CN102886528A - Preparation method for gold nanorod self-assembly material with optical rotation characteristic based on polymerase chain reaction - Google Patents

Abstract

The invention discloses a preparation method for a gold nanorod self-assembly material with an optical rotation characteristic based on a polymerase chain reaction, and belongs to the fields of nanometer materials and optics. The preparation method disclosed by the invention comprises the following main implementation steps of: (1) preparing gold nanorods; (2) performing directional modification on the gold nanorods with an upstream primer and a downstream primer, and performing the directional amplification of the PCR (polymerase chain reaction); and (3) performing optical and structural characterization on the PCR product. The chiral assembly material of the gold nanorods based on the polymerase chain reaction has a plasma resonance coupling property and represents the highest wavelength absorption intensity and a position dynamic change characteristic. The preparation method disclosed by the invention is strong in assembly specificity and controllability because of performing assembly for the gold nanorods by virtue of DNA (deoxyribonucleic acid) complimentary hybridization, and has the characteristics of being simple, efficient and the like because of performing controllable assembly for the gold nanorods by virtue of the PCR process; and the preparation method is the perfect combination of the traditional DNA amplification technology and a modern nanometre technology, and is a novel assembly strategy simultaneously.

Description

A kind of gold nanorods self-assembled material preparation method with optically-active feature based on the PCR

Technical field

The present invention is a kind of gold nanorods self assembly of PCR-based, has the new material preparation method of plasma optically-active feature.Belong to nano material and optical field.

Background technology

Development along with nanometer technology, the continuous interactive development of nanometer technology and other technologies, studying various nano materials and assembling character and application is great direction and the problem in science of current Nano-technology Development, the proposition of various novel problem in science and phenomenon is the very outstanding distinctive superior character of nano material that shows itself all, wherein study the small size effect of nano material self, skin effect, small-size effect, macroscopical building-up effect correlation properties that macro quanta tunnel effect shows become when one of previous important Disciplinary Frontiers, and the controlled assembling of nano particle provides effective means for this reason.Gold nanorods, because its special physical property, anisotropic plasma resonance essence has very widely application in fields such as biology sensor, materialogy and biomedicines.The character that the assembling of gold nanorods shows, special optical character particularly, it is the focus of nanocomposite optical area research in recent years, utilize the complementary hybridization of DNA to carry out the assembling of gold nanorods, have strong assembling specificity and controllability, and the characteristics such as the controllable groups harness that utilizes the PCR process to carry out gold nanorods has simply, and is efficient, being traditional DNA cloning technology and the perfect adaptation of modern nanometer technology, also is a kind of novel packaging strategy simultaneously therefore.

Summary of the invention

The purpose of this invention is to provide a kind of PCR-based reaction, able to programme, carry out easy, the controlled novel method of gold nanorods assembling efficiently, prepare a kind of novel optical material with plasma optically-active feature.

Technical scheme of the present invention: a kind of chirality assemble method of the gold nanorods based on polymerase chain reaction, has plasma resonance coupling character, and show maximum absorption wavelength intensity and position dynamic variation characteristic, comprise, the preparation of gold nanorods, the primer directed modification of gold nanorods, the controlled assembling of PCR, the sign of PCR product;

(1) preparation of gold nanorods

The preparation of gold nanorods is to utilize the gold seeds growth method, gold seeds synthetic: with the gold chloride HAuCl of the 0.0005M of 2.5mL ₄Be dissolved among the softex kw CTAB of 0.2M of 2.5mL, mix, with 0.3mL newly dispose, the sodium borohydride solution of the 0.01M of precooling is added in the mentioned solution fast, and powerful mixing 2min, then 25 ℃ of room temperatures are placed 2h, get gold seeds;

The growth of gold nanorods, the configuration of growth solution: with the AgNO of 0.15mL, 0.004M ₃, the HAuCl of 5mL, 0.001M ₄Solution is added among the CTAB of 5mL, 0.2M, then adds the Vc of the 0.0788M of 70 μ L, and abundant reductase 12 min adds the gold seeds of the above-mentioned configuration of 12 μ L, fully stir 20s after, leave standstill in 25 ℃, stand-by;

(2) the primer directed modification of gold nanorods

Carry out first the end face sulfhydryl compound and modify, this sulfhydryl compound is selected TGA, mercaptoethanol or two sulphur crisp sweets alcohol, and target is to carry out the end face sealing, second step carries out the side of primer to be modified, and is the prerequisite of controlled assembling;

Synthetic gold nanorods, concentration determination is 0.262nM, carry out the centrifugal 10min of 10000r/min, centrifugal sediment is resuspended with the CTAB solution of 0.005M, with trim two sulphur crisp sweets alcohol DTT with concentrated after gold nanorods with the ratio of mol ratio 5 ︰ 1, room temperature leaves standstill, modification reaction 10h, and purpose is modified the end face sealing of gold nanorods; The gold nanorods that above-mentioned end face is modified is centrifugal twice, and each centrifugal 10min of 7000r/min is resuspended in the CTAB solution of isopyknic 0.005M; The gold nanorods of this centrifugal treating is scattered in respectively in two PCR pipes, respectively with PCR in upstream primer and downstream primer respectively with the ratio of Yin Wu ︰ gold nanorods mol ratio 400 ︰ 1, carrying out the side of gold nanorods modifies, room temperature leaves standstill, modification reaction 10h, the gold nanorods of having modified is centrifugal twice, each centrifugal 10min of 7000r/min, be resuspended in the CTAB solution of isopyknic 0.005M, finally obtain golden rod-upstream primer and golden rod-downstream primer; Wherein

Upstream primer: 5 '-SH-TGGCTGACCCTGATGAGTTCG-3 ',

Downstream primer: 5 '-SH-GGGCCATGATTACGCCAGTT-3 ';

(3) the controlled assembling of PCR

This assembling process has dynamic optically-active and absorbent properties, different periods is set, increase along with period, assembly absorbability wavelength blue wave band moves, correspondingly garden two chromatograms also show the blue shift characteristic, and garden two chromatograms show the serpentine curve of feature, and these character are key characters of the controlled assembling of PCR;

The PCR process is take λ DNA as template, and the amplification base sequence is 5 '-TGGCTGACCCTGATGAGTTC GTGTCCGTAC AACTGGCGTA ATCATGGCCC-3 ' in the template, adopts the golden rod-upstream primer and the golden rod-downstream primer that make to carry out PCR;

PCR reaction system: dNTPs1mL, the Taq archaeal dna polymerase of 0.5 unit, λ dna profiling plasmid 0.5mL, reaction buffer 5mL gets respectively golden rod-upstream primer and each 4mL of golden rod-downstream primer, and adding aqua sterilisa to cumulative volume is 50mL; Amplification program is: 94 ℃ of denaturation 3min; 94 ℃ of sex change 30s, 60 ℃ of annealing 30s, 72 ℃ are extended 1min, and it is 2-15 time that cycle-index is set, and (proliferation time is 0,4,10,20,30min); 72 ℃ are extended 10min again; 10 ℃ of preservations get the PCR product;

(4) sign of PCR product

The PCR assembling product of gold nanorods carries out garden two chromatogram CD, the sign of ultraviolet spectra and projection Electronic Speculum; The PCR product is carried out optical signature characterize, wavelength is set to 200nm-1000nm;

(5) calculating of chirality dimer garden two chromatograms

Nonparallel between this assembly gold nanorods, namely there is angle, this also is the essence that there is chirality in this assembly, and what angled assembly plasma resonance showed as different gold nanorods excites the mutual coupling of dipole moment, thereby the rotation that produces light produces chirality;

The dimer spacing of gold nanorods utilizes three-dimensional reconstruction Electronic Speculum result to characterize, and distance is 17.2nm; Carry out simultaneously the theoretical calculating of chirality: this distance is set is the theoretical dimer spacing of calculating, utilize theoretical calculating, dimer angle 2 degree-20 degree are set, according to calculate and experiment go out the long degree of conformity of spike, obtain dimer angle and experiment the best and meet the result and be-9.5 degree; The setting of gold nanorods material obtains the spectral characteristic of this material by its real part of permittivity of definition and imaginary part; Calculate to come from the gold nanorods draw ratio and test that to use gold nanorods, draw ratio be 2.9;

(6) calculating of left-right rotary enantiomeric ratio

At first calculate mole garden two chromatogram ε, utilize computing formula (1) to calculate, wherein, c and l are respectively the light paths of the dimeric concentration of gold nanorods and cuvette, the dimeric concentration of gold nanorods can utilize formula (2) to calculate, obtain from original gold nanorods concentration and the dimeric productive rate of gold nanorods, while c and l can be used as a parameter m and carry out subsequent calculations, see formula (3); Then, Δ ε calculates by formula (4), and wherein: ε p and ε n are respectively the greatest measured values of posivtive spike and negative peak in the first chromatogram of mole, and Δ ε can calculate the collection of illustrative plates from garden two chromatograms simultaneously and calculate, aggregative formula (1)-(4) obtain final computing formula (5); The relative ratios k of left-right rotary enantiomer (+) and (-) _(+)/(-)Calculate by formula (6); Ratio η (the η of left-right rotary enantiomer ₍₊₎, η _(-)) calculate by formula (7)-(8), the ratio of obtaining is respectively: η ₍₊₎=59.1%, η _(-)=40.9%; Formula is seen lower:

$\mathrm{\ϵ}=\frac{\mathrm{CD}}{32.98\·c\·l}---\left(1\right),$ $c=\frac{c_{\mathrm{NRs}}\·y}{2}---\left(2\right),$

m=c·l (3)， Δε=|ε _p-ε _n| (4)，

$\mathrm{\Δ\ϵ}=\left|\frac{C_p-C_n}{32.98\·m}\right|---\left(5\right),$ ${\mathrm{\κ}}_{(-)/(+)}=\frac{{\mathrm{\Δ\ϵ}}_{\mathrm{Exp}}}{{\mathrm{\Δ\ϵ}}_{\mathrm{Sim}}}---\left(6\right),$

${\mathrm{\η}}_{(+)}=\frac{{1-\mathrm{\κ}}_{(-)/(+)}}{2}---\left(7\right),$ ${\mathrm{\η}}_{(-)}=\frac{{1+\mathrm{\κ}}_{(-)/(+)}}{2}---\left(8\right).$

Annotate: DNA used in the present invention all gives birth to worker's bioengineering Co., Ltd available from Chinese Shanghai, and carries out purifying by polyacrylamide gel electrophoresis.

Beneficial effect of the present invention: the present invention utilizes the complementary hybridization of DNA to carry out the assembling of gold nanorods, have strong assembling specificity and controllability, and the controllable groups harness that utilizes the PCR process to carry out gold nanorods has simply, the characteristics such as efficient, being traditional DNA cloning technology and the perfect adaptation of modern nanometer technology, also is a kind of novel packaging strategy simultaneously therefore.

Description of drawings

Fig. 1: the schematic diagram of gold nanorods assembling, the A:PCR schematic diagram that increase, B: dimer is assembled schematic diagram.

Fig. 2: the Electronic Speculum figure of gold nanorods assembling, A: dimer, B: polymer.

Fig. 3: the ultraviolet figure of the assembling of gold nanorods.

Fig. 4: garden two chromatograms of gold nanorods.

Fig. 5: the left-right rotary dimer schematic diagram of gold nanorods: A: levo form, B: d-isomer.

Fig. 6: the dimer spacing (A) of gold nanorods and different angles garden two chromatograms calculate (B).

Fig. 7: the dimeric experiment (A) of gold nanorods and (B) garden of calculating two chromatogram comparison diagrams.

Fig. 8: the surface charge distribution map of left-right rotary body under left and right sides circularly polarized light (LCP, RCP) excites.

The specific embodiment

Embodiment 1

(1) preparation of gold nanorods, the preparation of gold nanorods are to utilize the gold seeds growth method.

Gold seeds synthetic: with the gold chloride (HAuCl of the 0.0005M of 2.5mL ₄) be dissolved in the softex kw (CTAB) of the 0.2M of 2.5mL, mix, 0.3mL is newly disposed, the sodium borohydride solution of the 0.01M of precooling is added in the mentioned solution fast, and powerful mixing 2min, then 25 ℃ of placements of room temperature 2h, get gold seeds, stand-by.

The growth of gold nanorods, the configuration of growth solution: with the AgNO of 0.15mL, 0.004M ₃, the HAuCl of 5mL, 0.001M ₄Solution is added among the CTAB of 5mL, 0.2M, then adds the Vc of the 0.0788M of 70 μ L, and abundant reductase 12 min adds the gold seeds of the above-mentioned configuration of 12 μ L, fully stir 20s after, leave standstill in 25 ℃, stand-by.

(2) lateral orientation of gold nanorods is modified.

Synthetic gold rod, concentration determination is 0.262nM, carry out the centrifugal 10min of 10000r/min, centrifugal sediment is resuspended with the CTAB solution of 0.005M, gold rod with trim two sulphur crisp sweets alcohol (DTT) and after concentrating is with the ratio of mol ratio 5 ︰ 1, room temperature leaves standstill, modification reaction 10h, and purpose is modified the end face sealing of gold rod.The gold rod that above-mentioned end face is modified is centrifugal twice, and the centrifugal 10min of 7000r/min is resuspended in the CTAB solution of isopyknic 0.005M.The golden rod of this centrifugal treating is scattered in respectively in two PCR pipes, respectively with PCR in upstream primer and downstream primer respectively with the ratio of Yin Wu ︰ gold nanorods 400 ︰ 1, carry out the side of gold rod and modify, room temperature leaves standstill, modification reaction 10h.The gold rod of having modified is centrifugal twice, and the centrifugal 10min of 7000r/min is resuspended in the CTAB solution of isopyknic 0.005M, finally obtains golden rod-upstream primer and golden rod-downstream primer.Wherein upstream primer and downstream primer base sequence are respectively 5 '-SH-TGGCTGACCCTGATGAGTTCG-3 ', and 5 '-SH-GGGCCATGATTACGCCAGTT-3 '.

(3) PCR assembling process

The PCR process is with take λ DNA as template, and the amplification base sequence is 5 '-TGGCTGACCCTGATGAGTTC GTGTCCGTAC AACTGGCGTA ATCATGGCCC-3 ' in the template.Golden rod-upstream primer and golden rod-downstream primer that employing makes carry out PCR.The PCR reaction system: dNTPs 1mL, the Taq archaeal dna polymerase of 0.5 unit, λ dna profiling plasmid 0.5mL, reaction buffer 5mL gets respectively golden rod-upstream primer and golden rod-downstream primer 4mL, and adding aqua sterilisa to cumulative volume is 50mL; Amplification program is: 94 ℃ of denaturation 3min; 94 ℃ of sex change 30s, 60 ℃ of annealing 30s, 72 ℃ are extended 1min, and it is 0,4,10,20,30min that proliferation time is set; 72 ℃ are extended 10min again; 10 ℃ of preservations get the PCR product.

(4) sign of PCR product

The PCR assembling product of gold rod carries out garden two chromatograms (CD), the sign of ultraviolet spectra and projection Electronic Speculum.

(5) calculating of chirality dimer garden two chromatograms

The dimer spacing of gold nanorods utilizes three-dimensional reconstruction Electronic Speculum result to characterize, and distance is 17.2nm.Carry out simultaneously the theoretical calculating of chirality: setting changes distance and is the dimer spacing of theoretical calculating, and utilization is theoretical calculates, and dimer angle 2 degree-20 is set spends, and opens degree of conformity according to going out spike, and degree of obtaining meets the result with experiment the best and is-9.5 degree.The setting of gold bar material obtains changing the spectral characteristic of material by its real part of permittivity of definition and imaginary part.It is excellent with gold that calculating comes from experiment with golden excellent draw ratio, and draw ratio is 2.9.

(6) calculating of left-right rotary enantiomeric ratio

At first calculate mole garden two chromatograms (ε), utilize computing formula 1 to calculate, wherein, c and l are respectively the light paths of the excellent dimeric concentration of gold and cuvette, the excellent dimeric concentration of gold can utilize formula 2 to calculate, obtain from the excellent concentration of original gold and the excellent dimeric productive rate of gold, while c and l can be used as a parameter m and carry out subsequent calculations, see formula 3.Then, Δ ε can calculate by formula 4, and wherein: ε p and ε n are respectively the greatest measured values of posivtive spike and negative peak in the first chromatogram of mole, and Δ ε can calculate the collection of illustrative plates from garden two chromatograms simultaneously and calculate, and aggregative formula 1-4 obtains final computing formula 5.Relative ratios (the k of left-right rotary enantiomer ((+), and (-)) _(+)/(-)) can calculate by following formula 6.Ratio η (the η of left-right rotary enantiomer ₍₊₎, η _(-)) can calculate by formula 7-8, the ratio of obtaining is respectively: η ₍₊₎=59.1%, η _(-)=40.9%.Formula is seen lower:

$\mathrm{\ϵ}=\frac{\mathrm{CD}}{32.98\·c\·l}---\left(1\right)$ $c=\frac{c_{\mathrm{NRs}}\·y}{2}---\left(2\right)$

m=c·l (3) Δε=|ε _p-ε _n| (4)

$\mathrm{\Δ\ϵ}=\left|\frac{C_p-C_n}{32.98\·m}\right|---\left(5\right)$ ${\mathrm{\κ}}_{(-)/(+)}=\frac{{\mathrm{\Δ\ϵ}}_{\mathrm{Exp}}}{{\mathrm{\Δ\ϵ}}_{\mathrm{Sim}}}---\left(6\right)$

${\mathrm{\η}}_{(+)}=\frac{{1-\mathrm{\κ}}_{(-)/(+)}}{2}---\left(7\right)$ ${\mathrm{\η}}_{(-)}=\frac{{1+\mathrm{\κ}}_{(-)/(+)}}{2}---\left(8\right)$

Claims (1)

1. chirality assemble method based on the gold nanorods of polymerase chain reaction, it is characterized in that having plasma resonance coupling character, and show maximum absorption wavelength intensity and position dynamic variation characteristic, comprise, the preparation of gold nanorods, the primer directed modification of gold nanorods, the controlled assembling of PCR, the sign of PCR product;

(1) preparation of gold nanorods

The preparation of gold nanorods is to utilize the gold seeds growth method, gold seeds synthetic: with the gold chloride HAuCl of the 0.0005M of 2.5mL ₄Be dissolved among the softex kw CTAB of 0.2M of 2.5mL, mix, with 0.3mL newly dispose, the sodium borohydride solution of the 0.01M of precooling is added in the mentioned solution fast, and powerful mixing 2min, then 25 ℃ of room temperatures are placed 2h, get gold seeds;

The growth of gold nanorods, the configuration of growth solution: with the AgNO of 0.15mL, 0.004M ₃, the HAuCl of 5mL, 0.001M ₄Solution is added among the CTAB of 5mL, 0.2M, then adds the Vc of the 0.0788M of 70 μ L, and abundant reductase 12 min adds the gold seeds of the above-mentioned configuration of 12 μ L, fully stir 20s after, leave standstill in 25 ℃, stand-by;

(2) the primer directed modification of gold nanorods

Carry out first the end face sulfhydryl compound and modify, this sulfhydryl compound is selected TGA, mercaptoethanol or two sulphur crisp sweets alcohol, and target is to carry out the end face sealing, second step carries out the side of primer to be modified, and is the prerequisite of controlled assembling;

Synthetic gold nanorods, concentration determination is 0.262nM, carry out the centrifugal 10min of 10000r/min, centrifugal sediment is resuspended with the CTAB solution of 0.005M, with trim two sulphur crisp sweets alcohol DTT with concentrated after gold nanorods with the ratio of amount of substance mol ratio 5 ︰ 1, room temperature leaves standstill, modification reaction 10h, and purpose is modified the end face sealing of gold nanorods; The gold nanorods that above-mentioned end face is modified is centrifugal twice, and each centrifugal 10min of 7000r/min is resuspended in the CTAB solution of isopyknic 0.005M; The gold nanorods of this centrifugal treating is scattered in respectively in two PCR pipes, respectively with PCR in upstream primer and downstream primer respectively with the ratio of Yin Wu ︰ gold nanorods mol ratio 400 ︰ 1, carrying out the side of gold nanorods modifies, room temperature leaves standstill, modification reaction 10h, the gold nanorods of having modified is centrifugal twice, each centrifugal 10min of 7000r/min, be resuspended in the CTAB solution of isopyknic 0.005M, finally obtain golden rod-upstream primer and golden rod-downstream primer; Wherein

Upstream primer: 5 '-SH-TGGCTGACCCTGATGAGTTCG-3 ',

Downstream primer: 5 '-SH-GGGCCATGATTACGCCAGTT-3 ';

(3) the controlled assembling of PCR

This assembling process has dynamic optically-active and absorbent properties, different periods is set, increase along with period, assembly absorbability wavelength blue wave band moves, correspondingly garden two chromatograms also show the blue shift characteristic, and garden two chromatograms show the serpentine curve of feature, and these character are key characters of the controlled assembling of PCR;

The PCR process is take λ DNA as template, and the amplification base sequence is 5 '-TGGCTGACCCTGATGAGTTC GTGTCCGTAC AACTGGCGTA ATCATGGCCC-3 ' in the template, adopts the golden rod-upstream primer and the golden rod-downstream primer that make to carry out PCR;

PCR reaction system: dNTPs1mL, the Taq archaeal dna polymerase of 0.5 unit, λ dna profiling plasmid 0.5mL, reaction buffer 5mL gets respectively golden rod-upstream primer and each 4mL of golden rod-downstream primer, and adding aqua sterilisa to cumulative volume is 50mL; Amplification program is: 94 ℃ of denaturation 3min; 94 ℃ of sex change 30s, 60 ℃ of annealing 30s, 72 ℃ are extended 1min, and it is 2-15 time that cycle-index is set; 72 ℃ are extended 10min again; 10 ℃ of preservations get the PCR product;

(4) sign of PCR product

The PCR assembling product of gold nanorods carries out garden two chromatogram CD, the sign of ultraviolet spectra and projection Electronic Speculum; The PCR product is carried out optical signature characterize, wavelength is set to 200nm-1000nm;

(5) calculating of chirality dimer garden two chromatograms

Nonparallel between this assembly gold nanorods, namely there is angle, this also is the essence that there is chirality in this assembly, and what angled assembly plasma resonance showed as different gold nanorods excites the mutual coupling of dipole moment, thereby the rotation that produces light produces chirality;

The dimer spacing of gold nanorods utilizes three-dimensional reconstruction Electronic Speculum result to characterize, and distance is 17.2nm; Carry out simultaneously the theoretical calculating of chirality: this distance is set is the theoretical dimer spacing of calculating, utilize theoretical calculating, dimer angle 2 degree-20 degree are set, according to calculate and experiment go out the long degree of conformity of spike, obtain dimer angle and experiment the best and meet the result and be-9.5 degree; The setting of gold nanorods material obtains the spectral characteristic of this material by its real part of permittivity of definition and imaginary part; Calculate to come from the gold nanorods draw ratio and test that to use gold nanorods, draw ratio be 2.9;

(6) calculating of left-right rotary enantiomeric ratio

At first calculate mole garden two chromatogram ε, utilize computing formula (1) to calculate, wherein, c and l are respectively the light paths of the dimeric concentration of gold nanorods and cuvette, the dimeric concentration of gold nanorods can utilize formula (2) to calculate, obtain from original gold nanorods concentration and the dimeric productive rate of gold nanorods, while c and l can be used as a parameter m and carry out subsequent calculations, see formula (3); Then, Δ ε calculates by formula (4), and wherein: ε p and ε n are respectively the greatest measured values of posivtive spike and negative peak in the first chromatogram of mole, and Δ ε can calculate the collection of illustrative plates from garden two chromatograms simultaneously and calculate, aggregative formula (1)-(4) obtain final computing formula (5); The relative ratios k of left-right rotary enantiomer (+) and (-) _(+)/(-)Calculate by formula (6); Ratio η (the η of left-right rotary enantiomer ₍₊₎, η _(-)) calculate by formula (7)-(8), the ratio of obtaining is respectively: η ₍₊₎=59.1%, η _(-)=40.9%; Formula is seen lower:

$\mathrm{\ϵ}=\frac{\mathrm{CD}}{32.98\·c\·l}---\left(1\right),$ $c=\frac{c_{\mathrm{NRs}}\·y}{2}---\left(2\right),$

m=c·l (3)， Δε=|ε _p-ε _n| (4)，

$\mathrm{\Δ\ϵ}=\left|\frac{C_p-C_n}{32.98\·m}\right|---\left(5\right),$ ${\mathrm{\κ}}_{(-)/(+)}=\frac{{\mathrm{\Δ\ϵ}}_{\mathrm{Exp}}}{{\mathrm{\Δ\ϵ}}_{\mathrm{Sim}}}---\left(6\right),$

${\mathrm{\η}}_{(+)}=\frac{{1-\mathrm{\κ}}_{(-)/(+)}}{2}---\left(7\right),$ ${\mathrm{\η}}_{(-)}=\frac{{1+\mathrm{\κ}}_{(-)/(+)}}{2}---\left(8\right).$

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