The parent BSE from which several equations are derived or recalled below is [5]

$1=\frac{V}{{\left(2\pi \hslash c\right)}^3}\frac{1}{2}{\displaystyle \underset{\mu -k\theta }{\overset{\mu +k\theta }{\int }}{\text{d}}^{3}p}\frac{\tanh \left[\frac{1}{2kT}\left(p^2/2m_e-\mu -W/2\right)\right]}{p^2/2m_e-\mu -W/2},$ (4)

where V—which is non-zero only in the range of integration—is the same parameter as occurs in [N (0)V] in the BCS theory and has the dimensions of electronVolt-cm³. However, V/(2πħc)³ now plays the role of a propagator.

In terms of $\xi =p^2/2m_e-\mu$, we can recast (4) as

$1=\frac{\lambda }{2}{\displaystyle \underset{-k\theta }{\overset{k\theta }{\int }}\text{d}\xi \sqrt{1+\xi /\mu }}\frac{\tanh \left[\frac{1}{2kT}\left(\xi -W/2\right)\right]}{\xi -W/2},$ (5)

where

$\lambda =\frac{{\left(2m_e{c}^2\right)}^{3/2}{\mu }^{1/2}V}{4{\pi }^2{\left(\hslash c\right)}^3}.$ (6)

When T = T_c (t = 1), and W = 0, we label μ as μ₁ and λ as λ₁. We then have (5) as

$1=\frac{{\lambda }_1}{2}{\displaystyle \underset{-k\theta }{\overset{k\theta }{\int }}\text{d}\xi \sqrt{1+\xi /{\mu }_1}\frac{\tanh \left(\xi /2k{T}_c\right)}{\xi }},$ (7)

where

${\lambda }_1=\frac{{\left(2m_e{c}^2\right)}^{3/2}{\mu }_1^{1/2}V}{4{\pi }^2{\left(\hslash c\right)}^3}.$ (8)

Equation (7) is identical with the BCS equation for the T_c of an elemental SC when μ₁  kθ. Parametrizing μ₁ as

${\mu }_1=\rho k\theta$ (9)

and employing $x=\xi /2k{T}_c$, we obtain (5) in the form employed in this paper in the 1PEM scenario as

$1=\frac{{\lambda }_1}{2}{\displaystyle \underset{-\theta /2T_c}{\overset{\theta /2T_c}{\int }}\text{d}x\sqrt{1+\frac{2T_c}{\rho \theta }x}\frac{\tanh \left(x\right)}{x}}.$ (10)

When T = 0, we label μ as μ₀ and λ as λ₀ in (4), whence

${\lambda }_0=\frac{{\left(2m_e{c}^2\right)}^{3/2}{\mu }_0^{1/2}V}{4{\pi }^2{\left(\hslash c\right)}^3}.$ (11)

It follows from (8) and (11) that

${\lambda }_0={\lambda }_1\sqrt{q},$ (12)

where μ₀ has been parametrized as

${\mu }_0=q{\mu }_1=q\rho k\theta .$ (13)

In this case the equation for |W| follows from (5) by choosing the signatures of W as follows [5]

$W=+\left|W\right|\text{when}\xi <0\text{and}W=-\left|W\right|\text{when}\xi >0.$

Therefore, we have (5) as

$1=\frac{{\lambda }_1\sqrt{q}}{2}\left(I_1+I_2\right),$ (14)

where

$\begin{array}{l}{I}_1={\displaystyle \underset{-k\theta }{\overset{0}{\int }}\text{d}\xi \sqrt{1+\xi /q\rho k\theta }}\frac{\tanh \left[\frac{1}{2kT}\left(\xi -\left|W\right|/2\right)\right]}{\xi -\left|W\right|/2}\\ I_2={\displaystyle \underset{0}{\overset{k\theta }{\int }}\text{d}\xi \sqrt{1+\xi /q\rho k\theta }}\frac{\tanh \left[\frac{1}{2kT}\left(\xi +\left|W\right|/2\right)\right]}{\xi +\left|W\right|/2}.\end{array}$

When T = 0, the tanh in I₁ = −1 and the tanh in I₂ = +1. In terms of $x=\xi /q\rho k\theta$, we can compactly combine these equations to obtain (14) as

$1=\frac{{\lambda }_1\sqrt{q}}{2}{\displaystyle \underset{-1/q\rho }{\overset{1/q\rho }{\int }}\text{d}x}\frac{\sqrt{1+x}}{\left|x\right|+\left|W\right|/2q\rho k\theta },$ (15)

which is the equation employed below for |W| in the 1PEM scenario.

A composite SC characterized by two gaps has at least two ion-species that can potentially cause pairing, e.g., the Y and the Ba ions in YBCO (Bi-2212 has three such ion-species, viz., Bi, Ca and Sr). Operative in these SCs is not only the 1PEM, but also 2PEM, and in rare cases, even 3PEM. In the latter case, the equations for W₁, W₂ and W₃—which follow from (7) by replacing the propagator V/(2πħc)³ by [V_A/(2πħc)³ + V_B/(2πħc)³ + V_C/(2πħc)³]—are

$\begin{array}{l}F1\left(\rho ,T_c,{\lambda }_A,{\lambda }_B,{\lambda }_C\right)\\ \equiv 1-\frac{1}{2}\left({\lambda }_{A}I\left(A,\rho ,T_c\right)+{\lambda }_{B}I\left(B,\rho ,T_c\right)+{\lambda }_{C}I\left(C,\rho ,T_c\right)\right)=0\end{array}$ (16)

$\begin{array}{l}F2\left(\rho ,W_3,q,{\lambda }_A,{\lambda }_B,{\lambda }_C\right)\\ \equiv 1-\frac{\sqrt{q}}{2}\left({\lambda }_{A}J\left(A,\rho ,W_3\right)+{\lambda }_{B}J\left(B,\rho ,W_3\right)+{\lambda }_{C}J\left(C,\rho ,W_3\right)\right)=0,\end{array}$ (17)

where λ_A, λ_B and λ_C are the interaction parameters due to, respectively, the sub-lattices of the A, B and C species of ions at t = 1,

$I\left(i,\rho ,T_c\right)={\displaystyle \underset{-{\theta }_i/2T_c}{\overset{{\theta }_i/2T_c}{\int }}\text{d}x\sqrt{1+\frac{2T_c}{\rho {\theta }_i}x}\frac{\tanh \left(x\right)}{x}}$

$J\left(i,\rho ,q,W_2\right)={\displaystyle \underset{-1/q\rho }{\overset{1/q\rho }{\int }}\text{d}x}\frac{\sqrt{1+x}}{\left|x\right|+\left|W_2\right|/2q\rho k{\theta }_i}$

and i denotes the ion-species A, B, or C, with θ_A, θ_B and θ_C being their Debye temperatures. Note that when one of the λs is zero in (16) or (17), the equation pertains to the 2PEM, and when two of them are zero, it pertains to the 1PEM.

Equation (16) and Equation (17) are supplemented by (2) for v_F and (1) for ξ, written as

$F\text{3}\left(v_F,\rho ,q,\eta \right)\equiv 1-\frac{\sqrt{2\rho qk{\theta }_{SC}/\eta m_e}}{v_F}c=0\left(m_e\text{in units of electron-Volt}\right)$ (18)

with θ_SC being the Debye temperature of the SC, and

$F\text{4}\left(\xi ,W,\rho ,q,\eta \right)\equiv 1-\frac{\hslash v_F\left(\rho ,q,\eta \right)}{\pi W\xi }=0.$ (19)

Composite SCs are usually characterized in the literature by a single value of the Debye temperature. We recall that Born and Karman [7] [8] had pointed out a long time ago in the context of the molar heats of salts that elastic waves in such anisotropic materials travel with different velocities in different directions and hence are characterized by more than one Debye frequency or temperature. Because composite SCs are invariably anisotropic materials, we have incorporated this feature in our work via the double pendulum model [9] . In this model the Debye temperature θ of a binary SC A_xB_1-x is resolved into θ_A and θ_B via the following equations

$\theta \left(x\right)=x{\theta }_A+\left(1-x\right){\theta }_B$

$\frac{{\theta }_A}{{\theta }_B}={\left[\frac{1+\sqrt{m_B/\left(m_A+m_B\right)}}{1-\sqrt{m_B/\left(m_A+m_B\right)}}\right]}^{1/2},$ (20)

where A is the upper bob in the double pendulum and m_A (m_B) is the atomic mass of an A (B) ion. It is notable that the above equations are also applicable to an SC that has three ion-species that can potentially cause pairing. This is so because each of such species is found in a different sub-lattice and it is assumed that the Debye temperature of any sub-lattice equals the Debye temperature of the entire lattice. An example: θ (Bi-2212) = 237 K. The ions that can potentially cause pairing in this SC are: Ca, Bi and Sr, but Ca occurs in a sub-lattice that has no other occupant. Therefore θ_Ca = 237 K. Application of (20) to the BiO and SrO sub-lattices, respectively, yields, θ_Bi = 269 K (Bi as the upper bob; θ_O = 205 K) and θ_Sr = 286 K (Sr as the upper bob; θ_O = 188 K). The values of θ_O are not required in our work.

For these SCs, we solve (10), (15) and (19) simultaneously with the inputs of T_c, |W| and ξ to obtain the values of ρ, q and λ₁ which then fix μ₁, μ₀, λ₀ and v_F via the equations noted in the caption of Table 1 . Noted below are the conditions that the physically acceptable solutions must satisfy.

1) Both λ₁ and λ₀ must be non-negative and < 0.5 (Bogoliubov constraint).

2) λ₀ must be greater than λ₁.

3) μ₀ must be greater than μ₁, i.e., q must be > 1. (21)

The results of the above exercise for all the elemental SCs being dealt with are given in Table 1 where it can be seen that all the above conditions are met. Excepting η, the empirical values of the parameters in Table 1 have been taken from [11] ; those of η from [12] for Al, Sn, Cd and Pb and from [13] for Nb.

Unlike the elemental SCs for which the values of the empirical parameters in the list given at the beginning of the paper vary negligibly from one source to another, the values of these parameters for high-T_c SCs vary widely. This is so predominantly because they depend on the form of the SC (bulk, thin film, etc.) and how it is doped. There would still be no problem in carrying out our study if all the properties in the list were available for the same sample, which is never the case. This circumstance necessitates a discussion of both the choice of inputs into the GBCSEs for any SC and the interpretation of the results it leads to. This task is taken up below for each of the high-T_c SCs being dealt with.

For the sake of concreteness, we employ the values of the T_c, |W₁|, |W₂| and v_F for this SC given by Leggett [14] as

$T_c\approx 40\text{K},\left|W_1\right|\approx 2.4\text{meV},\left|W_2\right|\approx 7\text{meV},v_F\approx 5\times {10}^7\text{cm}/\text{s}$ (22)

For θ (MgB₂) which is not given in [14] , we employ the value 815 K which is the mean of its values given in the review by Buzea and Yamashita [15] , viz., 750 and 880 K. The values of θ_B and θ_Mg obtained by resolving θ (MgB₂) = 815 K via (20) are noted in column 1 of Table 2 .

We now proceed to show how the GBCSEs not only provide an explanation of the properties noted in (22), but also shed light on several other related parameters. To this end, we undertake to solve simultaneously the four equations noted against MgB₂ in column 2 of Table 2 and find that the values of the four parameters in (22) are insufficient to solve these equations—we also need the value of η (≡m*/m_e) which is not given in [14] or [15] . On searching, we found that this parameter has been assigned widely different values by different authors, as is exemplified by Yelland, et al. [16] where its value is quoted as between 0.44 and 0.68, but with the remark that it is in the same ballpark as 1.25, 1.57, ≈ 1.1, 0.47, 0.50 and 0.33 quoted by several other authors. If we add to these values the range of η given by Mazin and Antropov [17] as (1.08 - 1.20), we have a rather bewildering situation. To cope with it, we ran our program for several of these values, beginning with η = 1.57, which led to μ₀ ≈ 1116 meV, an unacceptable value because it differs widely from its generally believed value of ≈ 100 meV as given in, e.g., Alexandrov [18] . Continuing the process with different values of η, we found that we needed not only to lower its value, but also that of v_F from the one noted in (22) to 2.7 × 10⁷ cm/s given by Posazhennikova, et al., [19] .

In column 3 of Table 2 are given the final values of the input parameters employed to simultaneously solve the four equations noted in column 2. In column 4 are given the values of the four parameters obtained by solving these equations and, in column 5, the values the of six parameters following from the solutions in column 4. It is thus seen that the λs and the μs satisfy the requisite conditions noted in (21) and that the value of μ₀ = 116 meV is very close to its value given as 122 meV in [18] . While our value of ξ₂ = 8.1 nm differs from its value given as 5 nm in , or (1.6 - 3.6) nm given in , as also is the case for ξ₁ for which our value of 22.6 nm differs considerably from its value given as (3.7 - 12) nm in (only one value of ξ is quoted in ), we draw attention to values of ξ other than those given in or that can also be found in the literature. Notable among these are the values of ξ₂ given in several papers as 7 or 7.5 nm, e.g., [20] and that of (ξ₁/ξ₂) as 2.8 [21] . It is therefore seen that our value of ξ₂ is very close to the former values and that of (ξ₁/ξ₂) matches the latter exactly.

It has been shown that these SCs are characterized by a universal Fermi velocity in the range (2.7 ± 0.5) × 10⁷ cm/s [22] and $\eta \simeq 3$ [23] . In the following we shall employ the values of these parameters in accord with these results.

We now deal with YBCO. Unlike MgB₂ for which the data are given as {T_c, Δ₁, Δ₂}, we found it hard to find a source where the two gaps of this SC are unequivocally given as corresponding to the same value of T_c. The data for YBCO are typically reported as {T_c (K), Δ (meV)} with the following values: {92, 20}, {60, 9} [11] , {89, 29}, {79, 25} [24] , and {68, 15.8} [25] .

With θ (YBCO) = 410 K resolved via (20) into θ_Y and θ_Ba as noted in column 1 of Table 2 , we dealt with this SC via the four equations noted against it in column 2 and adopting for it the values of the parameters given in column 3. Among these, the value of T_c = 90 K is chosen because four pairs of values of ξ are listed against it in [11] —vide Table 3 below. Since only one value of the gap, viz., 20 meV, is given for this SC, the other value noted in column 3 is an assumed value. As concerns the value of v_F = 1 × 10⁷ cm/s in column 3, we note that the range of v_F for cuprates has also been obtained as (2 − 8) × 10⁶ cm/s besides the one in as noted above. The value employed by us is intermediate between the upper limit given in and the lower limit given in . Employed together with a value of η as specified above, it has the virtue of leading to μ₀ = 85.3 meV—as against its value obtained via an entirely different approach in [18] as 84 meV (for a sample of YBCO with T_c = 91.5 K)—and ξ₁ = 2.65 which matches exactly one of its listed values vide Table 3 . In addition, it also leads to the values of the λs and the μs that satisfy the conditions specified in (21).

We now take up Bi-2212, Bi-2223, Tl-2212 and Tl-2223. For the values of the θs of these in Table 2 , see [5] . The values of the T_c and |W| of each of these are taken from [11] , where only one value of |W| is given. It is seen from column 2 of Table 2 that while the rather large value of the gap of each of these SCs has been attributed to the 3PEM, the mechanism invoked for its T_c is the 2PEM. This is so because employment of the 3PEM for the |W| of these SCs was found to be imperative in order that the values of the λs corresponding to them be in accord with the Bogoliubov constraint. However, we found that the T_cs of these SCs could still be accounted for by the 2PEM, i.e., there was no compelling need to invoke the 3PEM for their explanation.

An important feature of the Bi- based SCs is that they are characterized by three values of λ, viz., λ_Ca, λ_Bi and λ_Sr, and likewise for the Tl-based SCs. This is unlike YBCO which is characterized by λ_Y and λ_Ba only. This implies that by employing the same set of equations as for YBCO, we can now find the values of the three λs and either ρ or q. We dealt with this situation by finding the values of the λs and q corresponding to an assumed value of ρ and then calculating the values of ξ via (19), and repeating the procedure by varying ρ until at least one of the values of ξ was obtained in close agreement with its listed value.

The {T_c (K), Δ (meV)} values for Bi-2212 and Bi-2223 given in [11] are {95, 38} and {105, 33}, respectively. It is seen here that the increase in the value of T_c in going from the former to the latter is accompanied by a decrease in the value of Δ, which is against the conventional wisdom about this feature. It therefore follows that if Δ = 38 meV for Bi-2212 is attributed to the 3PEM as we have done, then Δ = 33 meV for Bi-2223 cannot be attributed to the same mechanism. In other words, in the scenario of the 3PEM, Bi-2223 must have a gap-value greater than 38 meV which we found to be 51 meV.

For Tl-2212, (T_c = 114 K, |W₂| = 30 meV) values in Table 2 are taken from [11] , whereas |W₁| = 4.9 meV is an assumed value. For Tl-2223, we have employed the empirical data of Ponomarev, et al. [27] who have reported three gap-values for it. Of these, we have employed the values of the largest and the smallest gaps in our study. The maximum value of the intermediate gap that we are then led to via F2 (ρ, W, q, λ_Ca, λ_Tl, 0) is 28.3 meV (which leads to ξ₂ = 1.78 nm) but does not match 45 meV quoted by the authors.

For SCs characterized by three values of λ, given in Table 2 are three values of ξ. For each of these SCs, the inputs employed to obtain the values of the λs and q and thence of the ξs, in fact, lead to four additional values of ξ which are not given in Table 2 . This is so because for such SCs, in principle, seven different values of ξ can occur—three from the 1PEM, three from the 2PEM and one from the 3PEM. Among these, given in Table 2 are the values of ξ that are closest to the available empirical values as listed in [11] and/or [28] .

The procedure followed for these SCs is similar to the one followed for MgB₂ or YBCO. For v_F of these SCs, we chose a value in the range derived by Talantsev [29] as (2.5 − 3.8) × 10⁷cm/s, and for η the value 2.76 given by Durajski [30] which has been extensively employed in, e.g., [31] and [32] . The other inputs employed for these SCs and the results they lead to are given in Table 2 . The values of the input parameters for H₃S are the same as were employed by us in [10] . For LaH₁₀, the value of T_c = 246 K is taken from Sun et al. [33] . For the larger gap of this SC, we have employed a value in the range given by Ruangrungrote, et al., [34] as 4.32 ≤ 2Δ/kT_c ≤ 5.25 (i.e., 45.8 ≤ Δ₂ ≤ 55.6 meV), whereas the value of Δ₁ is an assumed value which is justified by the overall results it leads to. Insofar as the role of pressure in these SCs is concerned, we should like to note that since the inputs employed for these SCs are pressure-dependent, so should be considered the results that they lead to.

Both our values of ξ for H₃S are within its range following from the work of different authors. It is notable that, generally, only one value of ξ for this SC is quoted in the literature which varies between 1.2 and 3.0 nm. For references to several papers where these values have been obtained, we refer the reader to where nearly the same values of ξ were obtained as here. However, q was < 1 in , which led to complex-valued solutions of which the imaginary parts were neglected.

For compressed LaH₁₀ too, our value of ξ₁ = 1.56 nm is in good agreement with its value given in [33] as 1.514 nm corresponding to the T_c under consideration.