Modelling the power threshold and optimum thermal deformation of indirectly liquid-nitrogen cryo-cooled Si monochromators

A robust theoretical analysis of thermal management of cryo-cooled Si monochromators in hard X-ray beamlines is developed. This universal theory has been validated by extensive finite-element analysis studies, offering guidances to assess the heatload deformation quickly.

Finite-element analysis (FEA) studies are regularly carried out to assess the functionality of white-beam slits and DCMs at Diamond Light Source (DLS).Power (P) and power spatial density (P d ) absorbed by the optics will increase considerably on the upgraded machine Diamond-II (D-II) (Chapon et al., 2019).Installation of cryo-cooled or hybrid permanentmagnet undulators (CPMUs, HPMUs) with higher magnetic field will contribute to such an increase.Power management is key to conserving the photon source brightness on the loweremittance machines.
The design of suitable DCM cooling is a complex and multiparameter problem.Power levels are not constant on a given beamline, due to changing of settings, such as the insertion device gap, the angular fan of the incident beam, the optical layout, the presence of filters and the crystal Bragg angle.Exhaustive FEA is normally performed to study a few of these power scenarios and to finalize the cooling geometry.However, Si crystal temperature or thermal distortion do not follow a simple linear trend with P or P d , making interpretation and extrapolation of the FEA results complicated.In an earlier study an analytical model was developed (Khosroabadi et al., 2022), which describes the universal behaviour of cryocooled Si deformation and the transition between concave, flat and convex regimes.The results from the model agree both with available experimental data (Lee et al., 2001;Khosroabadi et al., 2022) and FEA data (Zhang et al., 2013(Zhang et al., , 2023;;Huang & Bilderback, 2012;Huang et al., 2014;Liu et al., 2016).
This article, which is an extension of the study already published by Khosroabadi et al. (2022), offers a practical and simplified treatment of parameters affecting crystal deformation.A threshold line is calculated for the space of possible parameters P and P d .Crystal deformation is acceptable below the threshold line, whilst it reaches a critical regime and is difficult to control above the line.In addition, the optimum temperature for the Si crystal base and the copper cooling block are provided; these ensure a minimized surface deformation for any values of P and P d below the threshold.

Theoretical model
The theoretical model (Khosroabadi et al., 2022) is summarized and extended to obtain a threshold power and the socalled 'sweet spot' condition.The crystal temperature distribution is calculated using the crystal base temperature T b as a boundary condition.T b can be either measured by a thermocouple attached to the crystal or derived analytically as shown below.For a circular footprint, we use the radial symmetry of the problem, and derive the temperature T(r) inside the crystal by solving the heat transfer conduction equation (Yener & Kakac ¸, 2008), where k Si is the thermal conductivity of Si, P(r) is the absorbed power, and A i = 2�r2 is the interface area at distance r.At low power, k Si is assumed to be constant, and T(r) is where where T p = T(0) is the crystal peak temperature and a is the radius of the beam footprint.For high power, k Si has an inverse quadratic temperature dependence, and so a complicated exponential function of temperature is derived (Khos-roabadi et al., 2022).For medium-energy synchrotron machines, with electron beam energy E e ' 3 GeV, the following linear equations are good approximations, where � 1 = 34 K and � 2 = 158 K. T Cu , A and k are, respectively, the average temperature of the Cu block, the contact area and the thermal conductance at the copper-silicon contact surface.Units used hereafter are W, W mm À 2 and K, for P, P d and T, respectively.The solution of equations 3(a)-3(c) shows that T p (which dictates the crystal deformation) has a complex dependence on power, beam footprint size, the cooling coefficients and finally the cooling geometry which determines the T Cu and T b values.
If P/2a = (P P d ) 1/2 < 100 W mm À 1 , then a compact firstorder expression of P/a is derived, where C ' 6 � 10 À 5 to 8 � 10 À 5 mm W À 1 K À 1 is a constant parameter dependent on Si material properties at cryogenic temperatures.A very similar function is obtained for elliptical beam footprints.This simple dependence of T p with the square root of absorbed power multiplied by power density has important consequences for the cooling of crystal monochromators.This will be further investigated in the remainder of the paper.The slope error � SE caused by thermal deformation can be estimated using the linear thermal expansion �L of silicon at cryogenic temperatures and can be found elsewhere (Middelmann et al., 2015).By assuming T ' T p in the footprint area and T ' T b at the depth d inside the crystal, we obtain �L(T) can be approximated as a parabolic function of temperature as shown in Fig. 1, � 2 and � 0 are constants, and T ze ' 127 K is the temperature of minimum thermal expansion of Si.A parabolic fit is sufficient compared with the previous fourth-order fitting (Khosroabadi et al., 2022).Units for �L were changed to the more practical nanometres.
Therefore equation ( 5) can be re-written as We define the second term in equation ( 7) as the f function, f is a universal function of cryo-cooled Si crystals, and the near-zero deformation conditions are found by setting it to zero.f has a complex dependence on P and P d , due to T b [equation (3c)], and the solution will be given in Section 4. Briefly, equation ( 8) can be solved for three different practical situations: (i) P and P d change due to the ring current ramp, ID gap or insertion of filters; (ii) P changes due to changed white-beam slit aperture, while P d is constant; (iii) P is constant while P d changes due to Bragg angle variation for instance.Numerical solutions will be discussed in Section 4; however, in the latter case, the threshold power density P d,c , below which high deformation is prevented, is ffi ffi ffi ffi ffi ffi ffi ffi

Power scenarios
The on-axis angular power emitted by an insertion device (ID) source is given by (Thompson, 2009) where I is the ring current.B, N and K ID are the ID parameters, i.e. the magnetic field, the number of periods and the deflection parameter, respectively, and G(K ID ) is a universal function with the value of >0.95 for K ID > 1.The beam apertures typically used on hard X-ray beamline at DLS are � H ' 140 mrad (horizontal) and � V ' 60 mrad (vertical).On D-II these will reduce to � H ' 80 mrad and � V ' 60 mrad.These are about five to six times the photon beam r.m.s.divergence from source.The angular power density of an undulator source is nearly constant in these typical apertures, and we can derive where d is the source-to-DCM distance and E is the monochromatic photon energy.The constants A 0 and A 1 and the power range are given in Table 1 for Si111 on the DLS and D-II machines.These are calculated at 4 mm (minimum) gap for 2 m-long CPMUs (N = 113, K ID = 2.2, B ' 1.4 T) and HPMUs (N = 106, K ID = 2, B ' 1.17 T).The power density is calculated assuming d = 30 m and energy from 2.1 to 25 keV.
The data agree well with accurate calculation using SPECTRA (Tanaka, 2021;Tanaka & Kitamura, 2001); however, it should be noted that the power absorbed by the first crystal is about 10-14% lower than calculated, due to scattering processes (Zhang et al., 2013).On several beamlines attenuation is performed by window and filter materials.The figures presented here are for the most severe power load scenarios.Finally, as in several other synchrotrons with upgraded photon sources, the issue is increased power density rather than total power.For instance, power density at lowest DCM energies of � 2 keV will surpass � 70 W mm À 2 for a CPMU: this is the worst-case power scenario on D-II hard X-ray beamlines.
For beamlines exploiting high-magnetic-field insertion device (wiggler) photon sources, and accepting large horizontal fans, the total power will instead increase considerably.The beam footprint area will also be the same order of magnitude as the crystal size.The analytical treatment presented would not apply to such scenarios.

Table 1
Values for the A 0 and A 1 coefficients on the DLS machine (E e = 3 GeV) and D-II (E e = 3.5 GeV) for I = 300 mA; maximum power P (W) and power density range P d (W mm À 2 ) for CPMU and HPMU sources are given.

Model validation and threshold power
Realistic power values on some upgraded DCMs are shown in Table 2.The inverse of the peak temperature calculated with equation ( 4) is plotted in Fig. 2 alongside FEA data from a variety of DLS and D-II scenarios.The symbols are FEA results for new DCMs installed in recent years on beamlines I18, I19, I22, I24 and VMXi with the parameters in Table 2.
Previously published FEA data (Zhang et al., 2013)  Schematic examples of deformation are given in Fig. 1.Generalized ranges of T b and T p at low, medium and high power are shown in this plot.The lengths of the coloured elliptical areas increase with power and power density as predicted by equation ( 4).The deformation, or slope error, is proportional to the effective temperature gradient in this diagram as per equation ( 6).The equation shows that both T b and T p contribute to the resulting thermal deformation.Crystal surface deformation is concave at low power (T b , T p < T ze ), nearly flat at medium power (T b < T ze < T p ) and convex at high power (T b , T p > T ze ).Smallest deformation, � SE ' 0, is achieved for T b and T p temperature values that are symmetric relative to T ze .This optimum, medium power regime is the socalled 'sweet spot'.The equation also illustrates that attempts to keep T p close to T ze do not work in practice.Normally this is achieved for higher T p , e.g. 150 K for T b ' 95 K.
Slope error estimation via the present model matches FEA data (Khosroabadi et al., 2022).However, more accurate data can be obtained by detailed FEA analysis in practice.Equations ( 6) and ( 8) describe the conditions for minimum � SE , i.e. either at very low power or close to the threshold power.Excessive thermal expansion is indicated in Fig. 1 by the elongated red ellipse and is responsible for a rapid deformation regime.We define this as the threshold regime to be avoided.
Recent DCMs at DLS (Sanchez-Navarro, 2021) have indirect side cooling and total contact area of A = 0.014 m 2 .

Table 2
List of FEA calculations for different source types and several beamlines (the symbols are the same as in the legend of Fig. 5).

Figure 2
The black solid line is the inverse of peak temperature calculated with equation ( 4) using C = 7 � 10 À 5 mm W À 1 K À 1 .The symbols are FEA results (see Table 2) and the green dashed line is from the literature (Zhang et al., 2013).Other lines are guides to the eye.
Examples of the f function [equation ( 8)] for these are shown in Fig. 3. Threshold power is found at f = 0, and the results are plotted in Fig. 4(a), using some typical values for the contact conductance corresponding to low and average thermal cooling (k = 850 and 2000 W m À 2 K, respectively).Threshold power density [equation ( 9)] is plotted in Fig. 4(b).These data and concepts are confirmed by FEA data performed for scenarios in Table 2.An acceptable degree of crystal deformation S was defined that would ensure conservation of photon beam brightness, spectral properties and focusing performance of the downstream optics.The result of this analysis is summarized in Fig. 5.The blue symbols represent power scenarios for which � SE calculated with FEA is < S. The red and orange symbols are for deformation at or above such a limit.The threshold power has an error bar due to the approximations used.Therefore, near these conditions the assessment should be more accurate.Below the threshold, equation ( 8) predicts the optimum T b temperature which minimizes the crystal surface deformation.This is shown in Fig. 6(a) and suggests that intentionally heating the crystal, or adjusting the flow rate of liquid nitrogen, can bring the temperature close to the 'sweet spot' (Khosroabadi et al., 2022;Sanchez-Navarro, 2022).The values plotted are for guidance only, as they depend on specific designs of cryo-cooled crystals.The copper block 'sweet spot' temperature can also be calculated from equation (3c), as shown in Fig. 6(b), using kA = 28 W K À 1 .These are very realistic values, being all above the boiling temperature of liquid nitrogen at 77 K.
In summary, the threshold power curve is a tool that can be used to decide whether a set of P and P d values are acceptable.The criterion has been used at DLS to choose suitable photon angular fan acceptance values, and to recommend additional filtering.The power regime above the threshold also represents a condition in which the deformation cannot be controlled as it increases steeply with increased power.

Summary and conclusion
A theoretical model has been developed to calculate the temperature distribution and surface deformation of an indirectly cryo-cooled Si crystal.Setting the conditions for lowest deformation leads to the definition of a threshold power level, above which the crystal deformation is unacceptably high.One practical result is the possibility to control the diffracted X-ray beam divergence or focal spot size at the sample position via intelligent cooling, keeping the crystal temperature within a small and well defined range.It has been shown that two characteristic temperatures, the peak and the base temperatures, have a unique relationship with (P P d ) 1/2 .FEA data have confirmed this behaviour.The threshold power curve is a function of contact conductance, crystal base temperature, power and power density.The model described here can be adapted to different optics geometries.
We propose to use this model as an intuitive and fast method to understand and limit (by improved designs) thermal deformation in cryo-cooled Si crystals.

Figure 1 (
Figure 1 (Top) Thermal length expansion of a 1 mm Si crystal relative to room temperature (squares) fitted by a T 2 polynomial (red line).The coloured ellipses show examples of T p and T b corresponding to low-, medium-and high-power regimes (green, yellow and red, respectively).T ze shows the temperature of minimum thermal expansion.(Bottom) Schematic representation of crystal deformation, switching from concave to flat, and to convex, when power is increased.

Figure 5
Figure 5Threshold power scenarios defined in this paper (solid line).Symbols indicate FEA data from Table2power scenarios and beamline specified in the legend.Blue symbols indicate power levels leading to acceptable crystal slope errors; orange and red are for deformation levels leading to decreased optical performance, such as decreased diffraction efficiency or the lensing effect.

Figure 3
Figure 3Function f for new DCM cooling geometry at DLS at several power density values.

Figure 4 (
Figure 4 (a) Threshold power derived from Fig. 3 (symbols) and guidelines (dashed), as a function of the power density for two different values of thermal contact conductance.The � 10% error bar is also shown.(b) Threshold power density calculated from equation (9) for different values of kA.

Figure 6
Figure 6Optimum temperature for the Si crystal base (a) and the Cu block (b) as a function of power and power density, ensuring the lowest deformation.
are also plotted for comparison.The data show a linear trend as predicted by the model and in good agreement with FEA data.The simple relationship 1/T p / (P P d ) 1/2 well describes the physical problem of cryo-cooled Si crystals.The vertical offset is caused by different T b values for different scenarios mentioned in Table2.The slight deviation in slope is due to the dependence of the C parameter [equation (4)] on P and P d .