6th Mar, 2020

University of Chittagong

Question

Asked 26th Jun, 2015

how to calculate nano crystallite size by Debye‐Scherrer equation using XRD ?

I think Scherrer’s equation is using to calculate **crystallite** size, not particle size. The crystallite size and particle size is not same. So, Scherrer’s equation is not valid for both. If I want to calculate the particle size, which formula is better?@ **Imane Mayouf** @ **Ahmed I. Osman**

1 Recommendation

Hi

Scherrer’s equation:

Particle Size = (0.9 x λ)/ (d cosθ)

λ = 1.54060 Å (in the case of CuKa1) so, 0.9 x λ = 1.38654

Θ = 2θ/2 (in the example = 20/2)

d = the full width at half maximum intensity of the peak (in Rad) – you can calculate it using Origin software.

To convert from angle to rad

Rad = (22 x angle) / (7 x 180) = angle x 0.01746

Example: if d = 0.5 angle (θ)

= (22 x 0.5)/ (7x 180) = 0.00873 rad

Have a very good day and best of luck

132 Recommendations

**Get help with your research**

Join ResearchGate to ask questions, get input, and advance your work.

When using the Debye-Scherrer equation for calculating particle size (D=Kλ/(β cos θ)

Select a approiate peak that is not overlapp with others, then be sure

D and λ have the same unit (e.g. nm )

FWHM is the full width at half maximum of the peak (not half of it) in rad.

then you can calculate.

I have attached my early papers.

3 Recommendations

Hi PRASANNA NITHIYA, use better a modified Williamson–Hall dependence, compare for instance:

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By applying Scherrer equation on the XRD pattern, the particle size can be calculated:

(D=Kλ/(β cos θ)

Where D is the mean size of crystallites (nm), K is crystallite shape factor a good approximation is 0.9, λ is the X-ray wavelength, B is the full width at half the maximum (FWHM) in radians of the X-ray diffraction peak and θ is the Braggs' angle (deg.)

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Scherrer equation D=Kλ/(β cos θ)

D is the average thickness in vertical direction of the crystal face

K is Scherrer constant. (if β is FMHM,K=0.89. if β is integral height to width of the diffraction peak, k=1 )

λ is the wavelength of X-ray

β is the half high width of the diffraction peak of the sample

θ is diffraction angle(deg)

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Scherrer formula is used to calculate the average size in vertical direction of crystal.

D=Kλ/(β cos θ).

For cubic crystal structure, K =0.94, λ is wavelength of X-ray.

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Hi

Scherrer’s equation:

Particle Size = (0.9 x λ)/ (d cosθ)

λ = 1.54060 Å (in the case of CuKa1) so, 0.9 x λ = 1.38654

Θ = 2θ/2 (in the example = 20/2)

d = the full width at half maximum intensity of the peak (in Rad) – you can calculate it using Origin software.

To convert from angle to rad

Rad = (22 x angle) / (7 x 180) = angle x 0.01746

Example: if d = 0.5 angle (θ)

= (22 x 0.5)/ (7x 180) = 0.00873 rad

Have a very good day and best of luck

132 Recommendations

Dear Fellows:

3 Recommendations

Dear Zia,

use the dominating peak for the phase that you want to calculate the particle size.

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Hi,

Be aware that Sherrer's formula not really applicable for nano size and also not calculate larger size more than 0.1-0.2 micrometer.

Mustafa

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Ahmed's answer explains the calculation quite well. I would like to add that this method of calculation, when applied to thin films, can sometimes yield the particle sizes which are of larger dimensions than the thickness of the films. This is not a discrepancy as the crystallites are assumed to be cylindrical plates in an extension to Scherrer's equation by Smilgies. [D.-M. Smilgies, J. Appl. Cryst. 42, 1030 (2009)]

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You can calculate FWHM using Origin software, but you should find the reference peak that reported in corresponding literature.

we have to used dominating peak for the phase that we want to calculate the particale size

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Dear Zia Ul, Use the peak which is dominating or better use the peak which is in between 30 to 45 degree.

I confirm Prof Ahmed I. Osman to convert from angle to rad

But for calculating the crystallize size with high precision

crystallites size (*Dv*) can be calculated by analyzing the XRD data using the Scherrer's formula, *Dv* = *kλ/βcos(θ)*, where *β *is the structural broadening, which is the difference in integral X-ray peak profile width between the sample and a standard (silicon) and is given by *β = [(β*_{abs})^{2}-(β_{std})^{2}]^{1/2}

4 Recommendations

However, if you used the Scherrer equation (carefully) instead, you may be able to relatively compare samples. There are too many assumptions involved with the Scherrer approach that make it untenable like:

- Shape factor
- Size factor
- Strain factor
- Preferred orientation factor
- Instrumental factor

You may need the help of the W-H technique to separate (deconvolute) some of the above mentioned factors. Learn more here:

This is the most common error while using many of the XRD "formulae". None of the "formulae" that I'm aware of, use the FWHM. Instead it is the "**INTEGRATED BREADTH"**. They may be close but are not the same. You'd need the Bragg profile shape to convert FWHM to INTEGRATED BREADTH (Beta, β). If your diffractogram is digital then computing integrated breadth may be easier than figuring out FWHM :-)

E.g., for a Gaussian distribution, the integral breadth, β, is related to the FWHM peak width, H, by β = 0.5 H (π / log_{e}2)^{1/2}.

- BTW Radians is angle as well!
- Using β
_{(hkl)}would yield the size estimate along the <hkl> direction. - https://www.researchgate.net/post/Please_tell_me_how_to_calculate_Crystal_Size_using_XRD_Data?_sg=NjLSQp9e1eBz06p6HBPRbJy3ty-47Yc-bviJ6dsBdhl1XBfRsDf0ku2yvtTvdU4VmV09LcPaOOzf5YUJ77KJ8colw3gJgVIv

- 449.57 KBHare and Tortoise.png

42 Recommendations

Only if you assume that the diffracting domains are spherical and/or if you are interested in size only along that crystallographic direction.

Post your XRD data as Excel file so others might help you too :-) I also suggest you include additional relevant topics up top in your question to share with a larger audience of XRD experts on RG. Refer to this discussion noted below to see some other topics to include. You may include up to 15.

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What are you talking about? This discussion is about XRD; the angle is the parameter which is being varied during the measurement.

1. Different diffraction angles will provide different FWHMs, when the crystal is not spherical. This is logical, as the number of lattice planes contributing to the diffraction determines the peak width. However, if you get different results from XRD software and from Origin, there must be a difference in data processing. Maybe Origin uses standard deviation instead of FWHM? You should check that.

2. Of course, FWHM has to be converted to radians. If you don't do that, you obtain a mismatch of dimensions in the formula.

3. λ must have the same dimension as the desired dimension of the crystal size. You can also fill in the wavelength in nanoinches and obtain the crystallite size in nanoinches as well.

5 Recommendations

Why u wanna use Scherrer Formula for powder XRD data instead modified WH-plot or WPPM ?

FIrst of all u have to substract the instrument broadening by using the SRM where u have to chose the same peak position as ur sample (exactly). And as rule of thumb u have to go to higher angle, but the peaks intensity decrease, broader and highly overlap. Yet u have to take into account the others contribution as mentioned by Dr. Ravi.

Unless u work with graphane kind of structure and thin film, Do not use Scherrer Formula manually, use software, use all the peaks and substract the instrument contribution (except those software that use FPA).

Look for advance method such as WPPM (size distribution), BGMN (isotropic or anisotropic mean size), Maud (mean and graphical size), GSAS (mean size), Topas (mean size), HSP (mean size), etc

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I agree with Dr. Ahmed I. Osman.

Moreover, you can use one of the following web sites:

Good luck!

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First Calculate the FWHM from XRD plot and then put the value in D=0.9λ/(β cos θ) . *β* is the line broadening at half the maximum intensity (FWHM) . *θ* is the Bragg angle. *λ* is the X-ray wavelength;

2 Recommendations

Please see this video

Andrew R Garcia - "**Ravi, it depends on your definition of calculate**."

No matter the etymology of "calculate", the phrase "Debye‐Scherrer equation" is a misnomer, isn't it? Yet, it seems to perpetuate past all these admonishments. Is it deliberate or sub-conscious or something else? Do folks on RG even know the "Edit" option exists? Amazingly fundamental! :-)

Here's another one: "**β is the line broadening at half the maximum intensity (FWHM)**" - *WRONG! It is "INTEGRATED BREADTH" not "FWHM"!* *Only for those who know not the difference are they the same. *You are obviously not even reading the past comments Himadri! Do yourself a favor and read prior posts when convenient. Avoid embarassment. Learn to use the "Edit" option as well. The words on RG posts are not PERMANANT :-)

Amend your words on your posts and make me retract mine.

I notice that most XRD users here on RG are still "groping in the dark" (still using a conventional point counter) without having the advantage of seeing and visualizing the XRD phenomena. Still practicing it as a "dark" art which it was never meant to be. The Braggs clearly visualized it a century ago.

Are you folks not even reading what Maykel has posted? Come on fellows be considerate. Don't be lulled by the comfort of reading only your own words. Let's wake up and face modernity and reality.

2 Recommendations

Particle Size = (0.9 x λ)/ (d cosθ)

λ = 1.54060 Å (in the case of CuKa1) so, 0.9 x λ = 1.38654

Θ = 2θ/2 (in the example = 20/2)

d = the full width at half maximum intensity of the peak (in Rad) – you can calculate it using Origin software.

To convert from angle to rad

Rad = (22 x angle) / (7 x 180) = angle x 0.01746

Example: if d = 0.5 angle (θ)

= (22 x 0.5)/ (7x 180) = 0.00873 rad

5 Recommendations

Scherrer’s equation: Particle Size Dxrd = (0.9 x λ)/ (d cosθ)

λ = 1.54060 Å (in the case of CuKalpha) ==> 0.9 x λ = 1.38654

note that Θ = 2θ/2

d = the full width at half maximum intensity of the peak (in Rad).

To convert from angle to rad; Rad = (22 x angle) / (7 x 180) = angle x 0.01746

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I asked that if any possible to find the size metal complexes by using scherrer's equation. In case what are the range of size in metal complexes compare with nanocrystalline form

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I agree with Dr. Ahmed ,Osman and Slimani. It is easy to find by following those.

In XRD pattern, You should select most relevant and high intense peak for the calculation. Then, You can use above methods to calculate. It will be easy.

2 Recommendations

to converts angstrom to nanometers divide to 10 .........1 λ /10

FWHM divide to 57.3 to convert to radias ........2 FWHM/ 57.3

then application Scherrer’s equation

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why do you keep answering this post? it is overly simplistic and outdated approach.

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I wouldn't say that, Maykel! The complex conversions from angstroms to nanometers and from degrees to radians maybe completely new concepts for some readers ;-).

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Particle Size Dxrd = (0.95 x λ)/ (d cosθ)

λ = 1.54060 Å (in the case of CuKalpha) ==> 0.95 x λ = 1.46357

Θ = 2θ/2

d = the full width at half maximum intensity of the peak (in Rad).

7 Recommendations

Scherrer introduced the equation taking into account the integral breadths (β) and never mentioned FWHM values. So, in order to "estimate" the crystallite size (not particle size!) you first must eliminate the instrumental contributions (veryyy important) to β_sample using the instrumental function,

β_sample = β_experiment - β_instrumental.

where β_instrumental is a function that are unique for a certain XRD diffracto-meter. So you should measure line position and line shape crystallographic standards for the powder diffraction in order to obtain the parameters that describe your function and so your diffractometer.

Finally, you can use Scherrer as follows:

L_hkl = k * lambda / β_sample * cos (theta).

FOLLOWING LINK ABSOLUTELY WILL BE HELPFUL REGARDING THE QUESTION:

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This is like facebook, when someone presents a stupid riddle asking for some number. People will keep on typing in the numbers for days and weeks, and if only to prove that they can c&p.

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Paramasivam Balasubramanian , Nice job making an excel

Alireza Bargahi

, Your excel needs to convert =cos(2theta/2) to =cos(radians(2theta/2))Thank you guys, nice discussion.

It is Scherrer’s equation, not Debye-Scherrerr:

Scherrer’s equation:

Particle Size = (0.9 x λ)/ (d cosθ)

λ = 1.54060 Å (in the case of CuKa1) so, 0.9 x λ = 1.38654

Θ = 2θ/2 (in the example = 20/2)

To convert from angle to rad

Rad = (22 x angle) / (7 x 180) = angle x 0.01746

Example: if d = 0.5 angle (θ)

= (22 x 0.5)/ (7x 180) = 0.00873 rad

10 Recommendations

It is Scherrer’s equation, not Debye-Scherrerr:

Scherrer’s equation:

Particle Size = (0.9 x λ)/ (d cosθ)

λ = 1.54060 Å (in the case of CuKa1) so, 0.9 x λ = 1.38654

Θ = 2θ/2 (in the example = 20/2)

To convert from angle to rad

Rad = (22 x angle) / (7 x 180) = angle x 0.01746

Example: if d = 0.5 angle (θ)

= (22 x 0.5)/ (7x 180) = 0.00873 rad

Dear Dr Fatameh Mostaghni

I had a look at your paper. Can you explain equation (1) in that paper in more details? Also, can this equation be used to calculate size of soft and spherical objects like biological cells, drug delivery vesicles, etc. ?

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If you have access to Origin software, then this tutorial video can give you a quick snap of what to do, and its easy. This way of analysis makes it more applicable.

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The Debye-Scherer formula in very simple way one may calculate:

Please see the attached file

- 209.69 KBThe Debye.pdf

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I just don't understand it, why STILL people post an answer to a question which has been answered YEARS ago. And - for the hundredth time: It is called SCHERRER formula, not Debye-Scherrer. How difficult can this be?

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See this video

then you can use this value for further calculation

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To calculate strains in a material using XRD I will suggest you to have a look at the following link which is a discription of Williamson-Hall Plot

Determination of Size and Strain

And to calculate crystallite size you can use Scherrer Equation and Modified Scherrer Equation for nano-crystallte sizes. Go through the following paper by Ahmad Monshi et al.

Hope this will help you. All the best.

1 Recommendation

you can see the following manuscripts..

- 2.97 MBIncorporation CdS with ZnS as nanocomposite and Using in Photo-Decolorizationof.pdf
- 1.35 MBBedour-Enhanced Photocatalytic Properties of Pure and Cr-Modified ZnS Powders Synthesized by Precipitation Method.pdf
- 319.09 KBPhotocatalytic Dehydrogenation of Aqueous Methano.pdf
- 808.63 KBImprovement the Photo Catalytic Properties of ZnS nanoparticle.pdf
- 6.14 MBLuma thesis 2013.pdf
- 2.87 MBRole of Platinum Deposited on TiO2 in Photocatalytic Methanol.PDF

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Scherrer equation: τ = K*𝜆/(β *cos(θ))

Where

𝜆 (wavelength): 0,154 [nm] (but depends on your equipment)

β (FWHM: line broadening at half the maximum intensity): ..... [rad] --> remember divide it into 2 because the angle recorded is 2θ

θ (angle): ...... [deegres]

K (shape factor): 0,9 [- ]

τ (particle size): ...... [nm]

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Judit Nuñez Welcome to the fun!

It is not possible to calculate the particle size of an amorphous compound by

Debye‐Scherrer eqN because XRD can only determine crystalline compounds or substances. Hence no data will be available if an amorphous compound will be analysed under XRD spectra

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Crystal size of crystalline materials can be calculated using Debye-Scherer equation if it is homogeneous then take (k=0.94) and if it heterogeneous (k=0.89). Thi can be indicated from SEM images for the investigated material. then beta(FWHM= β) converts from deg into radian scale and applying Scherer equation:

- D=(k
**λ**/**β**Cos Theta)

use all the peaks that you take from patterns and compare it with SEM or TEM in 100 nm range

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This is a new and improved method of Sherer's equation, I suggest studying it.

DOI:10.4236 / wjnse.2012.23020

Modified **Scherrer equation** to estimate more accurately nano-crystallite size using XRD

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Scherrer equation

T = Kλ/(β cos θ)

Where

2 θ = 7.210

So θ=3.6050

And

cos(3.605)=0.998

D=(0.9*0.154)/(0.00285 x cos(3.605))=48.729 nm

D=(0.9*0.154)/(0.00285*0.998)

D = 48.729 nm

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Siddique Shah You quote D = 48.279 nm. Do you know what the diameter of a hydrogen atom is? You should reconsider the number of decimal points you are stating.

Abbas Jassim Lafta If you've read the previous answers you'll know that there's no such animal as the 'Debye-Scherer equation' or even the Debye-Scherrer equation. It's solely the Scherrer equation with all its foibles.

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Dear Researchers,

The formula is not "Debye-Scherrer formula". Its simply "Scherrer formula"

Please refer the below linked nature article.

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I agree with Jayadheepan that there is NO such Debye-Scherrer formula and misleading so should be avoided.

Additionally while using Scherrer’s equation: Particle Size = (K x λ)/ (FWHM cosθ), few points to be considered:

(i) A care must be taken in converting degree into radians and also writing the final answer with errors, as without errors, the number may be misleading. Also if possible, this should be crosschecked with microscopic (TEM) measurements.

(ii) Usually K is considered constant and taken 0.9 (which is true for spherical particles), and for different shaped like cube and rod a smaller K value should be used.

The standard reference by J. I. Langford and A. J. C. Wilson "Scherrer after sixty years: A survey and some new results in the determination of crystallite size" *J. Appl. Cryst.* (1978). **11**, 102-113, should be used for better analysis.

Best

3 Recommendations

You can solve this automatically by Using Jade Software

By Jade You can calculate 2 theta, Intensity Peak ,HKL, Lattice parameters, Stress constant, and peak etc

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Hi, You may find this interesting:

"The equation is Dhkl = Kλ/(Bhklcosθ), where Dhkl is the crystallite size in the direction perpendicular to the lattice planes, hkl are the Miller indices *of* the planes being analysed, K is a numerical factor frequently referred to as the crystallite-shape factor5, λ is the wavelength *of* the X-rays ..."
Source: Holzwarth, U., & Gibson, N. (2011). The Scherrer equation versus the'Debye-Scherrer equation'. *Nature Nanotechnology*, *6*(9), 534.
Webpage: https://www.nature.com/articles/nnano.2011.145

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Hi, I find W-H plot more accurate method to calculate crystallite size. In Scherrer equation, we use the most intense xrd peak. In W-H plot, we use average 4-5 peaks.

I have made a tutorial video on W-H plot for XRD analysis to calculate crystallite size, strain and find the FWHM of each peaks using Gaussian Peak fitting method.

Link is here: https://youtu.be/BrZOpfvKUo0

hi, you can use the following manuscript...

- 1.35 MBBedour-Enhanced Photocatalytic Properties of Pure and Cr-Modified ZnS Powders Synthesized by Precipitation Method.pdf
- 2.87 MBRole of Platinum Deposited on TiO2 in Photocatalytic Methanol.PDF
- 808.63 KBImprovement the Photo Catalytic Properties of ZnS nanoparticle.pdf
- 319.09 KBPhotocatalytic Dehydrogenation of Aqueous Methano.pdf
- 416.96 KBIncorporation CdS with ZnS as nanocomposite and Using in Photo-Decolorization of Congo Red Dye.pdf
- 6.14 MBLuma thesis 2013.pdf

Debye was Scherrer's PhD mentor. He obtained his PhD in 1916, while he published his landmark equation in 1918. Debye-Scherrer equation does not exist.

4 Recommendations

@ Ravi Ananth

Preferred orientation factor or in order way to say is texture development, which is not expected commonly unless a special treatments applied.

Instrumental factor, should be the same form to batch to batch.

I agree, d = the full width at half maximum intensity is not integrated/integral Breadth, which is required for calculation.

I'am agree with **Jeyadheepan Karu** it's Scherrer formula

you need to be attention th the value of λ = 1.54060 Å, if you want to get the crystallite size directly in " nm " you must put λ = 0.154060 nm.

2 Recommendations

Guys have u ever read the recent publications on this ?

Are u really wanna calculate it manually just with a single information ?

Read the size-strain round robin article from IUCR website below as well as find all the related information, instead discussing this obsolete method.

Serch for whole powder pattern modeling as well as an advance recent method.

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I have calculated the crystalline size by MDI Jade 6.5 this version calculate the Size, lattices, hkl,angles values and many more

I have added a photo of demo material with its crystalline size

- 106.49 KBCapture.JPG

Dimitrios A. Giannakoudakis thank you for your fact.

Please see also. The Scherrer equation versus the 'Debye-Scherrer equation' Uwe Holzwarth & Neil Gibson.

Is there any English paper that we can cite the Scherrer formula?

@dogankaya There are thousands of papers. You can find them easily via Google Scholar

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I think Scherrer’s equation is using to calculate **crystallite** size, not particle size. The crystallite size and particle size is not same. So, Scherrer’s equation is not valid for both. If I want to calculate the particle size, which formula is better?@ **Imane Mayouf** @ **Ahmed I. Osman**

1 Recommendation

Chapter

- Aug 2007

The aim of this work is to determine mean diameter of poly- crystalline samples using the Scherrer equation. These calculations represent a direct connection between the mean diameter, crystal structure and morphology by applying the Bragg angle and shape factor. Two samples are investigated, first a nano-sized particles of MgO and second a CVD (ch...

Article

- Jan 1999

The Scherrer equation links the measured width of an X-ray diffraction peak (Scherrer width, SW) to the number of stacked cells (N) in the direction normal to the diffracting planes. The formula is only valid for one d-value occurring in the coherently diffracting domain. This equation can be modified for weakly swelling mixed-layer minerals. This...

Article

- Jan 2003

[Abstract] Bi2Te 3 nanoparticles of 15∼25nm were prepared via solvothermal processes in which pyridine, absolute ethanol and distilled water were used as reaction medium respectively. The XRD and TEM have been applied to analyze the samples, and the grain sizes have also been calculated by Scherrer Equation. The results showed that in these experim...

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