HK1067687B - Spiral watch spring and its method of production - Google Patents

Description

Description for the following Contracting States: AT, BE, BG, CH, CY, CZ, DK, EE, ES, FI, FR, GB, GR, IE, IT, LI, LU, ML, NL, PT, SE, SK, TR

The present invention relates to the control of the watch movement, called the spiral balance, and in particular concerns, on the one hand, a spiral spring for the purpose of equipping the balance with a mechanical watch movement and, on the other hand, a manufacturing process for this spiral.

The control of mechanical watches is made up of an inertial wheel, called a pendulum, and a spiral spring, called a spiral or spiral spring, fixed at one end to the pendulum's axis and at the other end to a bridge, called a cock, in which the pendulum's axis turns.

The spiral swing oscillates around its equilibrium position (or dead-end). When the swing leaves this position, it arms the spiral. This creates a boost torque which, when the swing is released, causes it to return to its equilibrium position. As it has acquired a certain speed, therefore kinetic energy, it exceeds its dead-end until the opposite torque of the spiral stops it and forces it to rotate in the other direction.

The spiral spring used in mechanical watches is an elastic metal blade with a rectangular section, which is coiled in a spiral of Archimedes and has 12 to 15 turns. ${\text{M = E/L (w}}^{\text{3}}\text{·t/12·ϕ)}$ with: E: Young's modulus of the blade [N/m2],t: thickness of the spiral,w: width of the spiral,L: length of the spiral,φ: angle of torsion (rotation of the pivot)

It is easy to understand that the constant of recall or rigidity of a spiral $\text{C = M/ϕ,}$ The torque used for the torque measurement of the torque in the torque converter must be as constant as possible, regardless of temperature and magnetic field.

At present, complex alloys are used, both by the number of components (iron, carbon, nickel, chromium, tungsten, molybdenum, beryllium, niobium, etc.) and by the metallurgical processes used.

The current metal spirals are difficult to manufacture. First, due to the complexity of the processes used to make the alloys, the intrinsic mechanical properties of the metal are not constant from one production to the next. Second, the setting, which is the technique to make the watch show the most accurate time at all times, is tedious and time consuming. This operation requires many manual interventions and many defective parts must be removed. For these reasons, production is expensive and maintaining a constant quality is a constant challenge.

JP-A-06117470 is also known. It describes a spiral-shaped spring made of monocrystalline silicon. It is sized to have a constant of recall, to provide a high-precision electrical measuring device. However, this document is silent about the thermal stability of the recall constant of this spring. It cannot therefore be used directly as a spiral spring in a watchmaking piece.

The present invention aims to overcome these drawbacks by proposing a coil which is minimally sensitive to thermal variations and magnetic fields, and which, thanks to manufacturing techniques which ensure perfect reproducibility, does not fluctuate in the quality of the coils supplied.

More specifically, the invention concerns a spiral spring intended to equip the balance of a mechanical watchpiece and consisting of a spiral bar resulting from the cutting of a plate {001} of monocrystalline silicon having the first (C1) and second (C2) thermal coefficients of its recall constant C, the spires of which have a width w and a thickness t.

According to the invention, this bar has a silicon core and an outer layer of ξ thickness formed around the silicon core and made of a material having a first Young's modulus thermal coefficient of the opposite sign to that of silicon.

The advantage is that the outer layer (14) is made of amorphous silicon oxide (SiO2).

The dimensional ratio ξ/w is defined to obtain a first thermal coefficient (C1) of its recall constant C of a predetermined or minimized value.

The outer layer is preferably made of amorphous silicon oxide (SiO2) and is about 6% of the w-width of the bar.

To minimize the second thermal coefficient (C2), the width w of the bar is modulated periodically according to the angle 9 defining the orientation of each of its points in polar coordinates.

In order to optimise the thermal behaviour of the spiral spring, the bar thickness t, its width w, modulated in the plane of the spiral, and the silicon oxide layer thickness ξ have values for which the thermal drift of the booster constant C is minimal in a given temperature range.

The invention also relates to a method for determining the optimum dimensions of the spiral spring just defined. mathematically express the stiffness of the spiral as a function of its thickness t, its width w, modulated in the plane of the spiral, the thickness ξ of the silicon oxide layer, the elastic anisotropy of the silicon and the temperature;calculate the thermal behaviour, in particular the first two thermal coefficients of the spiral spring recall constant (C1 and C2), for all possible combinations of values of the parameters t, w, ξ in a given temperature range;and retain the combinations t, w, ξ for which the thermal derivatives of these coefficients are minimal.

The invention will be better understood by reading the following description, made in relation to the attached drawing on which: Figure 1 shows a spiral spring according to the invention; Figure 2 shows a segment of this spiral, in longitudinal section a and in transverse section b, to illustrate the references of the parameters useful for the description; and Figure 3 shows the anisotropy of the Young's module in the plane {001} of silicon.

The coil according to the invention, shown in Figures 1 and 2, is a bar cut into a coil by machining, e.g. by plasma, a plate of monocrystalline silicon.

Unfortunately, it is noted that it is difficult to obtain a silicon spiral spring with constant C-boost constant because the Young E module of this rod is strongly influenced by temperature.

When modelling the temperature sensitivity of an elastic structure, it is usual to use the thermal coefficients of its recall constant C, as shown in a mathematical series of the type: ${\text{C = C}}_{\text{0}}{\text{(1 + C}}_{\text{1}}{\text{ΔT + C}}_{\text{2}}{\text{ΔT}}^{\text{2}}\text{...),}$ where C0 is the nominal value of the recall constant C and C1 and C2 are, respectively, its first and second thermal coefficients.

It will therefore be understood that in order to obtain a temperature-insensitive C recall constant, the thermal coefficients C1 and C2 are sought to be minimized.

It should be remembered that monocrystalline silicon has a crystalline anisotropy. In the plane {001}, the direction <110> is more rigid than the direction <100>, which of course influences the bending rigidity of the spiral 10.

The Young's modulus E (a) of the silicon plane {001} can be expressed, like the recall constant, by a mathematical series of the type: ${\text{E}}^{\text{(a)}}{\text{= E}}_{\text{0}}{\text{}}^{\text{(a)}}{\text{(1 + E}}_{\text{1}}{\text{}}^{\text{(a)}}{\text{ΔT + E}}_{\text{2}}{\text{}}^{\text{(a)}}{\text{ΔT}}^{\text{2}}\text{),}$ where E0 (a) is the nominal value of Young's modulus E (a) and E1 (a) and E2 (a) are, respectively, its first and second thermal coefficients.

The first thermal coefficient E1 ((a) of the Young's module is strongly negative (- 60 ppm/°C approx.) and the nominal value of the Young's module E0 ((a) is 148 GPa in the <100> direction of the plane {001}.

To compensate for this drift, the spiral 10 of the invention consists of a silicon core 12 and an outer layer 14 of SiO2, the first thermal coefficient of which E1 (b) is strongly positive.

This symmetrical trilam structure, obtained by thermal oxidation by any known process, thus allows the thermal stability of the overall rigidity of the planar-bending spiral to be affected.

It can be shown that for a spiral cut into the plane {001} there is an optimal minimization of the first thermal coefficient C1 of the spiral's booster constant when the thickness of the 14 oxide layer is about 6% of the width of the spiral spring.

According to the invention, the second thermal coefficient C2 can be minimized by modulating the width w of the spiral, which is the dimension in its plane of winding, according to the angle θ which characterizes the orientation of each of its points in polar coordinates.

As shown in Figure 1, modulation can be achieved by making the spiral thinner in the rigid direction <110> and thicker in the less rigid direction <100>. This makes it possible to compensate for the anisotropy of silicon and obtain local rigidity at constant bending.

In this particular case, if we call w0 a reference width of the spiral in the plane {001}, the width w varies according to the angle θ according to the relation: ${\text{w = w}}_{\text{0}}\sqrt[\text{3}]{\text{1-}\frac{\text{1-}\frac{{\overline{\text{s}}}_{\text{12.0}}}{{\overline{\text{s}}}_{\text{11.0}}}\text{-}\frac{\text{1}}{\text{2}}\frac{{\overline{\text{s}}}_{\text{44.0}}}{{\overline{\text{s}}}_{\text{11.0}}}}{\text{2}}{\text{sin}}^{\text{2}}\text{(2θ)}}\text{,}$ In which $\overline{\text{s}}$ 11 $\overline{\text{s}}$ 44 $\overline{\text{s}}$ 12 are the three elastic coefficients independent of silicon in the crystallographic axes, known to the professional, as defined in the publication by C. Bourgeois et al. Design of resonators for the Determination of the Temperature Coefficients of Elastic Constants of Monocrystalline Silicon (Proc. 51th Annual Frequency Control Symposium, 1997, 791-799).

More concretely, it is easy to understand that several parameters interact in an interdependent manner and that, for example, the improvement in thermal behaviour obtained with some modulation of the width w will not be the same for all oxide thicknesses and for all crystal orientations of the spiral.

To facilitate the determination of the optimum values of the various parameters, the method according to the invention consists, for example, in examining the variability of the thermal coefficients of the C recall constant of a coil as described above, according to these parameters.

The parameters involved in determining C are Young's modulus E (a) of silicon, Young's modulus E (b) of silicon oxide and the geometric quantities shown in Figure 2: t = thickness of the coil (constant) [m]w = width of the coil in the plane {001} [m]ξ = thickness of the oxide (constant) [m]

According to multi-blade theory, the Young E-equivalent bending module of a silicon rod coated with a silicon oxide layer can be modeled in a local section according to the following relation:

Like in the case of the spiral 10, $\frac{\text{ξ}}{\text{w}}$ 1 and $\frac{\text{ξ}}{\text{t}}$ <<1, the equation becomes:

The recall constant C of the spiral spring and its first two thermal coefficients C1 and C2 are determined by integrating over its entire length the local rigidity expression, itself a function of the expressions E, t, w and ξ.

It can then be shown that the first thermal coefficient C1 is essentially a function of ξ, while the second thermal coefficient C2 depends mainly on w.

Then, using a computer, the values of the thermal coefficients C1 and C2 for all possible combinations of values of the parameters t, w, ξ are calculated. The triplets t, w, ξ for which the thermal drift of the booster constant C of the spiral spring is minimal in a given temperature range are extracted from all possible combinations.

The triplet corresponding to the spiral can then be chosen, the C-reminder constant of which, determined by the formula already given, is best suited to the desired watchmaking application.

Finally, the spiral can be made according to the calculation.

The proposal is for a silicon coil with a minimum of temperature sensitivity, ready for use and requiring no special adjustment or manual operation.

The above description is only a particular and non-restrictive example of a silicon-based coil according to the invention, so that the sole thermal compensation provided by the oxide layer is already satisfactory for use in mid-range watches and the w-width modulation is optional.

Description for the following Contracting State: DE

The present invention relates to the control of the watch movement, called the spiral balance, and in particular concerns, on the one hand, a spiral spring for the purpose of equipping the balance with a mechanical watch movement and, on the other hand, a manufacturing process for this spiral.

Referring to the earlier application DE-A-101 27 733 lodged on 7 June 2001, the applicant, on its own initiative, limited the scope of the present application and submitted separate claims for Germany.

The control of mechanical watches is made up of an inertial wheel, called a pendulum, and a spiral spring, called a spiral or spiral spring, fixed at one end to the pendulum's axis and at the other end to a bridge, called a cock, in which the pendulum's axis turns.

The spiral swing oscillates around its equilibrium position (or dead-end). When the swing leaves this position, it arms the spiral. This creates a boost torque which, when the swing is released, causes it to return to its equilibrium position. As it has acquired a certain speed, therefore kinetic energy, it exceeds its dead-end until the opposite torque of the spiral stops it and forces it to rotate in the other direction.

The spiral spring used in mechanical watches is an elastic metal blade with a rectangular section, which is coiled in a spiral of Archimedes and has 12 to 15 turns. ${\text{M = E/L (w}}^{\text{3}}\text{·t/12·ϕ)}$ where:E: Young's modulus of the blade [N/m2],t: thickness of the spiral,w: width of the spiral,L: length of the spiral,φ: angle of torsion (rotation of the pivot)

It is easy to understand that the constant of recall or rigidity of a spiral $\text{C = M/ϕ,}$ The torque used for the torque measurement of the torque in the torque converter must be as constant as possible, regardless of temperature and magnetic field.

At present, complex alloys are used, both by the number of components (iron, carbon, nickel, chromium, tungsten, molybdenum, beryllium, niobium, etc.) and by the metallurgical processes used.

The current metal spirals are difficult to manufacture. First, due to the complexity of the processes used to make the alloys, the intrinsic mechanical properties of the metal are not constant from one production to the next. Second, the setting, which is the technique to make the watch show the most accurate time at all times, is tedious and time consuming. This operation requires many manual interventions and many defective parts must be removed. For these reasons, production is expensive and maintaining a constant quality is a constant challenge.

The document JP-A-06117470 is also known. It describes a spiral-shaped spring made of monocrystalline silicon. It is sized to have a constant of recall, to provide a high-precision electrical measuring apparatus. However, this document is silent about the thermal stability of the recall constant of this spring. It cannot therefore be used directly as a spiral spring in a watchmaking piece.

The present invention aims to overcome these drawbacks by proposing a coil which is minimally sensitive to thermal variations and magnetic fields, and which, thanks to manufacturing techniques which ensure perfect reproducibility, does not fluctuate in the quality of the coils supplied.

More specifically, the invention concerns a spiral spring intended to equip the balance with a mechanical watchpiece and consisting of a spiral bar, the spires of which have a width w and a thickness t. This bar has a silicon core and an outer layer of ξ thickness formed around the silicon core and is made of a material having a first heat coefficient of the Young's modulus of sign opposite to that of silicon.

According to the invention, the dimensional ratio ξ/w is defined in such a way as to obtain a first thermal coefficient (C1) of its recall constant C of a predetermined value.

The advantage is that the outer layer is made of amorphous silicon oxide (SiO2).

The dimensional ratio ξ/w is defined to minimize the first thermal coefficient (C1) of its booster constant C. Preferably, this ratio ξ/w is about 6% of the width w of the bar.

To minimize the second thermal coefficient (C2), the width w of the bar is modulated periodically according to the angle θ defining the orientation of each of its points in polar coordinates.

In order to optimise the thermal behaviour of the spiral spring, the bar thickness t, its width w, modulated in the plane of the spiral, and the silicon oxide layer thickness ξ have values for which the thermal drift of the booster constant C is minimal in a given temperature range.

The invention also relates to a method for determining the optimum dimensions of the spiral spring just defined. mathematically express the stiffness of the spiral as a function of its thickness t, its width w, modulated in the plane of the spiral, the thickness ξ of the silicon oxide layer, the elastic anisotropy of the silicon and the temperature;calculate the thermal behaviour, in particular the first two thermal coefficients of the spiral spring recall constant (C1 and C2), for all possible combinations of values of the parameters t, w, ξ in a given temperature range;and retain the combinations t, w, ξ for which the thermal derivatives of these coefficients are minimal.

The invention will be better understood by reading the following description, made in relation to the attached drawing on which: Figure 1 shows a spiral spring according to the invention; Figure 2 shows a segment of this spiral, in longitudinal section a and in transverse section b, to illustrate the references of the parameters useful for the description; and Figure 3 shows the anisotropy of the Young's module in the plane {001} of silicon.

The coil according to the invention, shown in Figures 1 and 2, is a bar cut into a coil by machining, e.g. by plasma, a plate of monocrystalline silicon.

Unfortunately, it is noted that it is difficult to obtain a silicon spiral spring with constant C-boost constant because the Young E module of this rod is strongly influenced by temperature.

When modelling the temperature sensitivity of an elastic structure, it is usual to use the thermal coefficients of its recall constant C, as shown in a mathematical series of the type: ${\text{C = C}}_{\text{0}}{\text{(1 + C}}_{\text{1}}{\text{ΔT + C}}_{\text{2}}{\text{ΔT}}^{\text{2}}\text{...),}$ where C0 is the nominal value of the recall constant C and C1 and C2 are, respectively, its first and second thermal coefficients.

It will therefore be understood that in order to obtain a temperature-insensitive C recall constant, the thermal coefficients C1 and C2 are sought to be minimized.

It should be remembered that monocrystalline silicon has a crystalline anisotropy. In the plane {001}, the direction <110> is more rigid than the direction <100>, which of course influences the bending rigidity of the spiral 10.

The Young's modulus E (a) of the silicon plane {001} can be expressed, like the recall constant, by a mathematical series of the type: ${\text{E}}^{\text{(a)}}{\text{= E}}_{\text{0}}{\text{}}^{\text{(a)}}{\text{(1 + E}}_{\text{1}}{\text{}}^{\text{(a)}}{\text{ΔT + E}}_{\text{2}}{\text{}}^{\text{(a)}}{\text{ΔT}}^{\text{2}}\text{),}$ where E0 (a) is the nominal value of Young's modulus E (a) and E1 (a) and E2 (a) are, respectively, its first and second thermal coefficients.

The first thermal coefficient E1 ((a) of the Young's module is strongly negative (- 60 ppm/°C approx.) and the nominal value of the Young's module E0 ((a) is 148 GPa in the <100> direction of the plane {001}.

To compensate for this drift, the spiral 10 of the invention consists of a silicon core 12 and an outer layer 14 of SiO2, the first thermal coefficient of which E1 (b) is strongly positive.

This symmetrical trilam structure, obtained by thermal oxidation by any known process, thus allows the thermal stability of the overall rigidity of the planar-bending spiral to be affected.

It can be shown that for a spiral cut into the plane {001} there is an optimal minimization of the first thermal coefficient C1 of the spiral's booster constant when the thickness of the 14 oxide layer is about 6% of the width of the spiral spring.

According to the invention, the second thermal coefficient C2 can be minimized by modulating the width w of the spiral, which is the dimension in its plane of winding, according to the angle which characterizes the orientation of each of its points in polar coordinates.

As shown in Figure 1, modulation can be achieved by making the spiral thinner in the rigid direction < 110> and thicker in the less rigid direction < 100>.

In this particular case, if we call w0 a reference width of the spiral in the plane {001}, the width w varies according to the angle θ according to the relation: ${\text{w = w}}_{\text{0}}\sqrt[\text{3}]{\text{1-}\frac{\text{1-}\frac{{\overline{\text{s}}}_{\text{12.0}}}{{\overline{\text{s}}}_{\text{11.0}}}\text{-}\frac{\text{1}}{\text{2}}\frac{{\overline{\text{s}}}_{\text{44.0}}}{{\overline{\text{s}}}_{\text{11.0}}}}{\text{2}}{\text{sin}}^{\text{2}}\text{(2θ)}}\text{,}$ In which $\overline{\text{s}}$ 11 $\overline{\text{s}}$ 44 $\overline{\text{s}}$ 12 are the three elastic coefficients independent of silicon in the crystallographic axes, known to the professional, as defined in the publication by C. Bourgeois et al. Design of resonators for the Determination of the Temperature Coefficients of Elastic Constants of Monocrystalline Silicon (Proc. 51th Annual Frequency Control Symposium, 1997, 791-799).

More concretely, it is easy to understand that several parameters interact in an interdependent manner and that, for example, the improvement in thermal behaviour obtained with some modulation of the width w will not be the same for all oxide thicknesses and for all crystal orientations of the spiral.

To facilitate the determination of the optimum values of the various parameters, the method according to the invention consists, for example, in examining the variability of the thermal coefficients of the C recall constant of a coil as described above, according to these parameters.

The parameters involved in determining C are Young's modulus E (a) of silicon, Young's modulus E (b) of silicon oxide and the geometric quantities shown in Figure 2: t = thickness of the coil (constant) [m]w = width of the coil in the plane {001} [m]ξ = thickness of the oxide (constant) [m]

According to multi-blade theory, the Young E-equivalent bending module of a silicon rod coated with a silicon oxide layer can be modeled in a local section according to the following relation:

Like in the case of the spiral 10, $\frac{\text{ξ}}{\text{w}}$ 1 and $\frac{\text{ξ}}{\text{t}}$ The equation is:

The recall constant C of the spiral spring and its first two thermal coefficients C1 and C2 are determined by integrating over its entire length the local rigidity expression, itself a function of the expressions E, t, w and ξ.

It can then be shown that the first thermal coefficient C1 is essentially a function of ξ, while the second thermal coefficient C2 depends mainly on w.

Then, using a computer, the values of the thermal coefficients C1 and C2 for all possible combinations of values of the parameters t, w, ξ are calculated. The triplets t, w, ξ for which the thermal drift of the booster constant C of the spiral spring is minimal in a given temperature range are extracted from all possible combinations.

The triplet corresponding to the spiral can then be chosen, the C-reminder constant of which, determined by the formula already given, is best suited to the desired watchmaking application.

Finally, the spiral can be made according to the calculation.

The silicon coil is therefore available with a minimum of temperature sensitivity and is ready for use without any special adjustment or manual operation.

The above description is only a particular and non-restrictive example of a silicon-based coil according to the invention, so that the sole thermal compensation provided by the oxide layer is already satisfactory for use in mid-range watches and the w-width modulation is optional.

Claims (11)

1. A hairspring intended to equip the balance wheel of a mechanical timepiece and in the form of a spiraled rod (10) cut from an {001} single-crystal silicon plate having a first thermal coefficient (C₁) and a second thermal coefficient (C₂) of its spring constant C, the turns of said hairspring having a width w and a thickness t, said rod comprising a silicon core (12) and an external layer (14) of thickness ξ formed around the silicon core and made of a material having a first thermal coefficient of the Young's modulus of opposite sign to that of the silicon, characterized in that the dimensional ratio ξ/w is defined so as to obtain a first thermal coefficient (C₁) of its spring constant C of predetermined value.
2. The hairspring as claimed in claim 1, characterized in that said external layer (14) is made of amorphous silicon oxide (SiO₂).
3. The hairspring as claimed in one of claims 1 to 2, characterized in that the dimensional ratio ξ/w is defined so as to minimize the first thermal coefficient (C₁) of its spring constant C.
4. The hairspring as claimed in claim 4, characterized in that the dimensional ratio ξ/w is about 0.06.
5. The hairspring as claimed in one of claims 1 to 4, characterized in that the width of said rod is modulated, periodically, as a function of the angle θ that defines the orientation of each of its points in polar coordinates in order to minimize the second thermal coefficient (C₂).
6. The hairspring as claimed in one of claims 1 to 5, characterized in that the width of said rod is modulated so that its local flexural stiffness is constant in order to minimize the second thermal coefficient (C₂).
7. The hairspring as claimed in claim 6, characterized in that the modulation is effected according to the formula: ${\text{w = w}}_{\text{0}}\sqrt[\text{3}]{\text{1-}\frac{\text{1-}\frac{{\overline{\text{s}}}_{\text{12.0}}}{{\overline{\text{s}}}_{\text{11.0}}}\text{-}\frac{\text{1}}{\text{2}}\frac{{\overline{\text{s}}}_{\text{44.0}}}{{\overline{\text{s}}}_{\text{11.0}}}}{\text{2}}{\text{sin}}^{\text{2}}\text{(2θ)}}$ in which $\overline{\text{s}}$ ₁₁, $\overline{\text{s}}$ ₄₄ and $\overline{\text{s}}$ ₁₂ are the three independent elastic coefficients of silicon along the crystallographic axes.
8. The hairspring as claimed in claim 5, characterized in that, in order to minimize the first thermal coefficient (C₁) and the second thermal coefficient (C₂), the thickness t of the rod, its width w in the {100} plane and the thickness ξ of the silicon oxide layer have values for which the thermal drift of the spring constant C of the hairspring is a minimum within a given temperature range.
9. A method for determining the optimum dimensions of the hairspring as claimed in claim 8, characterized in that it consists, in succession, in:

   - mathematically expressing the stiffness of the hairspring as a function of its thickness t, its width w modulated in the plane of the hairspring, the thickness ξ of the silicon oxide layer, the elastic anisotropy of the silicon and the temperature;

   - calculating the thermal behavior, in particular the first two coefficients (C1, C2) of the spring constant of the hairspring for all combinations of possible values of the parameters t, w and ξ within a given temperature range; and

   - adopting the t, w, ξ combinations for which the thermal drifts of said coefficients (C1 and C2) are minimal.
10. The method as claimed in claim 9, characterized in that it consists, finally, in calculating the width w of the spiral at any point from the formula: ${\text{w = w}}_{\text{0}}\sqrt[\text{3}]{\text{1-}\frac{\text{1-}\frac{{\overline{\text{s}}}_{\text{12.0}}}{{\overline{\text{s}}}_{\text{11.0}}}\text{-}\frac{\text{1}}{\text{2}}\frac{{\overline{\text{s}}}_{\text{44.0}}}{{\overline{\text{s}}}_{\text{11.0}}}}{\text{2}}{\text{sin}}^{\text{2}}\text{(2θ)}}$ in which $\overline{\text{s}}$ ₁₁, $\overline{\text{s}}$ ₄₄ and $\overline{\text{s}}$ ₁₂ are the three independent elastic coefficients of silicon along the crystallographic axes.