X ray的發現 X-ray的發現

X R Diff ti (XRD) X-Ray Diffraction (XRD)

斯頌平 斯頌平

中興大學物理系 中興大學物理系

1

X ray的發現 X-ray的發現

倫琴在1895年做陰極管實驗時，發現將陰極管用黑色紙包起來，在離管數英尺

外的螢光幕仍可看見發綠色螢光。倫琴發現X-ray可以穿透人體肌肉看見骨頭與y

內在的金屬。因此沒多久就用在檢查骨折與槍傷。

老鼠的膝蓋3D照片 電感3D照片 外包黑色塑膠

老鼠的膝蓋3D照片 電感3D照片，外包黑色塑膠

2

什麼是X-ray

X-ray 是電磁波

Å Å

X-ray波長範圍為0.1~ 100Å， 原子間鍵結距離約為數Å。

3

如何產生X-rays

1. 放射性物質衰變

常見的放射性核種所放出的輻射 常見的放射性核種所放出的輻射

核種 原子序 α 粒子 β粒子 γ射線

鈷-60 27 ★★ ★★

碘-131

(治療甲狀腺癌)

53 ★ ★

半衰期1.3天

碘 125 ★ ★

碘-125

(治療攝護腺癌

、腦瘤)

★ ★

半衰期60天 0.149MV

銫-137 55 ★ ★

氡-222 86 ★ ★

鐳 226 88 ★ ★

鐳-226 88 ★ ★

釷-232 90 ★ ★

鈾-238 92 ★ ★

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2. 同步輻射光源

根據電磁學的理論，帶電粒子的運動速度或方向改變時會放射出電磁波。當電 子以接近光速飛行，受到磁場的作用而發生偏轉時，便會因相對論效應沿著偏轉的 切線方向，放射出薄片狀的電磁波，這就是「同步輻射光」。此類電磁波具有高強 度與同相位的特點。同步輻射為一連續波段的電磁波，涵蓋紅外線、可見光、紫外 線及X光等，1947年首次在美國通用電器公司同步加速器上意外地被發現，因此命 名為「同步加速器輻射」，又簡稱為「同步輻射」。

5

同步輻射產生之光源和一般X光機產生之光源原因不同。不同之處 為X光機是用高速電子撞擊金屬靶，將內層電子游離出來後，外層 為X光機是用高速電子撞擊金屬靶，將內層電子游離出來後，外層 電子躍遷回 去放出的光；同步輻射是用電磁鐵讓電子或帶電粒子，

在一個固定的環內持續的做圓週運動，使能量累積，累積到一定程 度後，可控制電子或質子加速的速度，在切線的方向，動能就會以 度後，可控制電子或質子加速的速度，在切線的方向，動能就會以 光的方式釋放出來，所以加速可以控制放出光的強度和頻率。同步 輻射具有以下特性

— 強度極高

— 強度極高

— 波長連續

— 準直性佳

— 光束截面積小

— 具有時間脈波性與偏振性

若以X光為例，同步輻射在這個波段的亮度比傳統X光機還要強百萬 倍以上！過去需要幾個月才能完成的實驗，現在只需幾分鐘便能得 到結果。以往因實驗光源亮度不夠而無法探測的結構，現在藉由同 步輻射，都可分析得一清二楚，也因此於近年內許多新的研究領域 得以開發。目前同步輻射Powder and fiber X-ray diffraction beam lines 提供波長 1.333 Å與 1.03 Å ~ 0.37 Å光源做繞射研究。 Small and wide angle X-ray scattering for nano-materials and soft matter，

wavelength 2.48 Å ~ 0.54 Å。X-ray absorption spectroscopy beam lines ， k

energy range 2~33keV。

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3. 將粒子加速撞擊物質

當帶電粒子在減速的過程中，會釋放出電磁波，而在 當帶電粒子在減速的過程中，會釋放出電磁波，而在 減速過程中所放出之電磁波具有高能量，其波長可在X光 範圍。因此，當以高電壓加速之電子束撞擊陽極標靶，高 速電子受到標靶原子的阻擋急劇停止下來，電子在非彈性 速電子受到標靶原子的阻擋急劇停止下來，電子在非彈性 碰撞過程中能量損失部分轉變成X光子的能量，此X光稱 為連續輻射。此外，當電子束與標靶之原子碰撞時，原子 內層電子被打出，外層電子往內遞補，而由高能階轉變成

內層電子被打出 外層電子往內遞補 而由高能階轉變成

低能階的狀態，能量便藉由X-ray的方式產生。此X光稱為 特徵輻射。此時，若撞掉的K層電子由L層的電子遞補而產 生的X-ray稱為Ky _αα；若是由M層電子遞補則稱K_ββ。一般來 說，要產生1 Å波長的X光，需要上萬伏特的電壓。

7

連續輻射X光(Bremsstrahlung radiation，制動幅射)

8

X光特徵輻射(characteristic radiation)

9

典型X光光譜圖

10

Energy levels of Copper (Z = 29)

E_Kα1Kα1 = E_KK –E_L3L3 =8.048KeV E_Kα2 = E_K –E_L2 =8.028KeV E_Kβ= E_K – E_M2.3 = 8.905 keV

h c hf

E = hf = h λ E

K_α1=1.540562 Å K ₂=1 544390 Å K_α2 1.544390 Å

K_α1 K_α2

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X-Ray Tube

一般繞射用之X光管兩極間電壓，都維持在3至5萬伏特左右，

以供電子加速。電子的來源乃來自通以高電流之高熱燈絲(多為 鎢絲)所釋放出來的熱電子 這些熱電子受到兩電極間的高電壓 鎢絲)所釋放出來的熱電子，這些熱電子受到兩電極間的高電壓 加速，從陰極端射向陽極端集中撞擊在陰極靶的焦點(focal spot) 上。所產生的X光從焦點向X光管側面四周輻射，最後經由二至

四個窗口向外射出 為避免X光的損失 窗口材料多選用吸收X

四個窗口向外射出，為避免X光的損失，窗口材料多選用吸收X 光較少的元素材料，例如鈹。通常僅有1％之動能轉化成光其餘 99％則轉變成熱能，因此X光管外層必須以水冷卻避免過熱熔化。

12

X光管構造

一般 X光管功率為2000W(50kV ×40mA) ， 實際使用時約設定1200W (40kV ×30mA) 。

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Use Ni (Z 28) filter Use Ni (Z=28) filter

Incident beam I(K_α)/I(K_β)=7.5

Filter thickness for I(K( _αα)/I(K) ( _ββ)= 500 is 0.0008 in) I(K_α) trans./I(K_α)incident= 0.42

14

Fluorspar (CaF₂)

The most common colors are purple, blue, green, yellow, or colorless.

Less common are pink red white brown black The color of the fluorite Less common are pink, red, white, brown, black. The color of the fluorite is determined by factors including impurities, exposure to radiation, and

the size of the color centers. ¹⁵

CaF₂晶體結構 CaF₂ 晶體結構

鈣原子(藍色) 形成面心立方結構(face center cubic)。

氟原子(白色) 形成立方體結構(simple cubic) 。

Basis

Ca (0 0 0) Ca (0, 0, 0) F (1/4, 1/4, 1/4) F (3/4, 3/4, 3/4)

http://www.youtube.com/watch?v=Pu9BmA2YxQE

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NaCl晶體結構

NaCl 晶胞可以視為以鈉原子(綠色)與氯原子(藍色)為一組

形成面心立方結構(f ) 或是鈉原子與氯原子分別形成互

形成面心立方結構(fcc) ，或是鈉原子與氯原子分別形成互 相交叉的FCC結構。鈉原子或氯原子在彼此形成的立方體 對角線中間處。

17

18

十四種不同的三維晶格(lattice)

orthorhombic triclinic

rhombohedral tetragonal monoclinic

hexagonal cubic 19

晶體結構介紹 晶體結構介紹

1. 晶格(lattice)與基元 (basis)

a.理想晶體是由一群原子所組成的基元在空間中週期性不斷重複排列構成晶體。

b 基元中選取一特定點稱為晶格點(lattice point)，晶格點在空間當中週期排列的 b.基元中選取一特定點稱為晶格點(lattice point)，晶格點在空間當中週期排列的

方式不同就形成不同的晶格結構。

c.晶體=晶格+基元

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d. 二維晶格中如 a₁₁、a₂₂ 為晶格向量(lattice vector)，平面上每一晶( ) 格點可由向量

r_mn=m a₁+n a₂ 決定，其中m、n為整數。 三維晶格中 r_lmn= la₁ + m a₂ + n a₃ (l, m, n為整數)

r₃₂

21

2. 晶格點可組成primitive cell(原胞、基本晶胞) 或 conventional cell(晶胞、非原胞) a. primitive cell為由晶格點形成最小體積的cell， 而conventional cell 可以包含數個 a. primitive cell為由晶格點形成最小體積的cell 而conventional cell 可以包含數個

primitive cell 。定義primitive cell或 conventional cell的方式不是唯一的。

b. primitive cells的體積為V= |a₁ ( a₂ × a₃)|

c. fcc primitive cell 、 conventional cell 、Wigner-Seitz cell p g

Wigner-Seitz cell

V=a³ V=0.25a³

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d. bcc primitive cell 、 conventional cell 、Wigner-Seitz cell

primitive vectors:

1

1(? ˆ) 2

1( ? ˆ) aG = x+ −y z

G

2

3

( ? )

2

1(? ˆ) 2

a x y z

a x y z

= − + +

= − +

G

Wigner-Seitz cell

V=a³ V=0.5a³

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晶面與單晶方向

Miller index Miller index

1. 切割面與晶軸的切割點以晶格常數為單位表示為 (p, q, r) 2. 取(p, q, r)之倒數(1/p, 1/q, 1/r)後再取其最小公倍數(h, k, l) 3. 定義這個切割面為(h k l)面

4. (h k l)稱為Miller index ， [h k l]表示向量方向。立方晶系中晶面方向可以

[h k l]表示，非立方晶系[h k l] 向量方向不垂直於 (h k l)面， 因此不能以[h k l]

[h k l]表示 非立方晶系[h k l] 向量方向不垂直於 (h k l)面 因此不能以[h k l]

表示晶面方向。

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因立方體有對稱之關係將(1 0 0) ( )

，(0 1 0) ，(0 01) ， ，

， 表示為{1 0 0}面 0) 0 1 ( 0)

1 0 (

) 1 0 0 (

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Crystal directions in the cubic lattice.

26

單狹縫繞射

X-ray diffraction

單狹縫繞射

1,2,...

= m minima

mλ

= Dsinθ

光柵繞射

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X-ray diffraction ： Bragg’s Lawy gg 每一個原子都可當作散射光源

λ為波長，d為產生繞射峰的晶面距離，θ為入射光與晶面的夾角

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The X-ray diffractometer

繞射儀（diffeactometer）：是一種用來決定粉末試片發生繞射角 度的儀器，計數器被裝在可動的架上，其亦沿著O軸旋轉；它的 角度位置以2θ標在刻度上。可動架和試片是機械組合，如此試 片旋轉θ則計數器會伴隨旋轉2θ；此保證入射角和反射角彼此保 持相等。

從X-光管出來的光束包含了極強的K_α線以及相對較弱的K_β線、連續光 譜。濾片的材質為靶材的原子序減1 。偵測器以相對於試片2倍速度旋 轉。

29

Techniques in the XRD q

X P d Diff i (XRPD)

• X-ray Powder Diffraction (XRPD)

• Single Crystal Diffraction (SCD)

• Back-reflection Laue Diffraction (no acronym)

• Grazing Incidence Angle Diffraction (GIXD) Grazing Incidence Angle Diffraction (GIXD)

• X-ray Reflectivity (XRR)

S ll A l X S i (SAXS)

• Small Angle X-ray Scattering (SAXS)

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X-ray Powder Diffraction

A polycrystalline sample should contain thousands of crystallites. 

h f ll bl d ff k h ld b b d

Therefore, all possible diffraction peaks should be observed.

2θ 2θ 2θ

2θ 2θ 2θ

• For every set of planes, there will be a small percentage of crystallites that are properly oriented to diffract (the plane perpendicular bisects the incident and diffracted beams).

• Basic assumptions of powder diffraction are that for every set of planes there is an equal number of crystallites that will diffract and that there is a statistically relevant number of crystallites, not just one or two. ³¹

(111)

Au thick film on silica substrate Structure of Au is fcc

(200)

A Si (220) (311)

30 40 50 60 70 80

Au₅Si (220) (311)

30 40 50 60 70 80

2θ

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Phase Identification

The diffraction pattern for every phase is as unique as your fingerprint

1. Phases with the same chemical composition can have drastically different p y diffraction patterns.

2. Use the position and relative intensity of a series of peaks to match experimental data to the reference patterns in the database

33

XRD pattern of Na₅InSi₄O₁₂ crystal system: trigonal(rhombohedral)

a=21.7158(9) Å

Experiment

Si l t d Simulated

34

Crystal Structure of Na₅InSi₄O₁₂

12-ring channels

7-ring channels

InO SiO Na

對稱性降低增加繞射峰 InO₆ SiO₄ Na

35

36

X-ray diffraction patterns for

the heat treated Infiltrated/LiNO₃₃ gelg

37

Reciprocal lattice (倒置晶格)

The lattice points in the reciprocal lattice are mapped by vector

3 3 2

2 1

1b + v b + v b v

= G

The lattice points in the real space are mapped by vector

3 3 2

2 1

1a +n a +n a n

= R

Then the relationship exp(iG · R)=1 holds.

Vectors in the direct lattice have the dimensions of [length]; vectors in the reciprocal lattice have the dimension of [1/length]. The reciprocal lattice is a lattice in the Fourier space associated with the crystal

lattice is a lattice in the Fourier space associated with the crystal.

Wavevectors are always draw in Fourier space, so that every point in the reciprocal lattice may have a meaning as a description of a wave.

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Real Space Reciprocal Space

bcc Wigner-Seitz cell cell bcc Brillouin zone bcc Wigner-Seitz cell cell bcc Brillouin zone

fcc Wigner-Seitz cell cell fcc Brillouin zone ₃₉

What determines intensity of a peak I_hkl?

∑

= NS

= F

When the diffraction condition is satisfied, the scattering amplitude for N crystal cells is

晶胞內原子的排列

∑

j j j

G

G = NS = N f exp(-iG•r ) F

where S_G is called the structure factor. The atomic form factor f_jis written as 晶胞內原子的排列

)]

r - (r

• -iG exp[

) r - r ( dVn

=

f_j

∫

f b b

G b

The structure of Au is fcc the cubic cell has identical atoms (f is identical) at )

z - y x ),

z y - x ),

z y x -

fcc

ˆ ˆ + ˆ ( a / π 2

= b ˆ + ˆ ˆ ( a / π 2

= b ˆ + ˆ + ˆ ( a / π 2

= b

b v + b v + v

= G

3 2

1

1 1 1 1 1b1

The structure of Au is fcc, the cubic cell has identical atoms (f_j is identical) at (0 0 0), (0, ½, ½), (½, 0, ½), (½, ½, 0).

k]}

+ (h π exp[-i +

l)]

+ (h π exp[-i +

l)]

+ (k π -i exp[

+ 1 { f

= ) hkl ( S

If h, k, l are all even or all odd S=4f, otherwise S=0.

No refraction from {1 0 0} {1 1 0} planes, the first peak arises from {1 1 1} planes

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立方晶體中之X光繞射線選擇律

晶格 有繞射線

簡單立方 所有(h, k, l)

體心立方 h k l 偶數

體心立方 h + k + l=偶數

面心立方 h, k, l 全為偶數或奇數

鑽石立方 h, k, l 全為偶數或奇數,

鑽石立方 h, k, l 全為偶數或奇數,

且(h + k + l)不為奇數之兩倍

41

42

Calculate d value

n λ=2d

sin θ

) l + k + h 1 (

d = system 1

cubic

(

)

a

y d

2 2

2

1 l

+ ) k + h 1 (

t t l 1

l

+ c

) k + h a (

d = tetragonal

2 2 2 2

2 2

2

l

c + 1 b k

+ 1 a h

= 1 d

1 c rthorhombi

o d a b c

A (1 1 1) l 2θ 38 269^oλ 1 54 Å d 2 35 Å 4 07 Å Au (1 1 1) plane 2θ=38.269^o λ =1.54 Å

→

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JCPDS (Joint Committee on Powder Diffraction Standards)

S G - The three-dimensional space group symbol S.G. The three dimensional space group symbol Z - Number of chemical formula units per unit cell.

Ref. - Source of data

D_x- Density calculated from X-ray measurements by the NBS*AIDS83 program.

D - measured density D_m measured density

SS/FOM - Smith-Snyder figure of merit

Volume [CD] - Crystal Data Cell Volume is generated by NBS*AIDS83

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Meaning of Quality Symbol

"*“ high quality diffractometer data; the chemical composition is well characterized;

intensities have been measured objectively; errors and the average delta 2theta is less than 0.03 deg.

"I“ th tt h b i d d (th i l t t i l i l h ) Th i

"I“ the pattern has been indexed (thus is almost certainly single phase). There is a reasonable range and even spread in intensities; errors and the average delta 2theta is less than 0.06 deg.

“O“O diffraction data have been taken on poorly characterized material or that the diffraction data have been taken on poorly characterized material or that the data are known (or suspected) to be of low precision.

"B" A "Blank" quality mark is assigned to patterns which do not meet the "*", "I", or

"0" criteria0 criteria

"R" A "Rietveld" quality mark is used for patterns where it is clear that the d-values are directly the result of Rietveld refinement of the data. However, Rietveld

refinements are accepted only in unusual cases refinements are accepted only in unusual cases.

"C" A "Calculated" quality mark is assigned to those patterns which have been

calculated from single crystal data. C patterns generally have very precise d-values, but the intensities may not reflect what is obtained in an experimental pattern.

but the intensities may not reflect what is obtained in an experimental pattern.

Search-match program

"*“ "C" "I“ “N“

“Q“ questionable “D“ deleted

45

Q questionable D deleted

Information in a diffraction pattern

1 Peak positions give unit cell size and shape 1. Peak positions give unit cell size and shape.

2. Peak intensities tell you about the electron density inside the unit cell, i.e. where the atoms are located.

● Factors affecting the relative intensity of the diffraction peak

● Factors affecting the relative intensity of the diffraction peak

‧structure factor

‧multiplicity {h k l}planes

‧absorption p

‧temperature

‧ polarization ) 2

θ 2 cos + (1

2

θ 2 cos +

1 ²

‧lorentz factor

3. Systematic absences give type of unit cell and space group information θ)

cos θ sin

θ 2 cos + (1 ₂

4. Peak shapes and widths tell you about any deviation from a perfect crystal; crystallite size (<200 nm), microstrain.

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Effect of grain size on peak shape

For grain sizes less than ~100 μm, the larger the grain size the sharper the peak larger the grain size, the sharper the peak.

Grains << 1 μm have very broad peaks, eventually flattening out and creating a

‘hump’ in the background (‘amorphous’ p g ( p materials).

Scherrer formula λ 9 .

= 0 t

θB

cos

= B t

t : particle size

B:angular width at the half maximum intensity B:angular width at the half maximum intensity θ_B: angle to be used for calculation

47

Cerianite- - CeO₂

u.)ntensity (a.uI

23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

2θ (deg.)

Crystallite 22 ~ 30 Å for 2θ 28.5~95.4°y Average size: 25 Å

Standard Deviation: 3.4 Å

48

Effect of strain on peak shape

49

若是材料受到應變力(ε)作用時，則必需要考慮其所產生的寬化效應，綜合( ) 的效應可以用Williamson-Hall方程式來表示

由上述方程式在x及y軸分別以4sinθ和B cosθ 作圖，即可得到Williamson-Hall 圖。在圖中y軸的截距即為晶體大小；而斜率為材料所受的應變力。

β =Bcos θ/ λ β

Q=2sin θ/ λ

50

High resolution of XRD g

‧Rocking curve measurements made by doing θ scan at a fixed 2θ angle, the width of which is inversely proportionally to the dislocation density in the film and is therefore used as a gauge of the quality of the film.

‧ Superlattice measurements in multilayered heteroepitaxial

structures, which manifest as satellite peaks surrounding the main diffraction peak from the film. Film thickness and quality can be deduced from the data.

‧ Glancing incidence x-ray reflectivity measurements, which can d i h hi k h d d i f h fil Thi determine the thickness, roughness, and density of the film. This technique does not require crystalline film and works even with amorphous materials.

T t t Angle of incidence

‧ Texture measurements

Sample

Goebel mirror(crystal) X-ray source Diffraction angle 2θ

g

Sample rotation, φ

Diffraction vector Diffraction angle, 2θ

Sample inclination, ψ

Scintillation detector

Flat monochromator (crystal) φ

Normal direction

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Grain boundary and rocking curve

A B C

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Superlattice measurements

假設多層量子井結構的一週期厚度為d

d 假設多層量子井結構的一週期厚度為d

布拉格繞射定律

2dsinθ₁=nλ ………(1) 2dsinθ₂ = mλ ………..………(2)

GaAs InGaAs

2dsinθ₂ mλ ………..………(2) (1)-(2)時，可得下式:

2d [sinθ₁-sinθ₂ ] = (n-m) λ …………(3) 第(3)式移項可得

Buffer layer

(100) GaAs substrate

( )

(4) θ

sin d +

1 2

λ m) - n -(

= θ

sin _{2 1}

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圖中最高的波峰就是GaAs基 板，而其他的波峰就是InGaAs 的衛星波峰(satellite peaks)，因 為是多層量子井的結構所以會出 現由

於週期性的磊晶層排列所造成的 干涉波峰，這些波峰又稱為附屬 干涉圖樣

X ki

干涉圖樣(subsidiary diffraction pattern)或衛星波峰(satellite peaks) 。

θ sin 1 +

λ m) - n

= ( θ

X-ray rocking curve sin +sinθ

d - 2

= θ

sin _{2 1}

將最高peak角度當作θ₁，n=0 ， y=sin θ₂ x=m 斜率為λ/(2d) y sin θ₂, x m, 斜率為λ/(2d) 由斜率可計算出superlattice 膜厚的週期

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Grazing Incident Angle Diffraction (GIXD)

l ll d Gl i A l iff i (GA )

• also called Glancing Angle X-Ray Diffaction (GAXRD)

• The incident angle is fixed at a very small angle (<5°) so that X-rays are focused in only the top-most surface of the sample.

• GIXD can perform many of analyses possible with XRPD with the added ability to resolve information as a function of depth (depth-profiling) by collecting successive diffraction patterns with varying incident angles

– orientation of thin film with respect to substrate – lattice mismatch between film and substrate – epitaxy/textureepitaxy/texture

– macro- and microstrains – reciprocal space map

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Grazing Incident Angle Diffraction (GIXD)

10⁰ _α=2Θ/2

α=20^o

μm)

Gold, CuKα, μ ≈ 4000 cm^-1

10^-1

α=20 α=10^o α=5^o

ion depth (

10

α=2^o α=1^o

Penetrati

Symmetrical mode GIXD

0 20 40 60 80 100 120 140 10^-2

Diffraction angle (^o2Θ) Diffraction angle ( 2Θ)

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圖為針對多晶Ge薄膜成長在Si(100)基材，在入射角為

0.5度下所得到的繞射圖譜。我們可以看到所有的繞射峰都來 0.5度下所得到的繞射圖譜 我們可以看到所有的繞射峰都來 自Ge薄膜，沒有來自Si基材的訊號被偵測到。由於改變入射 角度可以控制X光的穿透深度，因此這個量測技術已被廣泛的 被利用在一些金屬矽化物之鑑定，如矽化鎳(NiSi)和二矽化鈷( ) (CoSi₂)。

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Texture Measurement (Pole Figure)

T t t d t d t i th i t ti di t ib ti f t lli i Texture measurements are used to determine the orientation distribution of crystalline grains in a polycrystalline sample. A material is termed textured if the grains are aligned in a

preferred orientation along certain lattice planes. One can view the textured state of a

material (typically in the form of thin films) as an intermediate state in between a completely material (typically in the form of thin films) as an intermediate state in between a completely randomly oriented polycrystalline powder and a completely oriented single crystal. The

texture is usually introduced in the fabrication process and affect the material properties by introducing structural anisotropy

introducing structural anisotropy.

A texture measurement is also referred to as a pole figure as it is often plotted in polar

coordinates consisting of the tilt and rotation angles with respect to a given crystallographic orientation A pole figure is measured at a fixed scattering angle (constant d spacing) and orientation. A pole figure is measured at a fixed scattering angle (constant d spacing) and consists of a series of φ -scans (in- plane rotation around the center of the sample) at

different tilt or Ψ -(azimuth) angles, as illustrated below.

The pole figure data are displayed as contour plots with zero angle in the center.

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Pole Figures

z-axis (sample normal)

(hkl) plane

f axis Equatorial

plane Projected position

of the (hkl) plane ψ

f-axis

South pole

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60

(a) (b)

Example: c-axis aligned superconducting thin films

φ φ

Biaxial Texture (105 planes) Random in-plane alignment Biaxial Texture (105 planes) Random in-plane alignment

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Rolling assisted biaxially textured substrates

pole figure of {111} cube texture {111} cube texture

Rolling

Annealing Annealing

Rolling

reduction >95% Recrystallization Grain growth

Liu Wei, Li xiaoling;Tsinghua University 62

X光反射率(X-ray Reflectometry)

光照射於薄膜樣品時 會由薄膜表面產生反射 光反射率為測量反射 光之強

X光照射於薄膜樣品時，會由薄膜表面產生反射，X光反射率為測量反射X光之強 度隨著入射角度變化情形。X光之所以會產生反射的原因，是因為大部份材料對於 X光的折射率會略小於1(空氣的折射率等於1)，而當X光經由空氣進入材料內部

時 由於是由高折射率介質進入低折射率介質 故當X光以一非常小的入射角進入

時，由於是由高折射率介質進入低折射率介質，故當X光以一非常小的入射角進入 材料表面時，會被材料表面完全反射，產生全反射。一般而言，入射角度在全反射 臨界角以上時，X光的反射率會開始下降得很迅速，在理想的情況下，一個完美平

整的界面 表層上沒有其它堆積層 基本上其反射率衰減的幅度是與散射波向量

整的界面，表層上沒有其它堆積層，基本上其反射率衰減的幅度是與散射波向量 q (=4πsinθ/λ)的-4次方成正比的關係，此處θ為入射X光與被測樣品表面的夾角，q 為散射波向量，λ為入射X光波長。若表面或界面具有一些粗糙度時，X光反射率則 下降得更快。如果所測樣品基板上有一層薄膜時，其X光反射率曲線就會呈現所謂 下降得更快。如果所測樣品基板上有一層薄膜時，其X光反射率曲線就會呈現所謂 的Kiessig干涉條紋。若當樣品為多層薄膜時，X光沿著每一界面反射和入射之X光 產生干涉，而使反射率強度曲線呈現多重週期，因此所測樣品之詳細的電子密度分 佈、薄膜厚度、界面及表層粗糙度等皆可以從X光反射率曲線上獲得。X光反射率

佈 薄膜厚度 界面及表層粗糙度等皆可以從X光反射率曲線上獲得 X光反射率

對於所測樣品之形態，並沒有特別限制，不論是多晶、單晶、或非晶質材料，甚至 於液態材料皆可測量。利用同步輻射X光光源可測的薄膜厚度在1 nm～1000 nm 之間，而且由於入射之X光波長大約為0.1 nm，故所測樣品厚度的準確度可達

之間 而且由於入射之X光波長大約為0.1 nm 故所測樣品厚度的準確度可達

0.1nm，是目前決定薄膜厚度最準確的方法之一。所以X光反射率是用來測量奈米 薄膜厚度、電子密度及其表面和界面粗糙度等結構參數的最有利工具。

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Diffraction from a Single Crystal

• X Rays striking a single crystal will produce diffraction spots in a sphere around the crystal.y

– The sample axis, phi, and the goniometer axes omega and 2theta are rotated to capture diffraction spots from at least one hemisphere

– Each diffraction spot corresponds to a single (hkl)Each diffraction spot corresponds to a single (hkl)

– The distribution of diffraction spots is dependent on the crystal structure and the orientation of the crystal in the diffractometer

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X-Ray 小角度散射 (SAXS)

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C₆₀(OH)₁₈在水溶液中的聚集對X光的小角度散射。虛曲線為公式對數 據的擬合結果。而虛直線為級數法散射(power law scattering)的特徵擬 據的擬合結果 而虛直線為級數法散射(power law scattering)的特徵擬 合。卡通示意插圖為C₆₀(OH)₁₈的碎形聚集，大約可用2ξ來圈出較密集 的單體碎形聚集區域。

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