Compton Scattering and Recoiling Electron Momentum

[CoffeeCrow]

Homework Statement

X-rays with wavelength 55pm are scattered from a graphite target. Consider an x-ray which is scattered from a valence electron at an angle of 180 degrees (back towards the x-ray source).

a. Is the wavelength of the X-ray greater or less than before the collision?
b. What is the momentum of the recoiling electrons?

Homework Equations

$\Delta\lambda=\frac {h} {mc}(1-cos(\theta))$
$p=\frac {h} {\lambda}$

The Attempt at a Solution

Part a. seems obvious. The X-ray collides with an electron, transfers some kinetic energy to it, losing some energy in the process. Less energy, longer wavelength, right?

Part b however, is giving me a few issues.
I start by finding the Compton shift for 180 degrees, which gives $\Delta\lambda=4.85pm$.
This means that the scattered X-ray should have a wavelength of 55+4.85=59.85pm, correct?
This is where things get a bit hairy. The solution for this problem however, states that the recoiling X-ray has a wavelength of 50.14 pm. This doesn't seem to make sense, as the energy of the X-ray has increased, despite transferring energy to the electron (these solutions have had issues in the past). Anyway,they then simply use $p=\frac{h}{\lambda}$ to calculate the momentum of the recoiling electron, yielding (with their wavelength), 1.32E-23 Ns.
I however have tried the following:
$p_{i}=p_{f}$
$\frac{h}{\lambda_{i}}=p_{electron}-\frac{h}{\lambda_{f}}$
$p_{electron}=h(\frac{1}{\lambda_{i}}+\frac{1}{\lambda_{f}})$
$p_{electron}=2.31E-23Ns$ (using my values for wavelength, 55pm and 59.85pm)
OR:
$p_{electron}=2.53E-23Ns$(using their values)
Which of course, doesn't agree with the solutions.
So, I was wondering, am I missing something fundamental and conceptual in regards to energy and wavelength shift and, is my momentum approach valid?

 

[mfb]

Part a. seems obvious. The X-ray collides with an electron, transfers some kinetic energy to it, losing some energy in the process. Less energy, longer wavelength, right?

Right.

This means that the scattered X-ray should have a wavelength of 55+4.85=59.85pm, correct?

I didn't check the number but the approach is right.
50.14 pm is clearly wrong.

 

[CoffeeCrow]

Alright, thanks, I was just worrying I'd managed to miss something fundamental!