Compton Effect

Dept. of Math and Physics, Alfred State Coll. SUNY, Alfred NY 14802

[Problem #12] Compton Effect. (a). Derive the expression for the shift in wavelength and plot the shift as a function of the scattering angle theta. (b). Derive the expression for the kinetic energy of the recoiled electron and plot it. (c). Plot a graph of the recoil angle as a function of the scattering angle. For (b) and (c), assume that the wavelength of the incident photon is 10 pm, 5 pm, and 2 pm. Note that the Compton wavelenth is h/(mc) = 0.0243 A = 2.43 pm.

[Solution] Ver. 6.01

a. Shift in wavelength.

> restart:

Conservation of energy. Note hf = pc for photon.

E(incident photon) + rest energy of electron = E( scatterd photon) + energy of the recoiled electron.

> eq3:=p0*c+m*c^2=p*c+sqrt(m^2*c^4+pe^2*c^2);

Conservation of momentum before and after the scattering . Substitute the expression in the equation above.

> subs(pe^2=p0^2+p^2-2*p0*p*cos(theta),eq3);

Momentum p is h / wavelength lambda, where h is Planck's constant.

> subs({p0=h/lambda0,p=h/lambda},%);

> lambda:=solve(%,lambda);

The shift in wavelength = h/(mc) (1-cos(theta)) is obtained, where theta is the scattering angle and

h/(mc) is the Compton wavelength ( = 0.0243 A).

> deltalambda:=simplify(lambda-lambda0);

Set h/(mc)=0.0243 A and plot deltalambda above in m.

> deltalambdanumber:=subs(h=0.0243e-10*c*m,%);

> plot(deltalambdanumber,theta=0..Pi,title=`Shift in wavelength`,labels=[`theta(rad)`,`shift(m)`]);

b. KE : the kinetic energy of the recoiled electron is the difference between the energy of the incident photon and

that of the scattered photon. Note pc = hf for photon. KE = hf - hf0 = hc /lambda - hc / lambda0 = hc/lambda - hc/ (lambda0+deltalambda).

> KE:=h*c/lambda0-h*c/(lambda0+deltalambda);

> KEnumerical:=(lambda0,theta)->h*c/lambda0-h*c/(lambda0+deltalambda);

Constants: h=6.626e-34 Js, m=9.11e-31 kg, lambda0=10e-12 m, c=3.0e8 m/s.

> h:=6.626e-34:m:=9.11e-31:c:=3.0e8:

The wavelength of the incident photon: 10 pm, 5 pm, and 2 pm.

> plot([KEnumerical(10e-12,theta),KEnumerical(5e-12,theta),KEnumerical(2e-12,theta)],theta=0..Pi, title=`KE of recoil electron`, labels=[` theta (rad)`,`KE (J)`],legend=["10 pm","5 pm","2 pm"]);

c. phi (recoil angle) vs. theta (scattering angle)

Conservation of momentum in x and y direction:

p0 - p cos(theta)+pe cos(phi) ... (1)

p sin(theta) = pe sin(phi) ... (2)

> eq3:=p0-p*cos(theta)=pe*cos(phi);

> eq4:=p*sin(theta)=pe*sin(phi);

> cosphi:=solve(eq3,cos(phi));

> sinphi:=solve(eq4,sin(phi));

> tanphi:=sinphi/cosphi;

> phi:=arctan(tanphi);

> phinumerical:=(lambda0,theta)->-arctan(h/(lambda0+deltalambda)*sin(theta)/(-h/lambda0+h/(lambda0+deltalambda)*cos(theta)));

>

> plot([phinumerical(10e-12,theta),phinumerical(5e-12,theta),phinumerical(2e-12,theta)],theta=0..Pi,title=`recoil angle vs. scattering angle`,legend=["10 pm","5 pm","2 pm"],labels=[` theta (rad)`,`phi (rad)`]);