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# Bragg

[brag]
Bragg, Braxton, 1817-76, Confederate general in the U.S. Civil War, b. Warrenton, N.C. A graduate of West Point, he fought the Seminole and in the Mexican War was promoted to lieutenant colonel for distinguished service at Buena Vista. He resigned from the army in 1856 and lived on his Louisiana plantation until the outbreak of the Civil War, when he was appointed a Confederate brigadier general and assigned to command the coast from Pensacola, Fla., to Mobile, Ala. Shortly after being promoted to major general (Jan., 1862), he assumed command of Gen. A. S. Johnston's 2d Corps, leading it in the battle of Shiloh (April). With Johnston's death, Bragg was made a general, and he succeeded (June) General Beauregard in command of the Army of Tennessee. His invasion of Kentucky (Aug.-Oct., 1862) was unsuccessful, ending in retreat to Tennessee after Gen. D. C. Buell caught up with him at Perryville. A reorganized Union army under Gen. W. S. Rosecrans was then sent against him and at Murfreesboro (Dec. 31, 1862-Jan. 2, 1863) forced him to withdraw again. In the Chattanooga campaign, Bragg, victorious in the battle of Chickamauga, laid siege to the Union army in Chattanooga, but in Nov., 1863, Gen. U. S. Grant thoroughly defeated him and forced him to retire into Georgia. Gen. J. E. Johnston took over his command (December) and Bragg went to Richmond, where he became military adviser to Jefferson Davis, with nominal rank as commander in chief of Confederate armies. After the war he was chief engineer of Alabama and later lived in Texas, where he died.

See biography by D. C. Seitz (1924, repr. 1971); study by G. McWhiney (2 vol., 1969-91).

Bragg, Sir William Henry, 1862-1942, English physicist, educated at King William's College, Isle of Man, and Trinity College, Cambridge. He served on the faculties of the Univ. of Adelaide in Australia (1886-1908), the Univ. of Leeds (1909-15), and the Univ. of London (1915-23). From 1923 he was Fullerian professor of chemistry in the Royal Institution and director of the Davy-Faraday research laboratory. He shared with his son W. L. Bragg the 1915 Nobel Prize in Physics for their studies, using the X-ray spectrometer, of X-ray spectra, X-ray diffraction, and of crystal structure. He became a Fellow of the Royal Society in 1906 and served as president of the society from 1935 to 1940. In 1920 he was knighted. Among his works are The World of Sound (1920), Concerning the Nature of Things (1925), An Introduction to Crystal Analysis (1929), and The Universe of Light (1933). With W. L. Bragg he wrote X Rays and Crystal Structure (1915, 5th ed. 1925).

See biography by Sir Kerr Grant (1952).

Bragg, Sir William Lawrence, 1890-1971, English physicist, b. Adelaide, Australia, educated in Australia and at Trinity College, Cambridge; son of W. H. Bragg. He was professor of physics at Victoria Univ., Manchester, from 1919 to 1937. From 1938 to 1953 he was professor of experimental physics at Cambridge and director of the Cavendish Laboratory. In 1954 he was made head of the Royal Institution. He shared with his father the 1915 Nobel Prize in Physics for their studies, with the X-ray spectrometer, of X-ray spectra, X-ray diffraction, and of crystal structure. In 1941 he was knighted. Among his works are The Structure of Silicates (1930, 2d enl. ed. 1932) and Atomic Structure of Minerals (1937). With his father he wrote X Rays and Crystal Structure (1915, 5th ed. 1925).

(b. July 2, 1862, Wigton, Cumberland, Eng.—d. March 12, 1942, London) British scientist, a pioneer in solid-state physics. With his son (William) Lawrence Bragg (1890–1971), he shared a 1915 Nobel Prize for research on the determination of crystal structures and Lawrence's discovery (1912) of the Bragg law of X-ray diffraction. The Bragg ionization spectrometer William designed and built is the prototype of all modern X-ray and neutron diffractometers; the two men used it to make the first exact measurements of X-ray wavelengths and crystal data.

Braxton Bragg, engraving by George E. Perine

(born March 22, 1817, Warrenton, N.C., U.S.—died Sept. 27, 1876, Galveston, Texas) U.S. and Confederate army officer. He graduated from West Point and served in the Seminole Wars and the Mexican War. When North Carolina seceded, he joined the Confederate army and fought in the American Civil War. He was promoted to general in 1862 at the Battle of Shiloh. As commander of the Army of Tennessee, he led his troops to victory at the Battle of Chickamauga. His forces besieged the Union troops at Chattanooga but were eventually routed. He was relieved of his command but appointed military adviser to Confederate Pres. Jefferson Davis.

(b. July 2, 1862, Wigton, Cumberland, Eng.—d. March 12, 1942, London) British scientist, a pioneer in solid-state physics. With his son (William) Lawrence Bragg (1890–1971), he shared a 1915 Nobel Prize for research on the determination of crystal structures and Lawrence's discovery (1912) of the Bragg law of X-ray diffraction. The Bragg ionization spectrometer William designed and built is the prototype of all modern X-ray and neutron diffractometers; the two men used it to make the first exact measurements of X-ray wavelengths and crystal data.

Braxton Bragg, engraving by George E. Perine

(born March 22, 1817, Warrenton, N.C., U.S.—died Sept. 27, 1876, Galveston, Texas) U.S. and Confederate army officer. He graduated from West Point and served in the Seminole Wars and the Mexican War. When North Carolina seceded, he joined the Confederate army and fought in the American Civil War. He was promoted to general in 1862 at the Battle of Shiloh. As commander of the Army of Tennessee, he led his troops to victory at the Battle of Chickamauga. His forces besieged the Union troops at Chattanooga but were eventually routed. He was relieved of his command but appointed military adviser to Confederate Pres. Jefferson Davis.

In physics, Bragg's law is the result of experiments into the diffraction of X-rays or neutrons off crystal surfaces at certain angles, derived by physicist Sir William Lawrence Bragg in 1912 and first presented on 1912-11-11 to the Cambridge Philosophical Society. Although simple, Bragg's law confirmed the existence of real particles at the atomic scale, as well as providing a powerful new tool for studying crystals in the form of X-ray and neutron diffraction. William Lawrence Bragg and his father, Sir William Henry Bragg, were awarded the Nobel Prize in physics in 1915 for their work in determining crystal structures beginning with NaCl, ZnS, and diamond.

When X-rays hit an atom, they make the electronic cloud move as does any electromagnetic wave. The movement of these charges re-radiates waves with the same frequency (blurred slightly due to a variety of effects); this phenomenon is known as the Rayleigh scattering (or elastic scattering). The scattered waves can themselves be scattered but this secondary scattering is assumed to be negligible. A similar process occurs upon scattering neutron waves from the nuclei or by a coherent spin interaction with an unpaired electron. These re-emitted wave fields interfere with each other either constructively or destructively (overlapping waves either add together to produce stronger peaks or subtract from each other to some degree), producing a diffraction pattern on a detector or film. The resulting wave interference pattern is the basis of diffraction analysis. Both neutron and X-ray wavelengths are comparable with inter-atomic distances (~150 pm) and thus are an excellent probe for this length scale.

The interference is constructive when the phase shift is a multiple to 2π; this condition can be expressed by Bragg's law:

$nlambda=2dcdotsintheta ,$

where

• n is an integer determined by the order given,
• λ is the wavelength of x-rays, and moving electrons, protons and neutrons,
• d is the spacing between the planes in the atomic lattice, and
• θ is the angle between the incident ray and the scattering planes

According to the 2θ deviation, the phase shift causes constructive (left figure) or destructive (right figure) interferences

Note that moving particles, including electrons, protons and neutrons, have an associated De Broglie wavelength.

## Reciprocal space

Although the misleading common opinion reigns that Bragg's Law measures atomic distances in real space, it does not. Furthermore, the $n lambda$ term demonstrates that it measures the number of wavelengths fitting between two rows of atoms, thus measuring reciprocal distances. Max von Laue had interpreted this correctly in a vector form, the Laue equation

$vec G = vec\left\{k_f\right\} - vec\left\{k_i\right\}$

where $vec G$ is a reciprocal lattice vector and $vec\left\{k_f\right\}$ and $vec\left\{k_i\right\}$ are the wave vectors of the incident and the diffracted beams.

Together with the condition for elastic scattering $|k_f| = |k_i|$ and the introduction of the scattering angle $2 theta$ this leads equivalently to Bragg's equation.

The concept of reciprocal lattice is the Fourier space of a crystal lattice and necessary for a full mathematical description of wave mechanics.

## Alternate Derivation

A single monochromatic wave, of any type, is incident on aligned planes of lattice points, with separation d, at angle θ, as shown below.

There will be a path difference between the 'ray' that gets reflected along AC' and the ray that gets transmitted, then reflected along AB and BC paths respectively. This path difference is:

$\left(AB+BC\right) - \left(AC\text{'}\right) ,$
If this path difference is equal to any integer value of the wavelength then the two separate waves will arrive at a point with the same phase, and hence undergo constructive interference. Expressed mathematically:
$\left(AB+BC\right) - \left(AC\text{'}\right) = nlambda ,$
Where the same definition of n and λ apply from the article above
Using the Pythagorean theorem it is easily shown that:
$AB=frac\left\{d\right\}\left\{sintheta\right\},$ and $BC=frac\left\{d\right\}\left\{sintheta\right\},$ and $AC=frac\left\{2d\right\}\left\{tantheta\right\},$
also it can be shown that:
$AC\text{'}=ACcdotcostheta=frac\left\{2d\right\}\left\{tantheta\right\}costheta,$
Putting everything together and using known identities for sinusoidal functions:
$nlambda=frac\left\{2d\right\}\left\{sintheta\right\}-frac\left\{2d\right\}\left\{tantheta\right\}costheta=frac\left\{2d\right\}\left\{sintheta\right\}\left(1-cos^2theta\right)=frac\left\{2d\right\}\left\{sintheta\right\}sin^2theta$
Which simplifies to:
$nlambda=2dcdotsintheta ,$
yielding Bragg's law.

## References

W.L. Bragg, "The Diffraction of Short Electromagnetic Waves by a Crystal", Proceedings of the Cambridge Philosophical Society, 17 (1913), 43–57.