Give the RB26/3 compensator its bellow plies back

This restores the thirteen plies of the RB26/3 axial compensator, which have
been absent since the commit that emptied the volume, and positions them so the
stack fits inside its own mother.

- 4b84a92 replaced every s3 with s2 in this positioning block, so
  RB26s3Bellow has been an empty vacuum tube since April 2019.
- Drawing LHCVC2A_0065 (EDMS 662891) gives 13 convolutions of 2 x 0.3 mm steel
  between R = 15.05 and 17.00 cm over 177 mm, which are the constants already in
  the file.
- Restoring the old block alone does not work. Its stack ran from -9.0084 to
  +9.0768 in a mother of half length 8.85, and 4b84a92's enlargement to
  8.99340 did not close that. Only 0.01 cm separates the bellow from the right
  welding tube, so the mother cannot grow.
- The mother is therefore unchanged and the ply pitch is the one that makes the
  fourteen roots span it exactly, 1.32692 cm, 1.5 percent shorter than
  4*PlieR - 2*PlieThickness. The webs are then 0.6501 and 0.6768 cm apart in
  turn rather than equidistant.
- The same change is applied to PipeRun4.cxx.
- CheckOverlaps(0.001) reports the same single pre-existing overlap as before.
  The pre-2019 geometry instead reports RB26s3Bellow extruded by 0.22680 cm and
  0.15840 cm.
- This adds material: 2.87 kg over the compensator subtree, at
  z = -1273.3 to -1231.0 cm and R = 15.05 to 17.00 cm, which is eta near 5.
  Measured against the pre-2019 geometry, the restored material is identical at
  the web and crest radii and 1.245 g/cm2 larger at the root radius, because two
  of the plies that stuck out of the mother were never seen by point location.

Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>

Author: sawenzel

Detectors/Passive/src/Pipe.cxx

@@ -2300,9 +2300,25 @@ void Pipe::ConstructGeometry()
   TGeoVolume* voRB26s3Bellow =
     new TGeoVolume("RB26s3Bellow", new TGeoTube(kRB26s3BellowRi, kRB26s3BellowRo, zBellowTot), kMedVacHC);
 
-  // RB26s3Bellow is left without plies. The wiggle stack that stood here was a copy of the
-  // RB26/2 code and filled voRB26s2Bellow a second time; filling it with the RB26/3 wiggle
-  // would change the material budget and is a separate fix.
+  // Positioning of the volumes.
+  //
+  // A thirteen-convolution bellow has fourteen inner roots, so one lower plie
+  // leads the thirteen wiggles. The pitch is not 4*PlieR - 2*PlieThickness: a
+  // torus-and-disc wiggle is longer than the convolution it stands for, and at
+  // that pitch the stack does not fit the bellow. There is no room to grow it
+  // either, since only 0.01 cm separates this bellow from the right welding
+  // tube. The pitch is therefore the one that makes the fourteen roots span the
+  // bellow exactly, which is 1.5 per cent shorter.
+  const Float_t kRB26s3PliePitch =
+    (2. * zBellowTot - 2. * kRB26s3PlieR) / kRB26s3NumberOfPlies;
+  const Float_t kRB26s3RootToWiggle = 3. * kRB26s3PlieR - 5. * kRB26s3PlieThickness / 2.;
+
+  z0 = -zBellowTot + kRB26s3PlieR;
+  voRB26s3Bellow->AddNode(voRB26s3WiggleL, 1, new TGeoTranslation(0., 0., z0));
+  for (Int_t iw = 0; iw < kRB26s3NumberOfPlies; iw++) {
+    Float_t zpos = z0 + (iw + 1) * kRB26s3PliePitch - kRB26s3RootToWiggle;
+    voRB26s3Bellow->AddNode(voRB26s3Wiggle, iw + 1, new TGeoTranslation(0., 0., zpos));
+  }
 
   voRB26s3Compensator->AddNode(voRB26s3Bellow, 1,
                                new TGeoTranslation(0., 0., kRB26s3WeldingTubeLeftL + zBellowTot));

Detectors/Passive/src/PipeRun4.cxx

@@ -2165,9 +2165,25 @@ void PipeRun4::ConstructGeometry()
   float zBellowTot = kRB26s3NumberOfPlies * (static_cast<TGeoBBox*>(voRB26s3Wiggle->GetShape()))->GetDZ();
   TGeoVolume* voRB26s3Bellow = new TGeoVolume("RB26s3Bellow", new TGeoTube(kRB26s3BellowRi, kRB26s3BellowRo, zBellowTot), kMedVacHC);
 
-  // RB26s3Bellow is left without plies. The wiggle stack that stood here was a copy of the
-  // RB26/2 code and filled voRB26s2Bellow a second time; filling it with the RB26/3 wiggle
-  // would change the material budget and is a separate fix.
+  // Positioning of the volumes.
+  //
+  // A thirteen-convolution bellow has fourteen inner roots, so one lower plie
+  // leads the thirteen wiggles. The pitch is not 4*PlieR - 2*PlieThickness: a
+  // torus-and-disc wiggle is longer than the convolution it stands for, and at
+  // that pitch the stack does not fit the bellow. There is no room to grow it
+  // either, since only 0.01 cm separates this bellow from the right welding
+  // tube. The pitch is therefore the one that makes the fourteen roots span the
+  // bellow exactly, which is 1.5 per cent shorter.
+  const float kRB26s3PliePitch =
+    (2. * zBellowTot - 2. * kRB26s3PlieR) / kRB26s3NumberOfPlies;
+  const float kRB26s3RootToWiggle = 3. * kRB26s3PlieR - 5. * kRB26s3PlieThickness / 2.;
+
+  z0 = -zBellowTot + kRB26s3PlieR;
+  voRB26s3Bellow->AddNode(voRB26s3WiggleL, 1, new TGeoTranslation(0., 0., z0));
+  for (int iw = 0; iw < kRB26s3NumberOfPlies; iw++) {
+    float zpos = z0 + (iw + 1) * kRB26s3PliePitch - kRB26s3RootToWiggle;
+    voRB26s3Bellow->AddNode(voRB26s3Wiggle, iw + 1, new TGeoTranslation(0., 0., zpos));
+  }
 
   voRB26s3Compensator->AddNode(voRB26s3Bellow, 1, new TGeoTranslation(0., 0., kRB26s3WeldingTubeLeftL + zBellowTot));