On the formulation and the calculation of the harmonic contributions to the Debye-Waller factor in metals (sodium)

The algebraic expressions for the anharmonic contributions to the Debye-Waller factor up to 0(A ) and 0 L% ) £ where ^ is the scattering wave-vector] have been derived in a form suitable for cubic metals with small ion cores where the interatomic potential extends to many neighbours. This has been achieved in terms of various wave-vector dependent tensors, following the work of Shukla and Taylor (1974) on the cubic anharmonic Helmholtz free energy. The contribution to the various wave-vector dependent tensors from the coulomb and the electron-ion terms in the interatomic metallic potential has been obtained by the Ewald procedure. All the restricted multiple whole B r i l l o u i n zone (B.Z.) sums are reduced to single whole B.Z. sums by using the plane wave representation of the delta function. These single whole B.Z. sums are further reduced to the •%?? portion of the B.Z. following Shukla and Wilk (1974) and Shukla and Taylor (1974). Numerical calculations have been performed for sodium where the Born-Mayer term in the interatomic potential has been neglected because i t is small £ Vosko (1964)3 • *n o^er to compare our calculated results with the experimental results of Dawton (1937), we have also calculated the r a t io of the intensities at different temperatures for the lowest five reflections (110), (200), (220), (310) and (400) . Our calculated quasi-harmonic results agree reasonably well with the experimental results at temperatures (T) of the order of the Debye temperature ( 0 ). For T » © ^ 9 our calculated anharmonic results are found to be in good agreement with the experimental results.The anomalous terms in the Debye-Waller factor are found not to be negligible for certain reflections even for T ^ ©^ . At temperature T yy Op 9 where the temperature is of the order of the melting temperature (Xm) » "the anomalous terms are found to be important almost for all the f i ve reflections.

The condition of Niagara Falls, and the measures needed to preserve them : eight letters published in the New York Evening post, the New York tribune, and the Boston daily advertiser, during the summer of 1882 /

Appendix: Extracts from Harper's weekly, New York herald, New York tribune and the Nation.

An Address of members of the House of Representatives of the Congress of the United States, to their constituents, on the subject of the war with Great Britain. --

Signed by George Sullivan and thirty-three other Federalist members.

At a special session of the General Assembly of the State of Connecticut, held at Hartford, on the twenty third day of February, A.D. 1809 ... An address of the General Assembly, to the people of Connecticut. --

Caption title.

Letter from George W. Campbell, esq., late Secretary of the Treasury, to the Chairman of the Committee appointed to inquire into the causes and particulars of the invasion of the City of Washington, and the neighboring town of Alexandria, in the month of August last. --

Laid before the House by the Chairman of the said Committee, and ordered to be printed January 2, 1815.

Message from the President of the United States, transmitting copies of the instructions given to the ministers of the United States appointed to negotiate a peace with Great Britain. --

13th Congress, 3d session. House. Doc. no. 8.

Message from the President of the United States, to both Houses of Congress, at the commencement of the third session of the thirteenth Congress. --

Document no. 1 in U.S. 13th Congress, 3d session, 1814-1815. House.

Message from the President of the United States to both Houses of Congress at the commencement of the second session of the twelfth Congress. --

November 4, 1812. Read, and ordered to be printed.

Proceedings of a court of inquiry, convened on board the United States frigate the President, in the harbour of New York, on the thirteenth day of August, 1811, pursuant to the following warrant : To Stephen Decatur, esquire, a captain in the Navy of the United States. --

Caption title.

Report of the Committee appointed to inquire into the Present condition and distribution of the flags, standards and colors, which have been taken by the forces of the United States from their enemies, and whether it would be expedient to make any provision in relation to them.--

Adam Seybet, Chairman.

The wars of the gulls; : an historical romance in three chapters; chap. I, Shewing how and why and with whom the gulls went to war: chap. II, Shewing how the gulls make the deep to boil like a pot: chap. III, Shewing how a certain doughty general of the gulls goes forth to play the game of hull-gull in Upper Canada.

Transcribed on front paste-down: W.G. Phelps Oct. 29 1890.

An Act to authorise the stock held by private parties in the Welland Canal to be purchased on behalf of the Province [Welland Canal Stock Bill], 1841

The bill is a proposal by the government to purchase Welland Canal stock held in private hands. The bill was read for the first time on Wednesday 4 August, 1841, and received second reading on 10th August, 1841. One hundred and fifty copies were printed of the bill. This example was addressed to Samuel Street at the Falls of Niagara, by William Hamilton Merritt, September 1841.