**PURPOSE**- Use a least-squares method to determine the parameters of envelope functions of the contrast transfer function from rotationally averaged power spectra images. Further info on CTF related operations in SPIDER.

**SEE ALSO****TF DNS**[Transfer Function - Delete noise background] **TF L**[Transfer Function - Generate CTF, in doc file] **TF D**[Transfer Function - Generate image showing effect of astigmatism on CTF] **CTF FIND**[Contrast Transfer Function - Estimation of CTF parameters] **TF DDF**[Transfer Function - Determine Defocus & amplitude contrast] **TF**[Transfer Function - Generate image showing effect of defocus on CTF] **TF LM4**[Transfer Function - Determine CTF envelope B-factor and noise parameters]

**USAGE**- .OPERATION: TF DEV

- .HOW MANY IMAGES IN THE SERIES?: 1

[Enter the number of images in the calculation.].WAVELENGTH LAMBDA [A]: 0.037

[Enter the wavelength of the electrons. The value used in this example corresponds to 100kV. A table of values is listed in the glossary under lambda.].CS [MM]: 2.7

[Enter the spherical aberration constant.].MAX SPATIAL FREQUENCY [1/A]: 0.10

[Enter the spatial frequency limit in units of 1 / Angstroms. The maximum spatial frequency is 1/(2*pixelsize), where pixelsize is the size of the pixel in Angstroms.].AMPLITUDE CONTRAST RATIO: 0.09

[Enter the amplitude contrast ratio estimated by 'TF DDF'.].SOURCE SIZE [1/A]: 0.003

[Enter an initial guess of the size of the illumination source in reciprocal Angstroms. This is the size of the source as it appears in the back focal plane of the objective lens. A small value results in high coherence; a large value, low coherence.].DEFOCUS SPREAD [A]: 200

[Enter estimated magnitude of the defocus variations corresponding to energy spread and lens current fluctuations.].CHARACTER OF THE FILM Kf [1/A]: 0.05

[Enter an initial guess for the parameter of the modulation transfer function. Actually it includes effects of the scanner, film, etc.].GAUSSIAN ENVELOPE HALFWIDTH: 0.05

[Enter initial guess of halfwidth of Gaussian envelope function.].IMAGE FILE: rod004

[Enter name of background-corrected 1D profile of power spectrum.].DEFOCUS [A]: 20000

[Enter defocus value calculated by 'TF DDF'. Convention: underfocus is positive, overfocus is negative.].FITTING REGION: 30, 230

[Enter fitting region in Fourier pixel units.].OUTPUT FILE: rop004

[Enter the name of file which will store simulated 1D profile using the parameters estimated above. Thus, you can compare how well they fit. If you input a series of files, the output files will be generated in the same order as the input. If you don't want an output file, type * to terminate.]

**NOTES**

- Theory and all definitions of electron optical parameters are according to:

Frank, J. (1973). The envelope of electron microscopic transfer functions for partially coherent illumination.

*Optik*, 38(5), 519-536.

and

Wade, R. H., & Frank, J. (1977). Electron microscope transfer functions for partially coherent axial illumination and chromatic defocus spread.

*Optik*, 49(2), 81-92.

Internally, the program uses the generalized coordinates defined in these papers. - The functions used for fitting the input profiles are:

f(A1,A2,A3,A4)=A1*SIN(X(KF))*E1(A2)*E2(A3)*E3(KFILM)*E4(A4)

SIN(X(K))=SIN(PI*(0.5*CS*LAMBDA**3*KF**4-DZ*LAMBDA*KF**2-Q) is the CTF for a monochronic wave;

E1 = EXP(-1*PI**2*A2**2*(K**3*CS*LAMBDA**3-DZ*K*LAMBDA)**2) is the partial spatial coherence envelope function, where A2 is the source size;

E2 = EXP(-PI**2*A3**2*K**4*LAMBDA**2/16LN2) is the partial temporal coherence function, where A3 is the energy spread;

E3 = 1/[1+(KF/KFILM)**2] is the envelope function for the recording;

E4 = EXP(-(KF/A4)**2) is the Gaussian evelope function, where A4 is the halfwidth of the Gaussian function. - The strategy to differentiate the effects of the partial spatial
coherence envelope function and the Gaussian envelope function is
to use a defocus series, because the partial spatial coherence
envelope function is defocus-dependent, but the Gaussian envelope
function is not.
- The partial temporal coherence function is not actually fitted, because it is not the main resolution-limiting factor in the spatial frequency range up to 0.15 1/A.

**SUBROUTINES**- ENVELOPE

**CALLER**- UTIL1

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