Influence of the Absorber Crystal Geometry on the Signal Height 85

The non-thermal and the thermal component (see section 4.2). As the thermal compo-nent of the signal is strongly suppressed, the following discussion focuses only on the amplitude of the non-thermal componentA_n.

In first approximation the signal height depends on the heat capacity of the ther-mometer C_e in the calorimetric mode and on the coupling of the thermometer to the

heat bath G_eb in the bolometric mode (see equations 4.29 and 4.31, respectively). In both operation modes the size of the target crystal does not affect the non-thermal component directly. However, in both cases the amplitude An is proportional to the fraction of non-thermal phonons absorbed in the thermometer ε. Two processes de-fine this fraction (see equation 4.18): The thermalization of non-thermal phonons in the absorber (described by the time constant τcrystal) and the thermalization in the thermometer film (described by time constantτ_{f ilm}).

The time constant τ_crystal is expected to scale as the ratio V_a/A_a, where V_a is the volume of the absorber and Aa is the surface area of the absorber. For two different crystals, crystal A and crystal B, differing in size and geometry, the respective ratios of volume and surface are (V_a/A_a)^A and (V_a/A_a)^B. The thermalization time of non-thermal phonons in the crystalτcrystal differs in the following way:

τ_crystal^B = (V_a/A_a)^B

(Va/Aa)^A ·τ_crystal^A =δ·τ_crystal^A , (5.1) where the superscriptsA andB denote the values in the respective crystal.

In order to keep the fraction of non-thermal phonons being absorbed by the ther-mometer constant, the time constant for the thermalization in the therther-mometerτ_{f ilm} has to be adapted with the same factorδ:

τ_{f ilm}^B =δ·τ_{f ilm}^A . (5.2)

Asτf ilm is proportional to Va/At (see equation 4.17), this is achieved by changing the area of the thermometerAtby the same amount as the area of the absorberAa:

A^B_t = A^B_a

A^A_aA^A_t . (5.3)

By this adaption of the thermometer area, the fraction of non-thermal phonons being absorbed by the thermometer is kept constant:

ε^B = τ_crystal^B

τ_crystal^B +τ_{f ilm}^B =ε^A. (5.4) Thereby, also the life-time of non-thermal phononsτ_n, which defines the pulse duration, is changed (see equation 4.15):

τ_n^B =δ·τ_n^A. (5.5)

Up to now, all alterations are independent of the operation mode of the detector. The effect on the signal amplitude, has to be discussed separately for a detector operated in the bolometric and the calorimetric mode.

Bolometric Detector

The decisive requirement for an operation in the bolometric mode is that the relation τ_in τ_n is valid. Thus, due to the change of the decay time of the pulse τ_n, also the thermal relaxation time of the thermometer τ_in, which is in the case of a bolometric

detector the rise time of the pulse, must be adapted, so that this relation is always fulfilled. Forτ_in the following relation is desired:

τ_in^B =δ·τ_in^A. (5.6)

Due to τin ≈ Ce/G_eb (see equation 4.27), the time constant is influenced by the pre-viously discussed change of the heat capacity of the thermometer, which is done by the variation of its areaAt (see equation 5.3). As the thickness of the thermometer is not intended to be changed, the thermal coupling of the thermometer to the heat bath G_eb must be changed in order to adapt τ_in to the new time constant τ_n. The thermal coupling must be modified in the following way:

G^B_eb= 1 δ ·A^B_a

A^A_a G^A_eb. (5.7)

According to equation 4.31 all the discussed changes result in the signal amplitude of the non-thermal component of a bolometric TESS_n,bol to scale in the following way:

S_n,bol^B = A^A_a

A^B_a ·S_n,bol^A (5.8)

This means, as long as the time constants and the thermometer are adapted in order to keep the collection efficiency of non-thermal phononsε constant, the amplitude of the non-thermal componentS_n,bol only scales with the surface of the crystal for a detector operated in the bolometric mode.

Calorimetric Detector

For an operation in the calorimetric mode the relationτnτin must be fulfilled. Also in this case,τin is adapted by a change of the thermal couplingG_eb. In the calorimetric modeG_eb is much weaker compared to the bolometric mode. However, the value does not influence the signal height in first approximation.

With only the alterations discussed above, the amplitude of the non-thermal com-ponent (see equation 4.29) in the calorimetric mode S_n,cal is expected to scale in the following way:

S_n,cal^B = A^A_a

A^B_a ·S_n,cal^A (5.9)

Although different parameters are dominating the signal height in the calorimetric detector, also in this mode the amplitude only scales with the absorber surface.

In contrast to a bolometric detector, in the calorimetric mode further optimizations are possible. An additional reduction of the thermometer areaA_tis expected to enlarge the signal further as the signal directly scales with the heat capacity of the thermometer Ce. However, also the fraction εis also influenced by the size of the thermometer via the time constant τ_{f ilm} as the latter scales with 1/A_t (see equation 4.17). A smaller thermometer area reduces the fractionε. However, the amount by whichε is reduced, depends on the ratio of τ_{f ilm} and τ_crystal, whereby at least the latter is not known a priori. These time constants and the resulting expectations for the signal height are investigated further in section 7.2.3.

cylindrical crystal cuboidal crystal small crystal CRESST-II phase 2 CRESST-II phase 2 CRESST-III (conventional modules) (stick modules)

d

h h

a b

h

a b

measures [mm] h=d= 40 h= 40, a=b= 32 h=a= 20, b= 10

mass [g] 300 249 24

Va [mm³] 50.3·10³ 40.96·10³ 4.00·10³

A_a [mm²] 7540 7168 1600

Va/Aa [mm] 6.7 5.7 2.5

Table 5.1:Comparison of different crystal geometries. The two large crystals were operated in CRESST-II phase 2 and are very similar in the given parameters. The small crystal is optimized for CRESST-III in order to provide an enhanced signal.

In conclusion, independent of the operation mode the amplitude of the non-thermal componentA_n inversely scales with the absorber surfaceA_a, when the thermometer is adapted accordingly. In the calorimetric mode a further increase of the signal might be accessible by an optimization of the thermometer. This can be achieved by a reduction of the thermometer area or by the usage of phonon collectors.

5.1.2 Optimized Crystal Geometry for a Large Signal

The lowest thresholds achieved in CRESST-II phase 2 are∼300 – 400 eV [74]. To reach a threshold of 100 eV the signal must be increased by a factor of 3 – 4. As discussed, this is expected for an absorber crystal with a surface reduced by a factor of 3 – 4 compared to the crystals utilized in CRESST-II phase 2.

Therefore, for CRESST-III a cuboidal shaped absorber crystal with a size of 20× 20×10 mm³ is planned to be used. This size corresponds to a mass of∼24 g. In table 5.1 the different crystal geometries used in CRESST-II phase 2 are compared to the geometry of the small crystal for CRESST-III.

The area of the small crystal is reduced by a factor of ∼4.5 to the cuboidal shaped crystal and ∼ 4.7 compared and the cylindrical crystal operated in CRESST-II. The mass of this crystal is smaller by a factor of ∼10 and∼12 compared to the cuboidal shaped and the cylindrical crystal, respectively.

With these changes, independent of the operation mode the signal amplitude S_n achievable with such a small crystal and an adapted TES structure is expected to be enhanced by a factor of∼4.5 compared to the large cuboidal shaped crystal and ∼4.7 compared to the large cylindrical crystal. Assuming the same noise conditions as before

such an increase of the signal height can possibly reduce the threshold from the lowest thresholds measured in CRESST-II of∼300 – 400 eV to thresholds of ∼60 – 90 eV.