Abstract—
The Earth surface inclinations recorded by the Precision Laser Inclinometer have been transformed to vertical oscillations using a specifically developed method. A method to calculate the space displacement of collider beams focuses is proposed to evaluate the effects of the surface waves propagation, taking into account: the length of the region of the beam focusing for collisions, the seismic wave frequency and the wave speed dependence on the frequency. It is shown that the beams focus divergence is measurable: for the LHC, with a 40-m long beam focusing path for collisions, in the frequency range above 1 Hz for the seismic surface wave; for the CLIC (first stage), with a 1.9-km long beam focusing path for collision, in the frequency range [0.1 Hz, 0.5 Hz], typical range of the micro-seismic peak. The data obtained shows the necessity of monitoring the seismic activity by using a network of Precision Laser Inclinometers.
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Notes
We mean here a specially design support for any high precision research equipment
The Love wave is a surface wave, which causes a displacement of the Earth surface in a horizontal plane in the direction perpendicular to the wave propagation direction.
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ACKNOWLEDGMENTS
The authors express their deep gratitude to V. Bednyakov and P. Jenni for their encouraging and supportive interest.
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Appendices
APPENDIX A
Figure 30 shows a plot of \(y = \sin x\) in the XY coordinate system.
Determination of the relationship between the angles α and β.
We need to determine the ratio between the angles α and β. As can be seen from Fig. 1, the angle α is determined from the relation
Angle β corresponds to the angle of the tangent CD at point A.
Accordingly, by definition, tan β to the tangent at point B is equal to
Consequently,
Given \(\alpha \ll 1^\circ \) and \(~\beta ~ \ll 1^\circ \) we get
APPENDIX B
The luminosity L of the colliding “Gaussian-profile” beams is determined by the expression [30, 31]
where \({{N}_{1}},~{{N}_{2}}\) are the number of particles in the bunches of the colliding beams, f is the collisions frequency, \({{N}_{{\text{b}}}}\) is the number of bunches, and \({{\sigma }_{x}},~{{\sigma }_{y}}\) are the diameters of the particle beam along the X and Y axes. In the LHC symmetric beams with a circular cross-section have been used so far.
To simplify the consideration, we determine the luminosity for the cylindrical colliding beams of same diameter with a uniform density:
where \({{S}_{0}} = \frac{{\pi {{z}^{2}}}}{4}\) is the beam cross-section area, а is the beam length, and N is the number of particles per bunch.
So, if the bunches completely overlap, the expression for the luminosity is
When the focuses are displaced, the number of colliding particles decreases, and so does the luminosity. When the beams coincide in space, the number of interactions in the colliding bunches is the largest and when the focuses are displaced by the distance δ, the interaction area decreases and accordingly the number of interacting particles in bunches decreases. (Fig. 31).
Dimensions of the cylindrical particle beams.
In this case the luminosity \({{L}_{{\text{S}}}}\) is determined by the mutual intersection area. Accordingly, with (2), formula (B.3) takes the form
Thus, we obtain the luminosity as a function of the area S of the partially overlapped beams. The relative luminosity change is
which (B.5) by can be rewritten using (B.4) as
To determine the relation between the relative luminosity variation and the displacement δ of the beams, we find the dependence between δ and S.
Overlapping area of the particle bunches displaced by the distance δ.
Figure 33 shows the cross-section of the two partially overlapping particle bunches. The distance between the centers of the bunches is δ. We can determine the dependence between δ and the intersection area S of the particle bunches.
Configuration of the interaction cross-section for the particle bunches shifted by the distance δ.
The desired area S is defined as twice the area of the circular segment DCm.
We define the area of the circular segment DCm as the difference between the area of the sector ADmC and the area of the triangle ADC. Thus, we obtain the formula for the area S
The relative luminosity variation \(\epsilon \) as a function of the displacement δ is
Figure 34 shows the dependence of the relative luminosity change \(\epsilon \) on the displacement δ of the particle bunches with the diameter σ = 17 µm in the focus.
Relative luminosity variation \(\epsilon \) as a function of the beam focus displacement δ for the particle bunches with the diameter of σ = 17 µm in focus.
Figure 34 shows that the luminosity change of 7% corresponds to a displacement of particle focuses δ = 0.5 µm.
Noteworthy is the strong dependence of the relative luminosity variation ε on the displacement δ of the particle bunch focuses. For example, a luminosity loss of 30% appears when focuses are displaced by 2.2 µm (13% of the particle bunch diameter). Thus, the luminosity is very sensitive to the displacement of focuses.
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Azaryan, N.S., Budagov, J.A., Lyablin, M.V. et al. Colliding Beams Focus Displacement Caused by Seismic Events. Phys. Part. Nuclei Lett. 16, 377–396 (2019). https://doi.org/10.1134/S154747711904006X
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DOI: https://doi.org/10.1134/S154747711904006X