Possible beat effect of specific heat due to impurities in semi-metal in external magnetic field

The simple model of a single-band semi-metal or degenerate semiconductor with impurities in an external quantizing magnetic field is discussed. The susceptibility and specific heat are studied in their dependence on the field-induced quasi-local levels, due to the impurities. These levels are shown to cause additional oscillations in the thermodynamical quantities as functions of the magnetic field. Their contribution to the susceptibility and specific heat is comparable to the values for the undoped system if the impurity concentration c_i fulfils the condition $\text{c}{}_{\mathrm{i}}\text{= (c}{}_{\mathrm{e}}\text{/4√π)(}\text{h}\text{?}\text{ω/ε}{}_{\mathrm{F}}\text{)}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$, with c_e the electron concentration.     

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.     

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.     

Critical scaling function for long-range interactions to order 1/n

A universal scaling function of the order-parameter correlation function for an n-vector system in d dimensions, with long-range interactions $\text{～1/z.sfnc}\text{r}\text{z.sfnc}{}^{\mathrm{d+\sigma }}$, is discussed on the basis of explicit zero-field calculations to order 1/n, for T?T_c and σ<d<2σ.     

Quantum theory of longitudinal dielectric response properties of a two-dimensional plasma in a magnetic field

An analysis of dynamic and nonlocal longitudinal dielectric response properties of a two-dimensional Landau-quantized plasma is carried out, using a thermodynamic Green's function formulation of the RPA with a two-dimensional thermal Green's function for electron propagation in a magnetic field developed in closed form. The longitudinal-electrostatic plasmon dispersion relation is discussed in the low wavenumber regime with nonlocal corrections, and Bernstein mode structure is studied for arbitrary wavenumber. All regimes of magnetic field strength and statistics are investigated. The class of integrals treated here should have broad applicability in other two-dimensional and finite slab plasma studies.The two-dimensional static shielding law in a magnetic field is analyzed for low wavenumber, and for large distances we find $\text{V(}\text{r}\text{) ～}\text{Q}\text{k}{}_0{}^2\text{r}{}^3$. The inverse screening length $\text{k}{}_0\text{=}\text{2πe}{}^2\text{??}\text{?ξ}$ (? = density, ξ = chemical potential) is evaluated in all regimes of magnetic field strength and all statistical regimes. k₀ exhibits violent DHVA oscillatory behavior in the degenerate zero-temperature case at higher field strengths, and the shielding is complete when $\text{ξ = r′}\text{l}\text{z.shtsls;}\text{ω}$, but there is no shielding when $\text{ξ ≠ r′}\text{l}\text{z.shtsls;}\text{ω}{}_{\mathrm{c}}$. A careful analysis confirms that there is no shielding at large distances in the degenerate quantum strong field limit $\text{l}\text{z.shtsls;}\text{ω}{}_{\mathrm{c}}\text{> ξ}$. Since shielding does persist in the nondegenerate quantum strong field limit $\text{l}\text{z.shtsls;}\text{ω}{}_{\mathrm{c}}\text{> KT}$, there should be a pronounced change in physical properties that depend on shielding if the system is driven through a high field statistical transition. (It should be noted that the static shielding law of semiclassical and classical models has no dependence on magnetic field in two dimensions, as in three dimensions.) Finally, we find that the zero field two-dimensional Freidel-Kohn “wiggle” static shielding phenomenon is destroyed by the dispersal of the zero field continuum of electron states into the discrete set of Landau-quantized orbitals due to the imposition of the magnetic field.     

On a least upper bound Tc of superconductors with paramagnetic impurities

We have obtained a least upper bound, $\text{k}{}_{\mathrm{B}}\text{T}{}_{\mathrm{c}}\text{? c(μ}{}^1\text{, t)A}$, on the critical temperature T_c of an isotropic superconductor with paramagnetic impurities described by the scattering matrix t for fixed values of μ¹. We have also obtained the corresponding optimal spectrum $\text{α}{}^2\text{F(m) = Aδ[ω?d(μ}{}^1\text{, A]}$. The numerical results for the functions $\text{c(μ}{}^1\text{, t)}\text{and}\text{d(μ}{}^1\text{, t)}$ are presented for $\text{α}{}^1\text{= 0.1}$ and 0.16 in the form of universal curves representing $\text{c(μ}{}^1\text{, t)}\text{and}\text{d(μ}{}^1\text{, t)}$ as functions of the reduced impurity concentration $\text{t}\text{= t/A}$. We have also established an upper limit to the reduced critical concentration $\text{t}{}_{\mathrm{crit}}$ for an arbitrary shape of α²F(ω)1.     

Anisotropic magnetic susceptibility and phase transitions in intercalated graphite: KC24

We measured the magnetic susceptibility of KC₂₄ from 4.2 to 300 K and found no anomalies near the phase transitions at 95 and 123 K as observed in the resistivity. We conclude that the transitions must be due to order- disorder transitions of the K atoms and not charge density wave formation. The susceptibility is anisotropic; at room temperature $\text{χ}{}_{\mathrm{g}}\text{(}\text{H}\text{6}\text{c}\text{)= + 1.50 × 10}{}^{-6}\text{emu g}{}^{-1}\text{± 2\%}$ and $\text{χ}{}_{\mathrm{g}}\text{(}\text{H}\text{⊥}\text{c}\text{)= + 0.045 × 10}{}^{-6}\text{emu g}{}^{-1}\text{± 50\%}$. This anisotropy is not understood in terms of simple rigid band extensions of the band structure of graphite.     

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.     

The reactions K−p → K1− + nucleon at 3.13 and 3.3 GeV/c

Results are presented of a 12 event/μb bubble-chamber experiment; the reactions discussed in detail are $\text{K}{}^{\mathrm{?}}\text{p}\text{→}\text{K}{}^1\text{(890)}{}^{\mathrm{?}}\text{p, K}{}^1\text{(1420)}{}^{\mathrm{?}}\text{p and K}{}^1\text{(890)}{}^{\mathrm{?}}\text{Δ}{}^{\mathrm{+}}$.The K¹ (890)^?p channel is dominated by the forward peak. The suggestion of flattering at cos θ = 1 is more pronounced in $\text{(?}{}_{11}\text{+ ?}{}_{\mathrm{1?1}}\text{)}\text{d}\text{σ}\text{d}\text{t}$; which is mainly natural-parity exchange. Pseudoscalar exchange contributes to $\text{?}{}_{00}{}^{\mathrm{J}}\text{d}\text{σ}\text{d}\text{t}$; this is more sharply peaked in t. The value of $\text{(?}{}_{11}\text{? ?}{}_{\mathrm{1?1}}\text{)}\text{d}\text{σ}\text{d}\text{t}$ is somewhat larger than the upper limit from the dominant natural-parity exchange. There is significant structure in $\text{?}{}_{00}{}^{\mathrm{H}}\text{d}\text{σ}\text{d}\text{t}\text{at}\text{t ≈ ?0.6 (}\text{GeV}\text{/c)}{}^2$.The K¹ (1420)^?p channel is much more pronounced at 3.3 GeV/c than at 3.13 GeV/c, but is not markedly peripheral. The width of the K¹ (1420) in the 3.3 GeV/c data is 42 ± 12 MeV/c².The cross section for $\text{K}{}^{\mathrm{1?}}\text{Δ}{}^{\mathrm{+}}$ agrees with that expected from $\text{K}{}^{\mathrm{+}}\text{p}\text{→}\text{K}{}^1\text{Δ}$, assuming a single t-channel exchange. Our measured density matrix elements are consistent with a strong pseudoscalar exchange.     

The correct extension of the Fortuin-Kasteleyn result to plaquette percolation

The standard s-state Potts model is generalized to a model defined of laquettes of $\text{Zz}{}^{\mathrm{d}}$, and the s → 1, 0 limits are shown to correspond to plaquette percolation and percolation of plaquette trees, respectively. The model is further extended to r-cells (cubes, hypercubes, etc.) on $\text{Z}{}^{\mathrm{d}}$ and the s → 1, 0 limits are exhibited.     

NMR study of the spin polarization in dilute Al-Mn alloys

The spatial dependence of the spin density oscillations σ(r) around Mn impurities in Al-Mn dilute alloys has been studied by ²⁷Al four pulse NMR experiments. The concentration dependence of the magnetic broadening $\text{ΔH}{}^{\mathrm{i}}\text{1}\text{2}$ of the ²⁷Al line is found slower than linear. This result, together with the temperature dependence of the magnetic broadening as measured by CW NMR, demonstrates both the preasymptotic depression of σ(r) in Al-Mn and the long range damping of σ(r) associated with the finite mean free path at high T.     

Scaling function for two-point correlations with long-range interactions to order 1/η

The two-point correlation function ? (q, ξ) is calculated in the critical region of momentum space q in terms of a suitable correlation lenght ξ, by means of perturbation expansion to order 1/n, for an n-vector system with long-range interactions decaying as $\text{|}\text{R}\text{/a|}{}^{\mathrm{?(d\; +\; \sigma )}}$, for $\text{|}\text{R}\text{/a|}\text{a}\text{?}\text{1}$, where a is the spacing on a d-dimensional lattice, σ < d < 2σ and 0 < σ ? 2 ? η_SR. The calculations are done in zero field for T ? T_c. Explicit expansions for long-range propagators are developed for $\text{σ « 1}$ and for the neighborhood of σ ? 2 ? η_SR, in terms of which a universal, cut-off independent scaling function is obtained over the whole range of $\text{x = |}\text{q}\text{| ξ}$, and it is shown that the amplitude of the correlation-length dependence of the susceptibility becomes a universal parameter. Both the exponents and the coefficients of the expansion for fixed q as $\text{(T ? T}{}_{\mathrm{<mtext>c</mtext>}}\text{)}\text{T}{}_{\mathrm{<mtext>c</mtext>}}\text{→0}$ are calculated explicitly. The former are shown to require the validity of the operator-product expansion and explicit logarithmic correction terms are obtained for $\text{d = d}{}^1\text{= 3σ/2}$. For these and other dimensionalities, the coefficients are shown to be finite functions of d and σ. The correction to the Ornstein-Zernike form is given explicitly, with non-integer powers of x that have finite coefficients.     

Magnetic monopoles in ferromagnetic materials

The interactions of slow $\text{(10}{}^{\mathrm{?5}}\text{? v/c ? 10}{}^{\mathrm{?2}}\text{)}$ magnetic monopoles with ferromagnetic materials are studied. The spin-flip cross section σ and the energy loss dE/dx are calculated for magnetic monopoles impinging parallel to the magnetization direction. In iron, these reach a maximum at v/c ～ 3 × 10^?4, where they take the value of 100 Ų and 100 MeV/cm respectively. The electromagnetic signal of a monopole passing through a ferromagnet and the generation of spin waves are discussed.     

Further investigations on the perturbative expansion for the anomalous magnetic moment

Continuing a previous analysis, the divergence of the QED perturbative series for the anomalous magnetic moment is further investigated. Taking into account graphs with three photon lines, with electron-positron vacuum polarization, the leading behaviour, at the (2n+1)th perturbative order is found to be the type $\text{α}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}\text{α}{}^{\mathrm{n}}\text{n!(}\text{ln}\text{n)}{}^2$. Further possible generalizations are considered.     

The effect of metal particle size on the accessibility of surface atoms to an impinging gas

As an example of similar calculations for any gas/metal combination, the reduced collision frequency Z_r (per second per surface atom) of CO molecules with Pd crystallites of various regular shapes and varied average diameter d_av, is calculated and normalized with respect to that for the close-packed plane Pd (111). Z_r¹¹¹. The ratio $\text{z =}\text{Z}{}_{\mathrm{r}}\text{Z}{}_{\mathrm{r}}{}^{111}$, which describes the relative accessibility of surface atoms to the impinging gas, is obtained for crystallites up to 5 nm in size. It is found that z increases rapidly above ~ 1 as d_av decreases below ~ 3 nm, in agreement with previously reported experimental data.     

Derivation of the classical theory of correlations in fluids by means of functional differentiation

The k-particle, infinite-volume distribution functions $\text{n}\text{?}{}_{\mathrm{k}}\text{(}\text{r}{}_1\text{, …,}\text{r}{}_{\mathrm{k?1}}\text{, γ)}$ and modified Ursell correlation functions $\text{U}\text{?}{}_{\mathrm{k}}\text{(}\text{r}{}_1\text{, …,}\text{r}{}_{\mathrm{k?1}}\text{, γ)}$ of a classical system of particles with the two-body potential $\text{q(}\text{r}\text{) + γ}{}^{\mathrm{\nu }}\text{K(γ}\text{r}\text{)}$ are considered. The limiting values of the functions $\text{n}\text{?}{}_{\mathrm{k}}\text{(}\text{r}{}_1\text{, …,}\text{r}{}_{\mathrm{k?1}}\text{, γ),}\text{n}\text{?}{}_{\mathrm{k}}\text{(}\text{S}{}_1\text{/γ, …,}\text{S}{}_{\mathrm{k?1}}\text{/γ, γ)}$ and $\text{γ(}{}^{\mathrm{1?k}}\text{)ν}\text{U}\text{?}{}_{\mathrm{k}}\text{(}\text{S}{}_1\text{/γ, …,}\text{S}{}_{\mathrm{k?1}}\text{/γ, γ)}$ in the limit γ → 0 are calculated, under fairly weak conditions on q and K, by a method involving functional differentiation. These limiting functions are used to describe the molecular structure of the various states of the system both in the range of the potential q(r) and in the rage of the potential γ^νK(γr). The direct correlation function c? (r, γ) is also considered and it is shown that for $\text{S}\text{≠ 0}$, $\text{lim}{}_{\mathrm{\gamma \to 0}}\text{γ}{}^{\mathrm{?\nu }}\text{c}\text{?}\text{(}\text{S}\text{γ}\text{, γ) = ?βK (}\text{S}\text{)}$, for all one-phase states, where β is the reciprocal temperature. Special cases of our results confirm those of other authors, including the well-known results of Ornstein and Zernike.     

Prediction of strong magnetic quantum oscillations of the nuclear spin relaxation in a quasi-two-dimensional metal

The nuclear spin lattice relaxation rate in a quasi-two-dimensional (2-D) metal under strong magnetic fields is studied in a special case where the electronic cyclotron mass is small compared with the free electron mass. In the pure limit (ω_cτ ? 1) and for sufficiently low temperatures ($\text{h}\text{?}\text{ω}{}_{\mathrm{c}}\text{> 2π}{}^2\text{k}{}_{\mathrm{B}}\text{T}$) we find remarkable quantum oscillations of the relaxation rate as a function of the magnetic field. The period of these oscillations is identical to that of the de Haas-van Alphen oscillations and their amplitude grows linearly with the magnetic field. The possibility of observing such oscillations experimentally in the quasi-2-D mercury chain compound Hg_3?δAsF₆ is discussed.     

On the mobility of very pure semiconductors at very low temperatures

The mobility μ of a very pure semiconductor at very low temperatures is investigated in terms of a model where electrons are scattered by charged impurities distributed uniformly in space, and the electron-electron interaction is taken into account by the Debye-Hueckel screening in the interaction potential. The equation for the current relaxation rate Γ, derived previously by the proper connected diagram expansion, incorporates the quasi-particle effect in a self-consistent manner. The solution of this equation at high carrier concentrations n yields the so-called Brooks-Herring formula. At lower concentrations, the solution deviates significantly from the latter. The solution is in general smaller than the standard expression for the rate based on the Boltzmann equation; and this is consistent with the existing conductivity data available. At the very low concentrations e.g. n = n₃ = 10¹³cm^?3 or lower for Ge, the mobility calculated is inversely proportional to the square-root of the impurity concentration n_s, and has a $\text{T}{}^{\mathrm{<mtext>1</mtext><mtext>4</mtext>}}$-dependence (T: temperature).

$\text{μ = 0.3597\&z.xl;h}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{k(k}{}_{\mathrm{B}}\text{T)}{}^{\mathrm{<mtext>1</mtext><mtext>4</mtext>}}\text{(ze)}{}^{\mathrm{?1}}\text{n}{}_{\mathrm{s}}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{m}{}^1{}^{\mathrm{<mtext>?3</mtext><mtext>4</mtext>}}$

, where k is the dielectric constant. The conductivity data directly comparable with this formula are not available at present. However, the quasi-particle effect which led to this peculiar concentration-dependence should also show itself in the cyclotron resonance width; there, experiment and theory both show the ${}_{\mathrm{n}}{}_{\mathrm{s}}$-dependence for very pure semiconductors.     

Study of the reactions 118Sn(d,t) and98Mo(d,t): (I). Spectroscopy of 117Sn and 97Mo

The vector analyzing power and differential cross section have been measured at a deuteron energy of 12.0 MeV for ${}^{118}\text{Sn}\text{(}\text{d}\text{,}\text{t}\text{)}$ transitions to six states of ¹¹⁷Sn (E_x = 0.0, 0.16, 0.31, 0.71, 1.02 and 1.18 MeV), for ${}^{98}\text{Mo}\text{(}\text{d}\text{,}\text{t}\text{)}$ transitions to eight states of ⁹⁷Mo (E_x = 0.0, 0.68, 0.72, 0.89, 1.12, 1.28, 2.39 and 2.52 MeV), and for ${}^{118}\text{Sn}\text{(}\text{d}\text{,}\text{d}\text{)}\text{and}{}^{98}\text{Mo}\text{(}\text{d}\text{,}\text{d}\text{)}$. Deuteron optical model potentials were obtained from analysis of the elastic scattering measurements, and were used in a DWBA analysis of the $\text{(}\text{d}\text{,}\text{t}\text{)}$ results. Comparison of the measurements and DWBA predictions for σ(θ) and for iT₁₁(θ) allows unambiguous determination of tl_n and j_n for all ¹¹⁸Sn(d, t) and most ⁹⁸Mo(d, t) transitions. Differences in the triton energy relative to the Coulomb barrier cause marked qualitative differences in the measured cross sections and analyzing powers between ${}^{118}\text{Sn}\text{(}\text{d}\text{,}\text{t}\text{)}\text{and}{}^{98}\text{Mo}\text{(}\text{d}\text{,}\text{t}\text{)}$ transitions of the same l_n and j_n.