飞秒泵浦-探测实验动力学数据处理

2013-11-09 15:00:10

基本过程

基态分子S0吸收266 nm的泵浦光子被激发到S2态, 然后吸收800 nm的探测光子被电离形成母体离子. S2激发态分子可能的弛豫过程是通过内转换到S0和S1态. 通过内转换弛豫到S1激发态的分子可以通过内转换弛豫到S0态, 也可以继续吸收探测光子被电离, 形成母体离子. 而根据Franc-Condon原理, 基态S0分子由于电离需要很高的能量, 不能被探测光子直接电离. 因此, 总的母体离子来源于激发态S1和S2的电离.

动力学方程

其中[Ion], [S2], [S1]和[S0]分别表示总的母体离子, 第二激发态, 第一激发态和基态高振动态的相对布局数, I(t)为检测到的信号. $\sigma_i$ 和 $\sigma_a$ 表示光电离截面和光吸收截面. \(I_{pump}(t)\) 和 $I_{prob}(t)$ 分别对应泵浦光和探测光的强度. $\alpha_1={1 \over \tau_1}, \alpha_2={1 \over \tau_2}$ 和 $\alpha_3={1\over \tau_3}$ 分别表示S2->S1, S1->S0 和S2->S0的内转换速率. 由于S2和S1激发态的光电离截面与它们本身和对应离子态的电子构型有关, 因此这两个态的光电离截面的相对比例是一个未知量, 设其为k.

特例

若忽略 $\sigma_a I_{pump}(t)[S_0]$ 这一项, 则动力学方程简化为:

此为线性微分方程组, 可求得通解

设初始条件为 $[S_2]_0=C_0, [S_1]_0=0$, 则特解为

探测光检测的是S2与S1电离的总信号, 检测信号 $I(t)$ 是[S2], [S1]之和与仪器响应函数 $G(t)$ 的卷积(其计算参看另一篇博文)

设仪器函数 $G(t)=\sigma e^{-\kappa t^2}$, 则

此种情形下, 对 $I(t)$ 进行非线性拟合, 可得到 $\alpha_{13}, \alpha_2, \kappa, I_0, P$, 但无法分别求得 $\alpha_1, \alpha_3$ 和 $k$, 除非假定 $\alpha_3$ 远小于 $\alpha_1$.

数据示例与拟合结果

参考

1. 张蓉蓉, 秦朝朝, 龙金友, 杨明晖, 张冰. “丙烯酸分子的激发态超快预解离动力学”, 物理化学学报, 28(3):522-527, 2012
2. Bumaliya Abulimiti, Rongshu Zhu, Jinyou Long, Yanqi Xu, Yuzhu Liu, Ahmed Yousif Ghazal, Minghui Yang, Bing Zhang. “Study of ultrafast dynamics of 2-picoline by time-resolved photoelectron imaging”, J. Chem. Phys. 134, 234301, 2011
3. Ying Tang, Yoshi-ichi Suzuki, Huan Shen, Kentaro Sekiguchi, Naoya Kurahashi, Kiyoshi Nishizawa, Peng Zuo, Toshinori Suzuki. “Time-resolved photoelectron spectroscopy of bulk liquids at ultra-low kinetic energy”, Chem. Phys. Lett., 494:111-116, 2010

代码

# Language: clike
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Jicun LI: Jerkwin@gmail.com
% 2013-11-09: Demo !!! Convolution NOT used !!!
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function PumpProb
	clc;

	[texp, Yexp] = textread('exp.dat', '%f %f');
	Ndat = size(texp);
	Yexp = Yexp/max(Yexp);

	LB = [0,   0,   0,   0,   0,   0  ];
	UB = [inf, inf, inf, inf, inf, inf];
	lsqoptions = optimset('lsqnonlin');
	lsqoptions.LM = 'on';
	lsqoptions.MaxFunEvals = 1000;
	lsqoptions.Tolx = 1e-6;                   % default is 1e-6
	lsqoptions.TolFun = 1e-6;                 % default is 1e-6
	lsqoptions.Display = 'iter';              % 'off' or 'final'
	lsqoptions.LineSearchType = 'cubicpoly';  % or 'quadcubic'

	a1 = 1; a2 = 1; a3 = 0; k = 1;
	Apump = 1; Spump= 70;

	fileID = fopen('exp_fit.dat', 'w');
	P0 = [ a1 a2 a3  k  Apump Spump ];
	Pfit = lsqnonlin(@(P)ObjFun(P, texp, Yexp), P0, LB, UB, lsqoptions);

	Pfit
	a1 = Pfit(1); a2 = Pfit(2); a3 = Pfit(3);
	k = Pfit(4); Apump = Pfit(5); Spump = Pfit(6);

	fprintf('T1= %f\nT2= %f\nT3= %f\n', 1/a1, 1/a2, 1/a3);
	fprintf('k = %f\n', k);
	fprintf('Apump= %f Spump= %f\n', Apump, Spump);

	Y0 = [0 0 1];
	options = odeset('RelTol',1e-6, 'AbsTol',[1e-6 1e-6 1e-6]);
	[t, Y] = ode45(@dY, texp, Y0, options, a1, a2, a3, Apump, Spump);
	Yfit = k*Y(:,1)+Y(:,2);
	Yfit = Yfit/max(Yfit);
	Y = Y/max(Yfit);

	fprintf(fileID,'%8s%12s%12s%12s%12s%12s\n', ...
		't','Yexp', 'Yfit', 'Y1','Y2', 'Yfit-Yexp');
	for i=1:Ndat;
		fprintf(fileID, '%8.3f%12.5f%12.5f%12.5f%12.5f%12.5f\n', ...
			t(i), Yexp(i), Yfit(i), k*Y(i,1), Y(i,2), Yfit(i)-Yexp(i));
	end
end
%%
function ObjFun = ObjFun(P, texp, Yexp)
	a1 = P(1); a2 = P(2); a3 = P(3);
	k = P(4); Apump = P(5); Spump = P(6);

	Y0 = [0 0 1];
	options = odeset('RelTol',1e-6, 'AbsTol',[1e-6 1e-6 1e-6]);
	[t, Y] = ode45(@dY, texp, Y0, options, a1, a2, a3, Apump, Spump);
	Yfit = k*Y(:,1)+Y(:,2);
	Yfit = Yfit/max(Yfit);

	ObjFun = Yexp-Yfit;
end
%%
function dY = dY(t, Y, a1, a2, a3, Apump, Spump)
	dY = zeros(3,1);
	Ipump = Apump*exp(-Spump*t^2);
	dY(1) = a1*Y(2) - a2*Y(1);
	dY(2) = -(a1+a3)*Y(2) + Ipump*Y(3);
	dY(3) = a3*Y(2) + a2*Y(1) - Ipump*Y(3);
end