Aula Raios X03

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- Para avaliar uma estrutura cristalina, é necessário usar as figuras de 
difração produzidas por ondas que interagem com os átomos e que 
possuem comprimentos de onda (l) comparáveis (da ordem ou menores) 
com a ordem de grandeza das distâncias interatômicas. 
- A estrutura cristalina pode ser estudada através da difração de fótons, 
elétrons de alta energia e neutrons. 
- A difração depende da estrutura cristalina e do comprimento de onda da 
radiação.
- Because X-rays have wavelengths similar to the size of atoms, they are 
useful to explore within crystals. 
Difração dos raios-X
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Filament Tubes
These were invented by Coolidge in 1913. They consist of an evacuated glass envelope which 
insulates the anode at one end from the cathode at the other, the cathode being a tungsten 
filament and the anode a water-cooled block of copper containing the desired target metal as a 
small insert at one end.
Tubos de raios-X
Esquema da secção transversal de um tubo vedado de raios-X 
Apenas 1% da energia 
cinética transferida ao 
alvo é convertida em 
raios X. O restante é 
convertido em calor.
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Tubo de raios-X
The x-ray tube must contain: 
(a) a source of electrons,
(b) a high accelerating voltage, and 
(c) a metal target. 
Furthermore, since most of the kinetic energy of the electrons is converted into 
heat in the target, the latter is almost always water-cooled to prevent its melting.
The x-ray tubes contain two electrodes:
- an anode (the metal target) maintained, with few exceptions, at ground potential, 
- a cathode, maintained at a high negative potential, normally of the order of 
30,000 to 50,000 volts for diffraction work. 
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Espectro Característico
Modelo de Bohr
Características das radiações
Target (alvo) l(Å) Ka l(nm) Ka E Ka (keV) E Kb (keV)
Mo 0,7107 0,07107 17,441 19,605
Cu 1,5418 0,15418 8,040 8,904
Co 1,7902 0,17902 6,924 7,648
Fe 1,9373 0,19373 6,398 7,057
Cr 2,2909 0,22909 5,411 5,946
Element
Ka
Wavelength 
(l) nm
Mo 0.07107
Cu 0.15418
Co 0.17902
Fe 0.19373
Cr 0.22909
X-rays wavelengths for commonly 
used target materials in X-ray tubes.
Metais –Raios atômicos e 
estrutura cristalina. 
Características das radiações
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FILTERS
Many x-ray diffraction experiments require radiation which is as closely monochromatic as possible. 
However, the beam from an x-ray tube operated at a voltage above V, contains not only the strong 
Ka line but also the weaker Kb line and the continuous spectrum. 
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Target 
(alvo) 
l(Å) Ka E Ka 
(keV)
E Kb 
(keV)
Filtro 
Kb
Espessura 
(µm)
Densidade 
g/cc
% Ka % Kb
Mo 0,7107 17,441 19,605 Zr 81 6,50 44 1
Cu 1,5418 8,040 8,904 Ni 15 8,90 52 2
Co 1,7902 6,924 7,648 Fe 12 7,87 57 3
Fe 1,9373 6,398 7,057 Mn 11 7,43 59 3
Cr 2,2909 5,411 5,946 V 11 6,00 58 3
Características do Filtros Kβ
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Conceitos de Max von Laue - 1912
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Difração de luz
Difração de luz em uma fenda – se uma onda incidir sobre uma
barreira que tem abertura de dimensões comparáveis ao
comprimento de onda, ela se espalhará na região além da
abertura.
Difração em duas fendas
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Difração de Raios-X
DIFRAÇÃO - Espalhamento coerente + Interferência
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Interferência e Espalhamento
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X-ray Diffraction
Since a beam of X-rays consists of a bundle of separate waves, the 
waves can interact with one another. Such interaction is termed 
interference. If all the waves in the bundle are in phase, that is 
their crests and troughs occur at exactly the same position (the 
same as being an integer number of wavelengths out of phase, nl, 
n = 1, 2, 3, 4, etc.), the waves will interfere with one another and 
their amplitudes will add together to produce a resultant wave that 
is has a higher amplitude (the sum of all the waves that are in 
phase.
If the waves are out of phase, being off by a non-integer 
number of wavelengths, then destructive interference will 
occur and the amplitude of the waves will be reduced. In an 
extreme case, if the waves are out of phase by a multiple of 
1/2l (n/2l ), the resultant wave will have no amplitude and 
thus be completely destroyed. 
Difração de Raios-X
Espalhamento Coerente (elástico) – λ1 = λ2 - DIFRAÇÃO 
Espalhamento Incoerente (inelástico) – λ2 > λ1 - EFEITO COMPTON
A onda incidente (fóton de raio-
X) possui campo elétrico e 
magnético que interage com o 
campo elétrico dos átomos.
Nesta interação pode ocorrer 
espalhamento inelástico (Efeito 
Compton) ou espalhamento 
elástico (reflexão).
DIFRAÇÃO - É a combinação de 
dois fenômenos: espalhamento 
coerente e interferência. 
Quando fótons de raios X de 
mesmo λ, espalhados 
coerentemente, interferem de 
modo construtivo, entre si, picos 
de difração (diffraction maxima) 
serão observados
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Os átomos do cristal interagem com os raios-X 
produzindo interferência. Esta interação pode 
ser vista como os átomos da rêde refletindo as 
ondas de raios-X. Mas, como a estrutura 
cristalina consiste de um arranjo ordenado de 
átomos, a reflexão ocorre a partir de um plano 
de átomos.
Um feixe de raios X incide sobre um conjunto 
de planos cristalinos, cuja distância interplanar 
é d. Os feixes refletidos por dois planos 
subseqüentes apresentarão o fenômeno da 
difração. Isto é, se a diferença entre seus 
caminhos óticos for um número inteiro de 
comprimentos de onda, haverá superposição 
construtiva (um feixe de raios X será 
observado); caso contrário, haverá 
superposição destrutiva, i.e. não se observará 
qualquer sinal de raios X. 
Interação Raios-X - cristal
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Diversos tipos de planos refletores 
numa rede cristalina cúbica simples. 
Os planos são indicados pelos 
respectivos indíces. 
A distância mais próxima entre dois 
planos paralelos tende a diminuir a 
medida que os indíces crescem. 
Logo, indíces de reflexão elevados 
necessitam de comprimentos de 
onda mais curtos. 
Interação Raios-X - cristal
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d (100)
d (110)
d (210)
d (310)
Difração de Raios X em cristais
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Two such X-rays are shown here, where 
the spacing between the atomic planes 
occurs over the distance, d.
Ray 1 reflects off of the upper atomic 
plane at an angle q equal to its angle of 
incidence.
Ray 2 reflects off the lower atomic plane 
at the same angle q.
While Ray 2 is in the crystal, however, it 
travels a distance of 2a farther than Ray 
1. If this distance 2a is equal to an 
integral number of wavelengths (nl), then 
Rays 1 and 2 will be in phase on their 
exit from the crystal and constructive 
interference will occur.
Difração de Raios X em cristais
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If the distance 2a is not an integral number of 
wavelengths, then destructive interference will 
occur and the waves will not be as strong as 
when they entered the crystal. Thus, the 
condition for constructive interference to occur is 
nl = 2a
but, from trigonometry, we can figure out what the distance 2a is in terms of the 
spacing, d, between the atomic planes.
a = d sin q
or 2a = 2 d sin q
thus, nl = 2d sin q
This is known as Bragg's Law for X-ray diffraction.
Difração de Raios X em cristais
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nl = 2d sin q
Bragg's Law for X-ray diffraction.
What it says is that if we know the wavelength ,l , of the X-rays going in to the crystal, and we 
can measure the angle q of the diffracted X-rays coming out of the crystal, then we know the 
spacing (referred to as d-spacing) between the atomic planes.
d = nl /2 sin q
Difração de Raios X em cristais
Então:
- conhecendo l , do raio-X que incide sobre o cristal e 
- medindo q do raio-X difratado do cristal, 
É possível conhecer a distância d entre os planos atômicos.
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Difração de Raios X em cristais
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A equação de Bragg nos 
permite calcularqualquer dos 
parâmetros, conhecidos 
outros dois: comprimento de 
onda da radiação, distância 
interplanar do cristal 
difratante ou ângulo de 
difração. n λ = 2d sen(θ)
Lei de Bragg
Difração de Raios X em cristais
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2q 2q 2q
Difração de Raios X em cristais
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William Henry Bragg
William Henry Bragg was born at Westward, Cumberland, 
on July 2, 1862. He was educated at Market Harborough 
Grammar School and afterwards at King William's College, Isle 
of Man. Elected a minor scholar of Trinity College, Cambridge, 
in 1881, he studied mathematics under the well-known 
teacher, Dr. E. J. Routh. He was Third Wrangler in the 
Mathematical Tripos, Part I, in June 1884, and was placed in 
the first class in Part II in the following January. He studied 
physics in the Cavendish Laboratory during part of 1885, and 
at the end of that year was elected to the Professorship of 
Mathematics and Physics in the University of Adelaide, South 
Australia. Subsequently he became successively Cavendish 
Professor of Physics at Leeds (1909-1915), Quain Professor of 
Physics at University College London (1915-1925), and 
Fullerian Professor of Chemistry in the Royal Institution.
His research interests embraced a great many topics and he was an adept at picking up a subject, almost casually, 
making an important contribution, then dropping it again. However, the work of Bragg and his son Lawrence in 
1913-1914 founded a new branch of science of the greatest importance and significance, the analysis of crystal 
structure by means of X-rays. If the fundamental discovery of the wave aspect of X-rays, as evidenced by their 
diffraction in crystals, was due to von Laue and his collaborators, it is equally true that the use of X-rays as an 
instrument for the systematic revelation of the way in which crystals are built was entirely due to the Braggs. This 
was recognized by the award of the Nobel Prize jointly to father and son in 1915.
July 2, 1862 - March 10, 1942 
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William Lawrence Bragg, son of William Henry Bragg, was born in 
Adelaide, South Australia, on March 31, 1890. He received his early 
education at St. Peter's College in his birthplace, proceeding to 
Adelaide University to take his degree in mathematics with first-class 
honours in 1908. He came to England with his father in 1909 and 
entered Trinity College, Cambridge, as an Allen Scholar, taking first-
class honours in the Natural Science Tripos in 1912. In the autumn of 
this year he commenced his examination of the von Laue
phenomenon and published his first paper on the subject in the 
Proceedings of the Cambridge Philosophical Society in November.
In 1914 he was appointed as Fellow and Lecturer in Natural Sciences 
at Trinity College and the same year he was awarded the Barnard 
Medal. From 1912 to 1914 he had been working with his father, and 
the results of their work were published in an abridged form in X-rays 
and Crystal Structure (1915). It was this work which earned them 
jointly the Nobel Prize for Physics in 1915, and from this year to 
1919, W. L. Bragg served as Technical Advisor on Sound Ranging to 
the Map Section, G.H.Q., France, receiving the O.B.E. and the M.C. in 
1918. He was appointed Langworthy Professor of Physics at 
Manchester University in 1919, and held this post till 1937.
William Lawrence Bragg, March 31, 1890 / July 1, 1971