Light composite fermions in a dynamical subquark model of pregauge interactions

The masses of composite leptons and quarks are discussed in a “dynamical subquark model of pregauge interactions”. In this model, the leptons and quarks are made of a spinor and scalar subquark with equal mass, M, and the gauge bosons and Higgs scalar of the SU(3)_c×SU(2)_L×U(1)_Y model are made of a subquark-antisubquark pair. The SU(2)_L×U(1)_Y symmetry is spontaneously broken by the composite Higgs scalar and the (scalar) subquark mass parameter is in turn bounded as $\text{M > 5.4}\text{TeV}\text{(=2π(}\text{2}\text{G}{}_{\mathrm{<mtext>F</mtext>}}{}^{\mathrm{?1}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{where}\text{G}{}_{\mathrm{<mtext>F</mtext>}}$ is the Fermi coupling constant). The spontaneously generated mass of a lepton or quark, m_i⁽ⁿ⁾ (i = 1, 2; n = 1 ～ N_g), is calculated to be: $\text{m}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{= r}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{= r}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{× (4+3N}{}_{\mathrm{<mtext>g</mtext>}}\text{)α}{}_{\mathrm{<mtext>e.m.</mtext>}}\text{(}\text{2}\text{G}{}_{\mathrm{<mtext>F</mtext>}}{}^{\mathrm{?1}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{/3}\text{6}\text{(=0.35r}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{(4+3N}{}_{\mathrm{<mtext>g</mtext>}}\text{)}\text{GeV}\text{),}\text{where}\text{r}{}_{\mathrm{i}}{}^{\mathrm{(n)}}$ are the parameters satisfying that 0 ? r_i⁽ⁿ⁾ ? 1 and Σ (r_i⁽ⁿ⁾)² = 1;N_g is the total number of generations of the leptons and quarks; α_e.m. is the fine structure constant. The appearance of light composite fermions is related to a specific mechanism of generating global chiral symmetries of the leptons and quarks. Global symmetries of scalar subquarks yield chiral symmetries of the leptons and quarks. Our model turns out to satisfy 't Hooft's anomaly conditions on massless composite fermions.     

Dielectric susceptibility and correlation length of one-dimensional CDW with impurties. II. weak disorder

Dependence of static dielectric susceptibility and correlation length of charge density waves (CDW) with weak defects on parameter of incommensurability with lattice is investigated. In almost commensurate phase (h?h_ch_c), $\text{χ ～ (h?h}{}_{\mathrm{c}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}\text{In}{}^{\mathrm{<mtext>-4</mtext><mtext>3</mtext>}}\text{h}{}_{\mathrm{c}}\text{/h?h}{}_{\mathrm{c}}$ and R_c ～ (h ? h_c)^{$\text{2}\text{3}$}. In${}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}\text{h}{}_{\mathrm{c}}\text{/h ? h}{}_{\mathrm{c}}$. Far from commensurability $\text{(h?h}{}_{\mathrm{c}}\text{) χ～ (a+h}{}^2{}_{\mathrm{c}}\text{/h}{}^2\text{)}{}^{\mathrm{<mtext>-2</mtext><mtext>3</mtext>}}$, $\text{R}{}_{\mathrm{c}}\text{～ (a + h}{}^2{}_{\mathrm{c}}\text{/h}{}^2\text{)}{}^{\mathrm{<mtext>-2</mtext><mtext>3</mtext>}}$, where a is the dimensionless ratio of random potential intensities, corresponding to backward and forward scattering impurities.     

Gravitational collapse in quantum mechanics with relativistic kinetic energy

It is proved that the quantum mechanical Hamiltonian $\text{H = Σ}{}_{\mathrm{i=1}}{}^{\mathrm{N}}\text{(p}{}^2\text{+ m}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{? κ Σ}{}_{\mathrm{i>j}}\text{|x}{}_{\mathrm{i}}\text{? x}{}_{\mathrm{j}}\text{|}{}^{\mathrm{?1}}$ for bosons (resp, fermions) is bounded from below if N ≤ c_bκ^?1 (resp. $\text{N ≤ c}{}_{\mathrm{f}}\text{κ}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext>}}$). H is unbounded from below if N ≥ c_b^lκ^?1 (resp. $\text{N ≥ c}{}_{\mathrm{f}}{}^{\mathrm{l}}\text{κ}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext>}}$). The constants c_b and c_b^l (resp. c_f and c_f^l) differ by about a factor 2 (resp. 4).     

“Forbidden” rotational spectra of symmetric-top molecules: PH3 and PD3

A millimeter-wave spectrometer having a sensitivity of 4 × 10^?10 cm^?1 in the 2-mm region has been constructed for observation of extremely weak millimeter-wave spectra of gases. It has been used to measure J → J, K = 0 ← 3 transitions in PH₃ and J → J, K = 0 ← 3 as well as K = ±1 ← ±4 transitions in PD₃. The B₀ and C₀ spectral constants (in MHz) are: for PH₃, B₀ = 133 480.15 ± 0.12 and C₀ = 117 488.85 ± 0.16; for PD₃, B₀ = 69 471.10 ± 0.03 and C₀ = 58 974.37 ± 0.05. The effective ground-state values obtained for the bond angle and bond length are: for PH₃, $\text{r}{}_0\text{(}\text{A}\text{?}\text{) = 1.420}{}_0$ and $\text{α}{}_0\text{(}{}^{\mathrm{o}}\text{) = 93.34}{}_5$; for PD₃, $\text{r}{}_0\text{(}\text{A}\text{?}\text{) = 1.417}{}_6$ and $\text{α}{}_0\text{(}{}^{\mathrm{o}}\text{) = 93.35}{}_9$. The corresponding zero-point-average values were calculated to be: for PH₃, $\text{r}{}_{\mathrm{z}}\text{(}\text{A}\text{?}\text{) = 1.4269}{}_9\text{± 0.0002}$ and $\text{α}{}_{\mathrm{z}}\text{(}{}^{\mathrm{o}}\text{) = 93.228}{}_7$; for PD₃, $\text{r}{}_{\mathrm{z}}\text{(}\text{A}\text{?}\text{) = 1.4226}{}_5\text{± 0.0001}$ and $\text{α}{}_{\mathrm{z}}\text{(}{}^{\mathrm{o}}\text{) = 93.256}{}_7\text{± 0.004}$. For both species, the equilibrium values are $\text{r}{}_{\mathrm{e}}\text{(}\text{A}\text{?}\text{) = 1.4115}{}_9\text{± 0.0006}$ and $\text{α}{}_{\mathrm{e}}\text{(}{}^{\mathrm{o}}\text{) = 93.32}{}_8\text{± 0.02}$.     

On the relativistic Thomas-Fermi model

The relativistic generalization of the Thomas-Fermi model of the atom is derived. It approaches the usual nonrelativistic equation in the limit Z ? Z_crit, where Z is the total number of electrons of the atom and $\text{Z}{}_{\mathrm{<mtext>crit</mtext>}}\text{=(}\text{3π}\text{4}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{α}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext>}}$ and α is the fine structure constant. The new equation leads to the breakdown of scaling laws and to the appearance of a critical charge, purely as a consequence of relativistic effects. These results are compared and contrasted with those corresponding to N self-gravitating degenerate relativistic fermions, which for $\text{N ≈ N}{}_{\mathrm{<mtext>crit</mtext>}}\text{=(}\text{3π}\text{4}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(m/m}{}_{\mathrm{<mtext>p</mtext>}}\text{)}{}^3$ give rise to the concept of a critical mass against gravitational collapse. Here m is the mass of the fermion and $\text{m}{}_{\mathrm{<mtext>p</mtext>}}\text{=(?c/G)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is the Planck mass.     

Lattice formulation of the CPn−1 non-linear σ models

We set up a general lattice version of non-linear σ models defined on homogeneous spaces. We then apply this to the CP^n?1 models which are the correct extension of the SU(2) σ model to SU(N). We exhibit their “confinement” property: the elementary multiplets Z_α used to describe the system do not appear as physical particles but only as bound states $\text{Z}{}_{\mathrm{\alpha }}\text{λ}{}_{\mathrm{\alpha \beta }}{}^{\mathrm{a}}\text{Z}{}_{\mathrm{\beta }}$. The method enable us to examine the “θ vacua” in the strong coupling limit by using a “dilute loop” approximation. We discuss the effect of the low activation energy for instantons which means that on a lattice, topological number is not conserved.     

Phenomenological SU(1, 1) supergravity

We investigate the structure of phenomenological supergravity models which permit the hierarchy problem to be “solved” in the sense that $\text{m}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ and m_W are determined dynamically to be exp [-O(1)/α] × m_P. Such models must have a flat hidden sector potential, which is only possible if the theory has an underlying SU(1, 1) invariance. Flat SU(1, 1) theories necessarily have a zero cosmological constant and the hidden sector is an Einstein space with . The SU(1, 1) invariance is necessarily broken down to U(1) by the gravitino mass. If $\text{m}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ is the only source of SU(1, 1) breaking then the tree-level gaugino masses are small and $\text{A =}\text{3}\text{2}$, while values of A up to 3 and non-zero gaugino masses are possible if other sources of SU(1, 1) breaking are tolerated. Yukawa couplings may scale as some power of $\text{m}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}\text{m}{}_{\mathrm{<mtext>P</mtext>}}$ in these models where $\text{m}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ is generated dynamically, which may explain the hierarchy of Higgs-fermion Yukawa couplings: $\text{m}{}_{\mathrm{<mtext>f</mtext>}}\text{m}{}_{\mathrm{<mtext>W</mtext>}}\text{=}\text{O}\text{(}\text{m}{}_{\mathrm{<mtext>W</mtext>}}\text{m}{}_{\mathrm{<mtext>P</mtext>}}\text{)}{}^{\mathrm{\lambda >0}}$? These models also permit the spontaneous violation of CP in the Yukawa coupling matrix. Numerical studies yield 20 GeV < m_t < 100 GeV in these phenomenological SU(1, 1) supergravity models. Speculations are presented about their relation to a fundamental theory based on extended supergravity.     

α(Q2) corrections to particle multiplicity ratios in gluon and quark jets

The order $\text{α(Q}{}^2\text{)}$ correction to the particle multiplicity ratio in gluon and quark jets is calculated in QCD. We find

$\text{r=}\text{9}\text{4}\text{1?}\text{αC}{}_{\mathrm{<mtext>A</mtext>}}\text{2π}\text{1}\text{3}\text{+}\text{N}{}_{\mathrm{?}}\text{3C}{}_{\mathrm{<mtext>A</mtext>}}\text{?}\text{2N}{}_{\mathrm{?}}\text{C}{}_{\mathrm{<mtext>F</mtext>}}\text{3C}{}^2{}_{\mathrm{<mtext>A</mtext>}}$

with r=〈n〉_{gluon jet}/〈n〉_{quark jet}. The method used is systematic and could be used for an order α(Q²) calculation.     

Quark masses

In quark gluon theory with very small bare masses, -$\text{ψ}$$\text{M}$ψ, spontaneous breakdown of chiral symmetry generates sizable masses M_u, M_d, M_s, … We find ($\text{M}{}_{\mathrm{<mtext>u</mtext>}}\text{+ M}{}_{\mathrm{<mtext>d</mtext>}}\text{) /2 ≈ m}\text{p}\text{/ √6 ≈ 312}\text{MeV}\text{,}\text{and}\text{M}{}_{\mathrm{<mtext>s</mtext>}}\text{≈ 432}\text{MeV}$. Scalar densities have well determined non-zero vaccum expectations $\text{〈0|u}{}^{\mathrm{a}}\text{|0〉 ≡ 〈0|ψ(x) (λ}{}^{\mathrm{a}}\text{/2)ψ(x)/0〉 ≈ ?}{}_{\mathrm{\pi }}{}^2\text{M}{}^{\mathrm{a}}\text{,}\text{i.e}\text{〈0? u}{}^{\mathrm{o}}\text{/vb0〉 ≈ 8 × 10}{}^{\mathrm{?3}}\text{(}\text{GeV}\text{)}{}^3$ at an SU(3) breaking of the vacuum c′ ≡ 〈0|u⁸|〉/〈0|u^o|0〉 ≈ ? 16%     

A Mössbauer study of amorphous Fe40Ni40B20

Amorphous Fe₄₀Ni₄₀B₂₀ (VITROVAC 0040) alloy has been investigated using ⁵⁷Fe Mössbauer Spectroscopy. The Curie temperature T_c is found to be well defined and is 695 ± 1 K. The quadrupole splitting just above T_c is 0.64 mm sec^?1. The crystallization temperature is 698 ± 2 K, close to but definitely above T_c. The average hyperfine field H_eff(T) of the glassy state shows a temperature dependence of $\text{H}{}_{\mathrm{<mtext>eff</mtext>}}\text{(0)[1 ? B}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}\text{(T/T}{}_{\mathrm{c}}\text{)}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}\text{? C}{}_{\mathrm{<mtext>5</mtext><mtext>2</mtext>}}\text{(T/T}{}_{\mathrm{c}}\text{)}{}^{\mathrm{<mtext>5</mtext><mtext>2</mtext>}}\text{? …]}$ indicative of the existence of spin wave excitations. The values of $\text{B}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ and $\text{C}{}_{\mathrm{<mtext>5</mtext><mtext>2</mtext>}}$ are found to be 0.40 and 0.06, respectively, for T/T_c ? 0.72. At temperatures close to T_c, H_eff(T) varies as (1 ? T/T_c)^β where β is one of the critical exponents and its value is found to be 0.29 ± 0.02.     

On the order of levels in charmonium

We prove a theorem concerning the energies of the 2S and 3D states in a potential $\text{V(r) =}\text{?g}{}^2\text{r}\text{+ V}{}_{\mathrm{<mtext>c</mtext>}}\text{(r)}$, where V_c is a non-singular confining potential. If $\text{(}\text{d}\text{d}\text{r)}{}^3\text{(r}{}^2\text{V}{}_{\mathrm{<mtext>c</mtext>}}\text{)}$ is positive, then the 3D state lies above the 2S state, provided

$\text{d}\text{d}\text{r}\text{1}\text{r}\text{d}\text{d}\text{r}\text{2V}{}_{\mathrm{<mtext>c</mtext>}}\text{+r}\text{d}\text{V}{}_{\mathrm{<mtext>c</mtext>}}\text{d}\text{r}\text{< 0, ?}{}_{\mathrm{r}}\text{>0.}$

For V_c = r^α, this corresponds to 0 < α < 2.     

Coriolis intensity perturbations in the SO2 molecule: Experimental determination of the relative signs of (∂μ∂Qi)

The Coriolis interactions between ν₁ and ν₃, and between ν₂ and ν₃ in SO₂ have been analyzed to obtain the signs of the products $\text{ζ}{}_{3.1}{}^{\mathrm{c}}\text{(}\text{?μ}{}^{\mathrm{a}}\text{?Q}{}_3\text{)(}\text{?μ}{}^{\mathrm{b}}\text{?Q}{}_1\text{)}$ and $\text{ζ}{}_{3.2}{}^{\mathrm{c}}\text{(}\text{?μ}{}^{\mathrm{a}}\text{?Q}{}_3\text{)(}\text{?μ}{}^{\mathrm{b}}\text{?Q}{}_2\text{)}$. It has been found that both of the signs of these products are positive. Then, relative signs of ($\text{?μ}\text{?Q}{}_1$) have been determined using the calculated values of the Coriolis zeta constants for the present definition of the normal coordinates. The obtained sign combination of ($\text{?μ}\text{?Q}{}_{\mathrm{i}}$) is ±(+?+), which agrees with the one predicted by the molecular orbital calculations. Using the sign combination (+?+), the polar tensors of S and O atoms were also calculated.     

Phenomenological consequences of N = 1 supergravity theories with non-minimal kinetic energy terms for vector superfields

It is shown that N=1 supergravity theories can have a GUT scale as large as the Planck scale if the kinetic energy terms for vector superfields are non-minimal. The canonical values for sin²θ_W (M_W), α₃ (M_W) and $\text{m}{}_{\mathrm{<mtext>b</mtext>}}\text{m}{}_{\mathrm{\tau }}\text{(M}{}_{\mathrm{<mtext>W</mtext>}}\text{)}$ are respected. In those theories masses of SU(3), SU(2) and U(1) gauginos may be different at the unification scale. Consequences for the low-energy particle spectrum are discussed in the extreme case where one of the gaugino masses is large while the other two vanish.     

On some recent results concerning the superconducting transition temperature

The Eliashberg gap equations relate the transition temperature T_c of an isotropic superconductor to its electron-phonon spectral function α²F(ω) and Coulomb pseudopotential parameter $\text{μ}{}^1$. Recently the Eliashberg theory has been used to derive some supposedly rigorous results bearing on the problem of attaining higher superconducting transition temperatures: Bergmann and Rainer derived an expression for the functional derivative $\text{δT}{}_{\mathrm{c}}\text{δα}{}^2\text{F(ω)}$; Allen and Dynes showed that in the asymptotic limit of very large $\text{λ(λ?10)k}{}_{\mathrm{B}}\text{T}{}_{\mathrm{c}}\text{=f(μ}{}^1\text{)(λ〈ω}{}^2\text{〉)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and Leavens proved that for any isotropic superconductor k_BT_c ?0.2309A, where A is the area under its electron-phonon spectral function. In this letter we show that the result of Allen and Dynes is not compatible with the other results and is, in fact, incorrect.     

Two theorems on the level order in potential models

We study the potentials of the form U(r)=?r^?1+λV(r), $\text{(}\text{d}\text{d}\text{r}\text{)(}\text{r}{}^2\text{d}\text{V}\text{d}\text{r}\text{)?0}$, and show that the energy levels satisfy the inequalities $\text{E(N}{}_{\mathrm{c}}\text{, l)?E(N}{}_{\mathrm{c}}\text{, l+1)}$ to first order in λ, where N_c denotes the coulombic principal quantum number and l the angular momentum. Similarly for potentials $\text{U(r)=r}{}^2\text{+λV(r), (}\text{d}\text{d}\text{r}{}^2\text{)}{}^2\text{V(r)?0}$, we prove to first order in λ that $\text{E}\text{?}\text{(N}{}_{\mathrm{H}}\text{,}\text{l}\text{)?}\text{E}\text{?}\text{(N}{}_{\mathrm{H}}\text{,}\text{l}\text{+2)}$, where NH denotes the harmonic oscillator quantum number. In the latter case, we give also quantitative restrictions on the relative positions at the lth and (l+1)th states.     

On the super symmetry charges in the theory of scalar and spinor free fields

It is shown that for spinorial charges Q^(L))_α (α = 1, 2, L = 1, …, S) satisfying the commutation relations

$\text{{Q}{}^{\mathrm{(L)}}{}_{\mathrm{\alpha }}\text{, Q}{}^{\mathrm{(M)}}{}_{\mathrm{\beta }}\text{} = ε}{}_{\mathrm{\alpha }}\text{βa}{}^{\mathrm{LM}}\text{Q,}$

$\text{{Q}{}^{\mathrm{(L)}}{}_{\mathrm{\alpha }}\text{, Q}{}^{\mathrm{(M)+}}{}_{\mathrm{\beta }}\text{} = cσ}{}^{\mathrm{\mu }}{}_{\mathrm{\alpha \beta }}\text{P}{}_{\mathrm{\mu }}\text{δ}{}^{\mathrm{LM}}\text{,}$

$\text{[Q}{}^{\mathrm{(L)}}\text{)}{}_{\mathrm{\alpha }}\text{, P}{}^{\mathrm{\mu }}\text{] = 0,}$

where Q is a scalar charge commuting with the spinor charges as well aswith the energy- momentum vector Pμ, there can exist several different multiplets for free massive scalar and spinor fields.     

Strong absorption model for the (α, 2α) reaction

The (α, 2α) reaction has been studied in the “diffraction model” for ¹⁶O, ²⁹Mg, ⁴⁰Ca and ⁶⁶Zn nuclei at 90 MeV incident energy. Using $\text{R}{}_{\mathrm{<mtext>c</mtext>}}\text{= 1.15(A}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}\text{+4}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}\text{)}\text{fm}$ for the strong absorption radius and Δ = 0.06R_c for the diffuseness of the absorbing surface the spectroscopic factors are extracted. These spectroscopic factors are in excellent agreement with the theoretical values. The positions of the minima and second maxima are also reproduced.     

Model calculation of the electronic structure and spectroscopy of Hg2

Energy curves and transition moments of the excited valence states of Hg₂ were obtained in a model calculation based on calculated Mg₂ energy levels and the assumption that the asymptotic spin-orbit matrix elements for the Hg atom are applicable to the molecular states. The spin-orbit and orbital-rotational interaction of the excited states of Hg₂ is analyzed in both a Hund's case (c) and (a) representation. The intermediate (a) → (c) transition moments are obtained as a function of the internuclear distance. The effect of the orbital-rotational interaction which introduces Hund's case (b) and (e) couplings is found to be small for transitions among excited states under the conditions normally encountered for populating excimer states.Using the energy level positions and transition moments, the observed spectra and predicted spectra are compared for both radiative transitions including the ground state and among the excited states. The lifetime of the $\text{1}{}_{\mathrm{u}}\text{(}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}\text{)}$ excimer state is calculated to be 1.4 μsec with the 335 nm band assigned to the $\text{1}{}_{\mathrm{u}}\text{→ X}{}^1\text{Σ}{}_{\mathrm{g}}{}^{\mathrm{+}}$ transition. The 485 nm bands cannot be assigned to any Hg₂ transitions. Strong bound-continuum absorptions are predicted for the 485 nm bands. On the other hand, the 335 nm emission is predicted to be absorbed by bound-bound transitions only.     

The A-dependence of J/ψ-particle inclusive distributions

The differential cross sections $\text{d}\text{σ}\text{d}\text{x}$ and $\text{d}\text{σ}\text{d}\text{p}{}_{\mathrm{t}}{}^2$ of inclusive J/ψ production by 43 GeV/c π^? off Be, Cu and W nuclei have been measured. Fitting $\text{d}\text{σ}\text{d}\text{p}{}_{\mathrm{t}}{}^2\text{～ A}{}^{\mathrm{\alpha }}\text{(p}{}_{\mathrm{t}}{}^2\text{)}$ we observed the increase of α with p_t².     

Functional derivative δTc/δα2 (Ω)F(Ω) for weak coupling superconductors

We present approximate analytic calculation of the functional derivative $\text{δT}{}_{\mathrm{c}}\text{δα}{}^2\text{(Ω)F(Ω)}$, where T_c is the superconducting critical temperature and $\text{α}{}^2\text{(Ω)F(Ω)}$ is the electron-phonon spectral function, within the “square-well model” for the phonon mediated electron-electron interaction and weak coupling limit $\text{ω}{}_{\mathrm{D}}\text{(2πT}{}_{\mathrm{c}}\text{)? 1 (ω}{}_{\mathrm{D}}$ is the Debye energy). It is found that $\text{δT}{}_{\mathrm{c}}\text{δα}{}^2\text{(Ω)F(Ω) = (1 + λ)}{}^{-1}\text{G(Ω)}$ where λ is the familiar electron-phonon coupling parameter and $\text{G(Ω)}$ is a universal function of the reduced frequency $\text{Ω}\text{=}\text{Ω}\text{T}{}_{\mathrm{c}}$. We compare this formula with accurate numerical results for several weak coupling superconductors. The overall agreement is good