIP is located at the front-end BM30-A of the European Synchrotron Radiation Facility (ESRF), a 6 GeV energy electron storage ring. The BM30-A X-ray source consists of a bending magnet with a "eld of 0.4 T (E "9.576 keV). A Multiwavelength Anomalous Di!raction (MAD)
X-ray experiment in biocrystallography requires a monochromatic beam with a variable wavelength chosen between 5 and 25 keV and a narrow relative energy resolution *E/E+10\. The remaining energy emitted by the source is useless and leads to heat loads on the optical devices of the beamline. The optics presented schematically in Fig. 1 consists of two platinum-coated mirrors and a double silicon crystal monochromator. The monochromator has been designed to work either with a pair of Si(1 1 1) or Si(3 1 1) crystals, with the di!racting crystallographic planes parallel to the re#ecting face. The energy of the quasi-monochromatic beam is determined by the modi"ed Bragg law
d 2d sin(H )"mj 1# , FIJ sin(H )
(1)
where d is the spacing of crystallographic planes FIJ of Miller indices hkl, and the refractive index of silicon is considered to be n"1!d!ib. The re#ected angle H is shifted from the Bragg angle H such as H "H #2d/sin(2H ) (H "H when n"1). The absorption of the full power emitted by the source is distributed between the "rst mirror, the "rst crystal of the monochromator, and the di!erent beryllium windows. The "rst mirror is
Fig. 1. Schematic view of the optic of the beamline FIP.
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165
a cut-o! "lter of high energy and then is used as a harmonics rejector (relieving the beam from its harmonic components, m'1 in Eq. (1)), its re#ectivity can be simpli"ed to a drastic step falling down from 1 to 0 above the cutting energy E . The cut-o! angle a is related to ! ! E through the platinum-coated mirror constant ! a E "83(mrad keV). The level E above which ! ! ! the energy is absorbed by the mirror is in practice chosen at of the value of the desired monochro matic beam energy E [1]. The remaining lowerH energy dissipated by the "rst crystal is then expressed as
P(E )"10 H
#H
e\I#R (E) dX dE.
(2)
Fig. 2. Power and #ux versus wavelength on the "rst crystal of the monochromator of FIP.
X
The brightness (E) is the number of photons emitted by the source per solid angle and time unit in 0.1% of spectral band width *E/E, k(E) is the X-ray absorption through a thickness `ta of beryllium. The solid angle is determined from the horizontal divergence of the source W "2 mrad, & and from the vertical divergence which can be approximated as W +mE\ . The total power 4 emitted by the source, 140 W for a current of 200 mA, is calculated using expression (2) over the whole energy spectrum (E "R) and with a zero H thickness of beryllium window. The power load on the "rst crystal P(E ) (with 1.5 mm of beryllium) is H presented in Fig. 2. The #ux U received by the "rst crystal is de"ned by the ratio of the power P(E ) H over the footprint area, which corresponds to the beam divergence with a photon energy running from 0 to 4E /3 after the "rst mirror. Fig. 2 shows H those #uxes versus wavelength for the two possible crystallographic orientations Si(1 1 1) and Si(3 1 1). The power of the white beam and the beam spot area on the "rst crystal are both increasing with the desired energy, and those antagonist e!ects lead to a maximum of #ux at about 0.7 As . 2.2. Thermal distortions, cryogenics versus water cooling In the case of an ideal crystal, the quasi-monochromatic beam extracted from the incident paral-
lel white beam is re#ected with a unique angle H and a spectral spreading corresponding to the Darwin width D. However, the high #uxes (above 10 W/m in Fig. 2) produce a thermal distortion which spreads the beam over an angular area *H detailed as *H"*H #*H #*H . U
(3)
The "rst and second terms on the right-hand side correspond to the spreading produced by the slope error in the crystallographic planes, whereas the last one is related to the thermal variation of re#ecting planes spacing. Those e!ects can be quickly evaluated for a non-thick crystal with respect to the spot size (c+10\ m) by a simple one-dimensional model, the heat is #owing from the re#ected face to the cooled back side with no volumic spread. A temperature gradient takes place between those two faces according to Fourier law: *¹"Uc/k. It produces strains in respect to the thermal expansion coe$cient a. The di!erential dilatation between the re#ecting and the back faces is responsible of the general bowing of the crystal. If ¸ is the beam foot-print size, the total spreading angle due to that thermal curvature is
Ua *H "¸ . k
(4)
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Table 1 Linear expansion and thermal conductivity of silicon extract from Ref. [2] Temperature (K)
a(k\)
k (W/mK)
"a"/k (W/mK)
80 125 300
!5;10\ 0 2.6;10\
1340 600 148
3.75;10\ 0 1.76;10\
The variation of #ux from 0 to U in the front face produces a thermal bump at the center of the spot. The height of this bump is approximated by summation of the dilatation lengths along the crystal thickness f"acU/2k. It is fair to consider a Gaussian shape to this bump with ¸ as full-width at half-maximum, the spreading corresponding the maximum slope is then
c *H "1.43 2¸
Ua . k
(5)
The spreading due to the thermal variation of dspacing, is given by the di!erential form of the Bragg law (Eq. (1)): *H "tan H(*d/d). The U relative lattice variation *d/d under the thermal gradient up to the X-rays attenuation length in silicon is weak, thus *H is negligible with BU respect to the other terms (4 and 5). The pertinent parameter describing the thermal distortion of the crystal is the ratio a/k (expressions (4) and (5)). Thus, the only way to minimize the thermal distortion is to increase the value of k and reduce the value of a. Those physical characteristics of the materials vary with temperature. The values presented in Table 1 [2] show that the thermal distortion of silicon is strongly reduced at low temperature. The idea of the advantage of using monochromators at cryogenic temperature was "rstly proposed by Rhen [3] and Bilderback [4] and the "rst test on synchrotron source has been done by Joksch [5]. The optimal temperature 125 K for which the ratio "a"/k falls down to zero is di$cult to reach in practice. However, a refrigeration below 125 K brings a very signi"cant gain of distortion of at least a factor of 40 compared to the room temperature (Table 1). Liquid nitrogen (LN2) is the most
Fig. 3. Comparison beween thermal angle spreading and Darwin widths (silicon 3 1 1 and 1 1 1), in the water and cryogenic cooling cases.
appropriate cryogenic #uid which is able to ful"ll this condition [6]. Some numerical estimations of the above model in di!erent cases demonstrate the usefulness of a cryogenic cooling. The values of the #uxes are extracted from Fig. 2, the expansion coe$cient and thermal conductivity from Table 1, the beam size ¸ is estimated from vertical divergence. The representation of the re#ecting spreading width curves versus wavelength (from 0 to 3 As ) are drawn in Fig. 3 and compared to the Darwin widths D for both Si(1 1 1) and Si(3 1 1) crystals. It clearly reveals that a cryogenic cooling system restricts the distortion to a negligible level compared to D. A water cooling system produces a spreading angle at least comparable to D around 1 As in the case of the Si(1 1 1), and the distortion of the Si(3 1 1) is unfavorable up to 1.5 As . The e$ciency of usual monochromator cooling systems reach its limit with the power supply by a bending magnet source at the ESRF. The thermal bump proportional to c (expression (5)) can be reduced using a very thin crystal, but it becomes too #imsy and subject to mechanical deformations. There is anyway nothing to do against the bowing e!ect by the way of usual cooling (Eq. (4)). The solution of the thermal problem requires necessarily a cryogenic coolant, and the considerable gain brought justi"es clearly the use of such a method.
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3. Cooling machine, principle and description 3.1. Thermodynamic considerations The running principle of the proposed cooling system is based on the force convection heat exchanges. The heat supplied to the crystal (hot point) is dissipated by boiling in a LN2 pool (cold point). The heat is carried between those two points by LN2 moved by a pump in a refrigeration loop (see the photograph in Fig. 4b). The main trouble is the mechanical vibrations of the "rst crystal that can be induced by LN2 #owing or boiling during transport, with dramatical consequences on the quality of the beam at the focus point. Thus, to avoid any boiling the heat must be stored in the LN2 as sensible heat (Table 2). The thermodynamic cycle of this system is schematically described in Fig. 4a, and involves four distinct phases. (1) Adiabatic LN2 compression +P , ¹ ,P +P #*P, ¹ ,. The LN2 is pumped from the cryogenic tank and pressurized inside the cooling loop. This adiabatic e!ect increments the #uid enthalpy in respect to the work supplied *h"=3/od (Table 2), but without signi"cant change of volume and temperature. (2) Isobaric heat absorption +P #*P, ¹ ,P +P #*P, ¹ #*¹,. The thermal load Q of the monochromator is transfered to LN2 through the exchanger as sensible heat P"odC *¹ (Eq. (2), . Table 2). (3) Isobaric heat rejection +P #*P, ¹ # *¹,P+P #*P, ¹ ,. The heat Q stored in the LN2 is evacuated through the sub-cooling coil by LN2 boiling in the bath with an evaporation rate P/¸o (Eq. (2), Table 2). (4) Isenthalpic LN2 expansion +P #*P, ¹ ,P +P , ¹ ,. The LN2 is expanded back in the cryogenic tank through a laminar expansion valve without exchanging energy. It produces an excess of enthalpy *h dissipated through boiling, but the evaporated fraction of expanded LN2 *h/¸+10\ is negligible. The LN2 is stored in the tank at the liquid}gas melting line (P "1.013 bar, ¹ "77.4 K). This open loop needs nitrogen mass transfer across the boundaries of the system: The evaporated nitrogen
Fig. 4. (a) Schematic representation of the thermodynamic open system; (b) Photograph of the cryogenic loop, with details: (A) pump; (B) LN2 transfer lines; (C) heat exchanger; (D) sub-cooler; (E) expansion valve; (F) LN2 bath; (H) monochromator; (RX) beam; (Q) heat load and (W) work supply.
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Table 2 Thermophysical properties of LN2 at 77.4 K and 1.013 bar extracted from Ref. [8] Density o (kg/m) Speci"c heat C (J/kg K) . Vaporization latent heat ¸ (J/kg) Viscosity k (N s/m) Thermal conductivity k (W/m K)
809 2064 199;10 1.52;10\ 0.135
in the tank is automatically replaced. The theoretical LN2 consumption only due to the heat dissipation in phase 4 is calculated between 0.5 to 2 l/h. In practice, there is in addition about 100 W of thermal losses along the loop, thus the LN2 real consumption of the machine is situated between 3 and 4 l/h. The e$ciency of the system is de"ned by the cryogenic circulator, a centrifugal pump BNCP43 for LN2 #uid manufactured by Barber}Nichols (Denver, CO, USA) [7]. The LN2 pressure increment in the loop *P in phase 1 is controlled by the head #ow curves of the pump (rotation frequency and #ow rate). The phase diagram of nitrogen [8] allows to know *¹ in phase 2, the
maximum increase of temperature without boiling for the given *P. The power dissipation e$ciency of the loop in phase 2 for di!erent running modes is presented in Fig. 5.
Fig. 5. Performance of the cooling loop at di!erent pump rotation frequencies and as a function of the #ow rate.
3.2. Heat absorber [9,10] The heat absorbers consist of two copper cooling jackets clamped to the crystal side faces via an indium foil in order to ensure a good thermal contact and to relax the thermal di!erential dilatation strains with silicon. Those absorbers are each composed of 28 cooling "ns regularly spaced and carved from the cooper block (see Fig. 6). The #ow is then divided in the ducts between the "ns to reduce the #uid velocity and then ensure a laminar #owing mode preventing the crystal from the turbulent #ow vibrations. The low e$ciency of heat transfer in laminar #owing mode is compensated by the extended "n surface, the e!ective heat exchange coe$cient of the crystal}exchanger interface is multiplied by 8. The heat transfer features
Fig. 6. Schematic representation of the crystal geometry and crystal-absorbers assembly.
obtained by the usual methods are summarized in Table 3. The indium foil and of the copper absorber plate produces a negligible gradient of temperature. The gradient induced by the heat exchange process at the LN2/absorber interface *¹" P/(2)S)h ) is estimated at about 5 K for an average heat load of 100 W. The limit of the e$ciency of this system is situated around 1 kW (*¹(50 K and Re(2100).
P. Carpentier et al. / Nuclear Instruments and Methods in Physics Research A 456 (2001) 163}176 Table 3 Heat exchanger characteristics Cooling "ns with size (section * length mm) 1*7*50 Rectangular duct size between "ns (mm) 1*7 Re: Reynold number (at 100 l/h or 0.136 m/s) 1280 Nu: (Nusselt number for rectangular duct 1;7 mm) 6 Prandtl number Pr"kC /K 2.32 H: Heat exchange coe$cient (W/m K) 460 h : E!ective heat exchange coe$cient (W/m K) 3600
4. Crystal design 4.1. Geometrical description The crystal is bulky but it is proposed to evacuate the heat in the volume and to decrease the
169
where q(r) is the heat generated at the point r in the crystal per unit of volume and time, which is nearly a surface e!ect owing to the X-ray absorption of silicon, q(r)+UH(a !"x")H(b !"y")d(z!c) (d and H are the Dirac and Heaviside functions). The boundary conditions are k ) urad(T) ) n" 0 in the free faces and symmetry planes, k ) urad(T) ) n"h(¹!¹ ) in the cooling faces, and k ) urad(T) ) n"U in the beam spot area. The vector n is the outward normal to the considered face. The method of separation of variables leads to the following solution to Eq. (6): ¹"X(x)>(y)Z(z). The homogeneous boundary conditions provide a solution expended in terms of Fourier cosine series for X and of Sturm}Liouville cosine series for >, whereas the non-homogeneity along z leads to an hyperbolic cosine series for Z,
U A cos(mpx/a)cos(k py/b)cos h(((m/a)#(k /b)pz) K L L L ¹"¹ # . k p((m/a)#(k /b)sin h(((m/a)#(k /b)pc) K L L L
gradient of temperature in the re#ecting face. Following the system of axes proposed in Fig. 6 with the origin at the center of the bottom face, the crystal is a parallelepiped delimited by !a(x(a, !b(y(b, and 0(z(c, with two planes of symmetry; x"0 and y"0. The beam spot is rectangular de"ned by !a (x(a ,!b (y(b and z"c. b is given by the vertical divergence of the beam, and a by the horizontal divergence and the Bragg angle. The heat is evacuated through the lateral faces y"$b, all the other surfaces are free.
The temperature gradient through the crystal in cryogenic conditions with the concerned #ux is small enough (*¹+Uc/k) to accurately describe the thermal steady state by the linear Poisson equation (k constant ) [13]. *¹"!q(r)/k,
In expression (7) m, n are positive integers, and the value k is the nth positive solution of the equation L resulting from the Robin boundary conditions (for large values k +n) L k (k p)"cotan(k p). L hb L The Fourier/Sturm}Liouville coe$cients are given by 2a sin(k pb /b) L A " , L ak p(sin(2k p)/2k p#1) L L L 4 sin(mpa /a)sin(k pb /b) L A " K L mk p(sin(2k p)/2k p#1) L L L
4.2. Thermal problem, analytic solution [11,12]
(6)
(7)
if m'0.
The distribution of cryogenic temperature in a quarter of the re#ecting face is depicted in Fig. 7 with a contour level plot drawn with steps of 0.1 K. The heat #ows (rather along y) from the center of the crystal to the exchangers, with "rst a spherical shape which becomes progressively planar. The
170
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Fig. 7. Surface temperature pro"le in the re#ecting face for the cryo-cooled Si(1 1 1) at j"1 As .
Table 4 Gradient of temperature along x, y and z, with di!erent cooling processes and crystallographic orientations
LN2 LN2 Water Water
Si (hkl)
¹ (K)
¹ (K)
¹ (K)
*¹ (K) V
*¹ (K) W
*¹ (K) X
(1 1 1) (3 1 1) (1 1 1) (3 1 1)
77.4 77.4 293 293
81.3 81.3 293.4 293.4
82.46 82.7 302.0 304.2
0.46 0.74 4.09 6.60
0.54 0.73 6.47 8.35
1.02 1.26 7.78 10.0
rectangle in the base represents the limits of a quarter of the beam spot. Expression (7) is applied to compare di!erent cooling modes and crystallographic orientations at 1 As . The results summarized in Table 4 demonstrate the high reduction of thermal gradient through the crystal brought by cryo-cooled silicon. The thermal gradient distribution inside the crystal leads to a bulk distortion. The slope error in the re#ecting face results in a combination of the bowing and bump e!ects (Eq. (3)), treated independently. The slope along the `ya-axis plays a secondorder role, and will be neglected in the calculation. The bump pro"le f (x, y) may be approximated XA as the simple summation along the `za-axis of all the length elements thermally expanded a(¹(x, y, z)!¹ ) dz from the node (x, y, z) to the node (x, y, c). The estimated slope relative to the
thermal bump in the re#ecting face is given by *f (x, y) *H +! XA *x Ua " k K L A mp sin(mpx/a)cos(k py/b) L ; K L . a((m/a)#(k /b))p L
(8)
The bowing e!ect results from the thermal di!erential expansion of the di!erent `xya sheets under the thermal gradient. The expansion f (y, z) along V to the `xa-axis, of a segment of initial length x is a approximated as the summation of the thermally expanded length elements a(¹(x, y,z)!¹ ) dx from the "xed node (0, y, z) to the node (x, y, z). The
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Fig. 8. Re#ecting face slope error along the `xa axis for the Si(1 1 1) case at j"1 As .
bowing slope error of the crystal along `xa may be evaluated as the "rst derivative of this expansion versus z, *f (y, c) *H + V *z Ua " k
A a mpx k py K L sin cos L . mp a b K L (9)
The distribution of the absolute value of the slope error *H""*H #*H " for a Si(1 1 1) crystal along the `xa-axis is drawn in Fig. 8 as a threedimensional map with a logarithmic scale, for both cryogenic and ambient cooling process, in a quarter of the re#ecting face. The calculation has been carried out on the basis of a beam spot corresponding to a wavelength of 1 As . The cryogenic cooling process provides a concave slope error nearly two order of magnitude below the convex one produced by the ambient cooling mode. This latter spreading angle is comparable to the Darwin width.
From 293 to 77.4 K the thermal linear expansion *¸/¸ of the copper exchangers is one order of magnitude above the silicon crystal. It produces, however, a negligible additional mechanical deformation. Firstly, the exchangers are clamped to the crystal via the faces `ya, whose dilatation has only a secondary e!ect on the slope error along the direction of beam propagation `xa (Fig. 6). Moreover, the indium foil between copper and silicon #ows and relaxes the di!erential dilatation between those two materials. Some simulations by Finite Element Analysis (FEA) have con"rmed those assumptions.
5. Monochromator performance 5.1. Theoretical consideration We can evaluate the consequence of the "rst crystal slope error *H(x, y) on the performance of the monochromator by ray tracing calculation, with a second crystal assumed without defects. The central ray (x"0, y"0) is de#ected by 2H and
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!2H inside the monochromator, to select the wavelength j . Any other ray falling at the point (x, y) on the "rst crystal area dx dy gives a slightly di!erent wavelength j with the glancing angle H "H #*H(x, y). The "rst crystal re#ected power inside a spectral band dj centered on j is dP(j)"I R(j!j ) dj. It is proportional to the re#ectivity R(j!j ) equated in the zero absorp tion approximation as [14]
4 D , R(H!H )" H!H $ (H!H )! D 4 (10) where D is the Darwin width de"ned as 2r jC"F(hkl)"e\+ D(hkl)" . (11)
If each ray (x, y) has an equal contribution to the intensity of the whole beam, the rocking curve R(q) may be expressed as the integration of the double power ratio carried out over the whole region of the beam spot.
? @ 1 R(q)" R (x, y,q) dx dy. (13) 4a b \? \@ Fig. 9a and b present the numerical results of the rocking curves estimated from Eq. (13) for both the Si(1 1 1) and Si(3 1 1) at a wavelength of 1 As . The three-dimensional graphical representations depict the evolution of the rocking curves for an absorbed typical power running from 0 to 200 W in the ambient temperature cooling case. Under the cryogenic cooling conditions, owing to the crystal
low distortion, the heat load has no visible e!ects on the rocking curves. The latter curves correspond strictly to those obtained under the ambient cooling conditions at the zero heat load limit. The single plot located in the back ground of each graph is the re#ecting power ratio of a perfect unique crystal with respect to the Darwin solution (Eqs. (10) and (11)), drawn for comparison. A contour level plot by step 0.1 relative to the re#ecting power ratio map is situated in the base of each graphs. Some typical numerical results of the rocking curves deterioration with the increasing power are summarized in Table 5; decrement of the maximum of re#ectivity and spreading of Full-Width at HalfMaximum (FWHM). The ideal rocking curves have been calculated using the Xop program [15] and a simulation based on Eq. (13) at 0 W, both results agree within 4%. 5.2. Experimental performance The tests of the present cryogenic cooling system and the performance of the monochromator have been conducted directly on the beam line FIP in its "nal working order, with the pair of Si(1 1 1) crystals and with 200 mA of current in the synchrotron ring. The veri"cation has "rstly concerned the beam position stability at focus point under di!erent working states of the cooling machine. The beam has been monitored through a direct X-ray camera and displayed on a video screen. We have noticed that there is no beam vibration up to a pump rotation speed of 4000 rpm. This observation has been con"rmed by rocking curves measurement. The curves of double re#ectivity remain unchanged up to 4000 rpm, and begin to broaden out above this limit. The weak vibrations appearing are due to the cryogenic pump which begins to work hard and to shudder. A perfect operating cycle, working smoothly, requires a pressure stability inside the cooling circuit kept better than 0.005 bar around the desired value, as it is clearly the case in a wide running range of the cryogenic circulator [7]. The practical running modes of the cooling machine which have been adopted on the beam line FIP are as soft as the thermal load requires, 2400 rpm near the low power density limit (at 1.75 As ) and 2700
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Fig. 9. Simulated rocking curves as a function of the power load at j"1 As : (a) Si(1 1 1); (b) Si(3 1 1).
rpm near the high #uxes limit (at 0.7 As ). The LN2 #ow rate is estimated from head #ow curves around 150 l/h. The stability of the beam position over a long period is also excellent owing to the constant
temperature of the cooling #uid (LN2) stored exactly at 77.4 K. The temperature of the crystal is continuously controlled by a thermocouple placed in the body of the crystal, the observed value
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Table 5 Comparison of rocking curves of Si(111) and Si(311) at di!erent heat loads (Values in parentheses: calculation from Xop [15])
Si(1 1 1) Si(3 1 1)
Unique crystal
Double crystal
0W
0W
100 W
200 W
D (lrad)
FWHM (lrad)
R
*
FWHM (lrad)
R
*
FWHM (lrad)
R
*
FWHM (lrad)
R
*
20.9 8.9
22.2 (22.9) 9.5 (9.6)
1 (0.97) 1 (0.97)
29.2 (30.4) 12.6 (12.8)
0.82 (0.79) 0.82 (0.76)
34 28
0.68 0.39
48 34
0.51 0.26
Fig. 10. Rocking curves of Si(1 1 1) as a function of wavelength: (䉫) experimental points; (- - -) computation for non-distorted crystal.
displays a perfect stability ((1 K) around 80 K. The pressure of LN2 inside the optic loop is also recorded by a captor located at the monochromator outgoing pipe. The absence of LN2 boiling is therefore carefully checked, a bubble formation is signed by two sudden consecutive jumps of pressure of opposite sign. Fig. 10 presents a comparison between the theoretical and the experimental Si(1 1 1) rocking curves. The double re#ectivity data has been collected at the focus point (see Fig. 1) with an ion chamber or/and with a photo-diode placed just behind. The measurements have been recorded at eight energies
sampled in the working range of the monochromator (see column 1 of Table 6). After the o!set subtraction from the whole experimental curves, the maxima positions have been scaled on the theory (see columns 2 and 3 of Table 6). One can unambiguously a$rm that the experimental double re#ectivity of the monochromator FIP takes the exact shape of the self-convoluted curve of a simple perfect silicon crystal re#ectivity at all energies. A slight discrepancy is however visible in the wings of the peaks, the observed values are lower than the theoretical ones. For the low values of intensity the used X-ray photons counters are probably less
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Table 6 Comparison between characteristics of theoretical and calculated Si(1 1 1) rocking curves at di!erent wavelengths FWHM(1): Darwin solution; FWHM(2): calculation from Xop [15]; FWHM(exp): measurement j (As )
R
*
H !H (lrad)
FWHM (1) (lrad)
FWHM (2) (lrad)
FWHM (exp) (lrad)
0.72 0.98 1.10 1.20 1.28 1.40 1.50 1.75
0.786 0.767 0.756 0.751 0.745 0.733 0.726 0.705
28.8 39.2 43.9 48.5 52.0 56.6 61.2 71.6
21.0 28.6 32.4 35.3 37.8 41.4 44.7 52.5
21.6 29.7 33.6 36.9 39.6 43.7 47.2 56.1
23.6 29.1 34.8 36.9 39.0 43.9 47.6 53.2
e$cient and unable to discriminate "ne variations, this e!ect is particularly pronounced at 1.75 As where the beam is less brilliant. Fig. 11 and Table 6 allow to compare the observed FWHM of the rocking curves to the expected ones. The observations agree with theory within 10% of deviation. The FWHM of double re#ectivity curves is (2 times the width of the simple re#ectivity curves in the Gaussian approximation. This latter width is itself 1.06 times the width of total re#ection D (Eq. (11)). Finally, one "nds that the double rocking curves width is given by, FWHM+1.5D. Taking Eq. (11) in the small Bragg angle approximation (sin(2H )+j/d ), D and then the FWHM of FIJ double re#ectivity are assumed to vary linearly with the wavelength. As depicted in Fig. 11, the experimental data (points and dashed line) behave as predicted (full line) with a "tted slope of 30 lrad/As close to the theoretical coe$cient. Moreover, the extrapolated residual value of the experimental FWHM at the zero wavelength is nearly null. The experiment has been repeated with a variable slits aperture (crystal heat load) and does not give a result signi"cantly di!erent. We can conclude that thermal and/or mechanical mounting distortion of the "rst crystal of the monochromator FIP is lower than its Darwin width. The re#ection Si(1 1 1) is not a!ected by the weak residual slope error (if it exists), and the double re#ectivity is set to the maximum. The geometry of the beamline FIP does not allow to pass the beam
Fig. 11. Rocking curves FWHM of Si(1 1 1) as a function of wavelength: (- -䢇- -) experimental points; (*-) computation for non-distorted crystal.
directly through the monochromator without a re#ection on the "rst mirror. Thus, it has not been possible to go further and to measure the "rst allowed harmonic Si(3 3 3), a sharper re#ection which is sensitive to "ne crystal deformations.
6. Discussion The analytical point of view shows that the slope error of the monochromator "rst crystal is driven by the expression U;+geometrical factor,;+a/k,. The #ux U emitted by a bending magnet source on a third generation synchrotron (up to 10 W/m) produces the heat load limit on optical elements for the e$ciency of usual cooling systems. For the Si(1 1 1) crystal, the deterioration of the monochromating process appears at short wavelengths expressed as a loss of intensity and energy resolution. As a typical example of protein crystallography X-ray experiment at the maximum #ux (around 0.7 As ), we quoted the use of the anomalous signal near the LIII edge of scatters belonging to the actinide series of atoms (Th, Pa and U). In an experiment requiring a monochromatic beam with a high-energy resolution (as X-ray spectroscopy
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studies) the monochromator can be equipped with crystals of higher Miller indices as the Si(3 1 1). The thermal distortion then becomes very problematic in the whole energy range, and "ne experimental details are lost due to the poor resolution of the apparatus. Up to now, one plays with the geometrical factor by devising more e$cient crystal shapes to maintain a reasonable thermal distortion, so as to go on working with an ambient thermalisation system. However, this restricted strategy comes up quickly to unrealistic complex or fragile silicon crystals leading to worst mechanical distortions. For the "rst time on a bending magnet source, we have chosen to act on the factor +a/k, using liquid nitrogen as coolant, to solve de"nitively the heat load problem. Thus, our monochromating crystal has a simple bulky parallelepipedic shape (nearly cubic) and then is free of mechanical distortion. For economical reasons, we have devised a new cryogenic cooling loop with a novel optimal design, made up of a reduced number of elements compared to the existing systems. The running principle is radically di!erent, based on an open thermodynamic system: the liquid nitrogen closed cooling circuit passes via the cryogenic tank, which is open to the external media and allows to evaporate nitrogen during the energy dissipation process and to feed the system with fresh #uid. The heat dissipation e$ciency is limited on one hand by the maximum #ow rate in laminar mode, and on the other by the maximum pressurisation o!ered by the pump which prevents no monochromator vibrations. This e$ciency limit is situated at about 1 kW, one order of magnitude above our needs. The experimental rocking curves recorded on the Si(1 1 1) re#ection con"rm that the "rst crystal slope error is less than the Darwin width. As expected, the performance of the cryocooled monochromator FIP is then optimum from the thermal load point of view in its full working range. The two crystals of the monochromator maintained at 80 and 294 K, respectively exhibit a di!erence in crystallographic plane spacing *d such as *d/d"!2.3;10\. It leads to the only nuisance that the two crystals are then not parallel according to the Bragg law. Thus, the direction of the monochromator exit beam walks with the Bragg angle by twice *H"!tan(H )*d/d. It corresponds to a shift of
the beam from 1.5 to 5 mm at the focus point which is, however, easily corrected by the second mirror to keep a "xed output. A prototype of this cryogenic cooling system is running on the beamline FIP (BM30-A) at the ESRF, Grenoble, since January 1999, and gives full satisfaction. Acknowledgements The authors would like "rstly to acknowledge G. Ardin from J.L.S. Inc. for the cryogenic realization of excellent quality. This work was performed with the technical support of the SERAS CNRS, we are grateful to J.P. Roux for his assistance, D. Grand for the technical drawings of the machine and P. Jeantey for the FEA simulation. From the ESRF optics group, we would like also to thank Dr. J. Hoszowska for fruitful discussions and J. P. Vassali for the perfect silicon crystals preparation. References [1] J.-L. Ferrer, J.-P. Simon, J.-F. Berar, B. Caillot, E. Fanchon, O. KamK kati, S. Arnaud, M. Guidotti, M. Pirocchi, M. Roth, J. Synchrotron Rad. 5 (1998) 1346. [2] Y.S. Touloukian, R.W. Powell, C.Y. Ho, P.G. Klemens (Vol. 1); Y.S. Touloukian, R.K. Kirby, R.E. Taylor, T.Y.R. Lee (Vol. 13), Thermophysical Properties of Matter, Vol. 1 and 13, IFI/Plenum Press, New York (recommended values), 1975. [3] V. Rhen, Proc. SPIE 582 (1985) 238. [4] D.H. Bilderback, Nucl. Instr. and Meth. A 246 (1986) 434. [5] S. Joksch, G. Marot, A. Freund, M. Krisch, Nucl. Instr. and Meth. A 306 (1991) 386. [6] G. Marot, Opt. Eng. 34 (2) (1995) 426. [7] Barber-Nichols (Denver, CO, USA), Cryogenic Circulator BNCP43 for LN2 #uid, 1998. [8] B.A. Younglove, J. Phys. Chem. Ref. Data 11 (Suppl 1) (1982), 62. [9] F.D. Michaud, Nucl. Instr. and Meth. A 246 (1986) 444. [10] G. Marot, M. Rossat, A. Freud, S. Joksch, H. Kawata, L. Zhang, E. Ziegler, L. Berman, D. Chapman, J.B. Hastings, M. Larocci, Rev. Sci. Instr. 63 (1) (1992) 477. [11] B.X. Yang, M. Meron, Y. Ruan, W. Schildkamp, Proc. SPIE 1997 (1993) 302. [12] G.S. Knapp, M.A. Beno, C.S. Rogers, C.L. Wiley, P.L. Cowan, Rev. Sci. Instr. 65 (9) (1994) 2792. [13] H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, Oxford University Press, Oxford, 1959. [14] R.W. James, The Optical Pricinples of the Di!raction of X-rays, Ox Bow Press, Woodbridge, CT, 1982. [15] M. Sanchez del Rio, R.J. Dejus, Proc. SPIE 3152 (1997) 14.