Predissociation and Λ-doubling in the even-parity Rydberg states of the nitrogen molecule

Predissociations in the y¹Π_g and $\text{x}{}^1\text{Σ}{}_{\mathrm{g}}{}^{\mathrm{?}}$ Rydberg states of N₂ (configurations $\text{1π}{}_{\mathrm{u}}{}^{\mathrm{?1}}\text{4pσ}$ and $\text{1π}{}_{\mathrm{u}}{}^{\mathrm{?1}}\text{3pπ}$, respectively) and their likely causes, are discussed. Peaking of rotational intensity at unusually low J values, without sharp breaking off, is interpreted as due to case c^? or case cⁱ predissociation. Λ doubling in the y state, attributed to interactions with the $\text{x}{}^1\text{Σ}{}_{\mathrm{g}}{}^{\mathrm{?}}$ state and with another, ¹Σ⁺, state of the same electron configuration as x, is analyzed. From this analysis the location of the (unobserved) ${}^1\text{Σ}{}_{\mathrm{g}}{}^{\mathrm{+}}$ state, here labeled x′, is obtained. It is concluded that the predissociation in the Π⁺ levels of the y state is an indirect one mediated by the interaction with x′ coupled with predissociation of x′ by a ${}^3\text{Σ}{}_{\mathrm{g}}{}^{\mathrm{?}}$ state dissociating to ${}^4\text{S +}{}^2\text{P}$ atoms: combined, however, with perturbation of the y state by the k¹Π_g Rydberg state (configuration $\text{3σ}{}_{\mathrm{g}}{}^{\mathrm{?1}}\text{4dπ}$), whose Π⁺ levels are completely predissociated.     

Lorentz contracted geometrical model

The model assumes that when two high energy particles collide each behaves as a geometrical object which has a Gaussian density and is spherically symmetric except for the Lorentz-contraction in the incident direction. Folding the two spatial distribution together we obtain the slope (b) of the elastic diffraction peak in terms of the c.m. velocities (β_i and β_j) and the sizes (A_i and A_j) of the two incident particles. These sizes are assumed to have the experimental s-dependence of σ_tot ∝ πA² for each reaction. The combined s-dependence of the σ_tot's and the β's gives the s-dependence of the elastic slope $\text{b}{}_{\mathrm{ij}}\text{(s) =}\text{1}\text{2}\text{(A}{}_{\mathrm{i}}{}^2\text{β}{}_{\mathrm{i}}{}^2\text{+ A}{}_{\mathrm{j}}{}^2\text{β}{}_{\mathrm{j}}{}^2\text{)}\text{σ}{}^{\mathrm{ij}}{}_{\mathrm{<mtext>tot</mtext>}}\text{(s)}\text{σ}{}^{\mathrm{ij}}{}_{\mathrm{<mtext>tot</mtext>}}\text{(∞)}$. This formula agrees with the experimental slope for p-p, $\text{p}$-p, K⁺-p, K^?-p and π^±-p elastic scattering from 3 to 1500 GeV/c, with only 3 parameters: A_π² = 6.1, A_K² = 3.3 and A_p² = 10.5 (GeV/c)^?2.     

Two emission systems of BS

Two new systems of emission bands near 2100 and 3100 Å have been produced by a microwave discharge in B₂S₃ vapor. From the known X²Σ⁺ and A²Π_i states of BS, these systems have been assigned as E²Σ⁺ → X²Σ⁺ and E²Σ⁺ → A²Π_i. Constants in cm^?1 for the new state are

$\text{E}{}^2\text{∑}{}^{\mathrm{\pm }}\text{: T}{}_{\mathrm{e}}\text{= 47 929.3, B}{}_{\mathrm{e}}\text{= 0.671 (λ}{}_{\mathrm{e}}\text{= 1.752 A), α}{}_{\mathrm{e}}\text{= 0.008}$

,     

An effective procedure to determine corrugation functions from atomic beam-diffraction intensities

A computational method is described, which, starting from given difraction intensities, approaches effectively the best-fit corrugation function $\text{ζ(}\text{R}\text{)}$. Because of the approximations involved, the procedure works well for smooth corrugations with amplitudes not exceeding ～10% of the lattice constant. The method rests on two crucial observations: (i) With the full knowledge of the scattering amplitudes $\text{A}{}_{\mathrm{G}}\text{= ¦A}{}_{\mathrm{G}}\text{¦}\text{exp}\text{(}\text{i}\text{?}{}_{\mathrm{G}}\text{)}$ (absolute values plus phases), the corrugation function can be calculated to a high degree of accuracy from $\text{ζ(}\text{R}\text{) = (2}\text{i}\text{k}{}_{\mathrm{i}}\text{)}{}^{\mathrm{?1}}$ In $\text{¦?ΣA}{}_{\mathrm{G}}\text{exp}\text{(}\text{i}\text{G·R}\text{)¦}$ which is derived easily from the hard corrugated wall scattering (HCWS) equation by approximating $\text{k}{}_{\mathrm{G}}$ by $\text{?}\text{k}{}_{\mathrm{i}}$ ($\text{k}{}_{\mathrm{i}}$ and $\text{k}{}_{\mathrm{G}}$ being the wavevectors of the incoming and diffracted beams, respectively), (ii) With only the $\text{¦A}{}_{\mathrm{G}}\text{¦'s}$ (or intensities) known, approximate solutions of the HCWS equation can be obtained with a rough estimate of the relative phases of only a few intense diffraction beams; the estimate is readily performed by investigating systematically a coarse mesh of phases. In this way, approximate corrugations are found with which a full set of phases can be generated, which allows the calculation of an improved ζ(R); this step is repeated in a loop, until optimum agreement between calculated and given intensities is obtained. The effectiveness of the procedure is demonstrated for three one-dimensional model corrugations described by several Fourier coefficients. The method is finally applied to the case of H₂ diffraction from the quasi-one-dimensional adsorbate corrugation Ni(110) + H(1 × 2).     

Transition moment variation in the A2Σ+ → X2Πi transition of HCl+, DCl+ and HBr+

Relative emission intensities of sixteen bands of HCl⁺ (A²Σ⁺ - X²Π_i), four bands of DCl⁺ (A²Σ⁺ - X²Π_i), and 5 bands of HBr⁺ (A²Σ - X²Π_i) have been made using both ion-beam excitation and microwave discharge sources. Intensities were determined by comparison with computer-generated spectra. Treatment of the data within the r-centroid approximation shows that in HCl⁺ the electronic transition moment decreases strongly at large $\text{r}{}_{\mathrm{v\prime v″}}$ [$\text{Re}\text{α}\text{exp}\text{(?3.6}\text{r}{}_{\mathrm{v\prime v″}}\text{)}\text{for 1.44}\text{A}\text{?}\text{< r}{}_{\mathrm{v\prime v″}}\text{< 1.82}\text{A}\text{?}$] but levels off at shorter $\text{r}{}_{\mathrm{v\prime v″}}$. DCl⁺ data agree quantitatively with HCl⁺. The variation in the HBr⁺ moment is similar, with $\text{R}\text{e}\text{α}\text{exp}\text{[?4.5}\text{r}{}_{\mathrm{v\prime v″}}\text{]}\text{for 1.58}\text{A}\text{?}\text{<}\text{r}{}_{\mathrm{v\prime v″}}\text{< 1.78}\text{A}\text{?}$.     

Inclusive pion distributions in e+e− annihilation from sum rules and duality

The threshold behaviour of pion production presented in our earlier work is successfully compared with the new SPEAR data. By using duality and sum rules we derive F_T^(π+)(x) ≈ F_L^(π+)(x) ≈ F_T^(π0)(x) ? F_L^(π0)(x) for x near 1. An accompanying results is $\text{σ}{}_{\mathrm{<mtext>\pi </mtext><mtext>A</mtext>}}{}_2\text{(s) ≈ 2σ}{}_{\mathrm{\pi \omega }}\text{(s) ≈ 4σ}{}_{\mathrm{\pi \pi }}\text{(s) ≈ 9(m}{}_{\mathrm{?}}{}^2\text{/s)}{}^3\text{σ}{}_{\mathrm{<mtext>\mu </mtext><mtext>\mu </mtext>}}$ for large s.     

Energies and orbital sizes for some Rydberg and valence states of the nitrogen molecule

Accurate SCF computations are reported on the Rydberg states of N₂ of electron configurations $\text{---1π}{}_{\mathrm{u}}{}^3\text{2π}{}_{\mathrm{u}}$, $\text{---1π}{}_{\mathrm{u}}{}^3\text{3σ}{}_{\mathrm{u}}$, and ---3σ_g2π_g, also on the valence states of the configuration $\text{---1π}{}_{\mathrm{u}}{}^3\text{1π}{}_{\mathrm{g}}$. The Rydberg state calculations supplement those of Lefebvre-Brion and Moser. A comparison is made between the $\text{---1π}{}_{\mathrm{u}}{}^3\text{2π}{}_{\mathrm{u}}$ states and the parallel set of states of the $\text{1π}{}_{\mathrm{u}}{}^3\text{1π}{}_{\mathrm{g}}$ configuration. This comparison shows a sharp difference in the ¹Σ⁺ states of the two configurations, the ¹Σ⁺ state being very high in the latter but relatively low in the former configuration. Recknagel coefficients are given for the several states of the two configurations; as expected, these are much smaller for the $\text{1π}{}_{\mathrm{u}}{}^3\text{2π}{}_{\mathrm{u}}$ configuration. Also, the ¹Δ state is relatively lower for the latter configuration.     

Search for 4n IN π− double charge exchange reactions with heavy nuclei

A search for the existence of the tetraneutron has been made using the double charge exchange reaction $\text{π}{}^{\mathrm{?}}\text{+}{}^{208}\text{Pb}\text{→}{}^4\text{n + π}{}^{\mathrm{+}}\text{+}$ residuals for ⁴n production and the capture process in the same target, ${}^{208}\text{Pb}\text{+}{}^4\text{n}\text{→}{}^{212}\text{Pb}\text{+ γ}$, for the ⁴n detection. No event has been found, giving an upper limit for the product of the production cross section σ_p, the detection cross section σ_d and the ⁴n lifetime τ. Assuming 10^?18 ≦ τ ≦ 10^?9 sec it follows that σ_pσ_dτ ≦ 2.5 × 10^?65cm⁴ sec with 90 % confidence, and for $\text{τ ≧ 10}{}^{\mathrm{?9}}\text{sec}\text{, σ}{}_{\mathrm{<mtext>p</mtext>}}\text{σ}{}_{\mathrm{<mtext>d</mtext>}}\text{≦ 2.5 × 10}{}^{\mathrm{?56}}\text{cm}{}^4$ with 90 % confidence. The magnitude of this value is comparable to the experimental limit of the ⁴He(π^?, π⁺)⁴n cross section.     

Observed and calculated interactions between valence states of the NO molecule

No perturbation between two valence states of NO has ever been identified, although many valence-Rydberg and several Rydberg-Rydberg perturbations have been extensively studied. The first valence-valence crossing to be experimentally documented for NO is reported here and occurs between the ¹⁵N¹⁸O B²Π (v = 18) and B′²Δ (v = 1) levels. No level shifts larger than the detection limit of 0.1 cm^?1 are observed at the crossings near $\text{J = 6.5 [B}{}^2\text{Π}\text{(F}{}_1\text{) ～ B′}{}^2\text{Δ}\text{(F}{}_2\text{)]}$ and $\text{J = 12.5 [B}{}^2\text{Π}\text{(F}{}_1\text{) ～ B′}{}^2\text{Δ}\text{(F}{}_1\text{)]}$; two crossings involving higher rotational levels could not be examined. Semi-empirical calculations of spin-orbit and Coriolis perturbation matrix elements indicate that although the electronic part of the $\text{B}{}^2\text{Π}\text{～ B′}{}^2\text{Δ}$ interaction is large, a small vibrational factor renders the ¹⁵N¹⁸O B (v = 18) ? B′ (v = 1) perturbation unobservable. Semi-empirical estimates are given for all perturbation matrix elements of the operators $\text{Σ}{}_{\mathrm{i}}\text{a}\text{?}{}_{\mathrm{i}}\text{l}{}_{\mathrm{i}}\text{·s}{}_{\mathrm{i}}$ and $\text{B(L}{}_{\mathrm{\pm }}\text{S}{}_{\mathrm{?}}\text{? J}{}_{\mathrm{\pm }}\text{L}{}_{\mathrm{?}}\text{)}$ which connect states belonging to the configurations $\text{(σ2p)}{}^2\text{(π2p)}{}^4\text{(π}{}^1\text{2p)}$, $\text{(σ2p)(π2p)}{}^4\text{(π}{}^1\text{2p)}{}^2$, and $\text{(σ2p)}{}^2\text{(π2p)}{}^3\text{(π}{}^1\text{2p)}{}^2$.     

The dipole moment of thioformaldehyde in its singlet and triplet π1 − n excited states

Laser-induced fluorescence excitation has been used to measure Stark splittings of selected lines in the $\text{A}\text{?}{}^1\text{A}{}_2\text{-}\text{X}\text{?}{}^1\text{A}{}_1$ and $\text{a}\text{?}{}^3\text{A}{}_2\text{-}\text{X}\text{?}{}^1\text{A}{}_2$ band systems of H₂CS in electric fields up to 13 kV/cm. The derived excited state a-axis dipole moments are 0.820 ± 0.007 D for the 4¹ level of the ¹A₂ state; 0.838 ± 0.008 D for the zeroth vibrational level of ¹A₂; and 0.534 ± 0.015 D for the zeroth vibrational level of the ³A₂ state. These results are compared with the corresponding values of H₂CO, and interpreted in terms of the changing localization of the π and $\text{π}{}^1$ orbitals accompanying electronic excitation.     

Inclusive strange particle production in π−p interactions at 6 GeV/c

The inclusive production of Λ, K_S⁰, Σ^±(1385) and $\text{K}{}^{\mathrm{1\pm }}\text{(892)}$ in π^?p interactions at 6 GeV/c has been studied. The observed cross sections are: σ(Λ)=0.94±0.06 mb, σ(K_S⁰)=0.98±0.06 mb, σ(Σ⁺(1385))=60±7 μb, σ(Σ^?(1385))=90±9 μb, $\text{σ(}\text{K}{}^{\mathrm{1+}}\text{(892))=216±28 μ}\text{b and}\text{σ(}\text{K}{}^{\mathrm{1?}}\text{(892))=41±8 μ}\text{b}$, respectively. The inclusive spectra of these particles are presented as functions of squared transverse momentum and Feynman scaling variable x. The polarization of Λ has also been investigated. It is found from a comparison with higher-energy data that the inclusive cross sections for Σ^±(1385) and the production ratios Σ^±(1385)/Λ in π^?p at 6 GeV/c have not reached the high-energy limiting values.     

Multiplicity distributions and double-scattering effects in π−d interactions at 360 GeV/c

Double-scattering effects are studied in π^?d interactions at 360 GeV/c. The partial cross sections $\text{σ}{}_{\mathrm{<mtext>N</mtext>}}\text{(π}{}^{\mathrm{?}}\text{d}\text{), σ}{}_{\mathrm{<mtext>N</mtext>}}\text{(“π}{}^{\mathrm{?}}\text{p}\text{”)}\text{and}\text{σ}{}_{\mathrm{<mtext>N</mtext>}}\text{(“π}{}^{\mathrm{?}}\text{n}\text{”)}$ are presented. The double-scattering probability per πd collision is found to be $\text{? = 0.15 ± 0.02}$. We have extracted the partial cross section X_N of the double-scattering plus interference contributions, and find that X_N obeys KNO scaling. The data are compared with various theoretical predictions.     

Mixed electrical conduction in Ce1−xCaxO2−x

The solid electrolyte Ce_1?xCa_xO_2-?x with the fluorite-type structure (Ca-doped CeO₂) is a mixed conductor. Conduction occurs predominantly by migration of O^2? ions via oxygen vacancies or by electrons, depending on the departure from stoichiometry. The ionic transference number σ_i/σ_i + σ_e was determined as a function of dopant concentration (0.07?x?0.15), temperature (400–800°C), and oxygen pressure by emf measurements with oxygen concentration cells. It is described by

The entropy term S¹(x) changes from 38.7k for x = 0.07 to 31.7k for x = 0.15; the enthalpy term, 5.42 eV, is independent of x and in excellent accord with semi-empirical calculations.     

Analysis of adsorbate-substrate configuration by admolecule-probe field emission microscopy

By means of field electron emission microscopy with Faraday collector and magnetic deflection the emission current fluctuations between two or more levels caused by single Cu-phthalocyanine admolecules have been recorded. Relative abundance distribution functions of the times of sojourn in distinct adsorption states are given. From measurements at various temperatures in case of stationary fluctuations Arrhenius plots of the mean time of sojourn were obtained. They yield the energy differences E_i between trough and saddle positions of single admolecules as well as the preexponential factors A_i and the entropy differences ΔS_i, respectively, according to the Frenkel equation

$\text{τ}{}_{\mathrm{i}}\text{= A}{}_{\mathrm{i}}\text{exp(}\text{E}{}_{\mathrm{i}}\text{/kT}\text{) = (}\text{h}\text{kT}\text{) exp(?}\text{ΔS}{}_{\mathrm{i}}\text{k}\text{) exp (}\text{E}{}_{\mathrm{i}}\text{kT}\text{)}$

. The average values and standard deviations of all measurements are E_i = 0.22 eV, σ_Ei = = 0.11 eV, and log A_i = ?6.0, σ_logA_i = 1.6.The use of Cu-phthalocyanine as a molecular probe to investigate and characterize the adsorbate-substrate configuration, is discussed.     

A study of the reactions d + p → τ + π0 and d + p → τ + η0

The reactions d + p → τ + π⁰ and d + p → τ + η⁰ were studied using the missing-mass method for incident deuteron momenta between 2.8 and $\text{3.8}\text{GeV}\text{c}$. The angular distributions show a peak in the forward direction ($\text{θ}{}_{\mathrm{\tau }}{}^1\text{= 0°}$) and are flat in the backward direction. The differential cross sections at $\text{θ}{}_{\mathrm{\tau }}{}^1\text{= 180°}$ for both reactions show a maximum for total center-of-mass energies near 3.46 GeV.     

Trapping probabilities and desorption energies of alkali atoms on a clean and an oxygen covered tungsten (110) surface

Alkali atoms were scattered with hyperthermal energies from a clean and an oxygen covered (θ ≈ 0.5 ML) W(110) surface. The trapping probability of K and Na atoms on oxygen covered W(110) has been measured as a function of incoming energy (0–30 eV) and incident angle. A considerable enhancement of trapping on the oxygen covered surface compared to a clean surface was observed. At energies above 25 eV there are still K and Na atoms being trapped by the oxygen covered surface. From the temperature dependence of the mean residence time τ of the initially trapped atoms the pre-exponential factor τ₀ and the desorption energy Q were derived using the relation: $\text{τ = τ}{}_0\text{exp}\text{(}\text{Q}\text{kT}{}_{\mathrm{<mtext>s</mtext>}}\text{)}$. On clean W(110) we obtained for Li: $\text{τ}{}_0\text{= (8 ±}\text{8}\text{4}\text{) × 10}{}^{\mathrm{?14}}\text{sec}$, Q = (2.78 ± 0.09) eV; for Na: τ₀ = (9 ± 3) × 10^?14 sec, Q = (2.55 ± 0.04) eV; and for K: τ₀ = (4 ± 1) × 10^?13 sec, Q = (2.05 ± 0.02) eV. Oxygen covered W(110) gave for Na: τ₀ = (7 ±3) × 10^?15 sec, Q = (2.88 ± 0.05) eV; and for K: $\text{τ}{}_0\text{= (1.3 ±}\text{0.9}\text{0.6}\text{) × 10}{}^{\mathrm{?14}}\text{sec}$, Q = (2.48 ±0.05) eV. The adsorption on clean W(110) has the features of a supermobile two-dimentional gas; on the oxygen covered W(110) adsorbed atoms have the partition function of a one-dimen-sional gas. The binding of the adatoms to the surface has a highly ionic character in the systems of the present experiment. An estimate is given for the screening length of the non-perfect conductor W(110):k_s^?1≈ 0.5 Å.     

Testing the nature of the electroweak breaking from polarized e+e- beams

We present a quite general expression of the longitudinal $\text{A}{}_{\mathrm{|}}$ and the transverse $\text{A}{}_{\mathrm{\cap }}$ asymmetries for the single and pair production electroweak (pseudo) scalars (Higgs, $\text{π}$, $\text{σ}$, S_μ …) from polarized e⁺e^- colliding beams via annihilation mechanism. We suggest that the measurements of $\text{A}{}_{\mathrm{|}}$ and $\text{A}{}_{\mathrm{\cap }}$ can reveal the nature of such spinless bosons and, then, the nature of the electroweak breaking, if these spinless bosons are produced at LEP and SLC energies.     

Spectrum and structure of the He2 molecule: Characterization of the singlet and triplet states associated with the UAO's 4s, 4dσ, 4dπ, and 4dδ

The emission spectrum of the He₂ molecule has been rephotographed in the ～4000–～5700 Å region and the $\text{4d(}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}\text{,}{}^3\text{Π}{}_{\mathrm{u}}\text{,}{}^3\text{Δ}{}_{\mathrm{u}}\text{) → 2pπ}{}^3\text{Π}{}_{\mathrm{g}}$, $\text{4d(}{}^1\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}\text{,}{}^1\text{Π}{}_{\mathrm{u}}\text{,}{}^1\text{Δ}{}_{\mathrm{u}}\text{) → 2pπ}{}^1\text{Π}{}_{\mathrm{g}}$, $\text{4s}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}\text{→ 2pπ}{}^3\text{Π}{}_{\mathrm{g}}$ and $\text{4s}{}^1\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}\text{→ 2pπ}{}^1\text{Π}{}_{\mathrm{g}}$ transitions analyzed. The 4dδj³Δ_u, 4dπj³Π_u, $\text{4dσj}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}$ and $\text{4sh}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}$ states have been characterized through v = 2 and the 4dδJ¹Δ_u, 4dπJ¹Π_u, $\text{4dσJ}{}^1\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}$, and $\text{4sH}{}^1\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}$ states for v = 0. The term levels for these perturbed and l-uncoupled states have been confirmed (a) by analyses of bands with common levels from Δv = 0, ±1 sequences and (b) by analyses of the transitions between the above states from 4d and 4s and the $\text{c}{}^3\text{Σ}{}_{\mathrm{g}}{}^{\mathrm{+}}$ and $\text{C}{}^1\text{Σ}{}_{\mathrm{g}}{}^{\mathrm{+}}$ states associated with 3pσ. Molecular constants are reported which have been partially corrected for the effects of l-uncoupling and the homogeneous perturbations between the state pairs J, H and j, h.     

Study of the photon spectrum in the decay KS0→π+π−γ

The branching ratio $\text{Λ(}\text{K}{}_{\mathrm{<mtext>S</mtext>}}{}^0\text{→π}{}^{\mathrm{+}}\text{π}{}^{\mathrm{?}}\text{γ)}\text{Λ(}\text{K}{}_{\mathrm{<mtext>S</mtext>}}{}^0\text{→π}{}^{\mathrm{+}}\text{π}{}^{\mathrm{?}}\text{)}$ has been determined to be (2.68±0.15)×10^?3 for photon energies $\text{E}{}_{\mathrm{\gamma }}{}^1$ greater than 50 MeV in the K_S⁰ rest frame. The decay K_S⁰→π⁺π^?γ is found to be dominated by the internal bremsstrahlung transition. The branching rato of a possible direct transition is found to be less than 0.06 × 10^?3 at 90% confidence level for $\text{E}{}_{\mathrm{\gamma }}{}^1\text{> 50}\text{MeV}$.     

An inequality for the trace of matrix products

In this paper we consider a product of n complex m×m matrices A_k (k=1,…,n) with singular values ∝^(k)_i ordered in decreasing magnitude. Using the spectral resolution for the operators $\text{A}{}^{\mathrm{dagger;}}{}_{\mathrm{k}}\text{A}{}_{\mathrm{k}}$, it is shown that $\text{|}\text{Tr}\text{A}{}_1\text{…A}{}_{\mathrm{n}}\text{|≤}\text{∑}\text{i=1}\text{m}\text{Φ}\text{i=1}\text{n}\text{α}{}^{\mathrm{(k)}}{}_{\mathrm{i}}\text{.}$This inequality is an extension of an inequality of von Neumann in the simple case that n=2. The necessary and sufficient condition for the equality sign to hold is established. Application of Hölder's inequality leads to further inequalities which can be useful in statistical mechanics.